フRheologyofLiquidsandSolids

7.1ELASTICANDVISCOUSBEHAVIOR

138

Thebehaviorofthematerialsdiscussedsofarconcernedtheirshort-termresponse

tostress.Thisresponseistheonlyoneneededtobeconsideredifthematerialisan

idealelasticsolid.However,inpractice,mostengineeringmaterialswillexhibitan

additionalcomponentinresponsetostresswhichistimedependent.Thisresponse

ischaracteristicofviscousmaterials,andthereforeasolidwhichexhibitsresponseto

stresswhichcombinesanimmediateelasticcomponentandatime-dependentvis-

couscomponentisreferredtoasaviscoelasticmaterial.Sincemostengineeringma-terialsexhibittime-dependentresponsesundercertainconditions,oneshould.

strictlyspeaking,definethemallasviscoelastic・However,inpractice､thistermisusedmainlyformaterialswhosetime-dependentresponseisparticularlylargeatroomtemperature,suchasasphaltsandpolymers.

Thedifferencebetweenelastic,viscous､andviscoelasticmaterialsmaybeseenbyreferringtoFig.7.1andconsideringtheresponseofthethreetypesofmaterialstothesameinstantaneousloadversustimecurve(wheretheloadisappliedinstanta-neouslyatt.andremovedsuddenlyatL).Foranelasticmaterial,allthestrainisinstan-

taneous;whentheexternalloadisremoved,allofthestrainisrecovered.Forapurelyviscousmaterial,thestrainincreasescontinuouslywithtimeunderloadandisnotrecoverable.Fortheintermediatecaseofaviscoelasticmaterial,thereisaninstanta-

neouselasticstrainwhenastressisapplied.However,thereisadditionalstrain,whichincreaseswithtimeunderloadandispartiallyrecoverablewhentheloadisremoved.

Elastic宮員山，口の

ＩTime

Viscous

=.~ざ

TimeTime

Figure7.1Responseofthreedifferenttypesofmaterialstothe

load-timecycleshown,

wheretheloadisappliedinstantaneouslyattandreleasedinstantaneouslvat

ヴ

t:.(a)elastic;(b)viscous;

Viscoelastic

(c)viscoelastic. Time

Theelasticresponseforanidealsolidhasalreadybeendiscussedandischar-acterizedbytherelationbetweenstressandstrain(Hooke'slaw)asfollows:

e=aE. (7.唾

whereeisthestrain,(丁isthestress,andEisthemodulusofelasticity.Theiﾉiscousresponsecanbedescribedbyananalogousrelationshipbetween

stressandstrainrate・Thatis,foraviscousmaterialloadedintension.

(72：s=a〃§

wheresistherateofstrain,aisthestress,anduisthecoefficientofviscosity.Simi-larly,foranideallyviscousmaterial(Newtonianfluid)loadedinshear,

(7.3）V=7777,

wherejistherateifshear,tistheshearstress,andγ）isthecoefficientofviscosity.Ifweassumefurtherthatviscousfluidsareincompressible,thenbyanalogy

withEq.5.22inChapter5,

E=2(1+v)G.

AssumingthatPoisson'sratiois0.5,wehave

メル=2(1+1/2)γ】

(5.22〉

(74：

139Sec.7.1ElasticandViscousBehavior

Finally,wecanshowthatthegeneralizedstrainrateequationsforNewtonianfluidshavethesameformasEq.5.25inChapter5:

。‐北当…)(7.5)竿ﾙーﾅ'…1隼ルーと…).

7.2SIMPLERHEOLOGICALMODELS

Themostconvenientwayofdepictingthebehaviorofviscoelasticmaterialsisbymeansofmechanicalmodels.Thesemodelsmaybebuiltupbyvariouscombinationsofthebasicrheologicalelements.ThethreebasicelementsconsideredhereareshowninFig.7.2.TheHookeiα〃element,orspring,isperfectlyelastic;alloftheenergyim-partedtothespecimenisstoredasstrainenergy.Itsstressversus,strainbehaviorisgivenbyo-=Ee,where,inthiscase,Erepresentsthestiffnessofthespring.TheNew-tomα〃element,ordashpot,isperfectlyviscous.Alloftheenergyimpartedtoitisdis-sipated,anditsstressversus,strainratebehaviorisgivenbyEq.7.2,cr="8．

EquationElemell［Name

E

HookeianelementﾄﾍﾉVWVV、ハー←(アSpring

ぴ＝E・ざ

(a

メル

Newtonianelement[ヨニトーーo

Dashpot

(1))

一旦牌

・Ｅ

Figure7.2

Thethreebasicrheologicalelements:(a)Hookeian(spring),(b)Newtonian(dashpot),(c)St.Venant.

