0195095030_stress and deformation etc

8/7/2019 0195095030_Stress and Deformation Etc

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Stress and Deformation:A Handbook on

Tensors in Geology

Gerhard Oertel

OXFORD UNIVERSITY PRESS


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S T R E S SA N DD E F O R M AT I O N


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S T R E S S A N DD E F O R M A T I O NA Handbook on Tensorsin Geology

Gerhard OertelUnivers i tyof C a l i f o r n i a ,L osAngeles

New York OxfordOXFORD U N I V E R S I T YPRESS

1996


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O x f o r dU n i v e r s i t yPress

Oxford N ew YorkAthens Auck land Bangkok Bombay

Ca lcu t t a C a p eTo w n D a r es Salaam D e l h iF l o r e n c e H o n gK o n g I s t a n b u lK a r a c h i

K u a l aL u m p u r M a d r a s M a d r i d M e l b o u r n eMex icoC i t y Nai rob i Pa r i s S ingapo re

Taipe i To k y o To r o n t o

and as soc i a t ed com pan ie si nB e r l i n I b a d a n

C o p y r i g h t 1996 b y O x f o r dUnivers i ty Press , Inc .

Pub l i shed by Oxfo rd Un ive r s i t y P re ss , I nc . ,19 8 Madi son Avenue .New Yo r k .New York 10016

Oxford i s a r e g i st e re d t r a d e m a r kof Oxfo rd Un ive r s i t y P res s

A ll rights reserved. No pan of thisp u b l i c a t i o nm a y b e r ep r o d u c e d ,s tored in a re t r ieva l sys tem,or t r a n s m i t t e d ,in any f o r mor by anym e a n s ,

e l e c t r o n i c ,m ech an ic a l , pho tocopy ing , r eco rd ing ,o ro t h e r w i s e ,w i t h o u tth e p r i o r p e r m i s s i ono f Oxfo rd U n i v e r s i t yPress.

L i b r a r yof CongressC a t a l o g i n g - i n - P u b l i c a t i o nDataOer t e l ,G. F.

Stressand d e f o r m a t i o n :a handb ook on t enso r sin geology / Gerhard Oer te l .

p . cm . Inc ludes b ib l i og ra ph ic a l r e f e r ences and i ndex .

I S B N0 - 1 9 - 5 0 9 5 0 3 - 01 .Vector ana lys is .2 . G e o l o g y M a t h e m a t i c s .3 . S t r a i n sand s t r e s s e s .I . Ti t l e .

Q E 3 3 . 2 . M 3 0 3 71 996 55 1'. 0 1 ' 51 5 6 3 d c 2 095-1862 1

1 3 5 7 9 8 6 4 2

Primed in the U n i t e d S t a t e s of A m e r i c aon a c i d - f r e epaper


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PREFACE

This book began as a set of notes for a graduate course taught by Dr. RonaldShreve,m y col leagueat the Univers i tyof Cali fornia ,Los Angeles;he first sawthat ou r geology students needed sucha course,and theau thorwas one of his"students" (asignif icantprop ort ion of the facu lty m em b ers of what was then theDepa r tmen tof Geologysat in, andworkedthe problems, whenthe coursew asofferedfor the first fewt im es). LaterI took overthe teachingof this courseandD r. Shreve's course notes , which consisted of a set of prob lem s, a set of con canswers, andseveral hand ou ts.I gradu ally addeda few newprob lemsb u t changedthe originalset hardlyat all . Fromthe form in which I received them,the noteshave evolved principallyb y elaborat ionsof Dr. Shreve 's handoutsand by theaddit ion of handou ts that I distr ibu tedafter each class session.They contained theworked answers and bec am e m ore explic i t over the years. Stu dents sug gesimprovemen t s ,and elegant solutions foundb y them replacedor supp lemen tedearlierversions. Especiallyhelpfulwere PeterA . Craig, TheresaL. Heirshberg,and mycolleagueDr. AnYin ,who useda firstdraftof this b ookas ahandout; they

turned ou t to be superb c o l labora tors b y d i scover ing nu m erou s m is takes shor tcomings .J. F. Nye ' s Physical Propertiesof Crystals, Clarendon Press, Oxford,in its

slightlyrevised 1964 edition, servedas the textbookfor that course.The at tent ivereaderwill recognize m any trai tsof Nye's approachin thepresent book.This, Ihope, wi ll b e u nders tood as an indic at ion of h is d idact ic suc cess. U pon DShreve's and la ter m y r e c o m m e n d a t i o n , m a n yof the s tuden ts usedI. S.Sokoln ikoffand R. M.Redheffe r ' sMathematics of Physics and Modern

Engineering, McGraw-Hil l , New York, 1958 orlater. Several of the prob lem shave b een inspired by that book. M y stud ents also foun d Y . C. F u ng'sFoundationsof Solid Mechanics,1965, andA FirstCourse in Continuum Mechanics,2nd edit ion,1977, b oth pu b l ished by Prent ic e-H al l , EnglewoodCliffs, lucid and helpful .

