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ThisDovereditionfirstpublishedin2004isanunabridgedrepublicationoftheeditionoriginallypublishedbytheLongmanGroupUKLimitedEssexEnglandin1980

LibraryofCongressCataloging-in-PublicationData

SpencerAJM(AnthonyJamesMerrill)1929-

ContinuummechanicsAJMSpencer

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OriginallypublishedLondonNewYorkLongman1980(Longmanmathematicaltexts)

Includesbibliographicalreferencesandindex

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1Continuummechanics1Title

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Tableof Contents

TitlePageCopyrightPagePreface1-Introduction2-Introductorymatrixalgebra3-Vectorsandcartesiantensors4-Particlekinematics5-Stress6-Motionsanddeformations7-Conservationlaws8-Linearconstitutiveequations9-Furtheranalysisoffinitedeformation10-Non-linearconstitutiveequations11-CylindricalandsphericalpolarcoordinatesAppendix-RepresentationtheoremforanisotropictensorfunctionofatensorAnswersFurtherreadingIndex

Preface

TheaimofthisbookistoprovideanintroductiontothetheoryofcontinuummechanicsinaformwhichissuitableforundergraduatestudentsItisbasedonlectureswhichIhavegivenintheUniversityofNottinghamduringthelastfourteenyearsIhavetriedtorestrictthemathematicalbackgroundrequiredtothatwhichisnormallyfamiliartoasecond-yearmathematicsundergraduateoramathematicallymindedengineeringgraduateeventhoughsomeofthetheorycanbedevelopedmoreconciselyandelegantlybyusingmoresophisticatedmathematicsthanIhaveemployedThematerialcoveredcomprisesintroductorychaptersonmatrixalgebraandonvectorsandcartesiantensorstheanalysisofdeformationandstressthemathematicalstatementsofthelawsofconservationofmassmomentumandenergyandtheformulationofthemechanicalconstitutiveequationsforvariousclassesoffluidsandsolidsCartesiancoordinatesandcartesiantensorsareusedthroughoutexceptthatinthelastchapterweshowhowthetheorycanbeexpressedintermsofcylindricalpolarandsphericalpolarcoordinatesIhavenotpursuedthevariousbranchesofthemechanicsofsolidsandfluidssuchaselasticityNewtonianfluidmechanicsviscoelasticityandplasticitybeyondthepointofformulatingtheirconstitutiveequationsTodosoinanymeaningfulwaywouldhaverequiredamuchlongerbookandthesesubjectsarefullydealtwithinlargerandmorespecializedtexts

IamofcoursegreatlyindebtedtomanyteacherscolleaguesandstudentswhohavecontributedtomyeducationincontinuummechanicsTheyaretoonumeroustomentionindividuallyratherthangivingaselectivelistIaskthemtoacceptacollectiveacknowledgementSimilarlyIhavefeltthatinanintroductorybookofthiskinditwouldbeinappropriatetogivereferencestooriginalworkbutitisobviousthatIhavemadeindirectuseofmanysourcesandIamgladtoacknowledgethecontributionofalltheauthorswhoseworkhasinfluencedme

ManyoftheproblemsaretakenfromexaminationpaperssetintheDepartmentofTheoreticalMechanicsintheUniversityofNottinghamandIacknowledgetheUniversityrsquospermissiontomakeuseofthese

FinallyIthankMargaretforthetyping

AJMSPENCERNottingham1979

1

Introduction

11ContinuummechanicsModernphysicaltheoriestellusthatonthemicroscopicscalematterisdiscontinuousitconsistsofmoleculesatomsandevensmallerparticlesHoweverweusuallyhavetodealwithpiecesofmatterwhichareverylargecomparedwiththeseparticlesthisistrueineverydaylifeinnearlyallengineeringapplicationsofmechanicsandinmanyapplicationsinphysicsInsuchcaseswearenotconcernedwiththemotionofindividualatomsandmoleculesbutonlywiththeirbehaviourinsomeaveragesenseInprincipleifweknewenoughaboutthebehaviourofmatteronthemicroscopicscaleitwouldbepossibletocalculatethewayinwhichmaterialbehavesonthemacroscopicscalebyapplyingappropriatestatisticalproceduresInpracticesuchcalculationsareextremelydifficultonlythesimplestsystemscanbestudiedinthiswayandeveninthesesimplecasesmanyapproximationshavetobemadeinordertoobtainresultsConsequentlyourknowledgeofthemechanicalbehaviourofmaterialsisalmostentirelybasedonobservationsandexperimentaltestsoftheirbehaviouronarelativelylargescale

ContinuummechanicsisconcernedwiththemechanicalbehaviourofsolidsandfluidsonthemacroscopicscaleItignoresthediscretenatureofmatterandtreatsmaterialasuniformlydistributedthroughoutregionsofspaceItisthenpossibletodefinequantitiessuchasdensitydisplacementvelocityandsoonascontinuous(oratleastpiecewisecontinuous)functionsofpositionThisprocedureisfoundtobesatisfactoryprovidedthatwedealwithbodieswhosedimensionsarelargecomparedwiththecharacteristiclengths(forexampleinteratomicspacingsinacrystalormeanfreepathsinagas)onthemicroscopicscaleThemicroscopicscaleneednotbeofatomicdimensionswecanforexampleapplycontinuummechanicstoagranularmaterialsuchassandprovidedthatthedimensionsoftheregionconsideredarelargecomparedwiththoseofanindividualgrainIncontinuummechanicsitisassumedthatwecanassociateaparticleofmatterwitheachandeverypointoftheregionofspaceoccupiedbyabodyandascribefieldquantitiessuchasdensityvelocityandsoontotheseparticlesThejustificationforthisprocedureistosomeextentbasedonstatisticalmechanicaltheoriesofgasesliquidsandsolidsbutrestsmainlyonitssuccessindescribingandpredictingthemechanicalbehaviourofmaterialinbulk

MechanicsisthesciencewhichdealswiththeinteractionbetweenforceandmotionConsequentlythevariableswhichoccurincontinuummechanicsareontheonehandvariablesrelatedtoforces(usuallyforceperunitareaorperunitvolumeratherthanforceitself)andontheotherhandkinematicvariablessuchasdisplacementvelocityandaccelerationInrigid-bodymechanicstheshapeofabodydoesnotchangeandsotheparticleswhichmakeuparigidbodymayonlymoverelativelytooneanotherinaveryrestrictedwayArigidbodyisacontinuumbutitisaveryspecialidealizedanduntypicaloneContinuummechanicsismoreconcernedwithdeformablebodieswhicharecapableofchangingtheirshapeForsuchbodiestherelativemotionoftheparticlesisimportantandthisintroducesassignificantkinematicvariablesthespatialderivativesofdisplacementvelocityandsoon

TheequationsofcontinuummechanicsareoftwomainkindsFirstlythereareequationswhichapplyequallytoallmaterialsTheydescribeuniversalphysicallawssuchasconservationofmassandenergySecondlythereareequationswhichdescribethemechanicalbehaviourofparticularmaterialstheseareknownasconstitutiveequations

TheproblemsofcontinuummechanicsarealsooftwomainkindsThefirstistheformulationofconstitutiveequationswhichareadequatetodescribethemechanicalbehaviourofvariousparticularmaterialsorclassesofmaterialsThisformulationisessentiallyamatterforexperimentaldeterminationbutatheoreticalframeworkisneeededinordertodevisesuitableexperimentsandtointerpretexperimentalresultsThesecondproblemistosolvetheconstitutiveequationsinconjunctionwiththegeneralequationsofcontinuummechanicsandsubjecttoappropriateboundaryconditionstoconfirmthevalidityoftheconstitutiveequationsandtopredictanddescribethebehaviourofmaterialsinsituationswhichareofengineeringphysicalormathematicalinterestAtthisproblem-solvingstagethedifferentbranchesofcontinuummechanicsdivergeandweleavethisaspectofthesubjecttomorecomprehensiveandmorespecializedtexts

2

Introductorymatrixalgebra

21MatricesInthischapterwesummarizesomeusefulresultsfrommatrixalgebraItisassumedthatthereaderisfamiliarwiththeelementaryoperationsofmatrixadditionmultiplicationinversionandtranspositionMostoftheotherpropertiesofmatriceswhichwewillpresentarealsoelementaryandsomeofthemarequotedwithoutproofTheomittedproofswillbefoundinstandardtextsonmatrixalgebra

AnmxnmatrixAisanorderedrectangulararrayofmnelementsWedenote

(21)

sothatAijistheelementintheithrowandthejthcolumnofthematrixATheindexitakesvalues12mandtheindexjtakesvalues12nIncontinuummechanicsthematriceswhichoccurareusuallyeither3x3squarematrices3times1columnmatricesor1x3rowmatricesWeshallusuallydenote3x3squarematricesbybold-faceromancapitalletters(ABCetc)and3x1columnmatricesbybold-faceromanlower-caseletters(abcetc)A1x3rowmatrixwillbetreatedasthetransposeofa3x1columnmatrix(aTbTcTetc)Unlessotherwisestatedindiceswilltakethevalues12and3althoughmostoftheresultstobegivenremaintrueforarbitraryrangesoftheindices

AsquarematrixAissymmetricif

(22)

andanti-symmetricif

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

Preface

TheaimofthisbookistoprovideanintroductiontothetheoryofcontinuummechanicsinaformwhichissuitableforundergraduatestudentsItisbasedonlectureswhichIhavegivenintheUniversityofNottinghamduringthelastfourteenyearsIhavetriedtorestrictthemathematicalbackgroundrequiredtothatwhichisnormallyfamiliartoasecond-yearmathematicsundergraduateoramathematicallymindedengineeringgraduateeventhoughsomeofthetheorycanbedevelopedmoreconciselyandelegantlybyusingmoresophisticatedmathematicsthanIhaveemployedThematerialcoveredcomprisesintroductorychaptersonmatrixalgebraandonvectorsandcartesiantensorstheanalysisofdeformationandstressthemathematicalstatementsofthelawsofconservationofmassmomentumandenergyandtheformulationofthemechanicalconstitutiveequationsforvariousclassesoffluidsandsolidsCartesiancoordinatesandcartesiantensorsareusedthroughoutexceptthatinthelastchapterweshowhowthetheorycanbeexpressedintermsofcylindricalpolarandsphericalpolarcoordinatesIhavenotpursuedthevariousbranchesofthemechanicsofsolidsandfluidssuchaselasticityNewtonianfluidmechanicsviscoelasticityandplasticitybeyondthepointofformulatingtheirconstitutiveequationsTodosoinanymeaningfulwaywouldhaverequiredamuchlongerbookandthesesubjectsarefullydealtwithinlargerandmorespecializedtexts

IamofcoursegreatlyindebtedtomanyteacherscolleaguesandstudentswhohavecontributedtomyeducationincontinuummechanicsTheyaretoonumeroustomentionindividuallyratherthangivingaselectivelistIaskthemtoacceptacollectiveacknowledgementSimilarlyIhavefeltthatinanintroductorybookofthiskinditwouldbeinappropriatetogivereferencestooriginalworkbutitisobviousthatIhavemadeindirectuseofmanysourcesandIamgladtoacknowledgethecontributionofalltheauthorswhoseworkhasinfluencedme

ManyoftheproblemsaretakenfromexaminationpaperssetintheDepartmentofTheoreticalMechanicsintheUniversityofNottinghamandIacknowledgetheUniversityrsquospermissiontomakeuseofthese

FinallyIthankMargaretforthetyping

AJMSPENCERNottingham1979

1

Introduction

11ContinuummechanicsModernphysicaltheoriestellusthatonthemicroscopicscalematterisdiscontinuousitconsistsofmoleculesatomsandevensmallerparticlesHoweverweusuallyhavetodealwithpiecesofmatterwhichareverylargecomparedwiththeseparticlesthisistrueineverydaylifeinnearlyallengineeringapplicationsofmechanicsandinmanyapplicationsinphysicsInsuchcaseswearenotconcernedwiththemotionofindividualatomsandmoleculesbutonlywiththeirbehaviourinsomeaveragesenseInprincipleifweknewenoughaboutthebehaviourofmatteronthemicroscopicscaleitwouldbepossibletocalculatethewayinwhichmaterialbehavesonthemacroscopicscalebyapplyingappropriatestatisticalproceduresInpracticesuchcalculationsareextremelydifficultonlythesimplestsystemscanbestudiedinthiswayandeveninthesesimplecasesmanyapproximationshavetobemadeinordertoobtainresultsConsequentlyourknowledgeofthemechanicalbehaviourofmaterialsisalmostentirelybasedonobservationsandexperimentaltestsoftheirbehaviouronarelativelylargescale

ContinuummechanicsisconcernedwiththemechanicalbehaviourofsolidsandfluidsonthemacroscopicscaleItignoresthediscretenatureofmatterandtreatsmaterialasuniformlydistributedthroughoutregionsofspaceItisthenpossibletodefinequantitiessuchasdensitydisplacementvelocityandsoonascontinuous(oratleastpiecewisecontinuous)functionsofpositionThisprocedureisfoundtobesatisfactoryprovidedthatwedealwithbodieswhosedimensionsarelargecomparedwiththecharacteristiclengths(forexampleinteratomicspacingsinacrystalormeanfreepathsinagas)onthemicroscopicscaleThemicroscopicscaleneednotbeofatomicdimensionswecanforexampleapplycontinuummechanicstoagranularmaterialsuchassandprovidedthatthedimensionsoftheregionconsideredarelargecomparedwiththoseofanindividualgrainIncontinuummechanicsitisassumedthatwecanassociateaparticleofmatterwitheachandeverypointoftheregionofspaceoccupiedbyabodyandascribefieldquantitiessuchasdensityvelocityandsoontotheseparticlesThejustificationforthisprocedureistosomeextentbasedonstatisticalmechanicaltheoriesofgasesliquidsandsolidsbutrestsmainlyonitssuccessindescribingandpredictingthemechanicalbehaviourofmaterialinbulk

MechanicsisthesciencewhichdealswiththeinteractionbetweenforceandmotionConsequentlythevariableswhichoccurincontinuummechanicsareontheonehandvariablesrelatedtoforces(usuallyforceperunitareaorperunitvolumeratherthanforceitself)andontheotherhandkinematicvariablessuchasdisplacementvelocityandaccelerationInrigid-bodymechanicstheshapeofabodydoesnotchangeandsotheparticleswhichmakeuparigidbodymayonlymoverelativelytooneanotherinaveryrestrictedwayArigidbodyisacontinuumbutitisaveryspecialidealizedanduntypicaloneContinuummechanicsismoreconcernedwithdeformablebodieswhicharecapableofchangingtheirshapeForsuchbodiestherelativemotionoftheparticlesisimportantandthisintroducesassignificantkinematicvariablesthespatialderivativesofdisplacementvelocityandsoon

TheequationsofcontinuummechanicsareoftwomainkindsFirstlythereareequationswhichapplyequallytoallmaterialsTheydescribeuniversalphysicallawssuchasconservationofmassandenergySecondlythereareequationswhichdescribethemechanicalbehaviourofparticularmaterialstheseareknownasconstitutiveequations

TheproblemsofcontinuummechanicsarealsooftwomainkindsThefirstistheformulationofconstitutiveequationswhichareadequatetodescribethemechanicalbehaviourofvariousparticularmaterialsorclassesofmaterialsThisformulationisessentiallyamatterforexperimentaldeterminationbutatheoreticalframeworkisneeededinordertodevisesuitableexperimentsandtointerpretexperimentalresultsThesecondproblemistosolvetheconstitutiveequationsinconjunctionwiththegeneralequationsofcontinuummechanicsandsubjecttoappropriateboundaryconditionstoconfirmthevalidityoftheconstitutiveequationsandtopredictanddescribethebehaviourofmaterialsinsituationswhichareofengineeringphysicalormathematicalinterestAtthisproblem-solvingstagethedifferentbranchesofcontinuummechanicsdivergeandweleavethisaspectofthesubjecttomorecomprehensiveandmorespecializedtexts

2

Introductorymatrixalgebra

21MatricesInthischapterwesummarizesomeusefulresultsfrommatrixalgebraItisassumedthatthereaderisfamiliarwiththeelementaryoperationsofmatrixadditionmultiplicationinversionandtranspositionMostoftheotherpropertiesofmatriceswhichwewillpresentarealsoelementaryandsomeofthemarequotedwithoutproofTheomittedproofswillbefoundinstandardtextsonmatrixalgebra

AnmxnmatrixAisanorderedrectangulararrayofmnelementsWedenote

(21)

sothatAijistheelementintheithrowandthejthcolumnofthematrixATheindexitakesvalues12mandtheindexjtakesvalues12nIncontinuummechanicsthematriceswhichoccurareusuallyeither3x3squarematrices3times1columnmatricesor1x3rowmatricesWeshallusuallydenote3x3squarematricesbybold-faceromancapitalletters(ABCetc)and3x1columnmatricesbybold-faceromanlower-caseletters(abcetc)A1x3rowmatrixwillbetreatedasthetransposeofa3x1columnmatrix(aTbTcTetc)Unlessotherwisestatedindiceswilltakethevalues12and3althoughmostoftheresultstobegivenremaintrueforarbitraryrangesoftheindices

AsquarematrixAissymmetricif

(22)

andanti-symmetricif

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

1

Introduction

11ContinuummechanicsModernphysicaltheoriestellusthatonthemicroscopicscalematterisdiscontinuousitconsistsofmoleculesatomsandevensmallerparticlesHoweverweusuallyhavetodealwithpiecesofmatterwhichareverylargecomparedwiththeseparticlesthisistrueineverydaylifeinnearlyallengineeringapplicationsofmechanicsandinmanyapplicationsinphysicsInsuchcaseswearenotconcernedwiththemotionofindividualatomsandmoleculesbutonlywiththeirbehaviourinsomeaveragesenseInprincipleifweknewenoughaboutthebehaviourofmatteronthemicroscopicscaleitwouldbepossibletocalculatethewayinwhichmaterialbehavesonthemacroscopicscalebyapplyingappropriatestatisticalproceduresInpracticesuchcalculationsareextremelydifficultonlythesimplestsystemscanbestudiedinthiswayandeveninthesesimplecasesmanyapproximationshavetobemadeinordertoobtainresultsConsequentlyourknowledgeofthemechanicalbehaviourofmaterialsisalmostentirelybasedonobservationsandexperimentaltestsoftheirbehaviouronarelativelylargescale

ContinuummechanicsisconcernedwiththemechanicalbehaviourofsolidsandfluidsonthemacroscopicscaleItignoresthediscretenatureofmatterandtreatsmaterialasuniformlydistributedthroughoutregionsofspaceItisthenpossibletodefinequantitiessuchasdensitydisplacementvelocityandsoonascontinuous(oratleastpiecewisecontinuous)functionsofpositionThisprocedureisfoundtobesatisfactoryprovidedthatwedealwithbodieswhosedimensionsarelargecomparedwiththecharacteristiclengths(forexampleinteratomicspacingsinacrystalormeanfreepathsinagas)onthemicroscopicscaleThemicroscopicscaleneednotbeofatomicdimensionswecanforexampleapplycontinuummechanicstoagranularmaterialsuchassandprovidedthatthedimensionsoftheregionconsideredarelargecomparedwiththoseofanindividualgrainIncontinuummechanicsitisassumedthatwecanassociateaparticleofmatterwitheachandeverypointoftheregionofspaceoccupiedbyabodyandascribefieldquantitiessuchasdensityvelocityandsoontotheseparticlesThejustificationforthisprocedureistosomeextentbasedonstatisticalmechanicaltheoriesofgasesliquidsandsolidsbutrestsmainlyonitssuccessindescribingandpredictingthemechanicalbehaviourofmaterialinbulk

MechanicsisthesciencewhichdealswiththeinteractionbetweenforceandmotionConsequentlythevariableswhichoccurincontinuummechanicsareontheonehandvariablesrelatedtoforces(usuallyforceperunitareaorperunitvolumeratherthanforceitself)andontheotherhandkinematicvariablessuchasdisplacementvelocityandaccelerationInrigid-bodymechanicstheshapeofabodydoesnotchangeandsotheparticleswhichmakeuparigidbodymayonlymoverelativelytooneanotherinaveryrestrictedwayArigidbodyisacontinuumbutitisaveryspecialidealizedanduntypicaloneContinuummechanicsismoreconcernedwithdeformablebodieswhicharecapableofchangingtheirshapeForsuchbodiestherelativemotionoftheparticlesisimportantandthisintroducesassignificantkinematicvariablesthespatialderivativesofdisplacementvelocityandsoon

TheequationsofcontinuummechanicsareoftwomainkindsFirstlythereareequationswhichapplyequallytoallmaterialsTheydescribeuniversalphysicallawssuchasconservationofmassandenergySecondlythereareequationswhichdescribethemechanicalbehaviourofparticularmaterialstheseareknownasconstitutiveequations

TheproblemsofcontinuummechanicsarealsooftwomainkindsThefirstistheformulationofconstitutiveequationswhichareadequatetodescribethemechanicalbehaviourofvariousparticularmaterialsorclassesofmaterialsThisformulationisessentiallyamatterforexperimentaldeterminationbutatheoreticalframeworkisneeededinordertodevisesuitableexperimentsandtointerpretexperimentalresultsThesecondproblemistosolvetheconstitutiveequationsinconjunctionwiththegeneralequationsofcontinuummechanicsandsubjecttoappropriateboundaryconditionstoconfirmthevalidityoftheconstitutiveequationsandtopredictanddescribethebehaviourofmaterialsinsituationswhichareofengineeringphysicalormathematicalinterestAtthisproblem-solvingstagethedifferentbranchesofcontinuummechanicsdivergeandweleavethisaspectofthesubjecttomorecomprehensiveandmorespecializedtexts

2

Introductorymatrixalgebra

21MatricesInthischapterwesummarizesomeusefulresultsfrommatrixalgebraItisassumedthatthereaderisfamiliarwiththeelementaryoperationsofmatrixadditionmultiplicationinversionandtranspositionMostoftheotherpropertiesofmatriceswhichwewillpresentarealsoelementaryandsomeofthemarequotedwithoutproofTheomittedproofswillbefoundinstandardtextsonmatrixalgebra

AnmxnmatrixAisanorderedrectangulararrayofmnelementsWedenote

(21)

sothatAijistheelementintheithrowandthejthcolumnofthematrixATheindexitakesvalues12mandtheindexjtakesvalues12nIncontinuummechanicsthematriceswhichoccurareusuallyeither3x3squarematrices3times1columnmatricesor1x3rowmatricesWeshallusuallydenote3x3squarematricesbybold-faceromancapitalletters(ABCetc)and3x1columnmatricesbybold-faceromanlower-caseletters(abcetc)A1x3rowmatrixwillbetreatedasthetransposeofa3x1columnmatrix(aTbTcTetc)Unlessotherwisestatedindiceswilltakethevalues12and3althoughmostoftheresultstobegivenremaintrueforarbitraryrangesoftheindices

AsquarematrixAissymmetricif

(22)

andanti-symmetricif

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

TheequationsofcontinuummechanicsareoftwomainkindsFirstlythereareequationswhichapplyequallytoallmaterialsTheydescribeuniversalphysicallawssuchasconservationofmassandenergySecondlythereareequationswhichdescribethemechanicalbehaviourofparticularmaterialstheseareknownasconstitutiveequations

TheproblemsofcontinuummechanicsarealsooftwomainkindsThefirstistheformulationofconstitutiveequationswhichareadequatetodescribethemechanicalbehaviourofvariousparticularmaterialsorclassesofmaterialsThisformulationisessentiallyamatterforexperimentaldeterminationbutatheoreticalframeworkisneeededinordertodevisesuitableexperimentsandtointerpretexperimentalresultsThesecondproblemistosolvetheconstitutiveequationsinconjunctionwiththegeneralequationsofcontinuummechanicsandsubjecttoappropriateboundaryconditionstoconfirmthevalidityoftheconstitutiveequationsandtopredictanddescribethebehaviourofmaterialsinsituationswhichareofengineeringphysicalormathematicalinterestAtthisproblem-solvingstagethedifferentbranchesofcontinuummechanicsdivergeandweleavethisaspectofthesubjecttomorecomprehensiveandmorespecializedtexts

2

Introductorymatrixalgebra

21MatricesInthischapterwesummarizesomeusefulresultsfrommatrixalgebraItisassumedthatthereaderisfamiliarwiththeelementaryoperationsofmatrixadditionmultiplicationinversionandtranspositionMostoftheotherpropertiesofmatriceswhichwewillpresentarealsoelementaryandsomeofthemarequotedwithoutproofTheomittedproofswillbefoundinstandardtextsonmatrixalgebra

AnmxnmatrixAisanorderedrectangulararrayofmnelementsWedenote

(21)

sothatAijistheelementintheithrowandthejthcolumnofthematrixATheindexitakesvalues12mandtheindexjtakesvalues12nIncontinuummechanicsthematriceswhichoccurareusuallyeither3x3squarematrices3times1columnmatricesor1x3rowmatricesWeshallusuallydenote3x3squarematricesbybold-faceromancapitalletters(ABCetc)and3x1columnmatricesbybold-faceromanlower-caseletters(abcetc)A1x3rowmatrixwillbetreatedasthetransposeofa3x1columnmatrix(aTbTcTetc)Unlessotherwisestatedindiceswilltakethevalues12and3althoughmostoftheresultstobegivenremaintrueforarbitraryrangesoftheindices

AsquarematrixAissymmetricif

(22)

andanti-symmetricif

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

2

Introductorymatrixalgebra

21MatricesInthischapterwesummarizesomeusefulresultsfrommatrixalgebraItisassumedthatthereaderisfamiliarwiththeelementaryoperationsofmatrixadditionmultiplicationinversionandtranspositionMostoftheotherpropertiesofmatriceswhichwewillpresentarealsoelementaryandsomeofthemarequotedwithoutproofTheomittedproofswillbefoundinstandardtextsonmatrixalgebra

AnmxnmatrixAisanorderedrectangulararrayofmnelementsWedenote

(21)

sothatAijistheelementintheithrowandthejthcolumnofthematrixATheindexitakesvalues12mandtheindexjtakesvalues12nIncontinuummechanicsthematriceswhichoccurareusuallyeither3x3squarematrices3times1columnmatricesor1x3rowmatricesWeshallusuallydenote3x3squarematricesbybold-faceromancapitalletters(ABCetc)and3x1columnmatricesbybold-faceromanlower-caseletters(abcetc)A1x3rowmatrixwillbetreatedasthetransposeofa3x1columnmatrix(aTbTcTetc)Unlessotherwisestatedindiceswilltakethevalues12and3althoughmostoftheresultstobegivenremaintrueforarbitraryrangesoftheindices

AsquarematrixAissymmetricif

(22)

andanti-symmetricif

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

(23)

whereATdenotesthetransposeofA

The3x3unitmatrixisdenotedbyIanditselementsbyδijThus

(24)

where

(25)

Clearlyδij=δjiThesymbolδijisknownastheKroneckerdeltaAnimportantpropertyofδijisthesubstitutionrule

(26)

ThetraceofasquarematrixAisdenotedbytrAandisthesumoftheelementsontheleadingdiagonalofAThusfora3x3matrixA

(27)

Inparticular

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

(28)

WithasquarematrixAthereisassociateditsdeterminantdetAWeassumefamiliaritywiththeelementarypropertiesofdeterminantsThedeterminantofa3x3matrixAcanbeexpressedas

(29)

wherethealternatingsymboleijkisdefinedasa eijk=1if(ijk)isanevenpermutationof(123)(iee123=e231=e312=1)b eijk=ndash1if(ijk)isanoddpermutationof(123)(iee321=e132=e213=ndash1)c eijk=0ifanytwoofijkareequal(ege112=0e333=0)

Itfollowsfromthisdefinitionthateijkhasthesymmetryproperties

(210)

TheconditiondetAne0isanecessaryandsufficientconditionfortheexistenceoftheinverseAndash1ofA

AsquarematrixQisorthogonalifithastheproperty

(211)

ItfollowsthatifQisorthogonalthen

(212)

and

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

(213)

Ourmainconcernwillbewithproperorthogonalmatricesforwhich

detQ=1

IfQ1andQ2aretwoorthogonalmatricesthentheirproductQ1Q2isalsoanorthogonalmatrix

22The summationconventionAveryusefulnotationaldeviceinthemanipulationofmatrixvectorandtensorexpressionsisthesummationconventionAccordingtothisifthesameindexoccurstwiceinanyexpressionsummationoverthevalues12and3ofthatindexisautomaticallyassumedandthesummationsignisomittedThusforexamplein(27)wemayomitthesummationsignandwrite

trA=Aii

Similarlytherelations(26)arewrittenas

δijAjk=AikδijAkj=Aki

andfrom(28)

δii=3

Usingthisconvention(29)becomes

(214)

Theconcisenessintroducedbytheuseofthisnotationisillustratedbytheobservationthatinfulltheright-handsideof(214)contains36=729termsalthoughbecauseofthepropertiesofeijkonlysixofthesearedistinctandnon-zero

Someotherexamplesoftheuseofsummationconventionarethefollowing

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index

a IfA=(Aij)B=(Bij)thentheelementintheithrowandjth3columnoftheproductABisAikBkjwhichiswrittenasAikBki

b Supposethatin(a)aboveB=ATThenBij=AjiandsotheelementintheithrowandjthcolumnofAATisAikAjkInparticularifAisanorthogonalmatrixQ=(Qij)wehavefrom(212)

(215)

c Alinearrelationbetweentwocolumnmatricesxandyhastheform

(216)

whichmaybewrittenas

(217)

IfAisnon-singularthenfrom(216)y=Andash1xInparticularifAisanorthogonalmatrixQthen

d ThetraceofABisobtainedbysettingi=jinthelastexpressionin(a)abovethus

(218)

Byadirectextensionofthisargument

trABC=AijBjkCki

andsoone Ifaandbarecolumnmatriceswith

thenaTbisa1times1matrixwhosesingleelementis

(219)

f Ifaisasin(e)aboveandAisa3x3matrixthenAaisa3x1columnmatrixandtheelementinitsithrowis

AirarwhichiswrittenasAirarg TwousefulrelationsbetweentheKroneckerdeltaandthealternatingsymbolare

(220)

ThesecanbeverifieddirectlybyconsideringallpossiblecombinationsofvaluesofijpqrandsActually(220)areconsequencesofamoregeneralrelationbetweenδijandeijkwhichcanalsobeproveddirectlyandis

(221)

From(214)and(221)wecanobtaintheusefulrelation

(222)

AnindexonwhichasummationiscarriedoutiscalledadummyindexAdummyindexmaybereplacedbyanyotherdummyindexforexampleAii=AjjHoweveritisimportantalwaystoensurethatwhenthesummationconventionisemployednoindexappearsmorethantwiceinanyexpressionbecausetheexpressionisthenambiguous

IntheremainderofthisbookitistobeassumedunlessthecontraryisstatedthatthesummationconventionisbeingemployedThisappliesinsubsequentchapterstoindiceswhichlabelvectorandtensorcomponentsaswellasthosewhichlabelmatrixelements

23Eigenvaluesande igenvectorsIncontinuummechanicsandinmanyothersubjectswefrequentlyencounterhomogeneousalgebraicequationsoftheform

(223)

whereAisagivensquarematrixxanunknowncolumnmatrixandλanunknownscalarIntheapplicationswhichappearinthisbookAwillbea3x3matrixWethereforeconfinethediscussiontothecaseinwhichAisa3x3matrixalthoughthegeneralizationtontimesnmatricesisstraightforwardEquation(223)canbewrittenintheform

(224)

andtheconditionfor(224)tohavenon-trivialsolutionsforxis

(225)

ThisisthecharacteristicequationforthematrixAWhenthedeterminantisexpanded(225)becomesacubicequationforλwiththreerootsλ1λ2λ3whicharecalledtheeigenvaluesofAForthepresentweassumethatλ1λ2andλ3aredistinctThenforexampletheequation

(Andashλ1I)x=0

hasanon-trivialsolutionx(1)whichisindeterminatetowithinascalermultiplierThecolumnmatrixx(1)istheeigenvectorofAassociatedwiththeeigenvalueλ1eigenvectorsx(2)andx(3)associatedwiththe

eigenvaluesλ2andλ3aredefinedsimilarly

Sinceλ1λ2λ3aretherootsof(225)andthecoefficientofλ3ontheleftof(225)is-1wehave

(226)

Thisisanidentityinλsoitfollowsbysettingλ=0that

(227)

NowsupposethatAisarealsymmetricmatrixThereisnoapriorireasontoexpectλ1andx(1)toberealSupposetheyarecomplexwithcomplexconjugates 1and (1)Then

(228)

Transposing(228)andtakingitscomplexconjugategives

(229)

Nowmultiply(228)ontheleftby (1)Tand(229)ontherightbyx(1)andsubtractThisgives

(230)

Sincex(1)isanon-trivialsolutionof(224) (1)Tx(1)ne0andsoλ1= 1Hencetheeigenvaluesofarealsymmetricmatrixarereal