｜トー一

(c)

St・Venantelement ぴmax＝ぴvield

TheSt.Vを"α"telementrepresentsablockthatresistsmotionunderstressbyvirtueofthefrictionbetweentheblockandthehorizontalsurfaceonwhichitrests.

Iftheappliedforceexceedstheforceoffriction,theblockmoves.Sincethiswould

applyanaccelerationoftheblockonceitovercamefriction,whichisunrealistic,the

St.Venantelementisusedonlyinconjunctionwithotherelements;insuchcombina-

tions,itrepresentsayieldstrengthwhichistimeindependent.Withonlythesethree

basicelements,increasinglycomplexrheologicalmodelsmaybebuiltupbysuitablecombinationsoftheelementstosimulatetheviscoelasticbehaviorofrealmaterials.

Thesimplemodelscanbecombinedtogetherinvariouscombinationstoac-

countforthetime-dependentresponseofsolidsandfluidstostress.Suchmodelsare

knownasrheologicalmodels,andthebranchofmechanicsdealingwithmodelingand

macroscopialcharacterizationoftime-dependentresponsetostressisknownasrhe-

ology.Itcoversbothsolidsandfluids,neitherofwhichbehaveinpracticeaccordingtotheidealmodelsforsolid(spring-Hookeian)orfluid(dashpot-Newtonian).Forcivil

140 Chap.7RheologyofLiquidsandSolids

engineeringapplications,thereisaneedtoaddressbothsolidsandfluids,asmanyofthemoreimportantconstructionmaterialsarebeingprocessedonsitewhiletheyarestillintheirfluidstate(e.g.portlandcementconcretesandasphaltconcretes).

7.3RHEOLOGYOFFLUIDS

oft Theevaluationofrelationsbetweenstressesandstrainsinfluids,whichisessentialtocharacterizetheirbehavior,isnotstraightforwardasinthecaseofsolids,whereloads(stresses)canbeapplieddirectlyanddeformations(strains)canbemeasuredbymountinggagesdirectlyontheloadedspecimen.Influidsthemeasurementsareindi-rectinnature,usinginstrumentswhicharecollectivelyknownasviscometers・Themorecommonviscometeristhecoaxialcylindertype,inwhichtheoutercylinderisro-

tatingatacontrolledangularvelocityandtheinnercylinderisstationary(Fig.7.3).Thetorquerequiredtokeeptheinnercylinderstationaryismeasuredasafunctionoftheangularvelocityoftheoutercylinder.Ifthegapbetweenthetwocylindersissuffi-cientlysmall,thenforanideallyNewtonianfluidthefollowingrelationcanbederived:

gap

篭襲い溌騨…

典一Ｅ一唖恥一唾

Ｉｌ即／ａｌ|為

R公

｢R,

…

Outer/|.一」ccylinderI一つ丁

『套榊催-点卜伽“｛7“whereTisthetorque,叩istheviscosityoftheHuid,R/andRotheinnerandouterradiirespectively,histheheightofthecylinder,andOistheangularvelocity.Thus,alinearcorrelationexistsbetweenTandfi,analogoustotheonebetweenshearstressand

rateofshearstrain,r=Tjy.Thus,measuredcurvesoftorqueagainstangularvelocityinacoaxialviscometercanprovideinformationofasimilarnaturetothatobtainedifthedirectstressandstrainratecouldbemeasuredinfluids.ForaNewtonianfluid,the

rversusflcurvewouldbelinearandpassthroughtheorigin;theslopeofthecurvecanenablethecalculationoftheviscositycoefficient17basedonEq.7.6.Curvesofthiskindareknownasflowcurvesandarethebasisforcharacterizationoffluids.

MeasuredflowcurvesindicatethatinmanyfluidsthebehaviorismorecomplexthantheonedefinedastheidealNewtonian.Therearefourdifferenttypesoffluids.threeofwhicharenon-Newtonian,asshowninFig.7.4.Intheshearthickeningfluid,theviscosityincreaseswithincreaseinrate,indicatingagreaterresistancetoflowasthestrainrateincreases.Inshearthinning,theviscositydecreasesathigherrates,sug-gestingthatbondsbetweenparticlesarebeingbrokenbytheshearstress,thusallow-ingforeasierflow.Thesetwotypesoffluidsaresometimescalledpseudoplastic,andtheirflowcurvecanberepresentedbythefollowinggeneralequation:

Figure7､3

Schematicdescriptionofarotationalcoaxialcylinderviscometerformeasuringtherheologicalpropertiesoffluids.

T=Aγ"(7.7)

whereAisaconstantcharacteristicofthefluidandnistheindexofflow;forn>1,

Slope=卜、轍吻の衿砦吻‐胃団の二②

ト、釣めの縄扇出国⑪［一的

apParen

V1ScosltV

Figure7.4Threetypesofrheologicalbehavioroffluidsasexhib-

itedbytheflowcurves:(a)shearthinning,(b)shearthickening,(c)yield-Bing-hambehavior.

Yield

Shearrate,7

(c)

Shearrate,y

(a)

Sheai･rate,y

(b)

1“Sec.7.3RheologyofFluids

thebehaviorisshearthickening,andforn<1,itisshearthinning.Althoughthesefluidsdonothaveasinglevalueofviscosity,theyareoftentreatedasNewtonianatagivenshearstepbydefininganapparentviscosity(Fig.7.4a).

Flowcurveswhichexhibityield(Fig.7.4c)arecharacteristicoffluidsinwhichinitialstressmustbeovercomebeforeflowcanstart.Iftheflowcurvebeyondtheyield

islinear,thenthebehaviorcanbedescribedbytheBinghammodelforfluids(Fig.7.5).Thedashpotandthefrictionalblockareinseries,andnostresscanbetransferredtothedashpotuntilthefrictionalblockyields.Binghambehaviorisshownonlybysolidsuspensions,suchascementpastesandconcretesintheirfresh,fluidstate.Theyieldstressrepresentsthemechanicalbreakdownofflocculatedstructures(seeFig.4.15).

ミーE捌Figure7.5Binghammodelforfluids

一

Theflowcurveisnotnecessarilyreversible;thatis,onreducingtheshearrate,thedownwardcurvemaynotnecessarilycoincidewiththeupwardbranchandahysteresisloopmayform.Thisischaracteristicofshearthinningfluids,inwhichincreasingshearinvolvesgradualbreakdownoftheflocculatedstructure,es-peciallyinparticulatesuspensions,wheretheinitialmixingandshearingseparatestheparticlesandreducestheattractiveforcesbetweenthem.ThisisshowninFig.7.6a.Ifthisbreakupismaintainedandcarriesintothedescendingbranch,theshearstressrequiredforthesamestrainrateislowerthanintheascendingcurveラasseeninFig.7.6a・Ifaftercompletionofthecycle,whenthefluidisatrest,thebondscanre-formandthenexttestcycleprovidesanidenticalcurve,thematerialissaidtobethixotropic(Fig.7.6b).Inanonthixotropicfluidthesecondtestwillresultinanascendingcurveidenticaltothedescendingbranchofthefirsttest(Fig.7.6c).

Incivilengineeringweoftenhavetodealwithsuspensionsratherthanpureliquids(e.g.,cementgrouts,freshconcrete,andasphaltcement),whoseflowcanbedescribedbythemodelsreviewedpreviously.Thepresenceofsuspendedsolidsaf-fectstherheologicalparameters;bothconcentrationandparticlesizeareimportant.Oneexampleofarelationshipofthiskindisasfollows:

”琴"{'一かwhereti=viscosityofsuspension

りo=viscosityofpurefluidp=volumefractionofsolidparticles

β",=themaximumvolumefractionwhentheparticlesareclosedpacked[ti]=constantrelatedtoviscosityofsuspensionwithalowconcentrationof

solids.

Figure7.6Illustrationofthecharac-

teristicsofthixotropicandnonthixotropicfluidasexhibitedbvflowcurves:

ヴ

(a)firstcycle-shearthin-ningandhysteresis,(b)sec-ondcycleinathixotropicfluid;(c)secondcycleinanonthixotropicfluid,

ト耐ぬ⑪潤一の穐毎の〔｛ぬ

卜”ぬの』一の揖毎の二の

Shearrate,y Sheai･rate.､ Shearrate,V

142 Chap.7RheologyofLiquidsandSolids

Non-Newtonianfluidscanalsobedescribedasviscoelasticmaterialsinwhich

theviscouscomponentisdominating.Therheologicalmodelsfordescribingsuchbehavioraresimilartothoseusedfortheviscoelasticsolidsdescribedinthefollow-ingsection.