Alsofrom(228)

(231)

andsimilarly

(232)

Nowtranspose(231)andsubtracttheresultingequationfrom(232)Thisgives

(233)

Hencetheeigenvectorsassociatedwithtwodistincteigenvaluesλ1andλ2ofasymmetricmatrixAhavethepropertyx(1)Tx(2)=0TwocolumnmatriceswiththispropertyaresaidtobeorthogonalIngeneraliftheeigenvaluesaredistinctthen

(234)

Byappropriatechoiceofthescalarmultipliertheeigenvectorx(1)canbenormalizedsothatx(1)Tx(1)=1Ingeneralwecannormalizetheeigenvectorssothat

(235)

Strictlyspeakingtheright-handsidesof(234)and(235)are1x1matricesbutformostpurposestheymaybetreatedasscalarsNowconstructa3x3matrixPwhoserowsarethetransposesofthenormalizedeigenvectorsx(1)x(2)x(3)

(236)

Thenitfollowsfrom(234)and(235)thatPPT=IandsoPisanorthogonalmatrixAlsousing(228)andanalogousrelationsforx(2)andx(3)

(237)

andhencefrom(235)(236)and(237)

(238)

ThusPAPTisadiagonalmatrixwiththeeigenvaluesofAastheelementsonitsleadingdiagonal

ItcanbeshownthatifAissymmetricandλ1=λ2neλ3thenthenormalizedeigenvectorx(3)isuniquelydeterminedandx(1)andx(2)maybeanytwocolumnmatricesorthogonaltox(3)Ifx(1)andx(2)arechosentobemutuallyorthogonalthentheresults(233)ndash(238)remainvalidIfλ1=λ2=λ3thenAisdiagonalAnycolumnmatrixwithatleastonenon-zeroelementisaneigenvectorandtheresultsremaintruethoughtrivialifx(1)x(2)andx(3)arechosenasanythreemutuallyorthogonalnormalizedcolumnmatrices

From(223)itfollowsthat

(239)

HenceifλisaneigenvalueofAandxisthecorrespondingeigenvectorthenλ2isaneigenvalueofA2andxisthecorrespondingeigenvectorMoregenerallyλnisaneigenvalueofAnandxisthecorrespondingeigenvectorIfAisnon-singularthisresultholdsfornegativeaswellasforpositiveintegersn

24The CayleyndashHamiltontheoremFrom(238)weseethat

trPAPT=λ1+λ2+λ3tr(PAPT)2=

NowsincePisorthogonalitfollowsfrom(215)that

Hence

(240)

From(225)and(226)

λ3ndash(λ1+λ2+λ3)λ2+λ2λ3+λ3λ1+λ1λ2)λndashλ1λ2λ3=0

Hencefrom(227)and(240)thecharacteristicequationcanbeexpressedintheform

(241)

TheCayleyndashHamiltontheoremstatesthatasquarematrixsatisfiesitsowncharacteristicequationthusforany3x3matrixA

(242)

ThetheoremmaybeprovedinseveralwaysProofswillbefoundinstandardalgebratexts

25The polardecompositiontheoremAmatrixAispositivedefiniteifxTAxispositiveforallnon-zerovaluesofthecolumnmatrixxAnecessaryandsufficientconditionforAtobepositivedefiniteisthattheeigenvaluesofAareallpositive

Thepolardecompositiontheoremstatesthatanon-singularsquarematrixFcanbedecomposeduniquelyintoeitheroftheproducts

(243)

whereRisanorthogonalmatrixandUandVarepositivedefinitesymmetricmatricesWeoutlinetheprooffor3x3matriceswhichisthecasewerequireThegeneralizationtontimesnmatricesisstraightforward

LetC=FTFandlet =FxThenCissymmetricandalso

But isasumofsquaresandsoispositiveforallnon-zerocolumnmatrices andhencexTCxispositiveforallnon-zeroxThusCispositivedefiniteandhaspositiveeigenvalueswedenotetheseby wherewithoutlossofgeneralityλ1λ2andλ3arepositiveBytheresultsofSection23if

PTdenotesthematrixwhosecolumnsarethenormalizedeigenvectorsofCthenPisorthogonaland

Wedefine

(244)

ThenUissymmetricandpositivedefiniteandalsosincePisorthogonal

(245)

WefurtherdefineR=FUndash1TheninordertoprovetheexistenceofthefirstdecompositionitisonlynecessarytoshowthatRisorthogonalNowfrom(243)and(245)

RTR=Undash1FTFUndash1=Undash1CUndash1=Undash1U2Undash1=I

andsoRisindeedorthogonalThematrixVisthendefinedbyV=RURT

ToproveuniquenesssupposethereexistsanotherdecompositionF=R1U1whereR1isorthogonalandU1ispositivedefiniteThen and

Hence

HowevertheonlyoneofthesematricesU1whichispositivedefiniteistheoneinwhichthepositivesignsaretakenHenceU1=UTheuniquenessofRandVthenfollowsfromtheirdefinitions

TheaboveproofproceedsbyconstructingthematricesURandVwhichcorrespondtoagivenmatrixFThusinprincipleitgivesamethodofdeterminingURandVInpracticethecalculationsarecumbersomeevenfora3x3matrixFFortunatelyforapplicationsincontinuummechanicsitisusuallysufficienttoknowthattheuniquedecompositionsexistanditisnotoftennecessarytocarrythemoutexplicitly

3

Vectorsandcartesiantensors

31VectorsWeassumefamiliaritywithbasicvectoralgebraandanalysisInthefirstpartofthischapterwedefinethenotationandsummarizesomeofthemoreimportantresultssothattheyareavailableforfuturereference

Weconsidervectorsinthree-dimensionalEuclideanspaceSuchvectorswill(withafewexceptionswhichwillbenotedastheyoccur)bedenotedbylower-casebold-faceitalicletters(abxetc)WemakeadistinctionbetweencolumnmatriceswhicharepurelyalgebraicquantitiesintroducedinChapter2andvectorswhichrepresentphysicalquantitiessuchasdisplacementvelocityaccelerationforcemomentumandsoonThisdistinctionisreflectedinouruseofromanbold-facetypeforcolumnmatricesanditalicbold-facetypeforvectors

Thecharacteristicpropertiesofavectorare(a)avectorrequiresamagnitudeandadirectionforitscompletespecificationand(b)twovectorsarecompoundedinaccordancewiththeparallelogramlawThustwovectorsaandbmayberepresentedinmagnitudeanddirectionbytwolinesinspaceandifthesetwolinesaretakentobeadjacentsidesofaparallelogramthevectorsuma+bisrepresentedinmagnitudeanddirectionbythediagonaloftheparallelogramwhichpassesthroughthepointofintersectionofthetwolines

Supposethereissetupasystemofrectangularright-handedcartesiancoordinateswithoriginOLete1e2e3denotevectorsofunitmagnitudeinthedirectionsofthethreecoordinateaxesThene1e2e3arecalledbasevectorsofthecoordinatesystemByvirtueoftheparallelogramadditionlawavectoracanbeexpressedasavectorsumofthreesuchunitvectorsdirectedinthethreecoordinatedirectionsThus

(31)

whereinthelastexpression(andinfuturewheneveritisconvenient)thesummationconventionisemployedThequantitiesai(i=123)arethecomponentsofainthespecifiedcoordinatesystemtheyarerelatedtothemagnitudeaofaby

(32)

InparticularavectormaybethepositionvectorxofapointPrelativetoOThenthecomponentsx1x2x3ofxarethecoordinatesofPinthegivencoordinatesystemandthemagnitudeofxisthelengthOP

Thescalarproductamiddotbofthetwovectorsabwithrespectivemagnitudesabwhosedirectionsareseparatedbyanangleθisthescalarquantity

(33)

Ifaandbareparallelthenamiddotb=abandifaandbareatrightanglesamiddotb=0Inparticular

Thatis

(34)

Thevectorproductatimesbofaandbisavectorwhosedirectionisnormaltotheplaneofaandbinthesenseofaright-handedscrewrotatingfromatobandwhosemagnitudeisabsinθIntermsofcomponentsatimesbcanconvenientlybewrittenas

(35)

whereitisunderstoodthatthedeterminantexpansionistobebythefirstrowByusingthealternatingsymboleijk(35)canbewrittenas

(36)

Thetriplescalarproduct(axb)middotcisgivenincomponentsas

(37)

32Coordinate transformationAvectorisaquantitywhichisindependentofanycoordinatesystemIfacoordinatesystemisintroducedthevectormayberepresentedbyitscomponentsinthatsystembutthesamevectorwillhavedifferentcomponentsindifferentcoordinatesystemsSometimesthecomponentsofavectorinagivencoordinatesystemmayconvenientlybewrittenasacolumnmatrixbutthismatrixonlyspecifiesthevectorifthecoordinatesystemisalsospecified

SupposethecoordinatesystemistranslatedbutnotrotatedsothattheneworiginisOprimewhereOprimehaspositionvectorx0relativetoOThenthepositionvectorxprimeofPrelativetoOprimeis

xprime=xndashx0

Inatranslationwithoutrotationthebasevectorse1e2e3areunchangedandsothecomponentsaiofavectoraarethesameinthesystemwithoriginOprimeastheywereinthesystemwithoriginO

Nowintroduceanewrectangularright-handedcartesiancoordinatesystemwiththesameoriginOastheoriginalsystemandbasevectorsē1ē2ē3ThenewsystemmayberegardedashavingbeenderivedfromtheoldbyarigidrotationofthetriadofcoordinateaxesaboutOLetavectorahavecomponentsaiintheoriginalcoordinatesystemandcomponentsāiinthenewsystemThus

(38)

NowdenotebyMijthecosineoftheanglebetweenēiandejsothat

(39)

ThenMij(ij=123)arethedirectioncosinesofēirelativetothefirstcoordinatesystemorequivalentlyMijarethecomponentsofēiinthefirstsystemThus

(310)

ItisgeometricallyevidentthattheninequantitiesMijarenotindependentInfactsinceēiaremutuallyorthogonalunitvectorswehaveasin(34)ēimiddotēj=δijHoweverfrom(34)and(310)

Hence

(311)

Sinceδij=δji(311)representsasetofsixrelationsbetweentheninequantitiesMijNowregardMijastheelementsofasquarematrixMThen(311)isequivalenttothestatement

(312)

ThusM=(Mij)isanorthogonalmatrixthatisthematrixwhichdeterminesthenewbasevectorsintermsoftheoldbasevectorsisanorthogonalmatrixForatransformationfromoneright-handedsystemtoanotherright-handedsystemMisaproperorthogonalmatrixTherowsofMarethedirectioncosinesofēiinthefirstcoordinatesystem

SinceMisorthogonalthereciprocalrelationto(310)is

(313)

andsothecolumnsofMarethedirectioncosinesoftheejinthecoordinatesystemwithbasevectorsēi

Nowfrom(38)and(313)

Thus

(314)

ThisgivesthenewcomponentsāiexclofaintermsofitsoldcomponentsajandtheelementsoftheorthogonalmatrixMwhichdeterminesthenewbasevectorsintermsoftheoldSimilarlyfrom(38)and(310)

(315)

InparticularifaisthepositionvectorxofthepointPrelativetotheoriginOthen

(316)

wherexiexclandxiarethecoordinatesofthepointPinthefirstandsecondcoordinatesystemsrespectively

Thetransformationlaw(314)and(315)isaconsequenceoftheparallelogramlawofadditionofvectorsandcanbeshowntobeequivalenttothislawThusavectorcanbedefinedtobeaquantitywithmagnitudeanddirectionwhich(a)compoundsaccordingtotheparallelogramlaworequivalently(b)canberepresentedbyasetofcomponentswhichtransformas(314)underarotationofthecoordinatesystem

IntheforegoingdiscussionwehaveadmittedonlyrotationsofthecoordinatesystemsothatMisaproperorthogonalmatrix(detM=1)Ifwealsoconsidertransformationsfromaright-handedtoaleft-handedcoordinatesystemforwhichMisanimproperorthogonalmatrix(detM=ndash1)thenitbecomesnecessarytodistinguishbetweenvectorswhosecomponentstransformaccordingto(314)andpseudo-vectorswhosecomponentstransformaccordingtotherule

(317)

Examplesofpseudo-vectorsarethevectorproductatimesboftwovectorsaandbtheangularvelocity

vectortheinfinitesimalrotationvector(Section67)andthevorticityvector(Section69)Thedistinctionbetweenvectorsandpseudo-vectorsonlyarisesifleft-handedcoordinatesystemsareintroducedanditwillnotbeofimportanceinthisbook

ItisevidentfromthedefinitionofthescalarproductamiddotbthatitsvaluemustbeindependentofthechoiceofthecoordinatesystemToconfirmthisweobservefrom(314)that

(318)

Aquantitysuchasaibiwhosevalueisindependentofthecoordinatesystemtowhichthecomponentsarereferredisaninvariantofthevectorsaandb

AsthevectorproductisalsodefinedgeometricallyitmusthaveasimilarinvariancepropertyInfactfrom(222)(310)and(314)wehave

(319)

providedthatdetM=+1

Thereaderwillobservetheadvantagesofusingthesummationconventioninequationssuchas(318)and(319)Notonlydoesthisnotationallowlengthysumstobeexpressedconcisely(forexamplethethirdexpressionin(318)representsasumof27terms)butitalsorevealsthestructureofthesecomplicatedexpressionsandsuggeststhewaysinwhichtheymaybesimplified

33The dyadic productTherearesomephysicalquantitiesapartfromquantitieswhichcanbeexpressedasscalarorvectorproductswhichrequirethespecificationoftwovectorsfortheirdescriptionForexampletodescribetheforceactingonasurfaceitisnecessarytoknowthemagnitudeanddirectionoftheforceandtheorientationofthesurfaceSomequantitiesofthiskindcanbedescribedbyadyadicproduct

ThedyadicproductoftwovectorsaandbiswrittenaotimesbIthastheproperties

(320)

whereαisascalarItfollowsthatintermsofthecomponentsofaandbaotimesbmaybewritten

(321)

WenotethatingeneralaotimesbnebotimesaTheformof(321)isindependentofthechoiceofcoordinatesystemfor

(322)

ThedyadicproductseiotimesejofthebasevectorseiarecalledunitdyadsInadditionto(320)theessentialpropertyofadyadicproductisthatitformsaninnerproductwitha

vectorasfollows

(323)

Sincethereisnopossibilityofambiguitythebracketsontheleft-handsidesof(323)maybeomittedandwecanwrite

(324)

Hence(324)canbewrittenintermsofcomponentsas

(325)

Formallyamiddotbmaybeinterpretedasthescalarproductevenwhenaorbformpartofadyadicproduct

TheconceptofadyadicproductcanbeextendedtoproductsofthreeormorevectorsForexampleatriadicproductofthevectorsabandciswrittenaotimesbotimescandcanbeexpressedincomponentformasaibjckeiotimesejotimesek

34CartesiantensorsWedefineasecond-ordercartesiantensortobealinearcombinationofdyadicproductsAsadyadicproductisby(321)itselfalinearcombinationofunitdyadsasecond-ordercartesiantensorAcanbeexpressedasalinearcombinationofunitdyadssothatittakestheform

(326)

Asaruleweshallusebold-faceitaliccapitalstodenotecartesiantensorsofsecond(andhigher)orderAstheonlytensorswhichwillbeconsideredinthisbookuntilChapter11willbecartesiantensorsweshallomittheadjectivelsquocartesianrsquoInChapters3-10thetermlsquotensorrsquomeanslsquocartesiantensorrsquo

ThecoefficientsAijarecalledthecomponentsofA(Whereverpossibletensorcomponentswillbedenotedbythesameletterinitaliccapitalsasisusedtodenotethetensoritself)BythemannerofitsdefinitionatensorexistsindependentlyofanycoordinatesystemHoweveritscomponentscanonlybespecifiedafteracoordinatesystemhasbeenintroducedandthevaluesofthecomponentsdependonthechoiceofthecoordinatesystemSupposethatinanewcoordinatesystemwithbasevectorsēiAhascomponentsĀijThen

(327)

Howeverfrom(313)

Hence

(328)

Thisisthetransformationlawforcomponentsofsecond-ordertensorsItdependsonthecomposition

rule(320)andcanbeshowntobeequivalenttothisruleThus(328)maybeusedtoformulateanalternativedefinitionofasecond-ordertensorInordertoidentifyasecond-ordertensorassuchitissufficienttoshowthatinanytransformationfromonerectangularcartesiancoordinatesystemtoanotherthecomponentstransformaccordingto(328)Incontinuummechanicstensorsareusuallyrecognizedbythepropertythattheircomponentstransforminthismanner

Moregenerallyacartesiantensoroforderncanbeexpressedincomponentsas

(329)

anditscomponentstransformaccordingtotherule

(330)

ThusavectorcanbeinterpretedasatensoroforderoneAscalarwhichhasasinglecomponentwhichisunchangedinacoordinatetransformationcanberegardedasatensoroforderzeroNearlyallofthetensorsweencounterinthisbookwillbeoforderzero(scalars)one(vectors)ortwo

Theinverserelationto(328)is

(331)

andtheinverseof(330)is

(332)

SupposethatA=Aijeiotimesej=Āpqēpotimesēqisasecond-ordertensorandthatAij=AjiThenfrom(328)

(333)

ThusthepropertyofsymmetrywithrespecttointerchangeoftensorcomponentindicesispreservedundercoordinatetransformationsandsoisapropertyofthetensorAAtensorAwhosecomponentshavethepropertyAiexclj=Aji(inanycoordinatesystem)isasymmetricsecond-ordertensorManyofthesecond-ordertensorswhichoccurincontinuummechanicsaresymmetric

SimilarlyifAij=ndashAjithenĀij=ndashĀjiandAisanantisymmetricsecond-ordertensor

Letusdenote =Ajiand Thenfrom(328)

(334)

HencethesetofcomponentsAjialsotransformasthecomponentsofasecond-ordertensorThusfromthetensorA=AijejotimesejwecanformanewtensorAjiexcleiotimesejwhichwedenotebyATandcallthetransposeofAThetensorA+ATissymmetricandthetensorAndashATisanti-symmetricSince

(335)

anysecond-ordertensorcanbedecomposedintothesumofasymmetricandananti-symmetrictensorandthisdecompositionisunique

35Isotropic tensorsThetensorI=δijeiotimesejiscalledtheunittensorIntermsofanothersetofbasevectorsēiwehavefrom(313)

ThusthetensorIhasthepropertythatitscomponentsareδijinanycoordinatesystemAtensorwhosecomponentsarethesameinanycoordinatesystemiscalledanisotropictensorItcanbeshownthattheonlyisotropictensorsofordertwoareoftheformpIwherepisascalarSuchtensorsaresometimescalledsphericaltensors

Similarlyitcanbeverifiedthatthealternatingtensor

(336)

isanisotropictensoroforderthreeprovidedthatonlycoordinatetransformationswhichcorrespondtoproperorthogonalmatrices(thatisrotations)areallowedAnythird-orderisotropictensorisamultipleof(336)Therearethreelinearlyindependentfourth-orderisotropictensorswhichmaybetakentobe

andsothemostgeneralfourth-orderisotropictensorhastheform

(337)

whereAμandνarescalars

36MultiplicationoftensorsLeta=aieiandB=Bijeiexclotimesejbeavectorandasecond-ordertensorrespectivelywithrespectivecomponentsaiandBijinacoordinatesystemwithbasevectorseiSupposethatinanewsystemwithbase

vectorsēi=MiexcljejaandBhavecomponentsāiand respectivelysothat

InadditionletCijk=aiBjkandconsiderthetensor

C=Cijkeiotimeseiotimesek

ThecomponentsofCreferredtobasevectorsēiexclare where

(338)

ThetensorCiscalledtheouterproductofthevectoraandthetensorB(inthatorder)andiswrittenaotimesBEquation(338)showsthatthecomponentsofCarerelatedtothoseofaandBinthesamewayinanycoordinatesystem

SimilarlyifAandBaresecond-ordertensorswithrespectivecomponentsAijandBijinthesystem

withbasevectorseithentheouterproductD=AotimesBisthefourth-ordertensorwithcomponentsDijkl=AijBklinthissystemandunderacoordinatetransformationthecomponentsofDtransformto

OuterproductsofthreeormoretensorsorvectorsareformedinasimilarwayandtheextensiontotensorsofhigherorderisdirectTheouterproductofatensorofordermwithatensorofordernisatensoroforderm+n(vectorsareregardedastensorsoforderone)Thedyadicproductoftwovectorsistheouterproductofthosevectors

ContractionNowconsiderathird-ordertensorCiexcljkeiexclotimesejotimesekThecomponentsCijktransformaccordingtotherule

Wenowsumonthelasttwoindicesof thatisweformthethreesums

FormallythisisaccomplishedbysettingthesecondandthirdindicesofCijkequaltoeachotherThen

(339)

ThusthecomponentsCprrtransformasthecomponentsofavectorMoregenerallyifDijpqrsarecomponentsofatensorofordernandwesumonanypairofitsindicessoastoformforexampleDijpprstheresultingquantitiesarethecomponentsofatensorofordernndash2ThisoperationofreducingtheorderofatensorbytwobysummingonapairofindicesiscalledcontractionofthetensorInparticularifAijarecomponentsofasecond-ordertensorthenAiiisascalar

AcontractionmaybeperformedonindicesoftwotensorswhicharefactorsinanouterproductThusifaiarecomponentsofavectoraandBijarecomponentsofasecond-ordertensorBthenaiBijarecomponentsofavectorandsoareBijajWecallthesevectorsinnerproductsofaandBandwrite

(340)

NotethatamiddotB=BmiddotaonlyifBisasymmetrictensor

Innerproductsofsecond-andhigher-ordertensorsareformedinasimilarwayLetAandBbesecond-ordertensorswithcomponentsAijandBijrespectivelyFromthemwecanformvariousinnerproductswhicharesecond-ordertensorsforexample

(341)

Wenoteforexamplethat

(AmiddotB)T=BTmiddotAT

AsaspecialcasethetensorsAandBmaybethesametensorThetensorAmiddotAisdenotedbyA2

IfthereexistsatensorAndash1suchthat

(342)

thenAndash1iscalledtheinversetensortoA

IfthetensorsATandAndash1areequalsothat

(343)

thenAissaidtobeanorthogonaltensor

Byusingthepolardecompositiontheorem(Section25)thecomponentsFijofasecond-ordertensorFcan(providedthatdet(Fij)ne0)bedecomposeduniquelyintheforms

Fij=RikUkjFij=VikRkj

whereRikareelementsofanorthogonalmatrixandUijandVijareelementsofpositivedefinitesymmetricmatricesWedefinethesecond-ordertensorsRUandVtobe

R=RijeiotimesejU=UijeiotimesejV=Vijeiotimesej

ThenRisanorthogonaltensorandUandVaresymmetrictensorsand

RmiddotU=RikUkjeiotimesej=Fijeiexclotimesej=F

and

VmiddotR=VikRkjeiexclotimesej=Fijeiexclotimesej=F

ThusthetensorFcanbedecomposedintoeitheroftheinnerproducts

(344)

37TensorandmatrixnotationRelationsbetweentensorquantitiesmaybeexpressedeitherindirectformasrelationsbetweenscalarsαβvectorsabandtensorsABorincomponentformasrelationsbetweenscalarsαβvectorcomponentsaibiandtensorcomponentsAijBijThedirectnotationhastheadvantagethatitemphasizesthatphysicalstatementsareindependentofthechoiceofthecoordinatesystemHoweverthisadvantageisnotentirelylostwhenthecomponentnotationisusedbecauserelationsincomponentnotationmustbewritteninsuchawaythattheypreservetheirformundercoordinatetransformationsThecomponentformusedinconjunctionwiththesummationconventionisoftenconvenientforcarryingoutalgebraicmanipulationsandinconsideringspecificproblemsitisalwaysnecessaryatsomestagetointroduceacoordinatesystemandcomponentsSomeexamplesoftheinterchangebetweenthedifferentformsaregiveninTable31Inthisbookweemploybothnotationsasconvenient

WhenitisnecessarytotransformcomponentsfromonecoordinatesystemtoanotheritisoftenconvenienttointroducematrixnotationSupposethataisavectorandAisasecond-ordertensorLetaandAhavecomponentsaiandAijrespectivelyinacoordinatesystemwithbasevectorseiandcomponentsāiandĀiexcljrespectivelyinacoordinatesystemwithbasevectorsēiwhereasin(310)ēi=MijejandMijareelementsofanorthogonalmatrixMThenthetransformationrules(314)and(328)forthecomponentsofaandAare

(345)

Thecomponentsaiandāimaybearrangedastheelementsoftwo3x1columnmatricesaandāthus

(346)

andthecomponentsAijandĀijmaybearrangedaselementsoftwo3x3matricesAandĀthus

(347)

Thenthetransformationrules(345)maybewritteninmatrixnotationas

(348)

SinceMisorthogonalweimmediatelyobtainthereciprocalrelations

(349)

MatrixnotationisalsousefulincarryingoutalgebraicmanipulationswhichinvolvecomponentsofvectorandtensorproductsInTable31welistanumberofexamplesofvectorandtensorequationsexpressedindirectnotationcomponentnotationandmatrixnotationInTable31αisascalaraandbarevectorswithcomponentsaiandbirespectivelyandABCDaresecond-ordertensorswithcomponentsAijBijCijDijrespectivelyAlsoaandbare3x1columnmatriceswithelementsaiandbirespectivelyandABCDare3times3matriceswithelementsAijBijCijDijrespectively

Table31Examplesoftensorandmatrixnotation

Direct tensor nota ti on Tensor componentnota ti on Matr ixnota ti on

α=abullb α=aibi (α)=aTb

A=aotimesb Aij=aibj A=abT

b=Abulla bi=Aijaj b=Aa

b=amiddotA bj=aiAij bT=aTA

α=abullAbull α=aiAijbj (α)=aTAb

C=AbullB Cij=AikBkj C=AB

C=AbullBT Cij=AikBjk C=ABT

D=AbullBbullC Dij=AikBkmCmj D=ABC

SinceAA-1=A-1A=IitfollowsthatifAisthematrixofcomponentsofAthenA-1isthematrixofcomponentsofA-1inthesamecoordinatesystemHencethetensorA-1existsonlyifdetAne0

ItisimportantnottoconfusethevectorawiththecolumnmatrixanorthetensorAwiththesquarematrixAInagivencoordinatesystemthematrixaservestodescribethevectoraHoweverthevectorisrepresentedbydifferentmatricesindifferentcoordinatesystemswhereasthevectoritselfisindependentofthecoordinatesystemSimilarlythematrixAdescribesthetensorAinagivencoordinatesystembutAhasdifferentmatrixrepresentationsindifferentcoordinatesalthoughAitselfisindependentofthecoordinatesystem

38Invariantsofa second-ordertensorLetAbeasecond-ordertensorwithcomponentsAijinthecoordinatesystemwithbasevectorseiandcomponentsĀijinthecoordinatesystemwithbasevectorsēi=MijejAlsoletA=(Aij)Ā=(Āij)andM=(Mij)SupposethatλisaneigenvalueofĀsothat

det(Ā-λI)=0

ThenĀ=MAMTandMisanorthogonalmatrixTherefore

detM(A-λI)MT=0

andhence

detMdet(A-λI)detM=0

HoweversinceMisanorthogonalmatrix(detM)2=1andso

det(A-λI)=0

HenceλisalsoaneigenvalueofAThustheeigenvaluesofthematrixofcomponentsofAareindependentofthecoordinatesystemtowhichthesecomponentsarereferredTheeigenvaluesareintrinsictothetensorAifAissymmetrictheyarerealnumbers(cfSection23)andtheyarethencalledtheprincipalcomponentsortheprincipalvaluesofAWedenotetheprincipalvaluesofAbyA1A2andA3IfA1A2andA3areallpositivethenAisapositivedefinitetensor

SupposethatAissymmetricIfA1A2andA3aredistinctthenthenormalizedeigenvectorsx(1)x(2)

andx(3)ofAareuniqueandmutuallyorthogonaland

Ax(i)=Aix(i)(i=123nosummation)

AlsosinceMisanorthogonalmatrixitfollowsthat

Henceifthevectorsxiaredefinedas

(350)

thenwehave

Amiddotxi=Aixi(nosummation)

LetusreferAtoacoordinatesysteminwhichxiarethebasevectorssothatwenowidentifyēiwithxiThenfrom(350)thematrixPofthetransformationfromcoordinateswithbasevectorseitocoordinateswithbasevectorsxiis(Pij)where

Therefore(cfSection23)from(238)and(348)

(351)

Thusthereexistsacoordinatesysteminwhichthematrixofcomponentsofasymmetricsecond-ordertensorAisadiagonalmatrixwhosediagonalelementsaretheprincipalvaluesofAThiscoordinatesystemhasbasevectorsxiItsaxesaretheprincipalaxesofAandtheirdirectionsaretheprincipaldirectionsofA

TheseresultsremainvalidifA1A2andA3arenotalldistinctIfA1=A2neA3thenthevectorx3isuniquelydeterminedandx1andx2maybetakentobeanytwounitvectorswhichareorthogonaltoeachotherandtox3IfA1=A2=A3thentheprincipalaxesmaybetakentobeanythreemutuallyorthogonalaxesandAisasphericaltensor

Ifforexampletheprincipalaxisdeterminedbyx3coincideswiththebasevectore3thenA13=0A23=0ConverselyifA13=A23=0thenthedirectionofx3isaprincipaldirection

Itfollowsfrom(239)thattheprincipalvaluesofA2are and MoregenerallytheprincipalvaluesofAnare and ThisholdsfornegativeaswellaspositiveintegersnprovidedthatA1A2andA3areallnon-zeroTheprincipalaxesofAncoincidewiththoseofA

ItwasemphasizedabovethattheprincipalvaluesofAareindependentofthechoiceofthecoordinatesystemtheyareinvariantsofthetensorAInvariantsplayanimportantroleincontinuummechanicsItcanbeshownthatifAissymmetricthenA1A2andA3arebasicinvariantsinthesensethatanyinvariantofAcanbeexpressedintermsofthemInmanyapplicationsitismoreconvenienttochooseasthebasicinvariantsthreesymmetricfunctionsofA1A2andA3ratherthantheprincipalvaluesthemselvesThreesuchsymmetricfunctionsare

(352)

Thesethreequantitiesareclearlyinvariantsandtheyareindependentinthesensethatnooneofthemcanbeexpressedintermsoftheothertwo

Theconvenienceoftheset(352)resultspartlybecausetheycanbecalculatedfromthetensorcomponentsinanycoordinatesystemwithoutgoingthroughthetediouscalculationofA1A2andA3We

seefrom(351)that

HoweversincePisorthogonal

(353)

Thusthefirstoftheinvariants(352)isequalinanycoordinatesystemtothetraceofthematrixofcomponentsofASimilarly

(354)

andinasimilarwayitfollowsthat

SincetrAisindependentofthechoiceofthecoordinatesystemwecanwithoutambiguitydefinetrA=trASimilarlywedefinetrA2=trA2andtrA3=trA3sothatthesetofinvariants(352)maybeexpressedas

(355)

Onlymatrixmultiplicationsareneededinordertocalculatetheset(355)

AnothersetofsymmetricfunctionsofA1A2andA3isI1I2I3where

(356)

TheseareclearlyinvariantquantitiesI2canbeexpressedintermsofcomponentsofĀasfollows

ForI3wehave

HencewithoutambiguitywemaydefinedetA=detA=I3andasetofthreeindependentinvariantsofA(andthesetusuallyusedinpractice)isI1I2I3where

(357)

From(242)weseethattheCayley-HamiltontheoremforAcanbeexpressedas

(358)

Bytakingthetraceof(358)andrememberingthattrI=3therefollowsanalternativeexpressionforI3=detA

(359)

39Deviatoric tensorsThetensor

(360)

hasthepropertythatitsfirstinvarianttrAprimeiszeroThusifAprimeissymmetricithasonlyfiveindependentcomponentsandonlytwoindependentnon-zeroinvariantsAtensorwhosetraceiszeroiscalledadeviatorictensorandAprimeiscalledthedeviatorofAItissometimesusefulincontinuummechanicstodecomposeatensorintothesumofitsdeviatorandasphericaltensorasfollows

(361)

Thetwonon-zeroinvariantsofAprimeare

(362)

Aftersomemanipulationitcanbeshownfrom(357)and(360)that

(363)

Thus and canbeexpressedintermsofI1I2andI3AlternativelyI2andI3canbeexpressedintermsofI1 and andsoI1I2 maybeadoptedasasetofbasicinvariantsforAwhichisequivalenttothesetI1I2I3

310VectorandtensorcalculusWeassumefamiliaritywithelementaryvectoranalysisandgiveonlyasummarywithoutproofofresultswhichwillbeneeded

Ifφ(x1x2X3)isascalarfunctionofthecoordinatesthen

(364)

isthegradientofφandisavectorgradφisavectorwhosedirectionisnormaltoalevelsurfaceφ(x1x2x3)=constantandwhosemagnitudeisthedirectionalderivativeofφinthedirectionofthisnormal

Ifa(x1x2x3)=ai(xj)eiisavectorfunctionofthecoordinatesthen

(365)

isthedivergenceofaandisascalarAlso

(366)

isthecurlofaandisavectorInthesymbolicdeterminantin(366)theexpansionistobecarriedoutbythefirstrow

Incontinuummechanicswemakefrequentuseofthedivergencetheorem(orGaussrsquostheorem)whichstatesthatifthevectorfieldahascontinuousfirst-orderpartialderivativesatallpointsofaregionℛboundedbyasurface then

(367)

wheredVanddSareelementsofvolumeandofsurfacearearespectivelyandnistheoutwardnormaltoIntermsofcomponents(367)takestheform

(368)

ThedivergencetheoremcanalsobeappliedtotensorsForexampleifAisasecond-ordertensorwith

componentsAijthen

(369)

andanalogousresultsholdfortensorsofhigherorder

4

Particlekinematics

41Bodiesandthe irconfigurationsKinematicsisthestudyofmotionwithoutregardtotheforceswhichproduceitInthischapterwediscussthemotionofindividualparticles(althoughtheseparticlesmayformpartofacontinuousbody)withoutreferencetothemotionofneighbouringparticlesThedeformationorchangeofshapeofabodydependsonthemotionofeachparticlerelativetoitsneighboursandwillbeanalysedinChapters6and9

WeintroduceafixedrectangularcartesiancoordinatesystemwithoriginOandbasevectorseiThroughoutChapters4to10allmotionwillbemotionrelativetothisfixedframeofreferenceandunlessotherwisestatedallvectorandtensorcomponentsarecomponentsinthecoordinatesystemwithbasevectorseiTimeismeasuredfromafixedreferencetimet=0Suppose(seeFig41)thatatt=0afixedregionofspaceℛ0whichmaybefiniteorinfiniteinextentisoccupiedbycontinuouslydistributedmatterthatiswesupposethateachpointofℛ0isoccupiedbyaparticleofmatterThematerialwithinℛ0att=0formsabodywhichisdenotedbyℬLetXbethepositionvectorrelativetoOofatypicalpointPowithinℛ0ThenthecomponentsXRofXinthechosencoordinatesystemarethecoordinatesofthepositionoccupiedbyaparticleofℬatt=0Eachpointoftheregionℛ0correspondstoaparticleofthebodyℬandℬistheassemblageofallsuchparticlesSupposethatthematerialwhichoccupiestheregionℛ0att=0movessothatatasubsequenttimetit

occupiesanewcontinuousregionofspaceℛandthatthematerialisnowcontinuouslydistributedinℛThisistermedamotionofthebodyℬWemaketheassumption(whichisanessentialfeatureofcontinuummechanics)thatwecanidentifyindividualparticlesofthebodyℬthatisweassumethatwecanidentifyapointofℛ(denotedbyP)withpositionvectorxwhichisoccupiedattbytheparticlewhichwasatP0atthetimet=0ThenthemotionofℬcanbedescribedbyspecifyingthedependenceofthepositionsxoftheparticlesofℬattimetontheirpositionsXattimet=0thatisbyequationsoftheform

(41)

forallXinℛ0andallxinℛIfxidenotethecomponentsofx(thatisthecoordinatesofpointsinℛ)then(41)maybewrittenincomponentformas

(42)

Figure41Referenceandcurrentconfigurationsofthebodyℬ

AgivenparticleofthebodyℬmaybedistinguishedbyitscoordinatesXRatt=0ThusthecoordinatesXRserveaslsquolabelsrsquowithwhichtoidentifytheparticlesofℬaparticularparticleretainsthesamevaluesofXRthroughoutamotionThecoordinatesxiontheotherhandidentifypointsofspacewhichingeneralareoccupiedbydifferentparticlesatdifferenttimesAccordinglythecoordinatesXRaretermedmaterialcoordinatesandthecoordinatesxiaretermedspatialcoordinatesThesetofpositionsoftheparticlesofℬatagiventimespecifiedaconfigurationofℬTheconfigurationofℬatthereferencetimet=0isitsreferenceconfigurationItsconfigurationattimetisitscurrentconfigurationatt

Asfaraspossibleweshalldenotescalarvectorandtensorquantitiesevaluatedinthereferenceconfigurationbycapitallettersandcorrespondingquantitiesevaluatedinthecurrentconfigurationbylower-caselettersOccasionallyweshallemploytheindexzero(asforexampleinρ0)forquantitiesevaluatedinthereferenceconfigurationThisconventionregardingtheuseofcapitalandlower-caseletterswillextendalsotoindicesofvectorandtensorcomponentsComponentsofvectorsandtensorswhichtransformwiththecoordinatesXRwillhavecapitalletterindices(ARCRSetc)andcomponentswhichtransformwiththecoordinatesxiwillhavelower-caseindices(aiTijetc)Occasionallytheconventionthatcapitalandlower-caselettersrelatetothereferenceandcurrentconfigurationsrespectivelywillconflictwiththenotationestablishedinChapter3thusinthisandsubsequentchaptersXisapositionvectordespitetheconventionthatvectorsarenormallyrepresentedbylower-caseitalicletters

Forphysicallyrealizablemotionsitispossibleinprincipletosolve(42)forXRintermsofxiandtwhichgivesequationsoftheform

(43)

Equations(43)givethecoordinatesXRinthereferenceconfigurationoftheparticlewhichoccupiesthepositionxiinthecurrentconfigurationattimet

ProblemsincontinuummechanicsmaybeformulatedeitherwiththematerialcoordinatesXRasindependentvariablesinwhichcaseweemploythematerialdescriptionoftheproblemorwiththespatialcoordinatesxiasindependentvariablesinwhichcaseweemploythespatialdescriptionOftenthetermslsquoLagrangianrsquoandlsquoEulerianrsquoareusedinplaceoflsquomaterialrsquoandlsquospatialrsquorespectivelyInthematerialdescriptionattentionisfocusedonwhatishappeningatorintheneighbourhoodofaparticularmaterialparticleInthespatialdescriptionweconcentrateoneventsatorneartoaparticularpointinspaceThemathematicalformulationofgeneralphysicallawsandthedescriptionofthepropertiesofparticularmaterialsisoftenmosteasilyaccomplishedinthematerialdescriptionbutforthesolutionofparticularproblemsitisfrequentlypreferabletousethespatialdescriptionItisthereforenecessarytoemploybothdescriptionsandtorelatethemtoeachotherInprincipleitispossibletotransformaproblemfromthematerialtothespatialdescriptionorviceversabyusing(42)or(43)Inpracticethetransitionisnotalwaysaccomplishedeasily

42Displacementandve loc ityThedisplacementvectoruofatypicalparticlefromitspositionXinthereferenceconfigurationtoitspositionxattimetis

(44)

InthematerialdescriptionuisregardedasafunctionofXandtsothat

(45)

andinthespatialdescriptionuisregardedasafunctionofxandtsothat

(46)

Therepresentation(45)determinesthedisplacementattimetoftheparticledefinedbythematerial

coordinatesXRTherepresentation(46)determinesthedisplacementwhichhasbeenundergonebytheparticlewhichoccupiesthepositionxattimet

ThevelocityvectorvofaparticleistherateofchangeofitsdisplacementSinceXRareconstantatafixedparticleitisconvenienttoemploythematerialdescriptionsothatfrom(45)

(47)

wherethedifferentiationsareperformedwithXheldconstantIntermsofthecomponentsviofv(47)maybewrittenas

(48)

Theresultofperformingthedifferentiation(47)or(48)istoexpressthevelocitycomponentsasfunctionsofXRandtthatistheygivethevelocityattimetoftheparticlewhichwasatXatt=0WefrequentlyneedtoemploythespatialdescriptioninwhichweareconcernedwiththevelocityatthepointxTodosoitisnecessarytoexpressviintermsofxibyusingtherelations(43)Thisisillustratedbythefollowingexample

Example41Abodyundergoesthemotiondefinedby

(49)

whereaisconstantFindthedisplacementandvelocityinboththematerialandspatialdescriptions

From(45)wehave

(410)

ThisgivesthedisplacementattimetinthematerialdescriptionToobtainthedisplacementinthespatial

descriptionwesubstituteforX1from(49)into(410)whichgives

(411)

Forthevelocitywedifferentiate(49)withrespecttotwithXRfixedtoobtaininthematerialdescription

(412)

ThisisthevelocityoftheparticlewhichoccupiedXatt=0ForthespatialdescriptionweeliminateX1from(49)and(412)

(413)

andthisgivesthevelocityoftheparticlewhichinstantaneouslyoccupiesthepointxattimet

43TimeratesofchangeSupposethatφissomequantitywhichvariesthroughoutabodyinspaceandintimeWecanregardφasafunctionoftandofeitherthematerialcoordinatesXRorthespatialcoordinatesxiThus

(414)

InconsideringratesofchangeofφweareusuallyinterestedinhowφvarieswithtimefollowingagivenparticleForexampleinSection44weshalldiscussaccelerationwhichistherateofchangeofvelocityofaparticleTheappropriatequantitytomeasuretherateofchangeofφfollowingtheparticleXRispartG(XRt)parttwhichgivestherateofchangeofφwithXRheldconstantOntheotherhandpartg(xit)parttdenotestherateofchangeofφwithconstantxi(thatisatafixedpointinspace)andthisisadifferentquantity

WeadopttheconventionalnotationsDφDtor fortherateofchangeofφfollowingagivenparticle

sothat

(415)

HoweverφmaybegiveninthespatialdescriptionsoitisnecessarytoexpressDφDtintermsofderivativesofg(xit)From(42)and(414)wehave

HencebydifferentiatingwithrespecttotwithXRconstant

Byusingthesummationconventionthisiswrittenconciselyas

(416)

Nowbyusing(48)DφDtmaybewritteninthesimplerform

(417)

oralternativelyinvectornotationas

(418)

wherethegradientistakenwithrespecttospatialcoordinatesxi

Figure42Thechangeofφfollowingaparticle

TheaboveisaformalderivationoftheformulaforDφDtTogiveitaphysicalinterpretationwerefertoFig42ConsiderthechangeinφfollowingaparticleSupposethatinthetimeintervalttot+δtφ(attheparticlewithcoordinatesxiatt)changesitsvaluefromφtoφ+δφDuringthistimeintervaltheparticlemovesfromxitoxi+υiδtwherevisthevelocityoftheparticleatsometimebetweentandt+δt(anynecessarycontinuityconditionsareassumedtobesatisfied)Thuswehavetocomparethevalueofφatxiandtgivenasg(xit)withitsvalueatxi+υiδtandt+δtwhichisg(xi+Viδtt+δt)Thus

Thenbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0intheusualwayitfollowsthat

whichis(417)

ThederivativeDφDtiscalledthematerialderivativeortheconvectedderivativeofφ

Althoughitislogicalin(414)tousethedifferentsymbolsGandgforthetwofunctionswhichdescribethedependenceofφonthetwosetsofindependentvariables(XRt)and(xit)itisfoundinpracticethatthisprocedurecanleadtoaconfusingproliferationofsymbolsInfutureweshalladopttheconventionofusingthesamesymboltodenoteadependentvariableandafunctionwhichdeterminesthatvariableandwherethereisapossibilityofconfusiontheargumentsoffunctionswillbeexplicitlyincludedtodemonstratewhichindependentvariablesarebeingemployedThusinplaceof(415)weshallwrite

(419)

andinplaceof(417)and(418)weshallwrite

(420)

Theexplicitinclusionoftheargumentsmakesitclearthatin(419)φisregardedasafunctionofXRandtandthatin(420)Φisregardedasafunctionofxiandt

44Acce lerationTheaccelerationofaparticleistherateofchangeofvelocityofthatparticlethatisitisthematerialderivativeofthevelocityWedenotetheaccelerationvectorbyfanditscomponentsbyfi

Thusinthematerialdescription

(421)

orinvectornotation

(422)

TheserelationsgivefinmaterialcoordinatesTofindtheaccelerationintermsofspatialcoordinatesitisnecessarytoexpressmaterialcoordinatesXRintermsofspatialcoordinatesxiFrequentlythisinformationisnotexplicitlyavailable

Although(421)givethesimplestexpressionsforfitheyarenotthemostgenerallyusefulbecauseitisoftenrequiredtoexpresstheaccelerationcomponentsintermsofderivativesofthevelocitycomponentswhenthevelocitycomponentsareexpressedinspatialcoordinatesxiexclThusfromtheresultsofSection43

(423)

ThephysicalinterpretationofthisexpressionisasfollowsInanincrementoftimeδttheparticlewhichattimethascoordinatesxkmovestoxk+υkδtHencethevelocitycomponentsofthisparticlechangefromυi(xkt)toυi(xk+υkδtt+δt)Thusthechangeinvataparticleisgivenby

and(423)followsbyapplyingthemean-valuetheoremandproceedingtothelimitδtrarr0Theexpression(423)givesfiintermsofthespatialcoordinatesxi

Example42Toillustratetheequivalenceoftheexpressions(421)and(423)forficonsiderthemotion(49)Thisgives(Example41)

Bytakingthefirstexpressionforυ1wefindfrom(421)that

(424)

Ifυ1isgiveninthespatialdescriptionas2x1a2t(1+a2t2)weobtainfrom(423)

(425)

Theexpressionsforf1givenby(424)and(425)arethesamebecausefrom(49)x1=X1(1+a2t2)

45SteadymotionPartic le pathsandstreamlinesAmotionissaidtobesteadyifthevelocityatanypointisindependentoftimesothatv=v(x)Conditionsapproximatingtosteadymotionareachievedinmanypracticalsituationsforexampleinflowofafluidthroughapipeatauniformrateorflowpastafixedobstaclewithuniformvelocityatalargedistancefromtheobstacle

AmotionmaybeunsteadyinrelationtoafixedcoordinatesystembutsteadyrelativetosuitablychosenmovingaxesForexampletheflowpastanaeroplanemovingatconstantvelocitythroughauniformatmosphereisunsteadyrelativetofixedcoordinatesbutissteadyrelativetoaxeswhicharefixedinrelationtotheaeroplaneandmovewithit

Theequations(42)xi=xi(XRt)givethesuccessivepositionsxioftheparticleXRwithtservingasaparameterThustheyareparametricequationsofthepathoftheparticleXRIndifferentialform(42)gives

andthiscanbeexpressedinspatialcoordinatesas

(426)

ThestreamlinesattimetarespacecurveswhosetangentsareeverywheredirectedalongthedirectionofthevelocityvectorThustheyaregivenintermsofaparameterτbytheequations

(427)

IngeneraltheparticlepathsandstreamlinesdonotcoincideHoweverifthemotionissteadysothat

visindependentoftthen(426)and(427)representthesamefamiliesofcurvesandthentheparticlepathsandstreamlinesarecoincident

46Problems1Amotionofafluidisgivenbytheequations

Findthevelocityandaccelerationof(a)theparticlewhichwasatthepoint(111)atthereferencetimet=0and(b)theparticlewhichoccupiesthepoint(111)attimetExplainwhythismotionbecomesphysicallyunrealisticastrarr1

2Thevelocityinasteadyhelicalflowofafluidisgivenby

υ1=ndashUx2v2=Ux1υ3=V

whereUandVareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

3Thevelocityatapointxinspaceinabodyoffluidinsteadyflowisgivenby

whereUVandaareconstantsShowthatdivv=0andfindtheaccelerationoftheparticleatxAlsodeterminethestreamlines

4Anelectromagneticfluidissubjectedtoadecayingelectricfieldofmagnitudeφ=r-1e-AtwhereandAisconstantThevelocityofthefluidis Determine(a)

therateofchangeofφatt=11oftheparticlewhichoccupiesthepointwithcoordinates(2ndash21)(b)theaccelerationofthesameparticleatthesametime(c)thepositionofthesameparticleatallsubsequenttimestWritedownthedifferentialequationsofthestreamlinesandshowthatateachinstantx2x3isconstantalongagivenstreamline

5Giventhevelocityfield

witha1a2b1b2andcconstantsshowthatthex2componentoftheaccelerationatt=0is(a1b1+b1b2

mdashb1)X1+( +b1a2mdashb2)X2whereXdenotesthepositionvectoratt=0Inthecasea1=Aa2=0b1=0b2=2Ac=3Aobtaintheparticlepathsandthestreamlinesandshowthatinthiscasetheycoincide

5

Stress

51Surface tractionInthischapterweconsidertheforcesactingintheinteriorofacontinuousbodySupposethatpartofabodyℬoccupiesaregionℛwhichhassurface asillustratedinFig51LetPbeapointonthesurfacenaunitvectordirectedalongtheoutwardnormalto atPandδStheareaofanelementof which

containsPWeassumethat andℛpossessanynecessarysmoothnessandcontinuitypropertiesforexampleitisassumedthatthenormalto isuniquelydefinedatP

ItisalsoassumedthatonthesurfaceelementwithareaδSthematerialoutsideℛexertsaforce

(51)

onthematerialinsideℛTheforceδpiscalledthesurfaceforceandt(n)themeansurfacetractiontransmittedacrosstheelementofareaδpfromtheoutsidetotheinsideofℛAsimilarforceequalinmagnitudebutoppositeindirectiontoδpandasimilarsurfacetractionequalinmagnitudebutoppositeindirectiontot(n)istransmittedacrosstheelementwithareaδSfromtheinsidetotheoutsideofℛClearlyt(n)willdependonthepositionofPandthedirectionofnItisfurtherassumedthatasδSrarr0

t(n)tendstoafinitelimitwhichisindependentoftheshapeoftheelementwithareaδSHenceforththesymbolt(n)isusedtodenotethelimit

(52)

andweomittheadjectivelsquomeanrsquoandcallt(n)thesurfacetractionatthepointPonthesurfacewithnormaln

TheassumptionsmadeaboveareplausiblebuttheyareofaphysicalnatureandcanonlybejustifiedtotheextentthatconclusionsbasedonthemagreewithobservationsofwhathappenstorealmaterialsItispossibleforcouplesaswellasforcestobetransmittedacrossasurfaceSuchcouplesareofinterest

butarebeyondthescopeofthisbookInpracticetheirinfluenceisrestrictedtoratherspecialsituations

Figure51Surfacetraction

Itisimportanttorememberthatingeneralt(n)doesnotcoincideindirectionwithnTheforcetransmittedacrossasurfacedoesnotnecessarilyactinthedirectionnormaltothesurface

52ComponentsofstressAtPthereisavectort(n)associatedwitheachdirectionthroughPInparticulargivenasystemofrectangularcartesiancoordinateswithbasevectorseithereissuchavectorassociatedwiththedirectionofeachofthebasevectorsLett1bethesurfacetractionassociatedwiththedirectionofe1fromthepositivetothenegativeside(thatist1istheforceperunitareaexertedonthenegativesideofasurfacenormaltothex1-axisbythematerialonthepositivesideofthissurfaceseeFig52)Surfacetractionvectorst2andt3aresimilarlydefinedinrelationtothedirectionsofe2ande3

Nowresolvethevectorst1t2andt3intocomponentsinthecoordinatesystemwithbasevectorseiasfollows

(53)

Figure52Thesurfacetractionvectort1

Theseequationsmaybewritteninmatrixformas

(54)

orusingthesummationconventionas

(55)

Sinceeimiddotej=δijitfollowsfrom(55)that

(56)

ThequantitiesTijarecalledstresscomponentsThecomponentT11forexampleisthecomponentoft1inthedirectionofe1T11ispositiveifthematerialonthex1-positivesideofthesurfaceonwhicht1acts(asurfacenormaltothex1-axis)ispullingthematerialonthex1-negativesideThematerialisthenintensioninthex1directionThematerialonthenegativesideofthesurfaceispullingintheoppositedirectiononthematerialonthepositivesideIfthematerialoneachsideofthesurfacepushesagainstthatontheotherT11isnegativeandthematerialissaidtobeincompressioninthex1directionThecomponentsT11T22andT33arecallednormalordirectstresscomponentsTheremainingcomponentsT12T13etcarecalledshearingstresscomponentsAllthestresscomponentscanbeillustratedasthe

componentsofforcesactingonthefacesofaunitcubeasshowninFig53

53The tractiononanysurfaceSupposethatthestresscomponentsTijareknownatagivenpointPWeconsiderhowwemaydeterminethesurfacetractiononanarbitrarysurfacethroughPForthisweexaminetheforcesactingontheelementarytetrahedronillustratedinFig54Wewishtofindthetractiont(n)onasurfacenormaltonatPInthetetrahedronshowninFig54PQ1PQ2PQ3areparalleltothethreecoordinateaxesandQ1Q2Q3isnormaltonWedenotebyndasht1ndasht2ndasht3themeansurfacetractionsonthefacesPQ2Q3PQ3Q1andPQ1Q2respectivelyTheminussignsarisebecausewewishtoconsidertheforcesactingonthetetrahedronsothatforexamplendasht1isthetractionexertedonthesurfacePQ2Q3bymaterialtotheleftofthissurfaceonmaterialtotherightofthesurfacethatisbythematerialoutsidethetetrahedrononthematerialinsidethetetrahedronSimilarlyt(n)denoteschemeansurfacetractiononQ1Q2Q3exertedbymaterialonthesidetowardswhichnisdirected(theoutsideofthetetrahedron)ontotheothersideLettheareaofQ1Q2Q3beδSandthevolumeofPQ1Q2Q3beδVThentheareasoftheotherfacesare

(57)

whereniarethecomponentsofnthatisniarethedirectiorcosinesofthedirectionofn

Figure53ComponentsoftheforcesonthreefacesofaunitcubeOppositeforcesactontheoppositefaces

Theforcesexertedonthetetrahedronacrossitsfourfacesare

ndasht1δS1ndasht2δS2ndasht3δS3t(n)δS

Figure54Forcesactingonanelementarytetrahedron

ItisalsosupposedthatthereisabodyforcewhosemeanvalueoverthetetrahedronisbperunitmassorρbperunitvolumewhereρisthedensityThemostcommonexampleofabodyforceisagravitationalforcebutthereareotherpossibilities

WenowassumethatforanypartofabodyandinparticularfortheelementarytetrahedronPQ1Q2Q3therateofchangeofmomentumisproportionaltotheresultantforceactingAlthoughthisisanaturalassumptiontomakeitisanewassumptionwhichisstrongerthanNewtonrsquossecondlawforNewtonrsquoslawappliesonlytobodiesasawholeMoreoveritisanassumptionwhichcannotbeverifieddirectlybyexperimentforitisimpossibletomakedirectmeasurementsofinternalsurfacetractionstheirexistenceandmagnitudescanonlybeinferredfromobservationsofotherquantitiesNeverthelesstheconsequencesofthisassumption(whichissometimescalledCauchyrsquoslawofmotion)aresowellverifiedthatitishardlyopentoquestion

ForthetetrahedronPQ1Q2Q3Cauchyrsquoslawgives

ndasht1δS1ndasht2δS2ndasht3δS3+t(n)δS+ρbδV=ρfδV

With(57)thismaybewrittenas

NowwithnandthepointPfixedletthetetrahedronshrinkinsizebutretainitsshapeThusδSrarr0andinthislimitallquantitiesareevaluatedatPsothatt1t2t3andt(n)becometractionsatPandρbandfareevaluatedatPAlsosinceδVisproportionaltothecubeandδSisproportionaltothesquareofthelineardimensionsofthetetrahedronδVδSrarr0asδSrarr0Thusinthislimit

(58)

wherethelastrelationmakesuseof(55)ThisgivesthetractiononanysurfacewithunitnormalnintermsofthestresscomponentsTijThecomponentstj(n)oft(n)aregivenby

(59)

Theeasiestwaytocalculatet(n)istouse(59)inthematrixform

(510)

AnumericalexampleisgiveninExample51inSection56

54TransformationofstresscomponentsThestresscomponentsTijweredefinedinSection52inrelationtothecoordinatesystemwithbasevectorseiThechoiceofadifferentcoordinatesystemwillleadtoadifferentsetofstresscomponentsWenowexaminetherelationshipbetweenthestresscomponentsTijassociatedwithbasevectorseiand

stresscomponents atthesamepointbutreferredtoanewcoordinatesystemwithbasevectorsēi

where

(511)

andM=(Mij)isanorthogonalmatrix

In(58)wemayasaspecialcasechoosentobeē1From(511)thecomponentsofē1referredto

basevectorseiareM11M12andM13Wedenoteby thetractiononasurfacenormaltoē1Thenfrom(58)(withni=M1i)and(511)

Wedefine and inasimilarwayandobtainsimilarrelationsforthemThegeneralrelationis

(512)

Howeverthestresscomponents referredtobasevectorseqaredefinedbytherelationanalogousto

(55)asthecomponentsof referredtobasevectorsēqsothat

(513)

Hencebycomparing(512)and(513)

(514)

Thisisjustthetransformationlaw(328)forthecomponentsofasecond-ordertensorHencethereexistsasecond-ordertensorT=TijeiotimesejwhosecomponentsareTijinthecoordinatesystemwithbasevectorseiand ijinthesystemwithbasevectorsēiTiscalledtheCauchystresstensoranditcompletelydescribesthestateofstressofabodySomeotherstresstensorswillbeconsideredbrieflyin

Section95butweshallnotusetheminthisbookandsoweshallrefertoTassimplythestresstensor

Equation(514)isanimportantresultbecauseitshowsthatTijarecomponentsofatensorsowebrieflyrecapitulatethestepswhichleadto(514)Theyarea defineTijby(53)usingbasevectorseib derivetheexpression(58)forthetractiononasurfacewithnormalnc takentobethenewbasevectorsē1ē2ē3inturnandsoobtain(512)d resolvethetractiononthenewcoordinatesurfacesinthedirectionsofthenewcoordinateaxesto

define asin(513)andcomparewith(512)

IfT=(Tij)and then(514)maybewritteninmatrixnotationas

(515)

ThusthecalculationofstresscomponentsinanewcoordinatesystemcanbecarriedoutbymatrixmultiplicationsandthisisusuallythemostconvenientwaytoperformsuchcalculationsAnumericalexampleisgiveninProblem1inSection510

SinceitisnowestablishedthatTijarecomponentsofatensorequation(59)canbeexpressedindirectnotationas

(516)

55EquationsofequilibriumWenowconsiderthatthebodyℬisinequilibriumThenotationofSection51isused(seeFig51)ℛisanarbitraryregioninℬand isthesurfaceofℛwithunitnormalnWeassumethatinequilibriumtheresultantforceandtheresultantcoupleaboutOactingonthematerialinℛarezeroTheforcesactingonthematerialinℛareoftwokindstherearethesurfaceforcesactingacross whoseresultantistheintegraloft(n)over andbodyforcesρbperunitvolumewhoseresultantistheintegralofρbthroughℛThustheconditionfortheresultantforcetobezerois

(517)

SimilarlytheresultantcoupleaboutOiszeroif

(518)

wherexdenotesthepositionvectorrelativetoO

Intermsofcomponents(517)and(518)maybewritten(withtheaidof(59))as

(519)

(520)

Wenexttransformthesurfaceintegralsintovolumeintegralsbyuseofthedivergencetheorem(Section310)ItisassumedthatTijhavecontinuousfirstderivativesThen(519)and(520)become

(521)

(522)

HowevertheserelationsmustholdineveryregionℛwhichliesinℬHencetheintegrandsmustbezerothroughoutℬforiftheywerenotitwouldbepossibletofindaregionℛforwhich(521)or(522)wasviolatedHencethroughoutℬ

(523)

(524)

Howeverpartϰppartϰr=δprandso(524)maybewrittenas

andbyusing(523)thisreducesto

ejpqTpq=0

whichimpliesthat

(525)

Equation(523)istheequationofequilibriumEquations(525)showthatinequilibriumthestresstensorisasymmetrictensorInSection75itwillbeshownthat(525)alsoholdsforabodyinmotionweanticipatethisresultandhenceforthtreatTasasymmetrictensorEquation(523)isgiveninfullin(537)

56Princ ipalstresscomponents princ ipalaxesofstressandstressinvariantsIngeneralthesurfacetractiont(n)associatedwithadirectionnthroughapointPwillnotactinthedirectionofthevectornthetractionwillhaveatangential(shearing)componentonthesurfacenormaltonaswellasanormalcomponentHoweveritmayhappenthatforcertainspecialdirectionsnthetractiont(n)doesactinthedirectionnWeinvestigatethispossibility

Ift(n)andnhavethesamedirectionthen

t(n)=Tn

whereTisthemagnitudeoft(n)From(516)rememberingthatTissymmetricthismaybewrittenas

nmiddotT=Tn

orincomponentsas

niTij=Tnj

thatis

(TijndashTδij)ni=0

Hence(Section38)TisoneofthethreeprincipalcomponentsT1T2andT3ofTandndeterminesthecorrespondingprincipalaxisLettheunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3IfthesethreeorthogonalvectorsaretakenasbasevectorsatPthenreferredtotheseaxesthematrixofthestresscomponentsisadiagonalmatrixwithdiagonalelementsT1T2andT3Theprincipalcomponentsaretherootsoftheequation

(526)

whereTijarethestresscomponentsreferredtoanycoordinatesystemIngeneraltheprincipaldirectionsvaryfrompointtopointsothatitisnotusuallypossibletofindarectangularcartesiancoordinatesysteminwhichthematrixofstresscomponentsisadiagonalmatrixeverywhere

LetT1T2andT3beorderedsothatT1geT2geT3ItisshowninExample52thatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentofthetractiononthesurfaceThispropertycanbeusedtogiveanalternativedefinitionoftheprincipalstresscomponentsandprincipalaxesofstress

If(526)hastwoorthreeequalrootstheabovestatementsremaintruebuttheprincipalaxesarenotuniquelydefined

Example51ThecomponentsofthestresstensoratapointParegiveninappropriateunitsby

Find(i)thetractiontatPontheplanenormaltothex1-axis(ii)thetractiontatPontheplanewhosenormalhasdirectionratios1ndash12(iii)thetractiontatPontheplanethroughPparalleltotheplane2x1ndash2x2ndashx3=0(iv)thenormalcomponentofthetractionontheplane(iii)(v)theprincipalstresscomponentsatP(vi)thedirectionsoftheprincipalaxesofstressthroughP

(i)Theplanenormaltothex1-axishasunitnormal(100)Hencethetractioncomponentsonthisplanearegivenby(510)as

(ii)Theunitnormalis(1ndash12)radic6Hence

(iii)Theunitnormalis Hence

(iv)Therequiredcomponentisnmiddott= 2times(-5)-2times(-10)-1times(-7)=

(v)Theprincipalcomponentsaresolutionsof

whichgivesT1=10T2=0T3=-4

(vi)TheprincipaldirectioncorrespondingtoforexampleT1=10isgivenbythesolutionof

whichgivethedirectionratiosn1n2n3=365Similarlythedirectionratiosoftheothertwoprincipaldirectionsarendash210and12ndash3(notethatthesedirectionsaremutuallyorthogonal)

Example52ProvethatastheorientationofasurfacethroughPvariesT1isthegreatestandT3istheleastnormalcomponentoftractiononthesurface(assumethatT1T2andT3arealldifferent)

ChoosethecoordinateaxestocoincidewiththeprincipalaxesofTsothatthematrixofstresscomponentstakestheform

ThenormalcomponentoftractiononasurfacewithunitnormalnisTijninjwhichwhenThasthegiven

diagonalformreducestoT=T1 +T2 +T3 HencewerequireextremalvaluesofTforvariations

ofn1n2andn3subjecttotheconstraint =1Theseextremaaregivenby

whereσisaLagrangianmultiplierThesolutionsoftheseequationsare(i)n=(plusmn100)TT=T1(ii)n=(0plusmn10)TT=T2(iii)n=(00plusmn1)TT=T3

SinceT1gtT2gtT3(i)givesthemaximumand(iii)givestheminimumvaluesofT

AsTisasymmetricsecond-ordertensorthediscussionofSection38showsthatThasthreeindependentinvariantsWedenotethesebyJ1J2andJ3where

NotethatthedefinitionofJ2isnotquiteconsistentwiththatofI2in(357)becausethereisadifferenceofsignwhichitisfoundconvenienttointroduce

57The stressdeviatortensorItisoftenusefultodecomposeTinthefollowingway

(528)

whereSisthestressdeviatortensorIfSijdenotethecomponentsofSthen

(529)

where

(530)

andhence

(531)

and

(532)

IfSij=0thenthestresshastheformTij=ndashpδijThisiscalledapurehydrostaticstateofstressandpisthehydrostaticpressureThenegativesignarisesbecauseweconventionallyregardpressurewhichcausescompressionaspositivebutwedefinecompressivestressasnegative

TheprincipalaxesofSarethesameasthoseofTIftheprincipalcomponentsofSareS1S2S3then

(533)

and

(534)