7.4RHEOLOGYOFVISCOELASTICSOLIDS

TherheologicalbehaviorofsolidscanbemodeledbydifferentcombinationsofthebasicelementsdescribedinFig.7.2.Withthem,increasinglycomplexrheologicalmodelsmaybebuilttoaccountforthebehaviorofrealengineeringmaterials.Thesimplestoftherheologicalmodelsarethosemadeupofonlytwoelementseach:theMaxwellmodel,theKelvinmodel,andthePrandtmodel.TheseareshowninFig.7.7二

Name EquationModel Stress-strain-timerelationships

一Er

qくぷ:…"二

Responsetoappliedstrain

ぴ一ECreep

Responsetoappliedstress

""..

‘源一：Responseto

appliedstress

Maxw釧卜w号匿”い）

。Ｅ些

挫十‐

汀一Ｆ』

－ヘハハハ/V、一

嘩i凹・Kelvin

Or

(Voigt)

(b)

計 L←ぴぴ=e+ノルE

I

Responseto

appliedstrain

一二

ﾚアa=Ee(forcr<cr)

(丁

〔γ→｡o(lorcr>a,)

Prandt

←一一一一

Fc

Figure7､7Two-elementrheologicalmodels.

7.4.1MaxwellMode

TheMaxwellmodelconsistsofaspringandadashpotinseries.Thesamestressacts

onbothelements,andsothetotalstrainisequaltothesumofthestrainsofthetwoelements.Theextensionofthespringisgivenbye,=alE;theextensionofthedash-potobeystherelationships,i=07/1.Differentiatings.withrespecttotime,andsumming,weget

． o 〔 丁

8s＋E〔ノーe＝一十一． （7.8）E 〃

NowconsidertheresponseoftheMaxwellmodeltotwolimitingloadingcases:con-slantstressandconstantdeformation.Underconstantstress,cr=0andsoEq､7.8becomes

ぴ一〃

Ⅳ

・ｅ 7.8a

143Sec.7.4RheologyofViscoelasticSolids

144

Thatis,therewillbeaninstantaneous(elastic)strain,givenbycrIE,whichisrecov-erable,followedbyalinearlyincreasingstrain,whichisirrecoverable,asshowninFig.7.7a.Thistypeofbehaviorisoftenreferredtoascreep(seeSec.7.5).Ontheotherhand,ifastrainissuddenlyappliedtothesystemandheldconstant,s-0,thenthestressasafunctionoftimeisgivenby

^+^=0.(7.8b》E〃

Solving,weget

ぴ=CTneI似. (7.8c)

Thismeansthatthereisanexponentialstressﾉ･ど/αxα"on,asshowninFig.7.7a.

7.4.2KelvinModel

TheKelvin(orVoigt)modelconsistsofaspringandadashpotinparα"el.Inthiscase,theelongationineachelementremainsthesame.Therefore,o=Ee,andcr,,=us,sothat

〔丁=cr,+cr,/=Es+メル8． (7.9：

Underaconstantstress,〔丁Oweagaingetcreepbehavior,withthesolutionofthedif-ferentialEq.7.9giving

*= (l-<r*n(7.9a)Thatis,thestraincr,,/,whichwouldbeobtainedinstantaneouslyintheabsenceofadashpot,isinsteadapproachedexponentially.Underaconstantstrain,e=0,thereissomestressrelaxation,andthenthestressremainsconstant,atcr=Es,as

showninFig.7.7b.Ifthematerialisgivenasuddendisplacementandthenreleased.thereisanexponentialstrainrelaxation,givenby

E=Ene-"* (7.9b)

7.4.3PrandtModel

InthePrandtmodel,thereisperfectlyelastic-plasticbehavior.Uptotheyieldstress.

thea-srelationshipisgivenbycr=Es:attheyieldstress,thedeformationcontin-

uesindefinitely,asshowninFig.7.7c,

7.4.4ComplexRheologicalModels

Clearly,itispossibletobuildupincreasinglycomplexrheologicalmodelstosimu-latemorecomplicatedtypesofmaterialbehavior,buttheyarebeyondthescopeof

thisbook.However,toillustratethisapproach,wedescribethesimplestcasewhichistheBinghammodel.