BecauseS1S2andS3satisfy(533)thereareonlytwobasicinvariantsofSThesearetakentobeand where

(535)

Theinvariants and canbeexpressedintermsofJ1J2andJ3byin(363)replacingI1I2I3

and byJ1ndashJ2J3 and respectivelyItissometimesconvenienttoadoptJ1 and asasetofbasicinvariantsofT

58ShearstressThenormalstresscomponentonasurfacenormaltothex1-axisisT11(seeFig53)TheshearstressonthissurfaceistheresultantoftheothertwocomponentsT12e2andT13e3ofthetractiononthesurfaceHencetheshearstresshasmagnitude andactsinadirectionwhichliesinthesurface

Forageneralsurfacewithunitnormalvectornthenormalcomponentofthetractiont(n)hasmagnitude

nmiddott(n)=ninjTijTheshearstressonthissurfaceisthecomponentoft(n)normaltonnamely

t(n)mdash(nmiddott(n))n=Trsnr(δsjmdashnsnj)ej

SupposethattheprincipalstresscomponentsareorderedsothatT1geT2geT3andletthecorrespondingunitvectorsinthedirectionsoftheprincipalaxesben1n2andn3ThenitcanbeshownthatasnvariesatpointPthemagnitudeoftheshearstressonthesurfacenormaltonreachesamaximumvalue (T1-T3)whennliesalongeitherofthebisectorsoftheanglebetweenn1andn3Theproofresemblesthatof

Example52andisleftasanexercise(Problem9)Notethat (T1-T3)= (S1-S3)andthatinahydrostaticstateofstressT1=T2=T3andthentheshearstressiszeroonanysurface

59Somesimple statesofstress(a)HydrostaticpressureSupposethat

Tij=ndashpδij

thatis

(536)

ThenwehaveastateofhydrostaticpressureThestresscomponentstaketheform(536)inanyrectangularcartesiancoordinatesystemandanythreemutuallyorthogonaldirectionsmayberegardedasprincipaldirectionsThisisthestateofstressinanyfluidinequilibrium(thatisinhydrostatics)orinaninviscidfluidwhetheritisinequilibriumornotThepressurepisingeneralafunctionofposition

Intheremainingexamplesbodyforceswillberegardedasnegligibleandweseekstressstateswhichsatisfytheequilibriumequations(523)whichare

(537)

SincethesearethreeequationsforthesixcomponentsofstresstheyareinsufficienttodeterminethesolutiontoanyproblemNeverthelesstheymustbesatisfiedforanybodyinequilibriumanditisofinteresttoexaminesomestressstateswhichsatisfythemWhenthebodyforceisneglectedtheyaresatisfiediftheTijareallconstantsinwhichcasethestressishomogeneousThenexttwoexamplesareinthiscategory

(b)Uniformtensionorcompressioninthex1directionisgivenby

(538)

whereσisconstantThisgivesthestressinauniformcylindricalbarwithgeneratorsparalleltothex1-axisnoforcesappliedtoitslateralsurfacesanduniformforcesσperunitareaappliedtoplaneendsnormaltothegeneratorsIfσispositivethebarisintensionandifσisnegativethebarisincompressionTheprincipalstressdirectionsarethex1directionandanytwodirectionsorthogonaltoeachotherandtothex1direction

(c)Uniformshearstressinthex1directiononplanesx2=constantarisesif

(539)

whereτisconstantThismayoccurforexampleinlaminarshearflowofaviscousfluidwhenthefluidflowsinthex1directionbyshearingontheplanesx2=constantTheprincipalaxesofstresshavethedirectionsofthex3-axisandthetwobisectorsofthex1-andx2-axes

(d)PurebendingLet

(540)

wherecisconstantThisapproximatesthestressinaprismaticbeamwithgeneratorsparalleltothex1-axiswhichisbentbyendcouplesappliedtoitsendsandactingaboutaxesparalleltothex3-axisThe

planex2=0ischosensothattheresultantforceoneachendiszeroIfcgt0theregionx2gt0ofthebeamisintensionandtheregionx2lt0isincompressionTheprincipalstressdirectionsareasin(b)above

(e)PlanestressIf

(541)

andT11T22andT12arefunctionsonlyofx1andx2wehaveastateofplanestressIntheabsenceofbodyforcestheequilibriumequationsreduceto

(542)

Thisistheapproximatestateofstressinathinflatplatelyingparalleltothex3-planeandsubjecttoforcesactinginitsplaneThex3directionisaprincipaldirectiontheothertwoprincipaldirectionsareintheplaneoftheplate

(f)PuretorsionSupposethat

(543)

wherer2 Thiscorrespondstothestateofstressinacircularcylindricalbarwhoseaxiscoincideswiththex3-axisandwhichistwistedbycouplesactingabouttheaxisofthecylinderandappliedtotheendsofthecylinderwithnoforcesactingonthecurvedsurfacesTheprincipaldirectionsaretheradialdirectionandthebisectorsofthetangentialandaxialdirections

510Problems1Thecomponentsofthestresstensorinarectangularcartesiancoordinatesystemx1x2x3atapointParegiveninappropriateunitsby

Find(a)thetractionatPontheplanenormaltothex1-axis(b)thetractionatPontheplanewhosenormalhasdirectionratios1ndash32(c)thetractionatPonaplanethroughPparalleltotheplanex1+2x2+3x3=1(d)theprincipalstresscomponentsatP(e)thedirectionsoftheprincipalaxesofstressatPVerifythattheprincipalaxesofstressaremutuallyorthogonal

Thecoordinates arerelatedtox1x2x3by

VerifythatthistransformationisorthogonalandfindthecomponentsofthestresstensordefinedaboveinthenewcoordinatesystemUsetheanswertochecktheanswersto(d)and(e)above

2Inplanestress(T13=T23=T33=0)showthatifthe -and -axesareobtainedbyrotatingthex1-andx2-axesthroughanangleαaboutthex3-axisthen

3Ifinappropriateunits

findtheprincipalcomponentsofstressandshowthattheprincipaldirectionswhichcorrespondtothegreatestandleastprincipalcomponentsarebothperpendiculartothex2-axis

4Acantileverbeamwithrectangularcross-sectionoccupiestheregionndashalex1leandashhlex2leh0lex3lelTheendx3=lisbuilt-inandthebeamisbentbyaforcePappliedatthefreeendx3=0andactinginthex2directionThestresstensorhascomponents

whereABandCareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforcesprovided2B+C=0(b)determinetherelationbetweenAandBifnotractionactsonthesidesx2=plusmnh(c)expresstheresultantforceonthefreeendx3=0intermsofABandCandhencewith(a)and(b)showthatC=ndash3P4ah3

5ThestressinthecantileverbeamofProblem4isnowgivenby

whereCandDareconstants(a)Showthatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)showthatthetractiononthesurfacex2=ndashhiszero(c)findthemagnitudeanddirectionofthetractiononthesurfacex2=handhencethetotalforceonthissurface(d)findtheresultantforceonthesurfacex3=lProvethatthetractiononthissurfaceexertszerobendingcoupleonitprovidedthatC(5l2ndash2h2)+5D=0

6Thestresscomponentsinathinplateboundedbyx1=plusmnLandx2=plusmnharegivenby

whereWandmareconstants(a)Verifythatthisstresssatisfiestheequationsofequilibriumwithnobodyforces(b)findthetractionsontheedgesx2=handx1=ndashL(c)findtheprincipalstresscomponentsandtheprincipalaxesofstressat(0h0)andat(L00)

7AsolidcircularcylinderhasradiusaandlengthLitsaxiscoincideswiththex3-axisanditsendslieintheplanesx3=ndashLandx3=0Thecylinderissubjectedtoaxialtensionbendingandtorsionsuchthatthestresstensorisgivenby

whereαβγandδareconstants(a)Verifythatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)verifythatnotractionactsonthecurvedsurfaceofthecylinder(c)findthetractionontheendx3=0andhenceshowthattheresultantforceonthisendisanaxialforceofmagnitudeπa2βandthattheresultantcoupleonthisendhascomponents( )aboutthex1-x2-andx3-axes(d)forthecaseinwhichbendingisabsent(γ=0δ=0)findtheprincipalstresscomponentsVerifythattwoofthesecomponentsareequalontheaxisofthecylinderbutthatelsewheretheyarealldifferentprovidedthatαne0Findtheprincipalstressdirectionwhichcorrespondstotheintermediateprincipalstresscomponent

8Acylinderwhoseaxisisparalleltothex3-axisandwhosenormalcross-sectionisthesquarendashalex1leandashalex2leaissubjectedtotorsionbycouplesactingoveritsendsx3=0andx3=LThestresscomponentsaregivenbyT13=partѱpartx2T23=mdashpartѱpartx1T11=T12=T22=T33=0whereψ=ψ(x1x2)(a)Showthatthesestresscomponentssatisfytheequationsofequilibriumwithnobodyforces(b)showthatthedifferencebetweenthemaximumandminimumprincipalstresscomponentsisandfindtheprincipalaxiswhichcorrespondstothezeroprincipalstresscomponent(c)forthespecial

showthatthelateralsurfacesarefreefromtractionandthatthecoupleactingoneachendfaceis32a69

9Letnbeaunitvectort(n)thetractiononthesurfacenormaltonandSthemagnitudeoftheshearstressonthissurfacesothatSisthecomponentoft(n)perpendiculartonProvethatasnvariesShasstationaryvalueswhennisperpendiculartooneoftheprincipalaxesofstressandbisectstheanglebetweentheothertwoProvealsothatthemaximumandminimumvaluesofSare

6

Motionsanddeformations

61Rigid-bodymotionsWeemploythenotationintroducedinSection41inwhichtheparticlesofabodyarelabelledbytheircoordinatesXRinareferenceconfigurationatthereferencetimet=0IfatalatertimettheparticleXRhascoordinatesxithentheequations

(61)

describeamotionofthebodytheygivethepositionofeachparticleattimetInChapter4weweremainlyconcernedwiththekinematicsofindividualparticlesInthischapterweconsiderhowaparticlemovesinrelationtoitsneighbouringparticles

Inarigid-bodymotionthebodyℬmoveswithoutchangingitsshapeThedistancebetweenanytwoparticlesofℬdoesnotchangeduringarigid-bodymotionneitherdoestheanglebetweenthetwolinesjoiningaparticletotwootherparticles

TranslationAtranslationisarigid-bodymotionofabodyinwhicheveryparticleundergoesthesamedisplacementthusthemotionisdescribedbytheequations

(62)

wherethevectorcisindependentofpositionanddependsonlyont

RotationConsideramotioninwhichℬrotatesintheanti-clockwisedirectionthroughanangleα(whichmaydependont)aboutthex3-axisThusinFig61theparticleinitiallyatatypicalpointP0movestothepointPsuchthatNP0=NPandtheanglebetweenNP0andNPisαThenbyelementarygeometry

(63)

orintensornotation

(64)

wherethecomponentsreferredtobasevectorseiofthetensorQaregivenby

(65)

ItiseasilyverifiedthatQisanorthogonaltensorandsowealsohave

(66)

Figure61Rotationaboutthex3-axis

NowconsideramoregeneralrotationinwhichℬrotatesaboutanarbitraryaxisthroughtheoriginOThedirectionoftheaxisisdefinedbyaunitvectornandtheangleofrotationisαinthesenseoftherotationofaright-handedscrewtravellinginthedirectionofnWerefertoFig62LetOQrepresenttheaxisofrotationandletXbethepositionvectorofatypicalpointP0inℬIntherotationtheparticlewhichisinitiallyatP0movestoPwithpositionvectorxHenceP0andPlieinaplanenormaltonsupposethatthisplaneintersectsOQatNThenNP0=NPandα=angP0NPandthepositionvectorofNrelativetoOiscnwherefromFig62

(67)

Figure62Rotationaboutanarbitraryaxis

Wealsodenotebyy0andythepositionvectorsofP0andPrespectivelyrelativetoNThus

(68)

Sinceyandy0havethesamemagnitudeitfollowsfromFig62that

y=y0cosα+ntimesy0sinα

Hencefrom(67)and(68)

(69)

Incomponents(69)maybewrittenas

(610)

oras

Xi=QiRXR

where

(611)

ItisevidentthatrotatingℬaboutagivenaxisthroughagivenangleisequivalenttoholdingℬfixedandrotatingthecoordinatesystemaboutthesameaxisthroughthesameanglebutintheoppositesenseThusitfollowsfromtheresultsofSection32thatifQisanyproperorthogonaltensortherelationx=QmiddotXandtheinverserelationX=QTmiddotxrepresentarigid-bodyrotationThecomponentsofanyproperorthogonaltensorcanberepresentedintheform(611)

Itcanbeshownthatanyrigid-bodymotionisacombinationofatranslationandarotationaboutanaxisthroughanypointInparticulariftheaxisofrotationpassesthroughOthenanyrigid-bodymotionisdescribedbyequationsoftheformor

(612)

wherec1(t)=ndashQT(t)c(t)

62Extensionofa material line e lementInageneralmotionabodywillchangeitsshapeaswellasitspositionandorientationAmotioninwhichachangeofshapetakesplaceiscalledadeformationabodywhichcanchangeitsshapeisdeformableincontrasttoarigidbodywhichcanonlyundergorigid-bodymotionsOneofthemainproblemsintheanalysisofdeformationistoseparatethatpartofamotionwhichcorrespondstoarigid-bodymotionfromthepartwhichinvolvesdeformation

Inadeformationtherearechangesindistancebetweenparticleswhereasinarigid-bodymotiontherearenosuchchangesWethereforebeginbyexaminingtheextensionorstretchofamateriallineelement

ConsiderasegmentP0Q0ofastraightlinelyinginthebodyℬinitsreferenceconfigurationsuchthatP0Q0haslengthδLandisalignedinthedirectionofaunitvectorA1asillustratedinFig63ThusifP0hascoordinates thenQ0hascoordinates TheparticleswhichlieonP0Q0attimet=0formasegmentofamaterialcurveandafteramotiontheseparticleswillingenerallieonanewcurveinspaceThemotionisdescribedbytherelations(61)andwewishtodeterminethelengthandorientationofthemateriallineelementafterthemotionSupposethatttheparticlesinitiallyatP0andQ0movetoPandQrespectivelyandthatthelinesegmentPQhaslengthδlandthedirectionofaunitvectoraThusifPhascoordinates thenQhascoordinates SincePwasinitiallyatP0itfollowsfrom(61)that(omittingtheargumentt)

andsinceQwasinitiallyatQ0itfollowssimilarlythat

Figure63Extensionofamateriallineelement

HencebyTaylorrsquostheoremsincetheARareoforderone

ThusinthelimitasδLrarr0

(613)

ThedifferentialcoefficientdldListheratioofthefinalandinitiallengthsofaninfinitesimalmateriallineelementinitiallysituatedat andinitiallyorientedinthedirectionofAThisratioiscalledtheextensionratioorstretchratioofthelineelementandisdenotedbyλHence(613)becomes

(614)

wheresince isageneralparticlewenowreplace byXRBysquaringeachsideof(614)andsummingontheindexiweobtain

Howeveraisaunitvectorsoaiexclaiexcl=1andtherefore

(615)

Whenλisdeterminedfrom(615)theorientationaofthelineelementinthedeformedconfigurationisthengivenby(614)

Ifthedeformationisdescribedbyequationsoftheform

XR=XR(xit)orX=X(xt)

whichgivethereferencecoordinatesXRoftheparticlewhichoccupiesxiattimettheninasimilarwaywemaydeterminethestretchratioλandtheorientationAinthereferenceconfigurationofalineelementwhichhasthedirectionainthedeformedconfigurationInessenceitisonlynecessarytointerchangeX

andxAandaandδLandδlintheaboveargumentDetailsarelefttothereader(Problem61)themainresultsare

(616)

(617)

63The deformationgradienttensorTheninequantitiespartxipartXRappearednaturallyintheanalysisofSection62TheyarecalledthedeformationgradientsItisclearthatthesequantitiesmustbeinvolvedinthedescriptionofhowaparticlemovesinrelationtoneighbouringparticlesandsotheyareofimportanceintheanalysisofdeformation

Wedenote

(618)

ThenFiRarecomponentsofasecond-ordertensorwhichiscalledthedeformationgradienttensorandisdenotedbyFToconfirmthatFiRarecomponentsofatensorweintroduceanewrectangularcartesiancoordinatesystembyarotationoftheaxesdefinedbytheorthogonalmatrixMTheninthenewsystemXandxhavecomponentsXRandxirespectivelywhere

Then

SincethecomponentsFiRconformtothetensortransformationlawFisasecond-ordertensorIngeneralFisnotasymmetrictensorBytheresultsofSection34FTisalsoasecond-ordertensorandsoisFndash1providedthatdetFne0(weshallshowinSection72thattherearephysicalreasonsforassumingthatdetFne0)Since

Fndash1isthetensorwhosecomponentsare where

(619)

ThemainresultsofSection62cannowbestatedindirecttensornotationEquation(614)maybeexpressedintheform

(620)

and(615)as

(621)

Similarly(616)and(617)maybewrittenrespectivelyas

(622)

(623)

ForthecalculationofaAandλitisoftenconvenienttousematrixnotationIfinafixedcoordinate

systemthecomponentsofAarewrittenasacolumnmatrixAthoseofaasacolumnmatrixathoseofFasasquarematrixFandthoseofFndash1asasquarematrixFndash1then(620)-(623)give

(624)

(625)

IfthereisnomotionthenXi=XiFIR=δiRandF=I

Thecomponentsofthedisplacementvectoruaregivenbyui=ximdashXiThedisplacementgradientsare

(626)

andsotheyarecomponentsofthetensorFndashIThistensoriscalledthedisplacementgradienttensorIfthereisnomotionthenitscomponentsareallzero

AlthoughthetensorFisimportantintheanalysisofdeformationitisnotitselfasuitablemeasureofdeformationThisisbecauseameasureofdeformationshouldhavethepropertythatitdoesnotchangewhennodeformationtakesplacethereforeitmustbeunchangedinarigid-bodymotionFdoesnothavethispropertyinfactintherigid-bodymotion(612)wehaveF=Q(t)

64F inite deformationandstraintensorsWedefineanewtensorCas

(627)

sothatthecomponentsCRSofCaregivenby

(628)

SinceCistheinnerproductofFTandFitisasecond-ordertensorthiscanalsobeverifieddirectlybyexaminingtheeffectofacoordinatetransformationonthecomponentsCRSFrom(628)itisevidentthatCRS=CSRsothatCisasymmetrictensor

From(615)and(621)theextensionratioofamateriallineelementwithdirectionAinthereferenceconfigurationisgivenby

(629)

ThusaknowledgeofCenablestheextensionratioofanylineelementtobecalculatedConsideranelementarymaterialtriangleboundedbythreemateriallineelementsKnowledgeofthestretchoftheselineelementscompletelydeterminestheshapeofthetriangle(thoughnotitsorientation)inadeformedconfigurationHencethecomponentsCRSataparticledeterminethelocaldeformationintheneighbourhoodofthatparticle

Fortherigid-bodymotion(612)F=Q(t)andso

(630)

HenceChastheconstantvalueIthroughoutarigid-bodymotionThusCisessentiallyconnectedwiththedeformationratherthantherigidmotionofabodyandisasuitablemeasureofthedeformationCiscalledtherightCauchy-Greendeformationtensor

CisnotauniquemeasureofdeformationTriviallyanytensorfunctionofC(suchasC2orCndash1)willserveassuchameasureItissometimesconvenienttoemploythemeasureCndash1whichisgivenintermsofFby

(631)

Thecomponents ofCndash1aregivenby

(632)

Anotherclassofdeformationmeasuresisbasedonthealternativeexpression(617)forλIfwedenote

(633)

thenBistheleftCauchy-GreendeformationtensorIfBandBndash1havecomponentsBijandrespectivelythen

(634)

and(617)becomes

(635)

HenceaknowledgeofBndash1orequivalentlyofBissufficienttodeterminethelocaldeformationintheneighbourhoodofapointinthedeformedconfigurationItiseasytoverifythatB=Iinarigid-bodymotion

TheLagrangianstraintensoryandtheEulerianstraintensorηaredefinedby2

(636)

(637)

BothofthesetensorsaresuitablemeasuresofdeformationTheyhavethepropertiesthatγ=0andη=0inarigid-bodymotionthatistheyreducetozerotensorswhenthereisnodeformation

Ifthedeformationisdefinedby(61)whichgivesthedependenceofxonXthenitisstraightforwardtocalculateFandnaturaltouseCBorγasadeformationmeasureThecomponentsofthesetensors

willthenbeobtainedasfunctionsofthematerialcoordinatesXRandsotheydescribethedeformationintheneighbourhoodofagivenparticleIfthedeformationisdescribedbyequationswhichgivethedependenceofXonxthenitiseasiertocalculateFndash1andthenaturaldeformationmeasuresareCndash1Bndash1andηthecomponentsofthesetensorsareobtainedasfunctionsofspatialcoordinatesxiandsotheydescribethedeformationwhichhastakenplaceintheneighbourhoodofagivenpoint

TheexpressionsforthecomponentsγRSofγandηijofηareoftengivenintermsofthedisplacementgradientsSince

u=xmdashX

wehave

Hencefrom(628)and(636)

(638)

sothatforexample

and

Similarly

anditfollowsfrom(634)and(637)that

(639)

andsoforexample

ThecalculationofthedeformationandstraintensorcomponentsforagivendeformationismosteasilycarriedoutusingmatrixoperationsWedenote

(640)

Thentheprincipalformulaeare

(641)

ThetensorsCCndash1BBndash1γandηareallsymmetricsecond-ordertensorssotheyallhaverealprincipalcomponentsandorthogonalprincipaldirectionsConsiderationoftheseisdeferredtoChapter9

65Somesimple f inite deformations(a)UniformextensionsSupposeabodysayalongbarofuniformcross-sectionisextendeduniformlyinthedirectionofthex1-axistoalengthλ1timesitsoriginallengthTheniftheparticleattheoriginis

fixedinpositionx1=λ1X1Thisdefinesauniformextensioninthex1directionIfthebodyundergoesuniformextensionsinallthreecoordinatedirectionsthedeformationisdescribedbytheequations

(642)

whereλ1λ2λ3areconstantsorpossiblyfunctionsoftSomespecialcasesof(642)areofinterestIfλ2=λ3thenthebodyundergoesauniformexpansionorcontractioninalldirectionstransversetothex1directionIfλ1=λ2=λ3thebodyundergoesauniformexpansionorcontractioninalldirectionsthisiscalledauniformdilationIfλ1= andλ3=1thenareasareconservedinplanesnormaltothex3directionandthedeformationisapureshear

Forthedeformation(642)wereadilyobtainfrom(640)and(641)

(643)

(b)SimpleshearInthisdeformationparallelplanesaredisplacedrelativetoeachotherbyanamountproportionaltothedistancebetweentheplanesandinadirectionparalleltotheplanesForexamplethesimplesheardeformationillustratedinFig64isdescribedbytheequations

(644)

HeretheplanesX2=constantaretheshearplanesandtheX1directionisthesheardirectionTheangleγisameasureoftheamountofshearNotethatasimpleshearinvolvesnochangeinvolumeofanyportionofthebodyForthedeformation(644)wefindfrom(640)and(641)that

(645)

Thecomponentsofγandηfollowfrom(641)

Figure64Simpleshear

(c)HomogeneousdeformationsThesearemotionsoftheformor

(646)

whereciandAiRareconstantsorfunctionsoftimeCases(a)and(b)abovearespecialcasesof(646)Inthemotion(646)F=ATheexpressionsforCRSBijandsoonfollowfrom(641)andweobserve

thatinahomogeneousdeformationallthedeformationandstraintensorsareindependentofthecoordinatesxiorXR

Homogeneousdeformationshaveanumberofpropertiesincludingthefollowing(i)Materialsurfaceswhichformplanesinthereferenceconfigurationdeformintoplanestwoparallelplanesdeformintotwoparallelplanes

(ii)Materialcurveswhichformstraightlinesinthereferenceconfigurationdeformintostraightlinestwoparallelstraightlinesdeformintotwoparallelstraightlines

(iii)Amaterialsurfacewhichformsasphericalsurfaceinthereferenceconfigurationisdeformedintoanellipsoidalsurface

TheproofoftheseandothersimilarresultsisstraightforwardAsanexampleweprove(i)TheequationsatisfiedbythematerialcoordinatesXRofparticleswhichinitiallylieonaplanewithunitnormalnandperpendiculardistancepfromtheoriginis

nmiddotX=p

AfterdeformationthesameparticleslieonasurfacesuchthattheirpositionvectorsxarerelatedtoXby(646)Hence

nmiddotAndash1middot(xmdashc)=p

ThisistheequationofaplanewhosenormalisinthedirectionofthevectornmiddotAndash1(itisassumedthatdetAne0)

(d)PlanestrainThedeformationdefinedby

x1=x1(X1X2)

x2=x2(X1X2)

x3=X3

iscalledaplanestrainTheplanesx3=constantarethedeformationplanesParticleswhichinitiallylieinagivendeformationplaneremaininthatplaneandtheirdisplacementisindependentoftheX3coordinateDeformationswhichapproximatetoplanestrainoccurinmanyproblemsofpracticalinterest

(e)PuretorsionThisdeformationismosteasilydescribedintermsofcylindricalpolarcoordinatesR

ΦZandrφzdefinedby

(647)

Thenapuretorsionisdefinedby

(648)

whereψisconstantorafunctionoftimeInthisdeformationplanesnormaltotheZ-axisrotateabouttheZ-axisbyanamountwhichisproportionaltoZThedeformationismosteasilyvisualizedintermsofthetwistingofacircularcylindricalrodwhoseaxisliesalongtheZ-axisTherearenovolumechangesandthedeformationisnothomogeneous

Figure65Pureflexure

(f)PureflexureThedeformationillustratedinFig65isdescribedby

(649)

ThisrepresentsthebendingofarectangularblockintoasectorofacircularcylindricaltubeThematerialsurfacesX1=constantwhichareparallelplanesinthereferenceconfigurationbecomeconcentriccircularcylindricalsurfacesinthedeformedconfigurationandthematerialplanesX2=constantaredeformedfromafamilyofparallelplanesintoafamilyofradialplaneseachcontainingthez-axis

66InfinitesimalstrainManycommonmaterialsexperienceonlysmallchangesofshapewhenforcesofreasonablemagnitudesareappliedtothemSuchmaterialsincludetheusualstructuralmaterialslikemetalsconcreteandwoodInapplicationsinvolvingmaterialsofthiskindagreatsimplificationcanbeachievedbyapproximatingthefiniteandexactstraintensorsintroducedinSection64bytheapproximateinfinitesimalstraintensor

Theapproximationweintroduceisthatallcomponentsofthedisplacementgradienttensor(whicharedimensionlessquantities)arenumericallysmallcomparedtooneThusweassume

(650)

andneglectthesquaresandproductsofthesequantities

Nowsinceui=xindashXi

Howeverbythebinomialexpansion

ImdashFmdash1=ImdashI+(FmdashI)mdash1=ImdashImdash(FmdashI)+(FmdashI)2mdash(FmdashI)3+

Hence

andsosinceFmdashI=(partuipartXR)

(651)

Thereforetofirstorderinthedisplacementgradientspartuipartxj≃partuipartXjanditisimmaterialwhetherthedisplacementgradientsareformedbydifferentiationwithrespecttomaterialcoordinatesXRortospatialcoordinatesxiTothisorderofapproximationitfollowsfrom(638)and(639)that

(652)

ThetensorEwhosecomponentsEijaredefinedas

(653)

iscalledtheinfinitesimalstraintensorThus

BothγandηreducetoEtotheapproximationinwhichsquaresproductsandhigherpowersofthedisplacementgradientsareneglectedFrom(626)itfollowsthat

(654)

ThisrelationisexactandinvolvesnoapproximationSinceFisasecond-ordertensorEisasecond-ordertensorandclearlyEissymmetric

ThetensorEcannotbeanexactmeasureofdeformationbecauseitdoesnotremainconstantinarigid-bodyrotationToillustratethisconsidertherotation(63)throughαabouttheX3-axisForthismotionwefindthat

ThusE11andE22arenotzeroHowevertheyareofsecondorderinthesmallangleαandsoareneglectedinthesmalldisplacementgradientapproximation

AlthoughtheinfinitesimalstraintensorisnotanexactmeasureofdeformationitoftenprovidesanexcellentapproximationtosuchameasureTypicallyfordeformationsofstructuralmaterialsEijareoforder0001orlessandtheapproximationneglectsthiscomparedwithoneTheclassicaltheoryoflinearelasticitywithitsnumeroussuccessfulapplicationsisconstructedonthebasisofthisapproximationTheadvantageoftheinfinitesimalstraintensoristhatunlikeγRSandηijthecomponentsEijarelinearinthedisplacementcomponentsuiThismeansthatthetechniquesoflinearanalysiscanbeappliedtothesolutionofboundary-valueproblemsinforexamplethelineartheoryofelasticity

ThegeometricalinterpretationofE11isillustratedinFig66ThelineelementP0Q0oflengthδLinitiallyliesparalleltotheX1-axisSincetherotationofthelineelementissmallitsextensiontofirstorderinδLis

(655)

HencetofirstorderE11istheextensionperunitinitiallengthofalineelementwhichisinitiallyparalleltotheX1-axis

AsimilargeometricalinterpretationofE23isillustratedinFig67SupposethatP0Q0andP0R0arelineelementswhichareinitiallyparalleltotheX2-andX3-axesThenbysimilarargumentstheanglesθ1andθ2showninFig67are

(656)

Hence2E23= istofirstorderthedecreaseduringthedeformationintheanglebetweentheinitiallyorthogonalmateriallineelementsP0Q0andP0R0

ThetensorEpossessestheusualpropertiessharedbyallsymmetricsecond-ordertensorsIthasanorthogonaltriadofprincipalaxesifthesearechosenascoordinateaxesthenthematrixofcomponentsofEhasdiagonalformThecorrespondingdiagonalelementsE1E2E3areprincipalcomponentsof

infinitesimalstrainSymmetricfunctionsofE1E2andE3areinvariantsoftheinfinitesimalstraintensor

Figure66GeometricalinterpretationofE11

Figure67GeometricalinterpretationofE23

BecausethecomponentsEijarederivedfromthethreedisplacementcomponentsuitheEijarenotfullyindependentbutmustsatisfyrelationsobtainedbyeliminatinguibetweenthemItcanbeverifiedbydirectsubstitutionfrom(653)thatEijsatisfythestraincompatibilityrelations

(657)

(658)

andthefoursimilarrelationsobtainedbycyclicpermutationsoftheindices123Thesesixcompatibilityrelationsarethemselvesnotcompletelyindependentforitcanbeverifiedagainbydirectsubstitutionthat

(659)

andtherearetwosimilarrelationsobtainedbycyclicpermutationoftheindices123ThefinitestraincomponentsγRSandηijarealsosubjecttocompatibilityconditionsbuttheseconditionsaremuchmorecomplicatedinform

67InfinitesimalrotationIn(69)and(610)wegaveformulaewhichdescribeafiniterigid-bodyrotationthroughtheangleαaboutanaxisnForaninfinitesimalrotationsinα≃αandcosα≃1andtothisorderofapproximation(610)gives

andhence

(660)

Thusaninfinitesimalrotationisdescribedbyananti-symmetrictensorWenotethatthisrotationisalsodescribedinmagnitudeanddirectionbythevectorαnandobservetheconnectionsbetweenthe

componentsofthevectorandthoseofthetensor

NowconsiderageneralinfinitesimalmotionwithdeformationgradienttensorFWedefinetheinfinitesimalrotationtensorΩanditscomponentsΩijasfollows

(661)

ClearlyΩisasecond-orderanti-symmetrictensorandsoitcanrepresentaninfinitesimalrotationThedisplacementgradienttensorFndashIisnowdecomposedintoitssymmetricandanti-symmetricpartsasfollows

(662)

ThisexpressesanyinfinitesimalmotionasthesumofaninfinitesimaldeformationrepresentedbyEandaninfinitesimalrotationrepresentedbyΩ

Theinfinitesimalrotationvectorωisdefinedby

(663)

Thenitfollowsfrom(661)and(663)that

(664)

(665)

FurtherdiscussionoftherotationwillbegiveninSection92

TheassumptionthatpartuipartXR≪1carriestheimplicationthatboththestrainandtherotationaresmallItispossibletoenvisageandtorealizesituationsinwhichthestraincomponentsareeverywheresmallbutsomematerialelementsundergolargerotationsThismayoccurforexampleinthebendingofalongthinflexiblerodIndividualelementsoftherodchangeshapeonlyslightlybuttherotationsanddisplacementscanbelargeSuchproblemsrequirecarefulformulationandwillnotbediscussedhere

68The rate-of-deformationtensorInmanyproblemsincontinuummechanicsthekinematicpropertyofgreatestinterestisnotthechangeofshapeofabodybuttherateatwhichthischangeistakingplaceThisisespeciallythecaseinfluidmechanicswhereitisusuallyrequiredtofindthefluidflowinaparticularregionofspaceandtheshapeofthebodyoffluidatareferencetimeisrarelyrelevant

WethereforebeginthissectionbyinvestigatingtherateofextensionofamateriallineelementthatistherateofchangeofλforafixedmateriallineelementThestartingpointisequation(615)

(666)

whichgivesλintermsofmaterialcoordinatesXRandthedirectioncosinesARofthelineelementinthereferenceconfigurationItisconvenienttobeginwith(666)despitethefactthateventuallywewishtoexpressDλDtintermsofspatialcoordinatesxiandthedirectioncosinesaiofthelineelementattimetinthecurrentconfiguration

Wedifferentiate(666)withrespecttotwithXRheldconstantSinceDxi(XRt)Dt=υi(XRt)thisgives

(667)

Tointroducederivativesofυiwithrespecttospatialcoordinatesweuserelationsoftheform

andtherebyexpress(667)intheform

Aninterchangeofthedummyindicesiandjinthefinaltermthengives

Nextwetwiceemploytherelation(614)tointroduceaiinplaceofARandsoobtain

(668)

Nowλmdash1DλDtistherateofextensionperunitcurrentlengthofamateriallineelementwithcurrentdirectioncosinesaiForanygivendirectionathisextensionrateisfrom(668)givenbyaiaiDijwhere

(669)

ThequantitiesDijarethecomponentsreferredtobasevectorseioftherate-of-deformationtensorD(othercommonnamesaretherate-of-strainorstrain-ratetensor)NotethatDijislinearinthevelocitycomponentsυiandthatthislinearityisexactandwehavenotmadeanyapproximationinderivingitWealsoobservethattherightsideof(668)involvesonlyquantitiesmeasuredinthecurrentconfigurationalthoughwehavemadeuseofareferenceconfigurationinordertoderive(668)

Therate-of-deformationtensorDhaspropertieswhichinalmosteveryrespect(butwithanimportantexceptionnotedbelow)areanalogoustothoseoftheinfinitesimalstraintensorEItisreadilyverifiedthatDisasecond-ordersymmetrictensorReferredtoitsprincipalaxesascoordinateaxesthematrixofcomponentsofDhasdiagonalformwithprincipalcomponentsD1D2andD3ThelargestandsmallestoftheprincipalcomponentsareextremalvaluesoftheextensionrateforvariationsofthedirectionaSymmetricfunctionsofD1D2andD3areinvariantsofDThecomponentsDijobeycompatibilityrelationswhicharepreciselyanalogoustotherelations(657)(658)and(659)satisfiedbyEijexceptthatdifferentiationmustbewithrespecttospatialcoordinatesxiandthesemaynotbereplacedbymaterialcoordinatesXR

ThetensorDdiffersfromthetensorEinthatitisanexactmeasureofdeformationratewhereasitwasemphasizedinSection66thatEcanneverbeanexactmeasureofdeformationThefactthatDijare

linearinthevelocitycomponentsisafortunatecircumstancewhichsimplifiesthesolutionofproblemsinfluidmechanics

69The ve loc itygradientandspintensorsThedeformation-ratetensorDcanbeidentifiedasthesymmetricpartofthevelocitygradienttensorLwhosecomponentsLijaregivenby

(670)

Theanti-symmetricpartofLisdenotedbyWandthecomponentsofWbyWijsothat

(671)

and

(672)

ItisstraightforwardtoverifythatLandWaresecond-ordertensors

ThetensorWiscalledthespinorvorticitytensorandithaspropertiesanalogoustothoseoftheinfinitesimalrotationtensorexceptthatnoapproximationisinvolvedinitsderivationoruseItisameasureoftherateofrotationofanelementtheexpressions(672)decomposeLintothedeformationrateDandthespinWThespinmayalsobedescribedbythevorticityvectorwdefinedby

(673)

Byrelationssimilarto(664)and(665)wehavethefollowingconnectionsbetweenWandw

(674)

Inarigid-bodyrotationwithangularspeedωaboutanaxisthroughOwithunitvectornthevelocityisgivenby

(675)

Henceinsuchamotionw=2ωnand

ThusDvanishesinarigid-bodyrotationMoreoverifageneralmotionismodifiedbysuperposingonittherigid-bodyrotation(675)thenDisthesameinthemodifiedmotionasitwasintheoriginalmotionThisconfirmsthatDisunaffectedbysuperposedrotationsandisthereforeasuitablemeasureofthedeformationrate

ThematerialtimederivativeofFiRisgivenby

Thus

(676)

InthecaseofsmalldisplacementgradientswehaveFndash1≃Iandthen

(677)

610Somesimple f lows

(a)SimpleshearingflowIftheplanesx2=constantaretheshearplanesandthex1directionisthedirectionofshearthen

υ1=sx2υ2=0υ3=0

wheresisconstantisasimpleshearingflowThefluidflowsinstraightlinesinthex1directionwithspeedproportionaltoitsdistancefromtheplanex2=0Forthisflow

(b)RectilinearflowInrectilinearflowthematerialflowsinparallelstraightlinesthismay(butdoesnotalways)occurinflowdownapipeofuniformcross-sectionorinflowbetweenparallelplatesIfthedirectionofflowisthatofthex3-axisthen

υ1=0υ2=0υ3=f(x1x2x3)

and

andtheremainingcomponentsDijandWijarezeroIfthevelocityisindependentofx3theninadditionD33=0

(c)VortexflowFlowintheneighbourhoodofavortexlinelyingalongthex3-axisisdescribedby

whereκisaconstantParticlestravelincirclesaroundthex3-axiswithspeedinverselyproportionaltothedistancefromtheaxisThecomponentsofDandWare

Thereisasingularityonthevortexline

(d)PlaneflowIfthevelocityisoftheform

υ1=υ1(x1x2t)υ2=υ2(x1x2t)υ3=0

theparticlesmoveinplanesparalleltox3=0andthevelocityisindependentofthex3coordinateThenon-zerocomponentsofDareD11D22andD12andthesearefunctionsofx1x2andtonlyTheonlynon-zerocomponentofWisW12=mdashW21andthevorticityvectorisinthedirectionofthex3-axisThesimpleshearingandvortexflowsdefinedabovearespecialcasesofplaneflow

611Problems1Provetheformulae(616)and(617)

2Abodyundergoesthehomogeneousdeformation

Find(a)thedirectionafterthedeformationofalineelementwithdirectionratios111inthereferenceconfiguration(b)thestretchofthislineelement

3FindthecomponentsofthetensorsFCBFmdash1Cmdash1Bmdash1γandηforthedeformation

x1=a1(X1+αX2)

x2=a2X2

x3=a3X3

wherea1a2a3andαareconstantsFindtheconditionsontheseconstantsforthedeformationtobepossibleinanincompressiblematerialAbodywhichinthereferenceconfigurationisaunitcubewithitsedgesparalleltothecoordinateaxesundergoesthisdeformationDeterminethelengthsofitsedgesand

theanglesbetweentheedgesafterthedeformationSketchthedeformedbody

4AcircularcylinderinitsreferenceconfigurationhasradiusAanditsaxisliesalongtheX3-axisItundergoesthedeformation

FindtheconditionsontheconstantsλμandψforthisdeformationtobepossibleinanincompressiblematerialAlinedrawnonthesurfaceofthecylinderhasunitlengthandisparalleltotheaxisofthecylinderinthereferenceconfigurationFinditslengthafterthedeformationFindalsotheinitiallengthofalineonthesurfacewhichhasunitlengthandisparalleltotheaxisafterthedeformation

5Showthattheconditionforamateriallineelementtobeunchangedindirectionduringadeformationis(FiRmdashλδiR)AR=0Deducethattheonlylineswhichdonotrotateinthesimplesheardeformation(644)arelineswhichareperpendiculartotheX2-axisForthedeformation

x1=μ(X1+X2tanγ)

x2=micromdash1X2

x3=X3(μne1)

showthattherearethreedirectionswhichremainconstantFindthesedirectionsandthecorrespondingstretches

6Provethatinthehomogeneousdeformation(646)particleswhichafterthedeformationlieonthesurfaceofasphereofradiusboriginallylayonthesurfaceofanellipsoidProvethatthisellipsoidisasphereofradiusaifa2AijAik=b2δjk

7Arodofcircularcross-sectionwithitsaxiscoincidentwiththex3-axisisgivenasmalltwistsothatitsdisplacementisgivenby

u1=mdashψx2x3

u2=ψx1x3

u3=0

whereψisconstantFindthecomponentsofinfinitesimalstrainandinfinitesimalrotationShowthatone

oftheprincipalcomponentsofinfinitesimalstrainisalwayszeroandfindtheothertwoprincipalcomponentsFindalsotheprincipalaxesoftheinfinitesimalstraintensor

8Forthedeformation

u3=CX3

whereABandCareconstantsfindthecomponentsofthetensorsFEandΩAlsofindtheprincipalvaluesandprincipalaxesofE

9ForthevelocityfieldsgiveninProblems2and3ofChapter4findthecomponentsofthetensorsLDandW

10Provethattherateofchangeoftheangleθbetweentwomateriallineelementswhosedirectioninthecurrentconfigurationaredeterminedbyunitvectorsaandbisgivenby

sinθ=(aiaj+bibj)Dijcosθmdash2aibjDij

Deducethatmdash2Dij(inej)istherateofchangeoftheanglebetweentwomateriallineelementswhichinstantaneouslyliealongthexi-andxj-axes

11AnincompressiblebodyisreinforcedbyembeddinginittwofamiliesofstraightinextensiblefibreswhosedirectionsinthereferenceconfigurationaregivenbyA1=cosβA2=plusmnsinβA3=0whereβisconstantThebodyundergoesthehomogeneousdeformation

x1=

x2=

x3=microX3

whereαandmicroareconstantsShowthatthecondition=1forinextensibilityinthefibredirectionrequiresthata2cos2(3+αmdash2sin2β=microDeducethat(a)theextenttowhichthebodycancontractinthex3directionislimitedbytheinequalitymicrogesin2β(b)whenthismaximumcontractionisachievedthetwofamiliesoffibresareorthogonalinthedeformedconfiguration

7

Conservationlaws

71ConservationlawsofphysicsManyofthelawsofclassicalphysicscanbeexpressedintheformofastatementthatsomephysicalquantityisconservedexamplesofsuchquantitiesaremasselectricchargeandmomentumLawsofthiskindaregeneralstatementsandarenotrestrictedintheirapplicationtoanyparticularmaterialorclassofmaterialsThemathematicalformulationsoftheselawsarethereforeequationswhichmustbealwayssatisfiedItisimportanttodistinguishsuchequationsfromequations(whichwecallconstitutiveequations)whichdescribethepropertiesofparticularmaterialsorclassesofmaterialsandwhicharethesubjectofChapters8and10

WenoteinpassingthatthesecondlawofthermodynamicsalthoughitisanimportantgenerallawofphysicsisratherdifferentfromtheconservationlawsmentionedaboveinthatitisexpressedasaninequalityContinuumthermodynamicsisoutsidethescopeofthisintroductorytextandweshallnotdiscussit

72ConservationofmassThelawofconservationofmasswillbeformulatedintwodifferentformsWefirstconsidertheeffectofafinitedeformationonavolumeelement

DeformationofavolumeelementThenotationofSections41and62-64isemployedConsideranelementarytetrahedroninthereferenceconfiguration(Fig71)suchthatitsverticesP0Q0R0S0havepositionvectorsX(0)X(0)+δX(1)X(0)+δX(2)X(0)+δX(3)withcoordinates

(71)

respectivelyThevolumeδVofP0Q0R0S0is

(72)

Figure71Deformationofavolumeelement

InadeformationtheparticlesinitiallyatP0Q0R0S0movetoPQRSwithpositionvectorsx(0)x(0)+

δx(1)etcandcoordinates etcrespectivelyThevolumeδυofthetetrahedronPQRSis

Thedeformationisdefinedbyequationsoftheformxi=xi(XRt)Hence

(73)

withthederivativesevaluatedat andsimilarrelationsholdfor and Thereforetheexpressionforδυbecomes

Byusingthealgebraicresult(222)thiscanbewrittenas

(74)

wherewehaveintroducedtheJacobian

Wenowproceedtothelimit (p=123)sothattheinitialvolumeofthetetrahedrontendstozeroThenfrom(72)and(74)

(75)

From(618)werecognizetheaboveJacobianasthedeterminantofthedeformationgradienttensorFsothat(75)canbewrittenas

(76)

IfthematerialisincompressiblethendυdV=1andhencedetF=1

ByexpandingdetFweobtain

Henceinthecaseofsmalldisplacementgradients

(77)

ThequantityEiiiscalledthedilatationandisdenotedbyΔFrom(77)ΔisthetraceoftheinfinitesimalstraintensorandsoisthefirstinvariantofthattensorThus

Δ=Eii=trE=E1+E2+E3

ForsmalldeformationsΔisameasureofthechangeofvolumeperunitinitialvolumeofanelement

ConservationofmassmdashLagrangianformNowsupposethatthematerialinthevolumeelementP0Q0R0S0hasmassδminthereferenceconfigurationConservationofmassrequiresthatthemassofthematerialinthematerialvolumeelementremainsconstantduringthedeformationHencetheinitialandfinaldensitieswhichwedenotebyρ0andprespectivelyare

Hence

(78)

andthisistherequiredstatementofthelawofconservationofmassWenotethat(78)justifiestheassumptionwhichwasmadeinSection63thatdetFne0forifdetF=0thenthedensityiseitherzerointheinitialconfigurationorinfiniteinthedeformedconfiguration

Conservationofmass-EulerianformEquation(78)expressesthelawofconservationofmassintermsofdeformationgradientsFormanypurposesitismoreconvenienttoexpressthelawintermsofthevelocitycomponentsForthisweconsideranarbitraryregionℛwithsurfaceSfixedinspaceinrelationtoafixedframeofreference(seeFig72)ThemassconservationlawisexpressedintheformthattherateatwhichthemasscontainedinℛincreasesisequaltotherateatwhichmassflowsintoℛoverSTherateatwhichmassflowsoveranelementofsurfaceofareadSispdSmultipliedbythenormalcomponentofvelocityHence

(79)

wherepartρparttistherateofincreaseofρatafixedpointinℛThenegativesignontheright-handsideappearsbecausendenotes

Figure72Theregionℛ

theoutwardnormaltoSByapplyingthedivergencetheoremtothesurfaceintegralweobtainfrom(79)

(710)

Sincetheregionℛisarbitrarytheintegrandin(710)mustbezeroeverywhereforotherwiseitwouldbepossibletoconstructaregionforwhich(710)wasviolatedHence

(711)

ThisequationisoftencalledthecontinuityequationByintroducingthecomponentsofυandx(711)isreadilyexpressedinthefollowingequivalentforms

(712)

(713)

(714)

whereasinSection43DpDtdenotesthematerialderivativeofρ

IfthematerialisincompressiblethenρisconstantatanyparticlesothatDρDt=0Itthereforefollowsfrom(714)thattheincompressibilityconditioncanbeexpressedinanyofthefollowingequivalentforms

(715)

ThedeviceofconvertingasurfaceintegralintoavolumeintegralbytheuseofthedivergencetheoremwillbeusedfrequentlyinthischapterNaturallytheresultsofdoingthisarevalidonlyiftheconditionsforthetheoremtobeapplicablearesatisfiedThemostimportantoftheseisthattheintegrandofthesurfaceintegralshouldbedifferentiableandthereforecontinuousProblemsdoariseincontinuummechanicsinwhichdensityvelocitystressandothervariablesarediscontinuousacrosscertainsurfaceswhichmaybestationaryorinmotionThissituationarisesparticularlyinstress-wavepropagationproblemsItisnotdifficulttoextendthetheorytodealwithsuchcasesandforsomeproblemsitisessentialtodosoHoweverinthistextitisalwaysassumedthatnecessarysmoothnessconditionsaresatisfied

73The materialt imederivative ofa volumeintegralSupposethatΦissomephysicalquantity(suchasmassorenergy)associatedwiththeparticlesofabodyandφistheamountofΦperunitmassThentheamountofΦperunitvolumeisρΦandtheamountofΦcontainedinafixedregionℛatagiventimetis

(716)

evaluatedattInanincrementoftimeδtthevalueof0atagivenpointoratagivenparticleinℛwill(ingeneral)changeandsomeparticleswilltravelacrossthesurfaceSofℛtransporting(DwiththemTherateofchangeoftheamountof(Dwhichisassociatedwiththeparticleswhichinstantaneouslyoccupy91attiscalledthematerialtimederivativeoftheintegral(716)andisdenotedas

(717)

Therateofincreaseoftheamountof4)withinthefixedregion91isequaltothesumoftherateofincreaseofassociatedwiththeparticlesinstantaneouslywithinℛtogetherwiththenetrateofinfluxofΦintoℛThus

Byapplyingthedivergencetheoremtothesurfaceintegralandrearrangingweobtain

(718)

Ifφ=1theintegral(716)representsthemasswithinℛandconservationofmassrequiresthatthematerialtimederivativeofthisintegraliszeroHencetheintegralontherightsideof(718)(withφ=1)musthavethevaluezeroforallregionsℛandsotheintegrandontherightsideiszeroThusweagainobtainthecontinuityequationintheform(711)

Forageneralquantityφtheintegrandoftherightsideof(718)maybewrittenas

(719)

Howeverby(420)andthecontinuityequation(711)theexpression(719)isjustρDφDtHence

(718)takestheform

(720)

74ConservationoflinearmomentumThelawofconservationoflinearmomentumforaparticleofmassmstatesthattherateofchangeofitslinearmomentumisequaltotheresultantforcepappliedtoitThus

ForacontinuumthisstatementisgeneralizedasfollowstherateofchangeoflinearmomentumoftheparticleswhichinstantaneouslyliewithinafixedregionℛisproportionaltotheresultantforceappliedtothematerialoccupyingℛThisresultantforceconsistsoftheresultantofthebodyforcesbperunitmassactingontheparticlesinℛtogetherwiththeresultantofthesurfacetractionst(n)actingonthesurfaceofℛHencethelawisexpressedintheform

(721)

Incomponentsaftermakinguseof(59)thistakestheform

wherenistheoutwardnormaltoS

Wenowuse(720)withφreplacedbyυjandapplythedivergencetheoremtothesurfaceintegralThisgives

BytheusualargumenttheintegrandiszeroandDυjDt=fjwherefistheaccelerationvectorHence

(722)

ThisistheequationofmotionforacontinuumItreducestotheequilibriumequation(523)whenthereisnoacceleration

75ConservationofangularmomentumForaparticlethelawofconservationofangularmomentumstatesthat

wherepistheresultantappliedforceandxisthepositionvectorfromanarbitrarilychosenoriginThegeneralizationforacontinuumanalogousto(721)is

orincomponents

(723)

Intheusualmannerweemploy(720)withφ=eijkxjυktransformthesurfaceintegraltoavolumeintegralandequatetheintegrandsoftheresultingvolumeintegralsonthetwosidesoftheequationThisgives

(724)

Now

and

Henceequation(724)canbewrittenas

(725)

Howevereijkυjυk=0andtheexpressionmultipliedbyxjin(725)iszerobytheequationofmotionandso(725)reducesto

(726)

Thusthelawofconservationofangularmomentumleadstotheconclusionthatthestresstensorisasymmetrictensor

Itshouldbementionedthatinwritingdown(723)itisimplicitlyassumedthatnodistributedbodyorsurfacecouplesactonthematerialinℛIfsuchbodyorsurfacecouplesdoacttheningeneralthesymmetryofTnolongerobtainsHoweverbodyandsurfacecouplesareofimportanceonlyinratherspecializedapplicationsandweshallnotconsiderthem

76ConservationofenergyThekineticenergyKofthematerialwhichinstantaneouslyoccupiesafixedregionℛisdefinedtobe

(727)

Thisisthenaturalextensiontoacontinuumoftheusualexpressionforthekineticenergyofaparticleorrigidbody

ThekineticenergyofacontinuumisonlypartofitsenergyTheremainderiscalledtheinternalenergyEwhichisexpressedintermsoftheinternalenergydensityeby

(728)

ThestatementweadoptofthelawofconservationofenergyisasfollowsthematerialtimederivativeofK+EisequaltothesumoftherateatwhichmechanicalworkisdonebythebodyandsurfaceforcesactingonℛandtherateatwhichotherenergyentersℛThelsquootherenergyrsquomaytakemanydifferentformsThemostimportantisenergyduetoheatfluxacross

SOtherpossibleformsareenergyarisingfromchemicalchangesinsideℛenergyarrivingbyradiationelectromagneticenergyandsoonWeshallconsideronlytheheatflux

TheabovestatementofthelawisnotparticularlyhelpfulonitsownbecauseitcanberegardedasbeingmerelyadefinitionofEItreallyonlybecomesusefulwhensomefurtherpropertiesofEorearespecifiedTodothisleadsintotheconsiderationofconstitutiveequationswhichwedeferuntilChapters8and10

Ifqidenotethecomponentsoftheheat-fluxvectorq(thatisqsdotnistheamountofheatflowinginthesenseoftheunitvectornacrossasurfacenormaltonperunitareaperunittime)thenthemathematicalformulationofthelawintheformstatedaboveis

(729)

ThenegativesigninthelasttermarisesbecausenistheoutwardnormaltoSandwerequiretheinfluxofheatontherightoftheequationByemploying(720)ontheleftsidetransformingthesurfaceintegraltoavolumeintegralandequatingtheintegrandsitfollowsfrom(729)bytheargumentwhichisnowstandardthat

(730)

NowDυiDt=fiHenceafterrearrangement(730)becomes

Theexpressioninbracketsiszerobytheequationofmotion(722)andso

(731)

ByinterchangingthedummyindicesiandjwehaveTjipartυipartxj=TijpartυjpartxiandsinceTissymmetricTjipartυipartxj=TijpartυiexclpartXjHenceby(669)

and(731)maybewrittenas

(732)

ThisistheenergyequationforacontinuumThetermTijDijcanbeinterpretedastherateofworkingofthestress

TomakefurtherprogressitisnecessarytoassignfurtherpropertiestoeandqForexampleitisoftenassumedthatagashasacaloricequationofstatee=e(ρT)whereTistemperatureTheheatfluxqisoftenassumedtoobeyFourierrsquoslawofheatconduction

(733)

whereKisthethermalconductivitySuchstatementsarenotgenerallawsbutareparticulartocertainmaterialsandarecertainlynotuniversallytrue

77The princ iple ofvirtualwork

TheprincipleofvirtualworkhasmanyapplicationsincontinuummechanicsAlthoughitisnotaconservationlawitisconvenienttointroduceithereSupposethereisdefinedintheregionℛastressfieldwithcomponentsTijwhichsatisfytheequilibriumequations

Alsosupposetobedefinedinℛavelocityfieldwithcomponentsυiwhicharedifferentiablewithrespecttoxiandlet

bethecomponentsofthedeformation-ratetensorderivedfromthevelocityfieldυi

ItisemphasizedthatTijandυineedbeinnowayconnectedTijmaybeanyequilibriumstressfieldandυianydifferentiablevelocityfield

WeformtheproductTijDijandintegrateitovertheregionℛThenusing(523)andthesymmetryrelationsTij=Tjiwehave

Finallybyanapplicationofthedivergencetheoremweobtain

(734)

whereniarethedirectioncosinesoftheoutwardnormaltothesurfaceSofℛandt(n)isthesurface-tractionvectoronSwhichcorrespondstothestresscomponentsTij

Equation(734)isthemathematicalexpressionoftheprincipleofvirtualworkforacontinuumItstatesthattherateofworkingofthestressfieldTijinthevelocityfieldυiisequaltothesumoftheratesofworkingofthesurfaceandbodyforcesassociatedwithTijinthesamefield

AnidenticalargumentmaybefollowedwithυireplacedbyinfinitesimaldisplacementcomponentsuiandDijreplacedbytheinfinitesimalstraincomponentsEij

Therelation(734)anditsanalogueintermsofinfinitesimaldisplacementandstrainformthebasisofanumberofvariationaltheoremsinparticularbranchesofcontinuummechanics

78Problems1ForanincompressibleNewtonianviscousfluidinwhichFourierrsquoslawofheatconductionissatisfiedTijqiandearegivenby

Tij=mdashpδij+2microDij

qi=mdashκpartTpartxi

e=CT

wheremicroκandCareconstantsandTisthetemperatureDeducethatinthiscasetheenergyequation(732)canbeexpressedintheform

2AsingularsurfaceisasurfaceacrosswhichthestressvelocityanddensitymaybediscontinuousByconsideringathincylindricalregionwhichenclosespartofasingularsurfaceshowthatinabodyatrestinequilibriumt(n)iscontinuousacrossastationarysingularsurfacewherenisthenormaltothesingularsurface

3SupposeasingularsurfacepropagatesthroughabodywithspeedVrelativetothebodyinthedirectionofthenormaltothesurfaceProvethatthequantitiesρVandρVυ+t(n)arecontinuousacrossthesingularsurface

4AsingularsurfacepropagatesinthedirectionofaunitvectornwithspeedυrelativetofixedcoordinatesShowthatifuiscontinuousacrossthesingularsurfacethenυiexcl+υnjpartuipartxjisalsocontinuousacrossthesingularsurface

8

Linearconstitutiveequations

81Constitutive equationsandidealmaterialsTheresultsgivensofarinthisbookapplyequallytoallmaterialsInthemselvestheyareinsufficienttodescribethemechanicalbehaviourofanyparticularmaterial

TocompletethespecificationofthemechanicalpropertiesofamaterialwerequireadditionalequationswhicharecalledconstitutiveequationsTheseareequationswhichareparticulartoindividualmaterialsorclassesofmaterialsandtheyservetodistinguishonematerialfromanotherThemechanicalconstitutiveequationofamaterialspecifiesthedependenceofthestressinabodyonkinematicvariablessuchasastraintensorortherate-of-deformationtensorNormallythermodynamicvariablesespeciallytemperaturewillalsobeinvolvedbutweshallmakeonlybriefreferencestotheseConstitutiveequationsarealsorequiredinotherbranchesofcontinuumphysicssuchascontinuumthermodynamicsandcontinuumelectrodynamicsbuttheseproblemsareoutsidethescopeofthisbookandweshallonlydiscussconstitutiveequationsforthestress

ThemechanicalbehaviourofrealmaterialsisverydiverseandcomplexanditwouldbeimpossibleevenifitweredesirabletoformulateequationswhicharecapableofdeterminingthestressinabodyunderallcircumstancesRatherweseektoestablishequationswhichdescribethemostimportantfeaturesofthebehaviourofamaterialinagivensituationSuchequationscanberegardedasdefiningidealmaterialsItisunlikelythatanyrealmaterialwillconformexactlytoanysuchmathematicalmodelbutiftheidealmaterialiswellchosenitsbehaviourmaygiveanexcellentapproximationtothatoftherealmaterialwhichitmodelsThemodelshouldbeselectedwiththeapplicationaswellasthematerialinmindandthesamerealmaterialmayberepresentedbydifferentidealmaterialsindifferentcircumstancesForexamplethetheoryofincompressibleviscousfluidsgivesanexcellentdescriptionofthebehaviourofwaterflowingthroughpipesbutisuselessforthestudyofthepropagationofsoundwavesthroughwaterbecauseforsound-wavepropagationamodelwhichtakesintoaccountthecompressibilityofwaterisessential

Historicallytheconstitutiveequationswhichdefinetheclassicalidealmaterials(linearelasticsolidsNewtonianviscousfluidsetc)havebeendevelopedseparatelyInapplicationsofthesetheoriesthisseparationisnaturalHoweverattheformulativestagethereareadvantagesinaunifiedapproachwhichclarifiesrelationsbetweenthedifferentspecialtheoriesAlsoitispossibletoformulatesomegeneralprincipleswhichshouldbefollowedintheconstructionofconstitutiveequations

AfirstrequirementwhichanyconstitutiveequationmustsatisfyisthatofdimensionalhomogeneitythedimensionsofalltermsinaconstitutiveequationmustbethesameSinceaconstitutiveequationalwaysincludesconstantsorfunctionswhichcharacterizethematerialunderconsiderationandthesequantitieshavedimensionsthedimensionalhomogeneityrequirementisusuallynotdifficulttosatisfy

Constitutiveequationsshouldnotdependonthechoiceofthecoordinatesystem(althoughtheymaybeexpressedintermsofcomponentsrelativetoanyselectedcoordinatesystem)Theythereforetaketheformofrelationsbetweenscalarsvectorsandtensors

Animportantrestrictiononmechanicalconstitutiveequationsistherequirementthatthestressresponseofabodytoadeformationisnotaffectedbyrigid-bodymotionssothatthestressinabodydependsonlyonthechangeofshapeofthebodyandisnotaffected(exceptforthechangeinorientationofthestressfieldrelativetofixedaxes)byasuperposedmotioninwhichthebodymovesasawholeToformalizethisrequirementwespecifythatifabodyundergoestwotime-dependentmotionswhichdifferfromeachotherbyatime-dependentrigid-bodymotionthenthesamestressresultsfromeachofthesemotionsThisisessentiallyequivalenttosayingthatconstitutiveequationsareinvariantundertranslationsandrotationsoftheframeofreferencetwoobserverseveniftheyareinrelativemotionwillobservethesamestressinagivenbody

MaterialsareusuallyregardedaseithersolidsorfluidsandfluidsaresubdividedintoliquidsandgasesWedonotattemptaprecisedefinitionofthisclassificationthedividinglinesarenotalwaysclearandtherearematerialswhichpossessbothsolid-likeandfluid-likepropertiesThecharacteristicpropertyofafluidisthatitcannotsupportashearingstressindefinitelysothatifashearingstressisappliedtoabodyoffluidandmaintainedthefluidwillflowandcontinuetodosoaslongasthestressremainsAsolidontheotherhandcanbeinequilibriumunderashearstressSomesolidspossessanaturalconfigurationwhichtheyadoptinastress-freestateandtowhichtheyeventuallyreturnifastressisimposedandthenremovedifanaturalconfigurationexistsitisusuallyconvenientthoughnotessentialtoadoptitasthereferenceconfigurationFluidshavenonaturalconfigurationandgivensufficienttimewilladapttotheshapeofanycontainerinwhichtheyareplaced

82MaterialsymmetryMostmaterialspossesssomeformofmaterialsymmetryThecommonestcaseisthatinwhichthematerialisisotropicanisotropicmaterialpossessesnopreferreddirectionanditspropertiesarethesameinalldirectionsItisimpossibletodetecttheorientationinspaceofasphereofisotropicmaterialbyperforminganexperimentonitManyrealmaterialsareisotropicornearlysotheseincludecommonfluidslikeairandwatermetalsintheirusualpolycrystallineformconcretesandinbulkandsoonOthercommonmaterialshavestrongdirectionalpropertiesanexampleiswoodwhosepropertiesalongitsgrainarequitedifferentfromthepropertiesacrossthegrainSinglecrystalsofcrystallinematerialshavedirectionalpropertieswhicharisebecausetheiratomsarearrangedinregularpatternsandthisgivesrisetothevariousclassesofcrystalsymmetryAmaterialwhichpossessesasinglepreferreddirectionateverypointissaidtobetransverselyisotropicAnexampleofsuchamaterialisacompositematerialwhichconsistsofamatrixreinforcedbyfibresarrangedinparallelstraightlinesOverlengthscaleswhicharelargecomparedtothefibrediametersandspacingssuchamaterialmayberegardedasmacroscopicallyhomogeneousandthefibresintroduceapreferreddirectionwhichisacharacteristicofthecompositematerial

Weconsidermaterialsymmetriesoftwotypesrotationalandreflectional

RotationalsymmetrySupposeasphericalvolumeelementundergoesthehomogeneousdeformationillustratedinFig81AtypicalparticleinitiallyatP0movestoP1andthedeformationisdescribedbytheequations

(81)

wheresincethedeformationishomogeneousthecomponentsFiRofFdependonlyont

Nowsupposethattheelementundergoesaseconddeformationwhichissimilartothefirstexceptthattheentiredeformationfield(butnotthebody)isrotatedthroughanangleαaboutanaxisnThusifQisthetensordefinedby(611)theparticlewhichisinitiallyatQsdotXmovesintheseconddeformationtothepointQsdotxwhere

(82)

Theseconddeformationisillustratedforthecaseinwhichn=e3inFig81(c)inittheparticleinitiallyatQ0movestoQ2where

angPoOQo=angP1OQ2=α

ThedeformedspherehasthesameshapeinthetwoconfigurationsbutthesecondisnotderivedfromthefirstbyarigidrotationAlthoughthetwodeformations(81)and(82)arerelatedtheyaredistinctandintheabsenceofappropriatematerialsymmetrytheywillgiverisetodifferentstressresponsesForexampletheforceswhichaccompanyagivenextensioninthedirectionOP0willbedifferentfromthoseassociatedwiththesameextensioninthedirectionOQ0HoweverforagivenmaterialitmayhappenthatforcertainrotationstheresultofrotatingthedeformationfieldthroughtherotationdefinedbyQistoproducethesamerotationofthestressfieldInthiscaseifthedeformation(81)givesrisetoastresstensorTthenthedeformation(82)givesrisetoastresstensorQTmiddotTmiddotQWethensaythatthematerialhasmaterialsymmetry(relativetothespecifiedreferenceconfiguration)fortherotationdeterminedbyQ