Theperfectlyelastic-plasticPrandtmaterial(Fig.7.7c)deformsinfinitelyoncetheyieldpointisreached,whichisclearlyunrealistic.Thisproblemcanbeovercome

byusingtheextendedBinghammodel(Fig.7.8),whichisthesimplestmodelthat

Chap.7RheologyofLiquidsandSolids

Figure7､8

ExtendedBinghammode)forsolidsexhibitingyield.

｜Ｖ

－ぴ

representstheflowofamaterialwhichpossessesayieldpoint・Itconsistsofthethreebasicelements-asprlng，adashpot，andafrictionblock-insenes・Pbl・

stresseslessthantheyieldstress,itbehaveselastically.Beyondtheyieldpoint,itgivesasteadilyincreasingstrain.Itsequationsareasfollows:

ぴ｜Ｅ

８ forびくぴ､ (7.1雌

.=2+<ぴ－αy)rfbrぴ＞αy・E〃

InFig,7,9,theresultofasimilaranalysisforacombinationofMaxwellandKelvinelementsinseriesisshown.Thiscurveshowsmanvofthecharacteristicsob-

servedincreepofsolids.

DelayedElastic

EkElasticViscous－，VVVVVVV－

一

一 手jLK

(a

Creep山紗口唱両揖］②

Time,t

ｂ面めの畑こめ

Figure7.9MaxwellandKelvinmod-

elsinseries(Burgers、

model).

Time,/

(b)

145Sec.7.4RheologyofViscoelasticSolids

7.5CREEPOFENGINEERINGMATERIALS

Theviscoelasticbehaviorofengineeringsolidswhichisofthegreatestpracticalsig-nificanceiscreep.Therheologicalmodelsdescribedintheprevioussectionprovidemathematicalformulationtodescribethecreepcurves.Thecharacteristicrheologi-calmodelandtheconstantsofthebasicrheologicalunitsofwhichitiscomposedare

usuallydeterminedbymatchingamodeltoanexperimentalcreepcurve.Therefore,tounderstandcreepbehavioritisessentialtoaddressexperimentalcreepcurvesaswellasthemechanismsbywhichcreepisgeneratedintheactualmaterial.Creepwillbediscussedingreaterdetailineachofthechaptersdealingwiththeindividualmaterials.Inthissectionanoverviewwillbegiven.

Generally,forallmaterials,threetypesofcreepcurvescanbeidentified(Fig.7.10b).Eachofthemcanoccurforeverymaterial,dependingonthestresslevelandtemperature.Thecurveisusuallydividedintothreestages(Fig.7.10a):thepri-marystage,alsoknownastransientcreep;thesecondarystage､alsoknownasthesteadystate(thecreeprate,sisconstant);andthetertiarystage,whichisterminatedwithfracture.Iftheloadingstressandtemperaturearesufficientlylow,onlythepn-marystagewouldoccur(i.e.,thecreeprateatthesecondarystagewouldbenil､asseeninthelowercurveinFig.7.10b).Athighenoughstressandtemperaturelevels.allthreestageswilloccur,resultinginfractureofthematerialattheendofthethird,tertiarystage.Ifthestressandtemperaturelevelsareintermediate,thetertiarystagemaybedelayedandoccurattimeperiodsgreaterthantheservicelifeofthemater-ial.Thephrasesu＃icientりん増hstressα"乱te"叩e'．α""ewasaddressedsofarinquali-tativetermsanddependsonthestrengthandtransitiontemperatureofthematerials

RuptureConstantstress

Constanttemperature

四声芦一Ｗこの

Creepstram 一 |

AE

4=CreeprateAri

Ar

-c

te言←i4First Secondsla

」(steadystastage

割一工也竺←；Elastic~ず

ostrain_L Stage‐

Time､； Rupturetlme

(錘

ｂ

山・巨三飼揖一の 、

ノIncrease

loador

temDerature久

Figure7.10Creepcurvescharacteristicofmaterialsloadedunder

differentconditions:(a)

generaldescriptionofthecurve;(b)effectofloadingandtemperaturecondi-tlons，

／Elastic-盲

stram-二

Time,f

(I))

146 Chap.7RheologyofLiquidsandSolids

Figure7.11Anillustrationofadisloca-

tionclimbawayfromob-stacks,(a)whenatomsleavethedislocationlineto

createinterstitialsortofill

vacancies,or(b)whenatomsareattachedtothe

dislocationlinebycreatingvacanciesoreliminatingin-terstitials,(fromD.R.Askeland,TheScienceα"。

E"g腕eer加gofMaterials,PWS-KentPublishing

Company,1985,p.62).