Figure81Rotationalsymmetry

AsasimpleexamplethetensorQwithcomponentsQiRwhere

representsananti-clockwiserotationofmagnitude abouttheX3-axisIfthematerialhasrotationalsymmetryforthisrotationthentheforcep1requiredtoproduceagivenextensionintheX1directionhasthesamemagnitudeastheforcep2requiredtoproducethesameextensionintheX2direction

ReflectionalsymmetryNowconsiderafurtherhomogeneousdeformationofthesphericalvolumeelementwhichisthemirrorimageofthedeformation(81)insomeplanewhichfordefinitenesswetaketobetheplaneX1=0Thisdeformationisdefinedby

(83)

or

(84)

wherethecomponentsofthetensorR1are

(85)

ThetensorR1representsareflectioninthe(X2X3)planeThedeformationisillustratedinFig82

Intheabsenceofmaterialsymmetrythedeformations(81)and(84)willgiverisetotwounrelatedstressresponsesHoweveriftheeffectofreflectingthedeformationfieldinthemannerdescribedistoreversethesignoftheshearstressontheplanex1=0wesaythatthematerialhasreflectionalsymmetry

withrespecttothisplanerelativetothechosenreferenceconfigurationIfthematerialhasthissymmetryandthedeformation(81)givesrisetothestressTthenthedeformation(84)givesrisetothestress bullTbullR1(thetranspositionofR1inthefirstfactorisredundantbecauseR1issymmetricbutisintroducedforconsistencywiththecorrespondingresultforrotationalsymmetries)

Figure82Reflectionalsymmetry

Moregenerallyareflectionintheplanethrough0normaltoaunitvectornisdefinedbyatensorRwithcomponentsRijwhere

R=Imdash2notimesnRij=δijmdash2ninj

ItiseasilyverifiedthatRisasymmetricimproperorthogonaltensor(thatisanorthogonaltensorwithdeterminantequaltomdash1)Amaterialhasreflectionalsymmetryforreflectionsintheplanesnormaltonifthedeformation

(86)

givesrisetothestressRTmiddotTmiddotRwhenthedeformation(81)givesrisetothestressT

ReflectionalsymmetrywithrespecttoplanesnormaltotheX1-axismeansthatthetangentialforcerequiredtoproduceasimpleshearin(say)thepositiveX2directionontheplanesX1=constantisequalinmagnitudebutoppositeindirectiontothatrequiredtoproduceashearofthesamemagnitudeinthenegativeX2directiononthesameplanes

SymmetrygroupsThesetoftensorssuchastherotationtensorsQandthereflectiontensorsRwhichdefinethesymmetrypropertiesofamaterialformagroup(inthetechnicalalgebraicsenseoftheterm)whichiscalledthesymmetrygroupofthematerial

Foranisotropicmaterialthesymmetrygroupincludesallrotationsaboutallpossibleaxesandreflectionsinanyplanethusitisthegroupofallorthogonaltensorswhichisthefullorthogonalgroupin

threedimensionsAmaterialwhosesymmetrygroupconsistsofallrotationsbutnoreflections(therotationgrouportheproperorthogonalgroupinthreedimensions)issaidtobehemitropicForourpurposethedistinctionbetweenisotropicandhemitropicmaterialsisnotimportant

MaterialswhichhavefewermaterialsymmetriesthananisotropicmaterialaresaidtobeanisotropicThesymmetrygroupforananisotropicmaterialisasubgroupofthefullorthogonalgroup

AmaterialwhosesymmetrygroupincludesallrotationsaboutaspecifiedaxisissaidtobetransverselyisotropicaboutthataxisVariousreflectionalsymmetriesmayormaynotbeaddedagainthedistinctionsarenotimportanthere

AmaterialwhichhasreflectionalsymmetrywithrespecttoeachofthreemutuallyorthogonalplanesissaidtobeorthotropicToagoodapproximationwoodisanexampleofsuchamaterial

ThesymmetrygroupforanorthotropicmaterialisafinitegroupcomposedoftheunittensorthreereflectiontensorsandtheirinnerproductsOtherfinitesubgroupsofthefullorthogonalgroupinthreedimensionsaresymmetrygroupsformaterialswithvariouskindsofcrystalsymmetryTherotationswhichoccurinthesesymmetrygroupsarerotationsthroughmultiplesof and Accountsofthecrystallographicgroupscanbefoundintextsoncrystallography

Forthemostpartweshallconcentrateonisotropywhichisthesimplestandmostimportantcaseandmakeonlyoccasionalreferencestoanisotropicmaterials

83Lineare lastic ityManysolidmaterialsandespeciallythecommonengineeringmaterialssuchasmetalsconcretewoodetchavethepropertythattheyonlyundergoverysmallchangesofshapewhentheyaresubjectedtotheforceswhichtheynormallyencounterTheyalsohaveanaturalshapetowhichtheywillreturnifforcesareappliedtothemandthenremoved(providedthattheforcesarenottoolarge)Thetheoryoflinearelasticityprovidesanexcellentmodelofthemechanicalbehaviourofsuchmaterials

Wedefinealinearelasticsolidtobeamaterialforwhichtheinternalenergyρoeperunitvolumeinthereferenceconfigurationhasthefollowingpropertiesa poeisafunctiononlyofthecomponentsEijoftheinfinitesimalstraintensorandisormaybe

adequatelyapproximatedbyaquadraticfunctionofthesecomponentsb ifKisthekineticenergy(727)andEistheinternalenergy(728)inanyregionℛthenthematerial

timederivativeofK+Eisequaltotherateatwhichmechanicalworkisdonebythesurfaceandbodyforcesactingonℛ

ItisconventionaltodenoteρoebyWandtocallWthestrain-energyfunctionThus(a)statesthatWhastheform

(87)

whereCijklareconstantsProperty(b)isarestatementofthelawofconservationofenergy(Section76)withheatfluxassumedtobeabsentorneglectedProperties(a)and(b)togetherstatethatallthemechanicalworkdoneonℛeithercreateskineticenergyorisstoredaspotentialenergy(whichiscalled

thestrainenergy)whichdependsonlyonthedeformationThesystemisconservativeinaclosedcycleofdeformationthestrainenergyisstoredandthenreleasedsothatnonetworkisdoneonthebody

ThemoregeneralcaseinwhichWisallowedtodependalsoontemperatureorentropyandinwhichheatfluxispermittedleadstothetheoryoflinearthermoelasticityWeshallnotdevelopthistheory

Itshouldbenotedattheoutsetthataconstitutiveequationbasedon(87)willnecessarilyfailtosatisfyoneoftherequirementsstatedinSection81foraswasshowninSection66thecomponentsEijdonotremainconstantinafiniterotationandsoWasdefinedby(87)mustchangewhenabodyrotateswithoutchangeofshapeThisisnotreasonablephysicallyHoweverifattentionisrestrictedtomotionsinwhichtherotationissmallthenthechangeinEijisofsecondorderintherotationcomponentsThetheoryoflinearelasticityisessentiallyanapproximatetheorywhichisvalidforvaluesofEijandΩijwhicharesmallcomparedtooneThetheoryisneverthelessveryusefulbecausetheapproximationisanexcellentoneinmanyapplicationsItisconsistentwiththeapproximationinvolvedinadopting(87)toneglectEijcomparedtooneandthiswillbedonewheneveritisconvenienttodoso

Supposewechangefromacoordinatesystemwithbasevectorseitoanewcoordinatesystemwithbasevectors suchthat

and(Mij)isanorthogonalmatrixThentheinfinitesimalstraincomponentsEijandĒijintheoldandnewsystemsarerelatedbytheusualtensortransformationrule

(88)

ThestrainenergyWcanalsobeexpressedasaquadraticfunctionofthecomponentsĒijas

(89)

HoweverWisascalarwhichisnotaffectedbyachangeofcoordinatesystemandsotheexpressions(87)and(89)arethesameHenceusing(88)

ThisisanidentityforallvaluesofĒijandso

HenceCijklarecomponentsofafourth-ordertensor

The34=81constantsCijklarecalledelasticconstantsTheyhavethedimensionsofstressandtheirvaluescharacterizeparticularlinearelasticmaterialsTheelasticconstantsarenotallindependentByinterchangingthedummyindicesiandjin(87)weobtain

HoweverEij=Ejiandso

ThusCijklmaybereplacedby whichissymmetricwithrespecttointerchangesofiandjHencewithoutlossofgeneralityCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsfirsttwoindicesSimilarlyCijklmaybeassumedtobesymmetricwithrespecttointerchangesofitsthirdandfourthindicesThus

(810)

Thesymmetries(810)reducethenumberofindependentelasticconstantsto36Furthermorebysimultaneouslyinterchangingtheindicesiandkandtheindicesjandltherefollows

HencenogeneralityislostbyassumingthatCijklalsohastheindexsymmetries

(811)

Thesymmetries(811)furtherreducethenumberofindependentelasticconstantsto21

AfurtherrequirementonWisthatthestoredelasticenergymustbepositivesothat(87)isapositivedefinitequadraticformintheEij

AnymaterialsymmetryfurtherreducesthenumberofindependentelasticconstantsWereturntothispointbelow

Sofarproperty(b)oflinearelasticsolidshasnotbeenemployedFrom(731)withereplacedbyWρoandtheheatfluxtermsneglectedwehave

(812)

Sinceby(77)and(78)ρρo=1+O(Eij)totheorderofapproximationusedinsmall-deformationtheorywemayreplaceρbyρoandwrite

ItwasshowninSection76thatTijpartυipartxj=TijDijandso

(813)

NowsinceWdependsonlyonEij(813)gives

and(677)thengivestotherequiredorderofapproximation

ThisisanidentitywhichholdsforallvaluesofDijandso

Howeverfrom(87)and(811)

Hence

(814)

andthisistheconstitutiveequationforalinearelasticsolidItisevidentthatthestresscomponentsarelinearfunctionsoftheinfinitesimalstraincomponents

AnalternativeformulationoflinearelasticitytheoryisbasedontheassumptionthatthestresscomponentsTijare(orcanadequatelybeapproximatedby)linearfunctionsoftheinfinitesimalstraincomponentsEijsothat(814)istakenasthestartingpointratherthanasaconsequenceof(87)InsuchaformulationthereisnolossofgeneralityingivingCijkltheindexsymmetries(810)but(811)doesnotobtainunlessfurtherassumptionsaremadeAmaterialwithconstitutiveequation(814)butlackingtheindexsymmetry(811)hastheunrealisticpropertythatworkcanbeextractedfromitinaclosedcycleofdeformationWethereforeprefertobasethetheoryon(87)fromwhich(811)followsautomatically

ThenumberofindependentelasticconstantsisfurtherreducedifthematerialpossessesanymaterialsymmetrySupposeforexamplethatthematerialhasthereflectionalsymmetrywithrespecttothe(X2X3)

planeswhichisassociatedwiththetensorR1whichisdefinedby(85)Since itiseasilyseenthattheeffectofreplacingthedeformation(81)bythedeformation(83)istoreplaceE12bymdashE12andE13bymdashE13whileleavingtheothercomponentsEijunalteredHoweverifR1belongstothesymmetrygroupWmustbeunchangedbythissubstitutionHenceifthematerialhasthissymmetrythen

(815)

andthisrelationmustholdidenticallyforallEijBywriting(87)infullwiththeabovetwosetsofargumentsorbyconsideringspecialcasesitfollowsfrom(87)and(815)that

C1112=C1113=C1222=C1223=C1233=C1322=C1323=C1333=0

OthermaterialsymmetriesimposefurtherrestrictionsontheelasticconstantsThevariouspossibilities

aredescribedintextsonlinearelasticityWeomitthedetailsandproceedtothecaseofisotropicmaterials

ThesymmetrygroupforisotropicmaterialsincludesallproperorthogonaltensorsQSupposeasbeforethatEijarethecomponentsofinfinitesimalstrainwhichcorrespondtothedeformation(81)ThenthecorrespondingstresscomponentsTijaregivenby(814)Theinfinitesimalstraincomponentswhichcorrespondtothedeformation(82)are

(816)

andtheassociatedstresscomponentsare

(817)

NowifQbelongstothesymmetrygroupthen

(818)

andhencefrom(816)(817)and(818)

(819)

Itfollowsbycomparing(814)and(819)that

(820)

andifthematerialisisotropicthismustholdforallorthogonaltensorsQHowever(820)thenbecomesastatementthatCijklarecomponentsofafourth-orderisotropictensor(Section35)Themostgeneralfourth-orderisotropictensorisgivenby(337)HenceCijkltaketheform

(821)

andtheconstitutiveequation(814)becomes

SinceEij=Ejinogeneralityislostbysettingν=microsothat

(822)

orequivalentlyintensornotation

T=λItrE+2microE

Equation(822)istheconstitutiveequationforanisotropiclinearelasticsolidsuchamaterialischaracterizedbythetwoelasticconstantsλandmicro

Weobservethattheform(821)possessestheindexsymmetryCijkl=CklijThusforanisotropicmaterialwearriveat(822)regardlessofwhetherweadopt(87)or(814)asthestartingpoint

84Newtonianviscousf luidsInexperimentsonwaterairandmanyotherfluidsitisobservedthatinasimpleshearingflow(Section610)theshearingstressontheshearplanesisproportionaltotheshearratestoanextremelygoodapproximationandoveraverywiderangeofshearratesThisbehaviourischaracteristicofaNewtonianviscousfluidoralinearviscousfluidThismodeloffluidbehaviourdescribesthemechanicalpropertiesofmanyfluidsincludingthecommonestfluidsairandwaterverywellindeed

Weconsiderfluidswithconstitutiveequationsoftheform

(823)

whereθisthetemperatureInafluidatrestDkl=0and(823)reducesto

(824)

whichistheconstitutiveequationemployedinhydrostaticswithp(ρθ)representingthehydrostaticpressureThus(823)specifiesthatinafluidinmotiontheadditionalstressoverthehydrostaticpressureislinearinthecomponentsoftherateofdeformationtensor

IfthefluidisisotropicthenargumentssimilartothoseusedinSection83toreduce(814)to(822)leadtotheconclusionthatBijklare(likeCijklforanisotropiclinearelasticsolid)thecomponentsofafourth-orderisotropictensorandthen(823)takestheform

(825)

orequivalently

T=mdashp(ρθ)+λ(ρθ)trDI+2micro(ρθ)D

Heretheviscositycoefficientsλ(ρθ)andmicro(ρθ)areofcoursenotthesameastheelasticconstantsλandmicrowhichwereintroducedinSection83Aparticularlinearviscousfluidischaracterizedbythetwocoefficientsλandmicro

ItwasshowninSection69thatDij=0inarigid-bodymotionandthatthesuperpositionofarigid-bodymotiononagivenmotiondoesnotchangethevalueofDijHencetheright-handsideof(825)isnotaffectedbyasuperimposedrigid-bodymotionThereforetheconstitutiveequation(825)hastherequiredpropertyofbeingindependentofsuperimposedrigid-bodymotionsThisisincontrasttotheconstitutiveequationoflinearelasticitytheorywhichitwasemphasizedinSection83isnecessarilyanapproximatetheoryandisvalidonlyforsmallrotationsanddeformationsEquation(825)isapossibleexactconstitutiveequationforaviscousfluidInpracticeitisfoundthat(825)servesextremelywelltodescribethemechanicalbehaviourofmanyfluids

InfluidmechanicstextsitisusualtoassumeaswehavedoneherethatthefluidisisotropicInfactitcanbeshownthatisotropyisaconsequenceof(823)andtherequirementthatthestressisnotaffectedbyrigid-bodymotionsandsoisotropyneednotbeintroducedasaseparateassumptionWeshalldemonstratethisinamoregeneralcontextinSection103ItdoesnotfollowthatallfluidsarenecessarilyisotropicFluidswithanisotropicpropertiesdoexistbuttheyrequiremoregeneralconstitutiveequationsthan(823)fortheirdescription

Severalspecialcasesof(825)areofinterestIfthestressisahydrostaticpressure(seeSection59)then

ItisoftenassumedthatinsuchastateofpurehydrostaticstressthestressdependsonlyonρandθandnotonthedilatationrateDkkIfthisisthecasethen =0andthisrelationisoftenadopted

Ifthematerialisinviscidthenλ=0andmicro=0andtheconstitutiveequationreducesto(824)Thestressinaninviscidfluidisalwayshydrostatic

IfthefluidisincompressiblethenρisconstantandDkk=0IncompressibilityisakinematicconstraintwhichgivesrisetoareactionstressThereactiontoincompressibilityisanarbitraryhydrostaticpressurewhichcanbesuperimposedonthestressfieldwithoutcausinganydeformationthispressuredoesnoworkinanydeformationwhichsatisfiestheincompressibilityconstraintSuchahydrostaticpressureisnotdeterminedbyconstitutiveequationsbutcanonlybefoundthroughtheequationsofmotionorofequilibriumandtheboundaryconditionsThusforanincompressibleviscousfluid(825)reducesto

(826)

wherepisarbitrarymicrodependsonlyonθandthetermλDkkhasbeenabsorbedintothearbitraryfunctionpWenotethatinthelimitasthematerialbecomesincompressibleDkkrarr0andλrarrinfininsuchawaythatλDkktendstoafinitelimit

Ifthefluidisbothinviscidandincompressible(suchafluidiscalledanidealfluid)then

(827)

wherepisarbitraryinthesensethatitisnotdeterminedbyaconstitutiveequation

85Linearviscoe lastic ityManymaterials(especiallymaterialswhichareusuallydescribedaslsquoplasticsrsquo)possessbothsomeofthecharacteristicsofelasticsolidsandsomeofthecharacteristicsofviscousfluidsSuchmaterialsaretermedviscoelasticThephenomenonofviscoelasticityisillustratedbycreepandstress-relaxationexperimentsForsimplicityconsiderthecaseofsimpletensionSupposeatensionFoisrapidlyappliedtoaninitiallystress-freeviscoelasticstringattimet=0andthenheldconstantasillustratedinFig83(a)ThecorrespondingrelationbetweentheelongationeandtimetmaybeoftheformshowninFig83(b)withaninitialelongationeo(suchaswouldoccurinanelasticmaterial)followedbyanincreasingelongationunderthemaintainedloadThisillustratesthephenomenonofcreepIfthematerialisaviscoelasticsolidtheelongationtendstoafinitelimiteinfinastrarrinfinifthematerialisaviscoelasticfluidtheelongationcontinuesindefinitely

Figure83Creepcurve

Alternativelysupposethatatt=0thestringisgivenanelongationeoandheldinthisposition(Fig84(a))TheresultingforceresponseisshowninFig84(b)theforcerisesinstantaneouslytoFoatt=0andthendecaysThisisstressrelaxationForafluidFrarr0astrarrinfininasolidFtendstoafinitelimitFinfinastrarrinfin

WeconsiderhereonlyinfinitesimaldeformationssothattheuseoftheinfinitesimalstraintensorisappropriateWiththebehaviourillustratedinFig84asmotivationweassumethatanincrementδEijinthestraincomponentsattimeτgivesrisetoincrementsδTijinthestresscomponentsatsubsequenttimestthemagnitudeoftheseincrementsdependingonthelapseoftimesincethestrainincrementwasappliedThus

(828)

whereweexpectGijkltobedecreasingfunctionsoftmdashτThesuperpositionprincipleisalsoassumedaccordingtowhichthetotalstressattimetisobtainedbysuperimposingtheeffectattimetofallthestrainincrementsattimesτlttThus

(829)

Figure84Stress-relaxationcurve

ThisistheconstitutiveequationforlinearviscoelasticityThefunctionsGijklarecalledrelaxationfunctionsIfthestrainwaszerointheremotepastsothatEklrarr0asτrarrmdashinfin(829)canbeexpressedinanalternativeformbycarryingoutanintegrationbypartsasfollows

(830)

Thestress-relaxationfunctionsGijk l(tmdashτ)havetheindexsymmetriesGijkl=Gjikl=GijlkbutnottheindexsymmetryGijkl=GklijunlessthisisintroducedasafurtherassumptionIfthematerialisisotropicthenGijklarecomponentsofafourth-orderisotropictensorandforexample(829)reducesto

(831)

andonlytworelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)arerequiredtodescribethematerial

Theinverserelationto(829)is

(832)

ThefunctionsJijkl(tmdashτ)areknownascreepfunctionstheyhavethesameindexsymmetriesasGijkl(tmdashτ)andarecomponentsofafourth-orderisotropictensorinthecaseinwhichthematerialisisotropic

Linearviscoelasticityhasthesamelimitationsaslinearelasticityitisnecessarilyanapproximatetheorywhichcanonlybeapplicablewhenthestrainandrotationcomponentsaresmall

InasenselinearelasticitycanberegardedasthelimitingcaseoflinearviscoelasticityinwhichtherelaxationfunctionsareindependentoftandaNewtonianviscousfluidasthelimitingcaseofanisotropiclinearviscoelasticmaterialinwhichtherelaxationfunctionsλ(tmdashτ)andmicro(tmdashτ)taketheformsλδ(tmdashτ)andmicroδ(tmdashτ)respectivelywhereλandmicroaretheviscositycoefficientsandδ(tmdashτ)istheDiracdeltafunction

86Problems1Alinearelasticmaterialhasreflectionalsymmetryforreflectionsinthe(X2X3)(X3X1)and(X1X2)planes(suchamaterialissaidtobeorthotropic)Showthatithasnineindependentelasticconstants

2ShowthatatransverselyisotropiclinearelasticsolidhasfiveindependentelasticconstantsandfindtheformofWforalinearelasticsolidwhichistransverselyisotropicwithrespecttotheX3-axis

3Fromtheconstitutiveequation(822)andtheequationofmotion(722)withb=0deriveNavierrsquosequationsforanisotropiclinearelasticsolid

4InsimpletensionofanisotropiclinearelasticsolidT11=EE11T22=T33=T23=T31=T12=0andE22=E33=mdashνE11whereEisYoungrsquosmodulusandνisPoissonrsquosratioProvethatE=micro(3λ+2micro)(λ+micro)

and Showthattheconstitutiveequation(822)canbeexpressedintheform

5ProvethatnecessaryandsufficientconditionsforWtobepositivedefiniteforanisotropiclinearelasticsolidaremicrogt0 gt0

6Inplanestressorinplanestraintheequilibriumequationsreduceto(542)ShowthattheseequationsareidenticallysatisfiedifthestresscomponentsareexpressedintermsofAiryrsquosstressfunctionXasT11= T22= T12=mdashpart2Xpartx1partx2ProvethatinplanestressorplanestrainofanisotropiclinearelasticsolidXsatisfiesthebiharmonicequation

7Fromtheconstitutiveequation(826)andtheequationsofmotion(722)derivetheNavier-StokesequationsforanincompressibleNewtonianfluid

8AVoigtsolidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationσ=E0 whereE0andtoareconstantsSketchthecreepandstress-relaxationcurvesforthismaterialShowthattherelaxationfunctionisE01+t0δ(tmdashτ)Giveathree-dimensionalgeneralizationoftheaboveconstitutiveequationforanincompressibleisotropicmaterial

9AMaxwellfluidisamodelviscoelasticmaterialwhichinuniaxialtensionhasthestress-strainrelationSketchthecreepandstressrelaxationcurvesShowthatthestressrelaxationfunctionis

E1expmdash(tmdashτ)t1Hencegiveathree-dimensionalgeneralizationforanisotropicincompressiblematerialintheintegralform(831)

9

Furtheranalysisoffinitedeformation

91Deformationofa surface e lementTheextensionofamateriallineelementinthedeformation(61)wasdiscussedinSection62andthechangeofvolumeofamaterialvolumeelementwasconsideredinSection72Insomeapplicationsitisimportanttoknowhowtheareaandorientationofamaterialsurfaceelementchangeinadeformationthisproblemarisesforexamplewhenspecifiedforcesareappliedtotheboundaryofadeformingbody

ConsideratriangularmaterialsurfaceelementwhoseverticesP0Q0andR0inthereferenceconfigurationhavepositionvectorsX(0)X(0)+δX(1)andX(0)+δX(2)respectivelyasshowninFig91LetthistrianglehaveareaδSandunitnormalvectorN3Thenbyelementaryvectoralgebra

(91)

Supposethatinthedeformation(61)theparticlesinitiallyatP0Q0andR0movetothepositionsPQandRwithrespectivepositionvectorsx(0)x(0)+δx(1)andx(0)+δx(2)andthatthetriangleP0Q0R0hasareaδsandunitnormalnThen

(92)

Wenowintroduce(73)andthesimilarrelationfor into(92)andsoobtain

Figure91Deformationofasurfaceelement

NextmultiplybothsidesofthisequationbypartxipartXRThisgives

Itthenfollowsfrom(222)and(91)that

(93)

InthelimitasδX(1)rarr0andδX(2)rarr0(93)becomes

(94)

SinceNisaunitvectoritfollowsfrom(94)that

(95)

andhencethat

(96)

Intensornotation(94)and(96)maybewrittenas

(97)

and

(98)

Equations(96)or(98)determinethearearatiodsdSintermsofthedeformationandthenormalninthedeformedconfigurationTheinitialnormalNisthengivenby(94)or(97)Theinverserelationsto(97)and(98)are

(99)

and

(910)

92Decompositionofa deformationBythepolardecompositiontheorem(Sections2536)thedeformation-gradienttensorFmaybeexpressedintheforms

(911)

whereRisanorthogonaltensorandUandVaresymmetricpositivedefinitetensorsSincedetF=ρoρitcanbeassumedthatdetFgt0andthenRisaproperorthogonaltensorForagiventensorFthetensorsRUandVareuniqueItfollowsimmediatelyfrom(911)that

(912)

Weconsiderfirstthecaseinwhichthemotionishomogeneoussothat

(913)

wherethecomponentsofFareconstantsSupposethatthebodyundergoestwosuccessivehomogeneousmotionsinwhichtheparticlewhichinitiallyhaspositionvectorXmovesfirsttothepointwithpositionvector andsecondlytothepointwithpositionvectorxwhere

(914)

Thenfrom(911)and(914)

x= =RsdotUsdotX=FsdotX

andthetwosuccessivemotions(914)areequivalenttothemotion(913)SinceRisorthogonalthesecondequationof(914)describesarotationofthebodyThefirstequationof(914)describesadeformationwhichcorrespondstothesymmetrictensorUThusthefirstequationof(911)showsthatanyhomogeneousdeformationcanbedecomposedintoadeformationwhichcorrespondstothesymmetrictensorUfollowedbytherotationRSimilarlythesecondequationof(911)showsthatalternativelyanyhomogeneousdeformationcanbedecomposedintothesamerotationRfollowedbyadeformationwhichcorrespondstothesymmetrictensorV

Ifthedeformationisnothomogeneous(913)maybereplacedbytherelation

dx=FsdotdX

betweenthedifferentialsdxanddXThenthedecompositions(911)canstillbemadebutRUandVarenowfunctionsofpositionInthiscasethedecompositionisregardedasoneintoalocaldeformationUfollowedbyalocalrotationRoralternativelyintoalocalrotationRfollowedbythelocaldeformationV

ThetensorRiscalledtherotationtensorThetensorsUandVarecalledtherightstretchandtheleftstretchtensorsrespectivelyThetensorsUandVarecloselyrelatedtothedeformationtensorsCandBforfrom(627)and(911)andsinceUissymmetricwehave

(915)

andfrom(633)and(911)wehave

(916)

BecauseUissymmetricandpositivedefinite(915)determinesthecomponentsofUintermsofthoseofCandconverselyThereforeUandCaremeasuresofthedeformationwhichareequivalenttoeachotherUhastheadvantageofpossessingthegeometricalinterpretationdescribedinthissectionHoweverforagivenFthedirectcalculationofUfrom(911)isinconvenientwhereasthecalculationofCfrom(627)isstraightforwardThereforeinapplicationstheuseofCisusuallytobepreferredtothatofUSimilarcommentsapplytothetensorsBandV

From(662)wehave

(917)

whereEissymmetricandΩisanti-symmetricInthecaseofsmallstrainsandrotationsweneglectsquaresandproductsofEandΩThen

U2=FTsdotF=(I+EmdashΩ)sdot(I+E+Ω)≃I+2E

andtothesameorderofapproximation

(918)

InasimilarwaywefindthatV≃I+EsothatbothUmdashIandVmdashIreducetotheinfinitesimalstraintensorinthecaseofsmalldeformationsAlsofrom(918)

(919)

andsofrom(911)(917)and(919)

(920)

ThusRmdashIreducestotheinfinitesimalrotationtensorΩinthecaseofsmallrotations

93Princ ipalstretchesandprinc ipalaxesofdeformationSupposethatFhasbeendecomposedintotheproductRsdotUasin(911)ThefactorRrepresentsarotationWenowconcentrateonthemotionwhichcorrespondstothesymmetrictensorU

Werecalltheresult(620)whichgivesthechangeoforientationofamateriallineelementinamotionForthemotionUthisresultbecomes

(921)

whereAandaareunitvectorsinthedirectionofthelineelementbeforeandafterthemotionUandλisthestretchoftheelement

SupposeaparticularlineelementwhoseinitialdirectionisgivenbyAstretchesbutdoesnotrotateduringthemotionThenforthislineelementAisequaltoaand(921)becomesor

(922)

ThusλisaprincipalvalueofUandAisaprincipaldirectionofUSinceUissymmetricandpositivedefiniteitsprincipalvaluesarerealandpositivewedenotethembyλ1λ2andλ3orderthemsothatλ1geλ2geλ3andcallthemtheprincipalstretchesAlsosinceUissymmetricithasatriadoforthogonalprincipaldirectionsgivenbyunitvectorsA1A2andA3whichareuniquelydeterminedifλ1λ2andλ3aredistinctThesevectorsdeterminetheprincipalaxesofU

IfthecoordinateaxesarechosentocoincidewiththeprincipalaxesofUthenthematrixofthecomponentsofUtakesthediagonalform

HencereferredtotheseaxesthedeformationUconsistsofextensionsalongthethreecoordinatedirectionswithnorotationofelementswhichliealongtheseaxesThereforethemotionwhichcorrespondstoF=RsdotUconsistsofthesethreeextensionsofmagnitudesλ1λ2andλ3alongthethreedirectionsA1A2andA3respectivelyfollowedbytherotationR

InasimilarwaythedecompositionF=VsdotRcanbeusedtoshowthatalternativelyFcanberegardedasarotationRfollowedbythreeextensionswhicharegivenbytheprincipalvaluesofValongthedirectionsoftheprincipalaxesofVHowevertheprincipalvaluesandprincipalaxesofUandVarerelatedSinceRTsdotR=Iitfollowsfrom(922)that

Rmiddot(UmdashλI)sdotRTsdotRsdotA=0

SinceRsdotIsdotRT=Ithisequationcanbeexpressedas

(RsdotUsdotRTmdashλI)sdotRsdotA=0

andhencefrom(912)as

(923)

Thustheprincipalstretchesλ1λ2andλ3ofUarealsotheprincipalvaluesofVandifA1A2andA3definetheprincipaldirectionsofUthenRmiddotA1RA2andRmiddotA3definetheprincipaldirectionsofVTheprincipaldirectionsofVareobtainedbyrotatingtheprincipaldirectionsofUthroughtherotationR

IfthedeformationishomogeneousthenUVandRareconstanttensorsandtheprincipalstretchesandtheprincipaldirectionsareuniformthroughoutthebodyInthegeneralcaseofanon-homogeneousdeformationtheprincipalstretchesλ1λ2andλ3andthevectorsA1A2andA3aswellastherotationRareallfunctionsofposition

BecauseC=U2andy theprincipaldirectionsofCandγcoincidewiththoseofUandtheirprincipalvaluesare and mdash1)(i=123)respectivelySimilarlytheprincipaldirectionsofBandηcoincidewiththoseofVandtheirprincipalvaluesare and (i=123)respectivelyForagivenFitismucheasiertocalculateCorBthanUorVandsotheeasiestwaytocalculatetheprincipalstretchesandprincipaldirectionsisbycalculatingtheprincipalvaluesandprincipaldirectionsofCorB

TheprincipalstretchesandprincipalaxesofthedeformationtensorscanbeinterpretedinanotherwayWerecalltheformula(629)

(924)

ForagiventensorCthisdeterminesanextensionratioλforeachsetofdirectioncosinesAsinthereferenceconfigurationWeenquireforwhatdirectionsAthisextensionratiotakesextremalvaluesthusweseekextremalvaluesofARASCRSsubjecttotheconstraintARAR=1Theseextremalvaluesaregivenbythesolutionsoftheequations

whereu2isaLagrangianmultiplierSincepartARpartAP=δRPandpartAspartAP=δSPthisequationreducesto

(925)

HencethedirectionsAforwhichA2isextremalaretwooftheprincipaldirectionsofCThereforethecorrespondingvaluesofλ2arethelargestandsmallestprincipalvaluesofCnamely and AsimilarprocedureappliedtothetensorBshowsthatλ2takesitsextremalvalues and fordirectionsinthedeformedconfigurationwhichcoincidewithtwooftheprincipaldirectionsofB

94StraininvariantsItfollowsfromthediscussionofSections38and93thattheprincipalstretchesλ1λ2andλ3areinvariantswhichareintrinsictothedeformationSinceλ1λ2andλ3areprincipalvaluesofUandVthreesymmetricfunctionsofλ1λ2andλ3maybechosenasthebasicinvariantsofUandVHoweveritispreferabletomakeuseofthefactthat and areprincipalvaluesofCandBandtodefinethestraininvariantsI1I2andI3asfollows

(926)

TheadvantageofthisprocedureisthatCandBaremuchmoreeasilycalculatedfromFthanareUandVThechoice(926)ofthestraininvariantsisofcoursenotuniquebutitisonewhichhasprovedtobeconvenient

Since and aretheprincipalvaluesofbothCandBtherefollowfrom(356)and(357)

(927)

AlternativeexpressionsforI3areobtainedbysubstitutingCandBforAin(359)

From(358)theCayleymdashHamiltontheoremforCandforBcanbeexpressedas

(928)

TheeigenvaluesofC-1andofB-1are and

Therefore

HenceweobtainthealternativeexpressionsforI2

(929)

Wenotealsothatfrom(78)

(930)

Ifthematerialisincompressiblethen(Section72)detF=1andsoI3=1Henceinanydeformationofanincompressiblematerialλ1λ2λ3=1

Example91UniformextensionsFortheuniformextensionsdefinedby(642)thepolardecompositionistrivialwehaveF=U=VR=ITheprincipalstretchesareλ1λ2andλ3andthecoordinateaxesaretheprincipalaxesofbothCandBThestraininvariantsare

Example92SimpleshearAsimpleshearingmotionisdefinedby(644)From(645)and(927)thestraininvariantsforthismotionare

I1=3+tan2γI2=3+tan2γI3=1

SinceI3=1asimpleshearingmotionispossibleinanincompressiblematerialasisobviousfromFig64BycalculatingtheeigenvaluesofthematrixofthecomponentsofthetensorCgivenin(645)wefindthat

λ1=secβ+tanβλ2=1λ3=secβ-tanβ

wheretanβ= γTheprincipaldirectionsofCaregivenbytheeigenvectorsofthematrixofthecomponentsofCtheseeigenvectorshavethefollowingcomponents

SimilarlythecomponentsoftheeigenvectorsofBare

ThecomponentsofthetensorRcanbecalculatedbyusingthepropertythatRrepresentstherotationwhichrotatestheorthogonaltriadofprincipalaxesofCintotheorthogonaltriadofprincipalaxesofBThusif

thenM2=RM1whereRisthematrixofcomponentsofRSinceM1isorthogonalitfollowsthatwhichgives

ThusRrepresentsarotationthroughβabouttheX3-axisThecomponentsofthetensorUarethendeterminedbytheequationU=RTFwhichgives

AnalternativeprocedureistocalculateUdirectlyfromtherelationU2=CandtousetherelationR=FU-1todetermineR

95Alternative stressmeasuresInSection52wedefinedthecomponentTijoftheCauchystresstensorTasthecomponentintheXjdirectionofthesurfacetractiononasurfaceelementwhichisnormaltothexidirectioninthecurrentconfigurationForsomepurposesitismoreconvenienttouseastresstensorwhichisdefinedintermsofthetractiononamaterialsurfacewhichisspecifiedinthereferenceconfiguration

ConsideranelementofamaterialsurfacewhichinthereferenceconfigurationisnormaltotheXR-axisandhasareaδSTheunitnormaltothesurfaceisthereforeeRinthereferenceconfigurationAfterthedeformation(61)thiselementhasarea8sandunitnormalnRwherefrom(99)

(931)

TheforceonthisdeformedsurfaceisdenotedbyπRδSThevectorπRisresolvedintocomponentsΠRisothat

(932)

ThusΠRirepresentsthecomponentinthexidirectionoftheforceonasurfacewhichisnormaltotheXR-axisinthereferenceconfigurationmeasuredperunitsurfaceareainthereferenceconfiguration

TorelateΠRitoTijwenotethattheforceonthedeformedsurfaceelementisalsoequaltonRTδsHencefrom(931)and(932)

(933)

Thereforebyequatingcomponentsoneithersideof(933)andtakingthelimitasδSrarr0weobtain

(934)

HenceΠRiarecomponentsofasecond-ordertensorIIwhere

(935)

andconversely

(936)

ThetensorIIisnotsymmetricWeshallcallitthenominalstresstensorItisoftenalsocalledthefirstPiola-KirchhoffstresstensorbutsomeauthorsreservethistermforitstransposeIIT

Byconsideringtheequilibriumofanelementarytetrahedronthreeofwhosefacesarenormaltothecoordinateaxesinthereferenceconfigurationitcanbeshownthatthetractiont(N)(measuredperunitareainthereferenceconfiguration)onamaterialsurfacewhichhasunitnormalNinthereferenceconfigurationisgivenby

(937)

Byconsideringtheresultantsurfaceandbodyforcesonanarbitraryregionofabodyandreferredtothebodyinitsreferenceconfigurationtheequationsofmotioncanbeexpressedintheform

(938)

ThesecondPiola-KirchhoffstresstensorPisdefinedas

(939)

Hence

(940)

ThetensorPissymmetricItdoesnothaveanysimpledirectinterpretation

ThetractiononasurfacedefinedinthecurrentconfigurationisnotdeterminedbyIIorPunlessFisalsogivenToleadingorderIIandPreducetoTinthecaseofinfinitesimaldisplacementgradientsWeshallnotuseIIorPin-thisbookexcepttopointoutinSection102thatcertainconstitutiveequationscanbeexpressedconciselyintermsofIIandP

96Problems1ForthedeformationdefinedinChapter6Problem2find(a)thedirectionofthenormaltoamaterialsurfaceelementinthedeformedconfigurationwhichhadnormaldirection(111)inthereferenceconfiguration(b)theratiooftheareasofthissurfaceelementinthereferenceanddeformedconfigurations(c)theprincipalstretches(d)theprincipalaxesofCandofB

2DetermineCRSforthedeformationgivenby

whereaandbareconstantsFindtheprincipalstretchesandtheprincipalaxesofC

3Forthedeformationdefinedby

whereAandλareconstantsfind Provethatthesquaresoftheprincipalstretchesareλ2andthetworootsofthequadraticequationμ2λ2-μ(A2r2+λ2A-2r-2)+1=0where HenceshowthatdetB-1=1

4Forthehomogeneousdeformation

x1=αX1+βX2x2=-αX1+βX2x3=μX3

whereαβandμarepositiveconstantsdeterminethecomponentsCRSandtheprincipalstretchesandfindRandUforthepolardecompositionF=RU

5Afluidmovessothattheparticleatthepointwithcoordinates(X1X2X3)attimet=0isatthepointwithcoordinates(x1(τ)x2(τ)x3(τ))attimet=τwhere

andαandβareconstantsObtainexpressionsforXiexcl(τ)intermsofthecoordinatesxioftheparticleattimetanddeterminethecomponentsofthetensorC(τ)definedby

ByexpandingC(τ)asapowerseriesins=t-τobtaintheRivlin-EricksentensorsA(n)(t)forallvaluesofnwhere

6TheRivlin-EricksentensorsA(n)satisfytherelations

Evaluatethesetensorsforthesteadyflowυ1=υ(x2)υ2=0υ3=0showingthat fornge3

10

Non-linearconstitutiveequations

101NonlineartheoriesInChapter8wediscussedsomeofthelineartheoriesofcontinuummechanicsLinearityofthegoverningequationsisalwaysagreatadvantageinthesolutionofboundary-valueproblemsbecauseitenablesthetechniquesoflinearanalysistobeemployedAsaresultofthisthelineartheoriesofcontinuummechanicshavebeenhighlydevelopedandappliedtonumerousproblemsManycommonmaterialsareadequatelymodelledbylinearconstitutiveequationsHowevertherearealsomanymaterialswhosemechanicalbehaviourisstronglynon-linearandtodescribethisbehaviouritisessentialtoformulateappropriatenon-linearconstitutiveequationsWegivesomeexamplesinthischapter

102The theoryoff inite e lastic deformationsThelineartheoryofelasticitywhichwasformulatedinSection83isveryeffectiveformanypurposesHoweverbecauseitisrestrictedtothecaseinwhichthedeformationgradientsaresmallithaslimitationsForexamplethelineartheoryisinadequatetodescribethemechanicalbehaviourofmaterialssuchasrubberwhicharecapableofundergoinglargedeformationsbut(toagoodapproximation)behaveelasticallyinthesensedescribedinSection83Tomodelthebehaviourofrubber-likematerialsandforotherpurposeswerequireatheoryoffiniteelasticdeformations

ToformulateatheoryoffiniteelasticdeformationswepostulateasinSection83theexistenceofastrain-energyfunctionW=p0ewhichdependsonlyonthedeformationandhastheproperty(b)(p111)Thusequation(812)remainsvalidinthefinitetheoryofelasticityHoweveritisnolongerassumedthatWmaybeapproximatedbyaquadraticfunctionoftheinfinitesimalstraincomponentsInsteadwepermitWtodependinanarbitrarymanneronthedeformationgradientcomponentsFiRsothat(87)isreplacedbythemoregeneralrelation

(101)

Then(676)(812)and(101)give

Thisrelationisvalidforallvaluesofpartυipartxiandso

(102)

Equation(102)isaformoftheconstitutiveequationforfiniteelasticityItsapparentsimplicityisdeceptivebecauseitrequiresWtobeexpressedasafunctionoftheninecomponentsFiRItwouldclearlybeimpracticabletoperformexperimentstodeterminethisfunctionforanyparticularelasticmaterial

Thevalueofthestrain-energyfunctionisnotchangedifarigid-bodyrotationissuperposedonthedeformationSupposethatatypicalparticleinitiallyhaspositionvectorXandthatinamotionitmovestothepointwithpositionvectorxInafurthersuperposedrigid-bodyrotationtheparticleoriginallyatXmovesto whereMisaproperorthogonaltensorLet

Then

(103)

Thenwerequirethat

(104)

forallproperorthogonaltensorsMEquation(104)isarestrictiononthemannerinwhichWmaydependonFTomakethisrestrictionexplicitweemploythepolardecompositiontheoremtoexpress(104)intheform

W(F)=W(M∙R∙U)

SincethisrelationholdsforallproperorthogonaltensorsMitholdsinparticularwhenM=RTHence

w(F)=W(U)

ThusWcanbeexpressedasafunctionofthesixcomponentsofthesymmetrictensorUHoweverthereisaone-to-onecorrespondencebetweenthetensorsUandC(Section92)andsoequivalently(andmoreconveniently)wemayregardWasafunctionofthesixcomponentsCRSofCConsequentlyanecessaryconditionforWtobeindependentofsuperposedrigid-bodymotionsisthatWcanbeexpressedintheform

(105)

whereofcoursethefunctionWisnotthesamein(105)asitisin(101)BecauseCdoesnotchangeitsvalueinasuperposedrigid-bodymotiontheform(105)isalsosufficienttoensurethatWremainsunchangedinasuperimposedrigid-bodymotionandsonofurthersimplificationscanbeachievedinthisway

WhenWisexpressedintheform(105)wehave

ByinterchangingthedummyindicesRandSinoneofthetermsontheright-handsidethisgives

(106)

In(106)andsubsequentlyWisregardedasasymmetricfunctionofCRSandCSRalthoughthesecomponentsareequaltoeachotherSincepartυipartxiisarbitrary(812)and(106)nowgive

(107)

Thisistherequiredgeneralformoftheconstitutiveequationforafiniteelasticsolid

Wenoteinpassingthattheconstitutiveequations(102)and(107)takesimplerformswhentheyareexpressedintermsofthenominalorPiola-KirchhoffstresstensorsSinceρ0ρ=detFwehavefrom(935)and(102)

ΠRi=partWpartFiR

andfrom(939)and(107)

AnymaterialsymmetrieswhichthematerialpossesseswillrestrictthemannerinwhichWmaydependuponCSupposeforexamplethattheproperorthogonalmatrixQdefinesarotationalsymmetryofthematerialTheeffectofreplacingthedeformation(81)bythedeformation(82)istoreplaceFbyQT∙F∙QandsotoreplaceC=FT∙FbyQT∙C∙QHoweverwhenQdefinesarotationalsymmetrythisreplacementleavesthevalueofWunchangedThus

(108)

forallrotationalsymmetriesQSimilarlyifRdefinesareflectionalsymnetrythen

(109)

Ifthematerialisisotropicthen(108)holdsforallrotationsQThen(108)canbeinterpretedasastatementthatWregardedasafunctionofCRStakesthesameforminanycoordinatesystemsothat(Section38)WisaninvariantofCThreeindependentinvariantsofCarethestraininvariantsI1I2andI3definedby(926)or(927)itcanbeshownthatanyinvariantofCcanbeexpressedasafunctionofI1I2andI3HenceforanisotropicmaterialWcanbeexpressedintheform

(1010)

whereagainthefunctionWisadifferentfunctionfromthatin(101)and(105)ItcanbeverifiedthatifWhastheform(1010)italsosatisfiesthecondition(109)forallreflectionsR

WhenWhastheform(1010)wehave

(1011)

From(927)itfollowsthat

(1012)

TheexpressionforpartI3partCRSismosteasilyobtainedbytakingthetraceof(928)whichgives

(1013)

andfromthisitfollowsthat

(1014)

Bysubstitutingfrom(1011)(1012)and(1014)into(107)weobtain

ThisisaformoftheconstitutiveequationforanisotropicfiniteelasticsolidItmaybeexpressedmoreconciselyusingtensornotationas

(1015)

wherewehaveusedtherelationI3=(ρ0ρ)2andforbrevitywehaveintroducedthenotations

(1016)

Equation(1015)maybefurthersimplifiedbynotingfrom(627)and(633)that

F∙FT=BF∙C∙FT=B2F∙C2∙FT=B3

andhencethat(1015)maybewrittenas

T=2(I3)- (W1+I1W2+I2W3)B-(W2+I1W3)B2+W3B3

Wenowuse(928)toeliminateB3Thisgives

(1017)

Alsobymultiplyingthesecondequationof(928)byB-1wehave

andsoB2canbeeliminatedfrom(1017)infavourofB-1whichgives

(1018)

Inpractice(1017)and(1018)aretheformsoftheconstitutiveequationforanisotropicelasticsolidwhicharefoundtobemostconvenient

FurthersimplificationarisesifthematerialisincompressibleInthiscaseI3=1butitisnotsufficienttosetI3=1intheconstitutiveequationbecauseinthelimitingcaseofanincompressiblematerialcertainderivativesofWtendtoinfinityThedifficultyismosteasilyavoidedbyintroducedanarbitraryLagrangianmultiplier andwritingWintheform

(1019)

Theanalysisleadingto(1017)and(1018)thengoesthroughasbeforebutI3takesthevalueoneandW3

isreplacedby PSincepisundeterminedtheothertermsmultiplyingIin(1017)and(1018)maybeabsorbedintopsothatforanincompressibleisotropicfiniteelasticsolidtheconstitutiveequationcanbeexpressedineitheroftheforms

(1020)

IncompressibilityisanexampleofakinematicconstraintThemechanicaleffectofsuchaconstraintistogiverisetoareactionstresswhichdoesnoworkinanymotionwhichiscompatiblewiththeconstraintInthecaseofincompressibilitythereactionstressisanarbitraryhydrostaticpressure-pIwhichisnotgivenbyaconstitutiveequationbutcanonlybedeterminedbyusingequationsofmotion(orequilibrium)andboundaryconditionsSuchanarbitraryhydrostaticpressuremustalwaysbeincludedaspartofthestressinabodyofanyincompressiblematerial

Theequationsoflinearelasticitytheorycanberecoveredfrom(107)byexpandingallquantitiesinpowersofthedisplacementgradientsanddiscardingtermsontheright-handsideof(107)whichareofdegreehigherthanthefirstinthesegradients

103Anon-linearviscousf luidInSection84weconsideredfluidswithconstitutiveequationsoftheform(823)inwhichT+pIislinearintherate-of-straincomponentsThistheoryprovestobeverysatisfactoryfordescribingthebehaviourofmanyfluidsincludingthecommonestfluidsairandwateroveraverywiderangeofratesofstrainHowevertherearealsofluidsincludingbloodandmanyfluidswhichareimportantinchemicalengineeringprocesseswhichexhibitphenomena(whichinsomecasesarequitespectacular)whichcannotbeexplainedonthebasisofthelinearmodelSuchfluidsaredescribedasnon-NewtonianfluidsFornon-Newtonianfluidstheassumptionthatthestressdependslinearlyonrateofstrainis

inadequateThereforeinthissectionwediscardlinearityandbeginwiththeassumptionthatTdependsinageneralwayondensitytemperatureandthevelocity-gradienttensorThus

(1021)

orintensornotation

(1022)

WefirstconsiderwhethertherequirementsthatTisindependentofsuperposedrigid-bodymotionsplacesanyrestrictionson(1022)Sinceby(672)L=D+Wwecanreplace(1022)by

(1023)

whereTrepresentsadifferentfunctionontheright-handsideof(1023)fromthefunctionwhichitrepresentsontheright-handsideof(1022)

Supposeabodyundergoesthemotion

(1024)

Consideranewmotionwhichdiffersfrom(1024)onlybyasuperposedtime-dependentrigidrotationsothatattimettheposition oftheparticleinitiallyatXisgivenby

(1025)

whereMisatime-dependentproperorthogonaltensorTheninthesecondmotionthevelocityis

(1026)

Thevelocity-gradientcomponentsinthesecondmotionaregivenby

orintensornotationas

Itfollowsthattherate-of-straintensor andthespintensorWforthesecondmotionaregivenby

(1027)

HoweversinceMisorthogonalM∙MT=IanditfollowsthatṀ∙MT+M∙ṀT=0Hence(1027)maybewrittenas

(1028)

IfTisthestresswhicharisesfromthefirstmotionthenindependenceofsuperposedrotationsrequiresthatthesecondmotiongivesrisetothestress Howeverfrom(1023)

(1029)

Hencefrom(1023)(1028)and(1029)

(1030)

andthefunctionTmustsatisfythisconditionidenticallyforallproperorthogonaltensorsM

Tomake(1030)explicitwesupposefirstthatM=IṀne0Then(1030)becomes

TDṀ+Wρθ=TDWpθ

HencethevalueofTisindependentofthevalueofWThereforetheargumentsWand maybeomittedin(1023)and(1029)DependenceofthestressontheninecomponentsofLcanbereplacedbydependenceonthesixcomponentsofD(thisresultwasimplicitlyassumedinSection84)WhentheargumentWisomitted(1030)reducestotheform

(1031)

forallorthogonaltensorsMAtensorfunctionTwiththeproperty(1031)issaidtobeanisotropictensorfunctionofDIfTisalinearfunctionofDasinSection84then(1031)impliesthatthestressisoftheform(825)sothatthefluidisnecessarilyisotropicThisjustifiesthestatementmadeinSection84thatitisnotessentialtointroduceisotropyasaseparateassumptionThesameistrueinthegeneralcasefor(1031)canbeinterpretedasastatementthatthematerialisisotropic

ItisshownintheAppendixthatthemostgeneraltensorfunctionTwhichsatisfies(1031)isoftheform

(1032)

wherepαandβarefunctionsofpθandinvariantsofDnamely

Amaterialwiththeconstitutiveequation(1032)iscalledaReiner-RivlinfluidIfthefluidisincompressiblethenρisconstantandtrD=0sothatαandβdependonlyonθandthesecondtwoinvariantsofDandprepresentsanarbitrarypressure

Althoughtheresult(1032)isofmathematicalinterestinpracticeithasbeenfoundthatmarkedlynon-Newtonianfluidshaveamorecomplexbehaviourthanispermittedbythemodeldefinedby(1021)Wediscussamoregeneralclassofmaterialsbrieflyinthenextsection

104Non-linearviscoe lastic ityInSection85weoutlinedthelineartheoryofviscoelasticityInaviscoelasticmaterial(whichmaybeasolidorafluid)thestressdependsnotonlyonthecurrentdeformationbutalsoonthepasthistoryofdeformationThematerialmaybesaidtohavealsquomemoryrsquoLinearviscoelasticityisgovernedbythesuperpositionprincipleaccordingtowhichtheeffectsofpastdeformationsmaybesuperposedtogivethepresentstressManynon-Newtonianfluidsandmanysolids(especiallypolymers)areviscoelasticinthatthestressdependsonthedeformationhistorybutthisdependenceismorecomplicatedthanadirectsuperpositionoftheform(829)Themodellingofsuchmaterialsrequiresthenonlineartheoryofviscoelasticity

Inanon-linearviscoelasticmaterialthestressataparticledependsnotonlyonthecurrentdeformationbutalsoonthehistoryofthedeformationThusformallytheconstitutiveequationmaybewrittenas

(1033)

ItcanbeshownthatifTisindependentofsuperposedrigid-bodymotionsthen(1033)canbereducedto

(1034)

andfurtherreductionscanbemadeifthematerialhasanymaterialsymmetry

Inthenon-linearcaseitisnolongerpossibletousethesuperpositionprinciplewhichleadstothecomparativelysimpleintegralrepresentation(829)forTThefunctionalin(1034)canberepresentedeitherexactlyorapproximatelyinvariouswaysbuttheresultingthoeriesaretooadvancedforconsiderationhere

105P lastic ityManymaterialsparticularlymetalsconformwelltothelineartheoryofelasticityprovidedthatthestressdoesnotexceedcertainlimitsbutiftheyaresubjectedtostressbeyondtheselimitstheyacquireapermanentdeformationwhichdoesnotdisappearwhenthestressisremovedSinceelasticityisareversiblephenomenonthisisclearlyinelasticbehaviourItisnotaviscoelasticphenomenonbecausetheviscoelasticstressdependsontherateofdeformationandtoagoodapproximationitisfoundthatalthoughthestressinametaldependsonthepreviousdeformationitisindependentoftherateatwhichthatdeformationtookplaceThephenomenoniscalledplasticitycharacteristically

Figure101Typicalstress-straincurveforaplasticsolid

itoccursincrystallinematerialsandinparticularitoccursinthesolidmetalswhichareineverydayusesuchassteelaluminiumandcopper

Figure101illustratesthemainfeaturesofthestress-straincurveinuniaxialtensionofatypicalmetaltheaxialstressisdenotedbyσandtheaxialstrainbyisinForsimplicityitisassumedthatthestrainissufficientlysmallfortheinfinitesimalstrainmeasuretobeadequate

ForthedeformationwhichcorrespondstothesectionOAofthecurvetherelationbetweenσandisinistoagoodapproximationlinearIfthestressisremovedbeforeσreachesthevalueσAthestrainreturnstozeroInthisrangethebehaviouristhatoflinearelasticitytheoryForstressgreaterthanσAthecurvedepartsfromastraightlineThestressσAiscalledtheinitialyieldstressintensionThechangeofslopeatAmaybeabruptorgradualIfthestressisincreasedtoσBgtσAandthenreducedtozerotheunloadingcurveBCisfollowedtoagoodapproximationBCisparalleltoOAWhenthestressiszerothereremainsaresidualstrainrepresentedbyOCthisisanexampleofaplasticdeformationOnreloadingthepathwillcloselyretraceCBandeventuallycontinuethecurveOAB

ItisclearfromFig101thatforthismaterialthereisingeneralnouniquerelationbetweenthestressandthestrainsothetheoryofelasticityisinappropriateThediscrepancycannotbeexplainedasaviscouseffectbecausethebehaviouris(exceptatveryhighratesofstrain)almostindependentofthespeedatwhichthedeformationisperformedFigure101alsosuggeststhattwophenomenaareinvolvedonebeingessentiallyelasticityandinvolvingdeformationswhichvanishonunloadingandtheothercalledplasticitygivingrisetorate-independentpermanentdeformationsThisideaissupportedbythedescriptionofthephenomenaonthemicroscopicscaleMaterialswhichexhibitthiskindofbehaviourareusuallycrystallinesolidsElasticdeformationonthemicroscopicscaleisexplainedassmallrecoverabledisplacementsoftheatomswhichformthecrystallatticefromtheirequilibriumpositionsPlasticdeformationiscausedmainlybypermanentslipofneighbouringplanesofatomsrelativetoeach

other

Toformulateathree-dimensionaltheoryofplasticitywerequirea ayieldconditionwhichdecideswhetheranelementofmaterialisbehavingelasticallyor

plasticallyatagiventimeb stress-strainrelationsforelasticbehaviourc stressmdashstrainrelationsforplasticbehaviour

YieldconditionThisisaninequalityoftheform

(1035)

whereƒ(Tij)istheyieldfunctionandkisaparameterwhichingeneraldependsonthedeformationhistoryIfƒ(Tij)ltk2thenthematerialbehaveselasticallyifƒ(Tij)=k2thenplasticdeformationmayoccurTheequationƒ(Tij)=k2canberegardedasrepresentingasurface(theyieldsurface)inthesix-dimensionalspaceofthestresscomponentsTijPlasticstressstateslieonthissurfaceelasticstatesinitsinteriorandstressstatesoutsidetheyieldsurfacearenotattainableforthecurrentvalueofk

Anymaterialsymmetryrestrictstheformofƒ(Tij)ForexampleforanisotropicmaterialtheyieldfunctionmustbeexpressibleasafunctionofthestressinvariantsJ1J2andJ3

FormanymaterialsparticularlymetalsitisfoundthattoagoodapproximationtheyieldingofthematerialisnotaffectedbyasuperposedhydrostaticstressThecomponentsSijofthestressdeviatortensor(Section57)areindependentofthehydrostaticpartofthestressandforthesematerials(1035)maybereplacedby

(1036)

Inthecaseofanisotropicmaterialtheyieldfunctionmaynowbeexpressedasafunctionofthetwoinvariants and ofS

ElasticstressmdashstrainrelationsBeforeanyplasticdeformationhasoccurredasforexampleonthesectionOAofthestressmdashstraincurveinFig101theusualelasticrelationsapplyforexampleforsmalldeformationsofanisotropicmaterialwehaveequations(822)

(1037)

ForsmallelasticdeformationsfollowingaplasticdeformationtherelationbetweenTandEisagainlinearbutthestateofzerostressdoesnotcorrespondtooneofzerostrainThusforanisotropicmaterial

(1038)

where representstheresidualstrainwhichwouldresultfromunloadingtozerostressandwhichdependsonthepreviousdeformationhistoryTheintroductionof canbeavoidedbyexpressingtheelasticstress-strainrelationintermsofstressandstrainincrementsorstressandstrainratesThus(1037)and(1038)canbereplacedby

(1039)

orby

(1040)

wherethesuperposeddotdenotesanappropriatetimederivativeForfinitedeformationsthesetimederivativesarenotunambiguousandtheyrequirecarefuldefinitionNosuchdifficultyarisesifattentionisrestrictedtoinfinitesimaldeformationsTheinverseof(1040)is

(1041)

PlasticstressmdashstrainrelationsTheformulationoftheseismoredifficultandcontroversialandwillnotbepursuedindetailTheclassicalapproachistoassumethattherateofdeformationcanbedecomposedintoanelasticpart andaplasticpart

(1042)

TheelasticpartisrelatedtothestressrateṪijby(1041)Fortheplasticpartthesimplesttheorypostulates(withsomejustification)thattheyieldfunctionservesasaplasticpotentialinthesensethat

(1043)

where isascalarfactorofproportionalitywhichdependsonthedeformationhistoryThenbycombining(1041)and(1043)weobtainthecompletestress-strainrelationsforanisotropicplasticmaterialnamely

(1044)

whereƒcanbeexpressedasafunctionof and

106Problems1Theunitcube0leX1le10leX2le10leX3le1ofincompressibleisotropicelasticmaterialundergoesthedeformationX1=λX1+αX2x2=λ-1X2x3=X3whereλandαareconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandshowthatpcanbechosensothatnoforcesactonthesurfacesX3=0andX3=1FindtheforcewhichmustbeappliedtothefaceinitiallygivenbyX2=1tomaintainthedeformationDeterminethenormalinthedeformedconfigurationtothefaceX1=1andthetractionwhichmustbeappliedtothisfacetomaintainthedeformation

2Aunitcubeofincompressibleisotropicelasticmaterialundergoesthefinitedeformation

x1=λX1x2=λ-1X2x3=X3

whereλisconstantThestrain-energyfunctionis

W=C1(I1minus3)+C2(I2minus3)

whereC1andC2areconstantsSketchthedeformedcubenotingthelengthsofitsedgesFindthestressandhencedeterminethetotalloadsF1F2andF3actingonthefacesnormaltotheX1X2andX3

directionsShowthatwhenC1gt3C2gt0therearethreevaluesofλforwhichthebodyisinequilibriumwithF1=F2=F3andfindthesevalues

3Showthattheconstitutiveequationforanelasticsolidcanbeexpressedintheform

4ForaparticulartransverselyisotropicelasticsolidwithpreferreddirectionthatoftheX1-axisWhastheform

whereαβγandδareconstantsFindtheconstitutiveequationforTandhencefindthestressinabodyofthismaterialsubjectedtotheuniformexpansion

x1=λX1

x2=λX2

X3=λX3

5SupposethatthestressinasolidisgivenbyarelationoftheformT=x(F)ShowthatifthestressisindependentofrotationsofthedeformedbodythenXmustsatisfytherelationx(M∙F)=M∙X(F)∙MT

forallproperorthogonaltensorsMVerifythatasufficientconditionforthisrelationtobesatisfiedisthatXcanbeexpressedintheformX=F∙Ψ(C)∙FTUsetherepresentationtheoremgivenintheAppendixtoobtainthemostgeneralsuchformforxinthecaseinwhichthematerialisisotropic

6DerivetheconstitutiveequationT=minuspI+2μEforincompressibleisotropiclinearelasticityasafirstapproximationforsmalldisplacementgradientstoequation(1020)

7ShowthatthemostgeneralincompressibleReinermdashRivlinfluid(1032)forwhichthestresscomponentsarequadraticfunctionsofthecomponentsDijhastheconstitutiveequationT=minuspI+α0D+β0D2whereα0andβ0areconstants

8Showthatavelocityfieldυ1=υ(x2)υ2=0υ3=0isapossibleflowineveryincompressibleReiner-Rivlinfluid(1032)Ifthisflowtakesplacebetweeninfiniteparallelplatesatx2=plusmnddeterminethepressuregradient(thatisminuspartT11partx1)requiredtomaintainthisflowandthetangentialforcesactingonunitareaofeachoftheplates

9ThestressinacertainReinermdashRivlinfluidisgivenbyT=minuspI+μ(1+αtrD2)D+βD2whereαβandμareconstantsDeterminethestressinthefluidarisingfromthevelocityfieldυ1=minusx2ω(x3)υ2=

x1ω(x3)υ3=0Showthatifω=Ax3+BwhereAandBareconstantstheequationsofmotionaresatisfiedonlyifA=0oriftheaccelerationtermscanbeneglectedInthelattercasefindvaluesofAandBcorrespondingtoflowbetweenparallelplatesatx3=0andx3=htheformerbeingatrestandthelatterrotatingaboutthex3-axiswithangularspeedΩ

10Thebehaviourofcertainviscousfluidsisoftenmodelledbytheconstitutiveequation

andkandnarepositiveconstants(andn=1correspondstoaNewtonianfluid)Suchapower-lawfluidundergoessimpleshearingflowbetweentwolargeparallelplatesadistancehapartsuchthatoneplateisheldfixedandtheothermoveswithconstantspeedUinitsplaneFindtheshearingforceperunitareaontheplatesandtheapparentviscosityμasafunctionoftheshearrateUh

11TheconstitutiveequationT=mdashpI+2μ0(2trD2)αDwhereμ0andαareconstantsmodelsaclassofReinermdashRivlinfluidsShowthatthesefluidscanundergothesteadyrectilinearshearflowυ1=υ(x2)υ2=0υ3=0providedp=p0+kx1wherep0andkareconstants

12DeterminethetensorsC(τ)andA(n)(t)whicharedefinedinProblem5ofChapter9forthemotionx1(τ)=X1x2(τ)=X2x3(τ)=X3+γτtanminus1(X2X1)whereγisaconstantThestressinafluidisgivenbyT=minuspI+μA(1)+σA(2)whereμandσarefunctionsoftrA(2)andpisarbitraryShowthatifpisafunctionofronly( )thentheequationsofmotionaresatisfiedprovidedthat