(e,g,,meltingtemperatureformetalsandglasstransitiontemperatureormeltingtemperatureforpolymers).Aroughestimateofthelevelsabovewhichcreepmaybecomesufficientlyimportantis～1/3ofthetransitiontemperatureandstrength.Above～1/2ofthetransitiontemperatureandstrength,creepmaybecomecriticalbecausethematerialmayenterthetertiarystageduringitsservicelife.

TheshapesofthecurvesshowninFig.7.10bcanbedescribedonthebasisoftherheologicalmodels.Theycanalsobedescribedbyempiricalrelations.Thelatterapproachiscommonformetalsandusesthefollowingequations:

Primary(transient)creep:

E=At", （7.11》

whereAisaconstantdependingonthematerial,load,andenvironmentalcondi-tions,andnformetalsisusually1/3.

Secondary(steady)statecreep:

=Sc『"exv(-EJRT), (7.12》

whereB,n,andE.areconstants.

Thecreepinthesteady-statestageistheonemostimportantfromanengi-neeringpointofviewbecauseitisthestageinwhichmuchofthecreepstrainisac-cumulatedduringtheservicelife.TheexponentialterminEq.7.12ischaracteristicofathermallyactivatedprocess,wherethevalueofE.istheactivationenergy.Theactualprocessesaredifferentforthevariousmaterialsandwillbebrieflyreviewed.

7.5.1CreepinMetals

Inmetals,themaincreepmechanismisthemovementofdislocations.Inthepri-marystage,theirmovementisgraduallysloweddownduetothepinningofdislo-cationsinvarioussites,asdescribedinChapter2.Thesesitescouldbepointdefects,intersectingdislocations,grainboundaries,orparticlesofsecondphase.Tocontinuetomovepasttheseobstacles,thedislocationsmustacquireadditionalen-ergy(toclimborjogovertheobstacle).AnillustrationofsuchaprocessisgiveninFig.7.11.Theactivationenergyisdirectlyrelatedtothatoftherateofdiffusionofdefectssuchasvacanciesandinterstitials・Thisdiffusionistemperaturedependent,asmightbeexpectedforanythermallyactivatedprocess.Itbecomesconsiderablyhighoncethetemperatureincreasesoverone-thirdofthemeltingtemperature,ac-countingforthesensitivityofcreepinmetalstotemperature.Theaccumulationof

』 ！(錘 (b:

Sec.7.5CreepofEngineeringMaterials 14#’

plasticdeformationinthesecondarycreepstagemayleadtoneckingandfracture.whichoccurattheendofthetertiarycreepstage.

7.5.2CreepinPolymersandAsphalts

Viscoelasticbehavior(creepandstressrelaxation)inpolymericandasphalticmate-rialsisalsoathermallyactivatedprocess.Itinvolvestheslidingofmacromoleculespasteachotherorslowextensionofindividualpolymericchainswhenkeptunderload.Thisextensionischaracteristicofanamorphouspolymerortheamorphous

partofapolymerchaininapartiallycrystallizedpolymer.Itinvolvesmovementofpolymersegmentsofapproximately50carbonunits,whentheyacquiresufficientthermalenergytoallowthemtomoveorrotatepastlocalobstacles.Theprobabilityforacquiringthisenergyisproportionaltotheexponentialtermintheequationde-scribingthermallyactivatedprocesses,likeEq.7.12.Inthecaseofpolymerandas-phaltmaterials,theactivationenergiesaremuchlowerthaninmetals,asexhibitedalsobytheirrelativelylowtransitiontemperatures(whichmaybeintherangeof100Cforamorphouspolymers).Thus,applyingtheruleofthumbthatcreep(and,forthatmatter,stressrelaxation)isbecomingimportantattemperaturesabove~l/3ofthetransitiontemperature,inpolymersonemustexpectconsiderablecreepatlevelsoftemperatureswhichareclosetoserviceconditions.

Itiscommontorepresentcreepandstressrelaxationdataforpolymersmarangeoftemperatures,usuallybymeansofcreepmodulusorrelaxationmoduluscurves.Thecreepmodulusisdefinedastheappliedstress(constantthroughoutthetest)dividedbythestrainatagiventime,whilerelaxationmodulusisdefinedasthestressmeasuredatgiventimedividedbythestrain(constantthroughoutthetest).Bothmodulidecreasewithtime,andbothtypesofcurvesprovideasimilarkindofinformation.

RelaxationmoduluscurvesareshowninFig.7.12a.Thetemperaturedepen-

denceoftherelaxationmoduluscanbeobtainedbvplottingthedataobtainedata

specificcreeptimeasafunctionoftemperature(Fig.7.12b).Thelattercurveresem-blesthemodulusofelasticity-temperaturecurvesforpolymers,aspresentedinChap-

ter15.Thissimilarityindicatesequivalenteffectsoftimeandtemperature,whichare

Relaxationmodulus Creepmodulusattimef2

的．三］己○（、巨如○目

Figure7､12Relaxationmoduluscurves

ofpolymericmaterialsatdifferenttemperatures(a)andtheeffectoftemoera-

ユ

tureonthecreepmodulusobtainedattimet,(b).

｜’I

f]hhU r.乃喝乃路'I¥,

Time,(Temperature,T

148 Chap.7RheologyofLiquidsandSolids

expectedtooccurinaprocesswhichisthermallyactivated:bothanincreaseintern-peratureandanincreaseintimeperiodwillincreasetheprobabilityofasegmentacquiringsufficientenergytoovercometheactivationenergybarriertomovement.

『Thissimilarityhasledtothedevelopmentofthetime-temperaturecorrespon-denceconcept,whichenablesthepredictionofarelaxationorcreepcurveataspec-ifiedtemperaturefromthecurveobtainedatanothertemperature.Thisisdonebymeansofashiftfactorinthetimescaleintherelaxationandcreeptest・Thisfactorisafunctionofthematerialandthetemperaturesofthetwocurves・TheshiftfactorA-rforamorphouspolymersisgivenby:

-Ci(r-rJ

'^ot^)' （7.13》

whereAj-istheshiftfactorbetweenthecurveattemperatureTandthetransitiontemperatureTC,andC2areconstantswhichchangeslightlyfromonepolymertotheother.ThecurveatTiscalledthereferencecurve.Therelationsbetweenthis

curveandtheothercurveattemperature7℃anbeformulatedasfollows:

(r,,f,,,)=E{T,t)

fref=tlAj.

(714〕

Withthisconceptitispossibletocalculatethecreep(relaxation)modulus-timecurveatonetemperatureifthecurveatanothertemperatureisknown.Thuswithasinglecurve,referredtoasmasterorreferencecurve,itispossibletodeter-minethewholefamilyofcurvesatdifferenttemperatures(Fig.7.13).Thisconceptisusuallyappliedtopredictthelong-termcreepandstressrelaxationexpectedatlowertemperatures,bycarryingoutashorttestatahighertemperature.Thisproce-dureisalsocommoninasphalts.Asimilarconcepthasbeenappliedinmetalstopre-dietthetimetofractureincreep,usingtheLarson-Millerparameter,whichisdefinedas:

T(C+lost,》 (715〉

whereCisaconstant(usuallyontheorderof20),TisthetemperatureintheKelvinscale,andfisthefracturelifetime.Thefracturelifetimeofagivenmetalmeasuredatsomespecificstresslevelwillvarywithtemperatureinorderthatthisparameterwillremainconstant.

Mastercurve

73n73乃弼

l－Data－lヨ

7， 乃10

冒寺－r,、

、

珂生さ）国帥○］

Figure7.13Stressrelaxationmodulus

mastercurvesandexpert-mentallymeasuredcurvesatvarioustemperatures

(afterAklonisandMack-night,/""oductiontoPoly-"zeﾉ．w““j“"α収Wiley,/983ﾉ．

- 113579

Logtimeinseconds

149Sec.7.5CreepofEngineeringMaterials

征一丁部－

鈴一言爺

｜

－

込冒宮

、、

、

、、~－－－__三一

～、ヨ、

ReferenceTemperature:7’ 、’