13Thestressinaparticularincompressiblenon-Newtonianfluidisgivenbydswheres=tmdashτandC(τ)isdefinedinProblem5ofChapter9

Determinethestressinthefluidduetothedisplacementfield

ifdƒdx2anddgdx2aresmallenoughfortheirsquarestobeneglected

11

Cylindricalandsphericalpolarcoordinates

111CurvilinearcoordinatesSofarwehaveusedonlyrectangularcartesiancoordinatesandthisisthesimplestwaytoformulatethegeneralequationsofcontinuummechanicsandtheconstitutiveequationsofvariousidealmaterialsHoweverforthesolutionofparticularproblemsitisoftenpreferabletoworkintermsofothersystemsofcoordinatesInparticularitisusuallydesirabletousecylindricalpolarcoordinatesforconfigurationswhichhaveanelementofsymmetryaboutanaxisandtousesphericalpolarcoordinateswhenthereissomesymmetryaboutapointItisthereforeusefultoexpressthemainequationsintermsoftheseothercoordinatesystems

ItispossibletodevelopelegantlytheequationsofcontinuummechanicsintermsofgeneralcurvilinearcoordinatesResultsinanyparticularcoordinatesystemcanthenbeobtainedbymakingtheappropriatespecializationsHoweverthisprocedurerequiresextensiveuseofgeneralcurvilineartensoranalysiswhichweprefertoavoidinthisintroductorytextAlsoitisonlyveryrarelythatcoordinatesystemsotherthanrectangularcartesiancylindricalpolarandsphericalpolarcoordinatescanbeemployedprofitablyAccordinglyweshallderivedirectlysomeresultsincylindricalandsphericalpolarseventhoughtheseresultscouldbeobtainedmoreconciselybytheuseofgeneraltensoranalysis

112CylindricalpolarcoordinatesCylindricalpolarcoordinatesrφz(0leφlt2π)arerelatedtocartesiancoordinatesx1x2x3by

(111)

(112)

ThebasevectorsoftherφzcoordinatesystemareunitvectorsdirectedintheradialtangentialandaxialdirectionsasillustratedinFig111TheyaredenotedbyereφandezandtheyaremutuallyorthogonalThus

(113)

(114)

Figure111Basevectorsforcylindricalpolarcoordinates

WedefinethematrixRtobe

(115)

andthen(113)and(114)maybewrittenas

(116)

ItiseasilyverifiedthatRisanorthogonalmatrix

Supposeavectorahascomponentsaiinthecoordinatesystemxiandcomponentsaraφazinthesystemrφzsothat

(117)

Let

(118)

bethecolumnmatricesformedfromthecomponentsofainthetwocoordinatesystemsThenfrom(114)and(117)

(119)

Asecond-ordertensorA=Aijeiotimesejcanbewrittenas

ormoreconciselyinmatrixnotationas

(1110)

where

(1111)

isthematrixofcomponentsofAreferredtorφzcoordinatesFrom(116)and(1110)therefollow

(1112)

whereA=(Aij)isthematrixofcomponentsofAreferredtoxicoordinatesFrom(1112)itfollowsthatifAisasymmetricmatrixthensoisAandifAisananti-symmetricmatrixthensoisASinceRisorthogonaltheeigenvaluesofAandAarethesamesotheprincipalvaluesofAaretherootsoftheequation

det(AminusAI)=0

MoreovertheinvariantsI1I2andI3ofAmaybewrittenas

(1113)

Referredtocylindricalpolarcoordinatesthegradientofascalarψ(rφz)andthedivergenceofavectora(rφz)arerespectively

(1114)

Thematerialderivativeofψ(rφzt)isthengivenby(418)as

(1115)

Ifυ=υrer+υφeφ+υzezisthevelocityvectorthenfrom(423)theaccelerationvectorfisgivenby

(1116)

SupposethatthematrixofcomponentsofthestresstensorreferredtorφzcoordinatesisTwhere

(1117)

andthatT=(Tij)Then

(1118)

Because(1118)areimportantrelationswegivetheminfullasfollows

(1119)

Letasurfacehavenormalnwhere

(1120)

Thenby(59)thetractionvectoronthesurfaceisniTijejandusing(1118)and(1120)thiscanbeexpressedas

(nrnφnz)T(ereφez)T

From(527)and(1118)thestressinvariantsJ1J2andJ3canbewrittenintheforms

(1121)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinthereferenceconfigurationhascylindricalpolarcoordinatesRΦZmovestothepositionwithcylindricalpolarcoordinatesrφzwhere

(1122)

(1123)

Themotioncanbedescribedbyequationsoftheform

(1124)

Let

(1125)

andinadditiontothematrixRdefinedby(115)introduceanorthogonalmatrixR0where

(1126)

Wealsoobservefrom(1122)that

(1127)

Thenitcanbeshownfrom(115)(1123)(1125)(1126)and(1127)afteralittlemanipulationthat

(1128)

SupposethatB=(Bij)=FFTisthematrixofcomponentsofBreferredtoxicoordinatesandletBbethematrixofcomponentsofBreferredtorφzcoordinatesThen

(1129)

HenceBisreadilycalculatedfrom(1124)and(1128)SimilarlyifC=(CRS)=FTFisthematrixofcomponentsofCreferredtoXRcoordinatesandCisthematrixofcomponentsofCreferredtoRΦZcoordinatesthen

(1130)

Wealsonoteforfuturereferencethat

(1131)

Forasmalldisplacementu=urer+uΦeΦ+uzezwehaveur=u1cosφ+u2sinφuφ=mdashu1sinφ+u2cosφuz=u3Thenpartuilpartxj≃partuilpartXjanditfollowsfrom(626)and(115)that

(1132)

andinthesmall-displacementapproximationthereisnoneedtodistinguishbetweenRφZandrφzin(1132)ThematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtocylindricalpolarcoordinatesarethengivenby

(1133)

SimilarlythematrixLofthecomponentsofthevelocitygradienttensorLreferredtocoordinates(r

φz)is

(1134)

Theexpression(1134)isexactThematricesDandWofthecomponentsreferredtorφzcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1135)

From(1128)wehavedetF=detFHencefrom(78)

(1136)

andinanincompressiblematerialdetF=1TheEulerianformofthemass-conservationequationisgivenby(711)andcanbeexpressedintermsofthecomponentsofυreferredtocylindricalpolarcoordinatesbyexpressingdiv(ρυ)inthesecoordinates

Theequationofmotion(722)canbeexpressedintermsofcylindricalpolarcoordinatesbyresolvingthebodyforceandaccelerationintocomponentsreferredtothesecoordinatesLet(brbΦbz)becomponentsofbandlet(brbΦbz)becomponentsoffincylindricalpolarcoordinatesThenfrom(722)and(119)

(1137)

From(111)wehave

(1138)

Byintroducing(1119)and(1138)into(1137)itfollowsaftersomemanipulationsthat

(1139)

Equations(1139)aretheequationsofmotionreferredtorφzcoordinatesTheseequationscanalsobederivedbyconsideringtheforcesactingonanelementaryregionboundedbythecoordinatesurfaces

ConstitutiveequationsaremosteasilyexpressedintermsofcylindricalpolarcoordinatesbymultiplyingtheappropriateexpressionforthematrixT=(Tij)ofstresscomponentsontheleftbyRandontherightbyRTForexampleforanisotropiclinearelasticsolidweobtainfrom(822)

RTRT=λRRTtrE+2microRERT

HoweverRTRT=TRRT=ItrE=trEandRERT=Eandso

(1140)

whereλandmicroareelasticconstantsSimilarlytheconstitutiveequation(825)foraNewtonianviscousfluidcanbeexpressedintheform

(1141)

wherethepressuremdashpandtheviscositycoefficientsλandmicroarefunctionsofthedensityandthetemperature

From(927)(929)(1129)and(1130)itfollowsthatthestraininvariantsI1I2andI3canbeexpressedas

(1142)

Theconstitutiveequation(1018)foranisotropicelasticsolidgives

whichafterusing(1118)(1129)and(1131)takestheform

(1143)

Ifthematerialisalsoincompressiblethisbecomes

(1144)

Inasimilarmannertheconstitutiveequation(1032)foraReinerndashRivlinfluidcanbeexpressedintheform

(1145)

wherepaandβarefunctionsofdensitytemperatureandoftrD anddetD

113SphericalpolarcoordinatesSphericalpolarcoordinatessθφ(0leθleπ0leφlt2π)arerelatedtocylindricalpolarcoordinatesrφzby

(1146)

(1147)

andtocartesiancoordinatesx1x2x3by

(1148)

(1149)

VectorandtensorequationscanbeexpressedintermsofsphericalpolarcoordinatesinasimilarmannertothatwhichwasemployedinSection112forcylindricalpolarcoordinatesalthoughthealgebrainvolvedisslightlymorecomplicatedItisoftenconvenienttoemploycylindricalpolarcoordinatesasanintermediatestagebetweencartesiancoordinatesandsphericalpolarcoordinatesAstheapproachisanalogoustothatofSection112weomitsomedetailsofthederivationsoftheresultspresentedbelow

ThebasevectorsofthesθφsystemaredenotedeseθandeφandareillustratedinFig112TheyaremutuallyorthogonalThen

(1150)

where

(1151)

ThematricesRrsquoandRrdquoareorthogonalmatricesandofcourseRrdquo=RrsquoR

Figure112Basevectorsforsphericalpolarcoordinates

Ifthevectorahascomponentsasaθaφinthesystemsθφthen

a=ases+aθeθ+aφeφ

andifadenotesthecolumnmatrix(asaθaφ)Twehave

(1152)

Thesecond-ordertensorAcanbeexpressedintheform

(1153)

where

(1154)

isthematrixofthecomponentsreferredtosphericalpolarcoordinatesofthetensorAThen

(1155)

TheprincipalvaluesofAaretherootsofdet(AndashAI)=0andtheinvariantsI1I2andI3ofAmaybewrittenas

(1156)

Referredtosphericalpolarcoordinatesthegradientofthescalarψ(sθφ)andthedivergenceofthevectora(sθφ)are

(1157)

(1158)

Thematerialderivativeofψisgivenby(418)as

ψ=partψpartt+υgradψ

andtheaccelerationvectorfisgivenintermsofthevelocityvectorυas

164Cylindricalandsphericalpolarcoordinates

LetthematrixofcomponentsreferredtobasevectorseseθeφofthestresstensorTbeTwhere

(1159)

Thenfrom(1155)

(1160)

andthestressinvariantsJ1J2andJ3canbewrittenas

(1161)

NowconsiderafinitedeformationinwhichatypicalparticlewhichinitiallyhassphericalpolarcoordinatesSΘΦmovestothepositionwithsphericalpolarcoordinatessθφThemotioncanbedescribedbyequationsoftheform

s=s(SΘΦ)θ=θ(SΘΦ)φ=φ(SΘΦ)

InadditiontothematricesRlsquoandRldquodefinedby(1151)weintroduceorthogonalmatricesRrsquo0andRrdquo0where

(1162)

Thenaftersomemanipulationweobtain

(1163)

ThenthematricesofthecomponentsreferredtosphericalpolarcoordinatesofBandCare

(1164)

(1165)

Forasmalldisplacementu=uses+uθeθ+uφeφwehave

(1166)

Henceforsmalldisplacements mdashI=Rrsquo RrsquoT-I≃

(1167)

ThenthematrixEofinfinitesimalstraincomponentsandthematrixΩofinfinitesimalrotationcomponentsreferredtosphericalpolarcoordinatesaregivenby

(1168)

SimilarlythematrixLofthecomponentsreferredtos0φcoordinatesofthevelocitygradienttensorLisobtainedfrom(1167)byreplacingFndashIbyLusuθanduφbyυsυθandυφrespectivelyandS andΦbysθandφrespectivelyTheexpressionisexactThematricesDandWofthecomponentsreferredtos0φcoordinatesoftherate-of-deformationtensorDandthevorticitytensorWarethengivenby

(1169)

From(1163)detF=detFandsofrom(78)

(1170)

Byresolvingthebodyforceandaccelerationintocomponentsreferredtobasevectorseseθandeφtheequationsofmotioncanbeexpressedas

(1171)

Alternativelytheseequationscanbederivedbyconsideringtheforcesactingonanelementaryregionboundedbythesurfaces

Byargumentsanalogoustothosewhichleadto(1140)and(1141)theconstitutiveequationsforanisotropiclinearelasticsolidandforaNewtonianviscousfluidcanbeexpressedas

(1172)

and

(1173)

respectivelywherein(1172)λandμareelasticconstantsandin(1173)pλandμhavethesamemeaningasin(1141)

ThestraininvariantsI1I2andI3canbeexpressedas

(1174)

Theconstitutiveequationforanisotropicelasticsolidcanbewrittenas

(1175)

orinthecaseofanincompressiblematerialas

(1176)

TheconstitutiveequationforaReiner-Rivlinfluidcanbeexpressedintheform

(1177)

wherepαandβcanbeexpressedasfunctionsofdensitytemperaturetrD (trD)2ndashtrD2anddetD

114Problems1Steadyhelicalflowisdefinedbytheequations

r=Rφ=Φ+tω(R)z=Z+tα(R)

whereωandαarefunctionsonlyofR

(a)Sketchthepathfollowedbytypicalparticle(b)findthevelocityoftheparticleat(rφz)attimet(c)findthevelocityoftheparticlewhichwasat(RΦZ)att=0(d)findtheaccelerationoftheparticleat(rφz)attimet(e)findthedivergenceofthevelocityvector(f)findthecomponentsofLDandΩreferredto(rφz)coordinates

2Ifv=υ(rt)υφ=0υz=0showthattheaccelerationvectorisdirectedintherdirectionandhasmagnitudepartυpartt+υpartυpartr

3Ifincylindricalpolarcoordinates

findthevelocityandaccelerationintermsofrφzandt

4Forthedeformationdefinedby

whereABandCareconstantsdeterminethematrixBandshowthattheinvariantsI1I2I3areconstants

5IfAistheunitvectorAReR+AΦeΦ+AzezandAisthematrix(ARAΦAZ)TshowthattheextensionofalineelementwhichhasthedirectionAinthereferenceconfigurationisgivenby(λ2)=ATCAHencedeterminetheinitialdirectionsofallthelineelementswhoselengthdoesnotchangeinthepuretorsiondeformation

r=Rφ=Φ+ψZz=Zwhereψisconstant

6ProvethattheeigenvaluesofCarethesameasthoseofCandthatifyisaneigenvectorofCthenR0yisaneigenvectorofCHencefindtheprincipalstretchesforthepuretorsiondeformationofProblem5

7ProvethatifF1=RFthen

andthatB=F1F1TC=F1TF1

8Provethatthestressresultinginacompressibleisotropicelasticsolidfromthepuretorsion

deformationofProblem5willnotingeneralsatisfytheequationsofequilibrium

9Acircularcylinderofisotropicincompressiblematerialundergoestheextensionandtorsiondeformation

z=λZr=λndash Rφ=Φ+ψZ

whereλandψareconstantsFindthestresscomponentTφzandhencedeterminetheendcouplerequiredtomaintainthedeformationifW=C1(I1ndash3)+C2(I2ndash3)whereC1andC2areconstants

10ThematrixF2isdefinedasF2=FR0TProvethatB=F2F2TC=F2TF2andthat

AnisotropicincompressibleelasticbodyisinitiallyboundedbythesurfacesR=AR= Φ=plusmnαZ=plusmnBwhereABandaareconstantsItundergoesthedeformation

Sketchthebodyinitsreferenceanddeformedconfigurations

Showthatthedeformationispossibleinanincompressiblematerialanddeterminethestressinthedeformedbody

11Thebehaviourofanincompressiblenon-Newtonianfluidisgovernedbytheconstitutiveequation

T=ndashpI+2μ(1ndash2εtrD2)D+4βD2

whereμεandβareconstantswithεlaquo1Determinethestresscomponentsincylindricalpolarcoordinateswhenthefluidisundergoingtheflow

υr=0

υφ=0

υz=w(r)

Verifythatthisiscompatiblewiththeincompressibilityconditionandshowthatinordertosatisfytheequationsofmotionw(r)isgivenby

wherecisanarbitraryconstantandk=mdashpartppartzBywriting

w(r)=w0(r)+εw1(r)+ε2w2(r)+

obtainanexpressionforw(r)correcttotermsoforderεwhichgivesthevelocitydistributionforaxialflowalongacircularpipeofradiusaunderaconstantpressuregradientk

12Therelations

s3ndasha3=ndash(s3ndashA3)θ=πndash φ=Φ

whereAandaareconstantsdescribetheeversion(turninginside-out)ofasphereFindFandBforthisdeformationHencedeterminethestressinanincompressibleisotropicelasticsolidwithstrain-energyfunctionW=C(I1ndash3)whereCisconstant

Appendix

RepresentationtheoremforanisotropictensorfunctionofatensorSupposethatTandDaresecond-ordertensorssuchthatthecomponentsofTarefunctionsofthecomponentsofDthus

T=T(D)

Thenif

(A1)

forallorthogonaltensorsMwesaythatT(D)isanisotropictensorfunctionofDWeconsiderthecaseinwhichTandDaresymmetrictensorsanddenote

(A2)

TheoremTisanisotropictensorfunctionofDifandonlyif

(A3)

whereαβγarescalarfunctionsoftrDtrD2andtrD3

Proof(a)SufficiencySinceMisorthogonaltrD=tr trD2=tr 2andtrD3=tr 3Henceαβand

γareunchangedifDijarereplacedby ij

Assume(A3)holdsThenfrom(A2)

(b)NecessityAssumethat(A1)issatisfiedandchoosethexicoordinatesystemsothatthecoordinateaxesaretheprincipalaxesofDTheninthesecoordinates

(A4)

and

(A5)

Choose

Then

(A6)

(A7)

However(A1)and(A6)requirethat =TijHenceT12=0T13=0SimilarlybyanotherchoiceofMitcanbeshownthatT23=0Thusif(Diexclj)isadiagonalmatrixsois(Tij)thatisDandThavethesameprincipalaxesThereforewecannowwrite

(A8)

Nextchoose

Then

andso(A1)gives

(A9)

HenceT1T2andT3canbeexpressedintermsofthesinglefunctionF(D1D2D3)as

(A10)

Finallychoose

Then

andthen(A1)gives

(A11)

Nowtheequations

(A12)

havesolutionsforαβandγasfunctionsofD1D2andD3AlsobecauseF(D1D2D3)hasthesymmetryexpressedby(A11)equations(A12)areunalteredifanypairofD1D2andD3areinterchangedHenceαβandγaresymmetricfunctionsofD1D2andD3Itfollowsfromatheoreminthetheoryofsymmetricfunctionsthatαβandγcanbeexpressedasfunctionsof

(A13)

Alsofrom(A10)and(A12)

whichwith(A13)isequivalentto(A3)

Answers

Chapter41 (a)υ1=υ2=υ3=1+2tf1=f2=f3=2

(b)υ1=υ2=υ3=(1+tndash2t2)(1ndasht3)f1=f2=f3=2(1ndasht)l(1ndasht3)

Astrarr1allparticlesapproachthesamelinex1=x2=x32 f1=ndashU2x1f2=ndashU2x2f3=0

Helicesgivenparametricallybyx1=AcosUt+BsinUtx2=AsinUtmdashBcosUtx3=Vt+CwhereABandCareconstants

3 mdash2U2a4(x12+x22)ndash3(x1e1+x2e2)streamlinesr=r0sinθVr03(θndash sin2θ)=2Ua2(zndashz0)wherex1=rcosθx2=rsinθ

4 (a)ndash( Andash )endashA(b)f=ndash2e1ndash12e2+6e3

(c)x1=2exp(1ndashtndash1)x2=ndash2tndash2x3=tndash2

dx1dx2dx3=x1x3 x2x3tHencedx2dx3=x2x35 x1=X1(1+t)Ax2=X2(1+t)2Ax3=X3(1+t)3A

Chapter51

(a)3e1+2e2+2e3(b)(e1ndash10e2+6e3)(14)

(c)(13e1+10e2+8e3)(14) (d)036

(e)directionratios2mdash1mdash21mdash22221

3Principalcomponents21-3Directionratiosofprincipaldirections20101010ndash24

(b)A+Bh2=0(c)ndash4ah(A+ Bh2)e25

(c)ndash Ch3e2ndash Calh3e2

(d) Calh3e26(b) WπmL-1e1sin( πχ1L)coshmhmdash cos sinhmh coshmx2

(c)Wm2sinhmh sinhmhe1e2e3 (e1plusmne2)7(c)-αx2e1+αx1e2+(β+γx1+δx2)e3

(d)0 Principalstressdirectionforintermediateprincipalstressdirectionistheradialdirection

8(b)directionratiospartψpartx1partψpartx20(iethenormalstothesurfacesψ=constant)

Chapter62(a)directionratios7radic2radic2-1radic2+13

Lengthsa1 a3angles cos-1

4λmicro2=1 5Stretchesmicro1micro-1Directionratios100001microtanγmicro-1-micro07

Principalcomponents0 directionratiosofprincipalaxesx1x20-x2x1x2-x1

8

(ΩiR)=0 CdirectionratiosofprincipalaxesX2-X10X1X200019

Chapter8

22W=λEiiEkk+2microEikEik+2αEiiE33+4βEi3Ei3+ (severalequivalentalternativeformsexist)

8Siexclj=2micro0(Eiexclj+t0Dij)

9Tij=-pδij+2micro1

Chapter91

(a)

(b) (c)

(d)100010001 2

Principalstretchesa2abDirectionratiosofprincipalaxesX1X20-X2X100013

4

5

6

Chapter101Edgelengthsλ 1

T11=-p+2W1(λ2+α2)-2W2λ-2T12=2(W1+W2)αλ-1T13=0T22=-p+2W1λ-2-2W2(λ2+α2)T23=0T33=-p+2(W1-W2)λ(T12e1+T22e2) (T11-αλT12)e1+

2Edgelengthsλλ-11

T11=-p+2λ2C1-2λ-2C2T22=-p+2λ-2C1-2λ2C2T33=-p+2C1-2C2T23=T31=T12=0F1=λ-1T11F2=λT22F3=T33λ=12C2λ=C1-C2plusmn

4Tij=(ρρ0)(partxipartXR)(partxjpartXs)4αCPPδRS+4βCRS+4γC11δ1Rδ1S+δ(C12δ1Rδ2S+C12δ1Sδ2R+C13δ1Rδ3S+C13δ1S53R)T11=4λ(3α+β+γ)T22=T33=4λ(3α+β)T23=T31=T12=0

5χ=αI+βB+γB2whereαβγarefunctionsoftrBtrB2andtrB3

8 plusmnα(υrsquo2)υrsquowhereαisafunctionofυrsquo2

9T11=

T23= T13= T12= A=ΩhB=0

10k(Uh)nk(Uh)(n-1)12

13T11=T22=T33=-pT23=T31=0

Chapter111

(b)rω(r)eφ+α(r)ez(e)0

(c)Rω(R)eφ+α(R)ez

(d)-rω2(r)er(f)

34

5Az=0or

69Tφz=2(λC1+C2)rψπa4ψ(λC1+C2)whereaisthefinalradius10T11=-p+4W1x1A-W2Ax1T33=-p+2(W1-W2)T22=-p+W1Ax1-4W2x1AT23=T31=T12=0I1=I2=2x1A+A2x1

11Trr=Tzz=-p+βwlsquo2

Tφφ=-p

Trφ=Tφz=0

Trz=micro(1-εwlsquo2)wrsquow=-k(r2-a2)4micro-εk3(r4-a4)32micro3

12

Furtherreading

ChadwickPContinuumMechanicsConciseTheoryandProblemsGeorgeAllenandUnwin1976

EringenACMechanicsofContinuaWiley1967

HunterSCMechanicsofContinuousMediaEllisHorwood1976

MalvernLEIntroductiontotheMechanicsofaContinuousMediumPrenticeHall1969

RivlinRSNon-linearContinuumTheoriesinMechanicsandPhysicsandTheirApplicationsEdizioniCremonese1970

TruesdellCSTheElementsofContinuumMechanicsSpringer1966

Inadditiontotheabovetextswhichareconcernedwithcontinuummechanicsingeneraltherearemanybookswhichdealwithparticularbranchesofcontinuummechanicssuchaselasticityviscousfluidmechanicsviscoelasticityandsoon

Index

accelerationAiryrsquosstressfunctionalternatingsymbolalternatingtensorangularmomentumangularvelocityvectoranisotropicmaterialarearatio

basevectorsbendingbiharmonicequationbodybodyforce

caloricequationofstatecantileverbeamcartesiantensorseetensorCauchystresstensorCauchy-GreendeformationtensorsCauchyrsquoslawofmotionCayley-Hamiltontheoremcharacteristicequationcompatibilityrelationscompressionconfigurationconservation

lawsofangularmomentumofenergyoflinearmomentumofmass

constitutiveequationscontinuityequationcontractionconvectedderivativecoordinatetransformationcreepfunctionscrystalsymmetrycurvilinearcoordinatescylindricalpolarcoordinates

decompositionofadeformationdeformablebodydeformation

gradienttensorgradientshomogeneoustensors

densitydeterminant

deviatordilatationdilationdimensionalhomogeneitydirectstresscomponentsdirectioncosinesdisplacement

gradientsgradienttensor

divergence

ofavectortheorem

dummyindexdyadicproduct

eigenvalueseigenvectors

normalizedelasticconstantselasticityenergy

conservationofequationinternalkinetic

equationofmotionequilibriumequationsEulerian

descriptionstraintensor

eversionextension

rateofratio

finite

deformationtensorselasticdeformationsstraintensors

flexurefluidsFourierrsquoslawofheatconduction

gasesGaussrsquostheoremgradientofascalar

heat

conductionflux

helicalflowhemitropicmaterialhomogeneousdeformationhydrostaticpressurehydrostatics

ideal

fluidmaterials

incompressiblematerialindexsymmetriesinfinitesimal

rotationrotationtensorrotationvectorstrain

initialyieldstressinnerproductinternalenergyinvariantinversetensorinviscidfluidisotropic

materialtensortensorfunction

kinematicconstraintkinematicskineticenergyKroneckerdelta

Lagrangian

descriptionstraintensor

leftCauchy-Greendeformationtensorleftstretchtensorlinearelasticitylinearmomentumlinearthermoelasticitylinearviscoelasticitylinearviscousfluidliquids

material

coordinatescurvederivativedescriptionsymmetry

matrix

algebraantisymmetriccolumndiagonalelementofinverseofnormalizedcolumnorthogonalpositivedefiniterowsquaresymmetrictraceoftransposeofunit

Maxwellfluidmotion

equationofsteady

Navier-StokesequationsNavierrsquosequationsNewtonrsquossecondlawNewtonianviscousfluidnominalstresstensornon-linearviscoelasticitynon-linearviscousfluidnon-Newtonianfluidnormalstresscomponents

orthogonal

matrixtensorvectors

orthogonalityorthotropicmaterialouterproduct

particle

kinematicspaths

Piola-Kirchhoffstresstensorsplane

flowstrainstress

plastic

deformationpotential

plasticitypolardecompositionpower-lawfluidpressureprincipal

axesofdeformationstretches

principleofvirtualworkpseudo-vectorspureshear

rate-of-deformationtensorrate-of-straintensorreactionstressrectilinearflowreference

configurationtime

reflectionalsymmetryReiner-RivlinfluidrelaxationfunctionsresidualstrainrightCauchy-Greendeformationtensorrightstretchtensor

rigidbody

motionrotation

Rivlin-Ericksentensorsrotation

ofcoordinatesystemtensorvector

rotationalsymmetry

scalarproductsheardirectionplaneshearingflowstresssimpleshearsingularsurfacesolidsspatialcoordinatesdescriptionsphericalpolarcoordinatestensorspintensorsteadymotionstrain

compatibilityrelationsenergyinvariantsplanetensors

strain-ratetensorstreamlinesstress

componentsofdeviatortensorhomogeneousplane

stresstensor

invariantsofprincipalaxesof

principalcomponentsof

symmetryofstressrelaxation

functionstretch

ratiotensors

summationconventionsuperpositionprinciplesurface

elementforcetraction

symmetricfunctionssymmetry

groupreflectionalrotational

tension

tensor

alternatinganti-symmetriccomponentscontractionofdeviatoricinnerproductinvariantsofinverseisotropicmultiplicationnotationsorthogonalouterproductpositivedefiniteprincipalaxesprincipalcomponentsprincipaldirectionsprincipalvaluessecond-ordersphericalsymmetrictransformationlawtransposeunit

thermalconductivitythermoelasticitytimeratesofchangetorsiontrace

ofamatrixofatensor

tractiontransformation

ofcoordinatesofstresscomponentsoftensorcomponentsofvectorcomponents

translationtransverselyisotropicmaterialtriadicproducttriplescalarproduct

uniform

compressionextensionsshearstresstension

unit

dyadtensorvector

vector

basecomponentsorthogonal

productunit

velocity

gradienttensorvirtualworkviscoelasticityviscositycoefficientsviscousfluidVoigtsolidvolumeelementvortexflowvorticitytensorvorticityvector

yield

conditionfunctionstresssurface

1 TheuseofAtodenoteavectorinthereferenceconfigurationisanotherexceptiontoourgeneralrulethatvectorsaredenotedbylower-caseletters

2 Theuseofγandηtodenotestraintensorsisadeparturefromourconventionofdenotingsecond-ordertensorsbybold-faceitaliccapitalletters

3 TheuseofNtodenoteavectorisanotherdeparturefromtheconventionthatvectorsaredenotedbylower-caseletters

• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 - Introduction
• • 11 Continuum mechanics
  • • 2 - Introductory matrix algebra
    • • 21 Matrices
      • 22 The summation convention
      • 23 Eigenvalues and eigenvectors
      • 24 The CayleyndashHamilton theorem
      • 25 The polar decomposition theorem
      • • 3 - Vectors and cartesian tensors
        • • 31 Vectors
          • 32 Coordinate transformation
          • 33 The dyadic product
          • 34 Cartesian tensors
          • 35 Isotropic tensors
          • 36 Multiplication of tensors
          • 37 Tensor and matrix notation
          • 38 Invariants of a second-order tensor
          • 39 Deviatoric tensors
          • 310 Vector and tensor calculus
          • • 4 - Particle kinematics
            • • 41 Bodies and their configurations
              • 42 Displacement and velocity
              • 43 Time rates of change
              • 44 Acceleration
              • 45 Steady motion Particle paths and streamlines
              • 46 Problems
              • • 5 - Stress
                • • 51 Surface traction
                  • 52 Components of stress
                  • 53 The traction on any surface
                  • 54 Transformation of stress components
                  • 55 Equations of equilibrium
                  • 56 Principal stress components principal axes of stress and stress invariants
                  • 57 The stress deviator tensor
                  • 58 Shear stress
                  • 59 Some simple states of stress
                  • 510 Problems
                  • • 6 - Motions and deformations
                    • • 61 Rigid-body motions
                      • 62 Extension of a material line element
                      • 63 The deformation gradient tensor
                      • 64 Finite deformation and strain tensors
                      • 65 Some simple finite deformations
                      • 66 Infinitesimal strain
                      • 67 Infinitesimal rotation
                      • 68 The rate-of-deformation tensor
                      • 69 The velocity gradient and spin tensors
                      • 610 Some simple flows
                      • 611 Problems
                      • • 7 - Conservation laws
                        • • 71 Conservation laws of physics
                          • 72 Conservation of mass
                          • 73 The material time derivative of a volume integral
                          • 74 Conservation of linear momentum
                          • 75 Conservation of angular momentum
                          • 76 Conservation of energy
                          • 77 The principle of virtual work
                          • 78 Problems
                          • • 8 - Linear constitutive equations
                            • • 81 Constitutive equations and ideal materials
                              • 82 Material symmetry
                              • 83 Linear elasticity
                              • 84 Newtonian viscous fluids
                              • 85 Linear viscoelasticity
                              • 86 Problems
                              • • 9 - Further analysis of finite deformation
                                • • 91 Deformation of a surface element
                                  • 92 Decomposition of a deformation
                                  • 93 Principal stretches and principal axes of deformation
                                  • 94 Strain invariants
                                  • 95 Alternative stress measures
                                  • 96 Problems
                                  • • 10 - Non-linear constitutive equations
                                    • • 101 Nonlinear theories
                                      • 102 The theory of finite elastic deformations
                                      • 103 A non-linear viscous fluid
                                      • 104 Non-linear viscoelasticity
                                      • 105 Plasticity
                                      • 106 Problems
                                      • • 11 - Cylindrical and spherical polar coordinates
                                        • • 111 Curvilinear coordinates
                                          • 112 Cylindrical polar coordinates
                                          • 113 Spherical polar coordinates
                                          • 114 Problems
                                          • • Appendix - Representation theorem for an isotropic tensor function of a tensor
                                            • Answers
                                            • Further reading
                                            • Index