150

7.5.3CreepinPortlandCementConcreteandWood

Thecreepmechanismsinportlandcementconcreteandwoodareassociatedwiththemovementofwater.Bothmaterialsareporousandcontainwaterinthepores,aswellasintinyconfinedspacesbetweenmoleculesorparticles,whereconsiderablebondingofthewatertothesurfaceofthesolidiseffective.Theloadappliedresultsinstressesinthesolidandinthewater.Thewaterrespondstothisstressbyflowing

slowlytospaceswherestresseswouldbesmaller・Asaresult,creepstrainoccursbe-causeadditionalstressisimposedonthesolidparticlesasthewatermovesout.Therateofthisprocesswilldependonthediffusioncharacteristicsofthewaterintheporestructure(whichisamaterialsproperty)aswellasonthedrivingforcetothediffusion,whichisthestressappliedonthewaterinthematerial.Becausediffusionisalsoathermallyactivatedprocess､onemayexpectdependenciesontemperaturesimilartotheonesobservedformetalsandpolymers,butinthiscasetheywouldnotberelatedtotransitiontemperatures.However,amuchmoreimportantenviron-mentalconditionforthesematerialsishumidity,Ifexternaldryingconditionsexist,

theywillprovideanadditionaldrivingforceforwatermovement，thusenhancingthetime-dependentstrains.Dryingalone,withoutanyloading,cancausecontrac-tionaswaterisdiffusingfromtheporesandfromthespacesbetweentheparticles.causingthemtoapproacheachother.Thisstrainisreferredtoasshrinkage,anditcomesontopofthecreep.Inpractice,thetwooccursimultaneouslybecausethewoodorportlandcementconcreteareusuallysubjectedtoloadanddryingatthesametime.Althoughthecreepandshrinkagearetheresultofsimilarprocesses,theirstrainsmaynotnecessarilybeadditive,asdiscussedinChapter11,

BIBLIOGRAPHY

Aklonis,J,J.andMacKnight,W.J.,ﾉ""o血c"ontoPo〃"icrViscoelasticity,John

WileyandSons.1983,

Askeland,D,R..TheScienceandEngineeringofMate"α酌,3rdedition.PWS-Kent

PublishingCo.Boston,1984.

Banfill,P.FG.(editor),RheologyofFreshCementα"〃Concre花,E&FNSpon，UK,1991.

Bazanl,Z.P.andCarol,I.(editors)､Cre印α"ds/"加えαgeofConcrete,E&FNSpon,UK,1993.

Illston,J.M.(editor).ConstructionMα花"αIs,E&FNSpon,UK,1994.

SuperpavePerformanceGradedAsphaltBinderSpecificationsand!bsting.Asphalt

InstituteSuperpaveSeriesNo.1,(SP-1),AsphaltInstitute,Lexington,Kentucky，USA,1994.

Tattersall,G.H.,Wo'"b"伽α"dQuα"〃Co""olofConcrete,E&FNSpon,UK､1991.

PROBLEMS

7.1Discussthedifferencesinthecreepprocessesofcementitiousmaterialsandpolymers.

7.2Dryandwetwoodspecimensareexposedtocreepunderload.Inwhichwillthecreepstrainsbehigher？Explain．

Chap,7RheologyofLiquidsandSolids

7.3Twopolymerswhicharesimilarintheirgeneralstructurehaveadifferentglasstransitiontemperature.Ifbotharesubjectedtoaconstantloadatasimilarser-vlcetemperature,whichwillexperiencehighercreepstrai､？Explain、

7.4Creepofengineeringmaterialscanfollowabehaviorwhichcanbedescribedbvプ

aKelvinmodelinserieswithaspringoraMaxwellmodel.（a）Fromanengineerlngpointofviewwhichbehaviorispreferred？Explain(b)Whatloadingandenvironmentalparameterscancauseashiftinthecreep

behaviorofagivenmaterialtobechangedfromonewhichcanbedescribedbyaKelvinmodelinserieswithaspringtoaMaxwellmodel.

7.5Inordertopredicttheviscoelasticbehaviorofanasphaltmaterialisitneces-sarytocarryoutalaboratorytestattheactualservlcetemperature？Ifnot、wouIdyourecommenditoutatahigherorlowertemperature？Explain、

7.6TwofreshconcretemixeswitharheologicalbehaviorwhichcanbedescribedbyaBinghammodelhavethesameapparentviscositywhichismeasuredinthemixerataspecifiedrotationvelocity.（a）Dothemixesnecessarilyhavethesamerheologicalproperties？(b)Ifnot,whatcouldbethedifferences,andwhichrheologicalbehaviorwould

bepreferre｡？

7.7Discusswhichrheologicalbehaviorwouldbepreferredforafreshmortarwhichlsappliedonaverticalsurface:NewtonlanorBingham::う

Sec.7.5 CreepofEngineeringMaterials 151