B ype rmis s ionof McGraw-Hi l l Book Company Inc . , P rob lems1 to 9, 12 to14, 19 to 23, and 2 5 of this book w ere reproduc ed, som e with m odific at ions,fromI. S. S o k o l n i k o f f& R. M. Redheffer, Mathematics of Physics and ModernEngineering, M c G r a w - H i l l ,N ew Yo r k , 1 9 5 8 . B y pe rmis s ionof OxfordUniversi tyPress, Prob lem s39, 40, 57, 66, 69 to 71, and 98, andFigures4.3 and 5.2


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v i STRESSA ND DEFORMATION

of this book were reproduced, some with modifications, from J. F. Nye, PhysicalProperties of Crystals, Clarendon Press, Oxford, 1957. By permiss ion of

P r e n t i c e Hall, Inc ., Figures 4.1 and 5.5 were redrawn f rom Y. C. Fung,Foundations of Solid Mechanics, 1965.

Los Angeles

July 1995 G. O.


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To THEREADERGeologistsat various stagesof their careers,b u t especiallyas graduate studentsready to begin theirfirst own research project,may f indthat their mathematical

t ra ininghas left them unpreparedto deal withthe notationand conceptsofc o n t i n u u m m e c h a n i c sand other branchesof physics dealing with tensorquantities. Tryingto read literature pertinentto their research topic,theym ay b ebewilderedb y symbols decoratedb y amul t ip l ic i tyof letteror number subscr ip t s .When they seekout theappropriate textbooks, their bewildermentm ay growbecause such booksarewri t tenfor specialistsand cover much more subject mat terthan a geolog ist needs. Even if the text of such abookis lucid enou gh, stud entseagerlyreading forwardto thetopic theyhopethemselvesto apply, com m onlyendu p unab le to solve thei r prob lem s. M erely reading abou tthe f u n d a m e n t a loperat ions wi thou t pract ic ing them , theytend to forget m u c hof what theyeventual ly needto know. Typic al ly, theyf ind the t ex tbook increas ing lyconfusingand give up before reaching the topic of interest.

In theauthor 's experience,the apparentdifficultiesof cont inuum mechanicsand oftensor and matr ix nota t ionare manageab leto anys tudentwho comesprepared with some ins t ruct ionin calculusand the r u d i m e n t sof different ia l

equa t ionsand has notdiscardedthe old textbooks,so t ha t memorym a y b erefreshedb y someof thedetailsin these sub jects. However,the s tudent mustb epreparedto spend the t i m eand effort necessaryto solvea sequenceof prob lemsthat lead him gradu ally from the sim ple to the com plex and from the intu it iobviousto the, som etim es, co u nter intu i t iveb u t t rue .

The problem s presentedin this book start witha reviewof the mathema t i c sof vectorsand lead from thereto themathema t i c sof stressand s t ra in , includingfinite s t ra in . They are arrangedso that the so lu t ionsto success ive problems

depend on comprehens ionof ear l ier solut ions . A l thou gh m anyof the laterp r o b l e m sare notint r ins ical ly m oredifficultthan the earlier ones,I r ecommendonly to a well versed reader to skip the seem ingly sim ple prob lem s in the eachapters.

The main purposeof theproblemsis to t rain the s tudentin mathema t i ca lm ethods and not to cover any partic u lar field of m athem atical physics. I hopethe same, thatthe topicson whichthe problemsarebasedwill interestthe studentin theirown r ightand that theym ayi l lumina teone or theother branchof physics

p e r t i n e n t to severa l d iv i s ionsof geology, inc lud ing s t ruc tura l geo logy,


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v i i i S T R E S SA N DD E F O R M AT I O N

glaciology, crystallography, crystal physics,the geophysicsof heat flow, andothers. Thesesub jects, however,are notcovered systematically.

C hap ter 1 leads the readerfrom the conventional to the subscript notation forvectorsand their ar i thm et ic . Chapter2 deals with scalarand vector fields.Chapter3 introduces tensorsof the second rank describingthe propert iesofanisotropic materials,the so-called m atter tensors. C hapters4 and 5deal withtwo field tensors of the second rank, the stress andinfini tesimalstrain tensors,tensors the effects of which can becombinedb y sim ple addit ion (theyaresuperposable). Chapter6 providesthe methods necessaryto calculateth e effectsoff ini te s t ra ins , whichare not superposable . Chapte r 7 contains var ious

applicationsof con t inuum mechan ic sto geologyand glaciology. Chapter8 ,finally,in t roducesthe conceptof a strain historyand explainsa method offactoringa finitestrain intofiniteincrements.The most important partof thesechapters is the problems, which I encourage the reader to solveinitiallywithaslittlehelp as possiblefrom the Answers that followafter the end ofChapter8.

Theseanswersare mean tprincipal lyto allowthe reader to check whetheraprob lem has been solved correctly . If, however, a prob lem is fou nd to be diff icul tafter a serious attem pt at solution, the stu dent should allow no m orthan a glanceat theanswer, enoughto bepointedin theright direction,and thenre turn to independent work.Too much relianceon theprepared solution willdefeat the purpose of this book.Thus ,look at any of the answers onlyafterfinishing a solution or at leastafter having triedseveralapproaches to one.

A list of sym bols precedes C hapter 1. The Su m m ary ofFormulaeat the end ofthe book is m eant to spare the reader m em orization and to be consu ltedfreely. Itis a co l lect ion of im po rtan t iden t i t ies and work ingfo rmulaso c c u r r i n gthrou gh ou t the b ook. A ppen dix A contains a tab le of pairin gs of elasparametersfor anisotropicmaterial,and AppendixB listsunitsof stress.


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CONTENTS

Symbolsand Notation x

Chapter1. Vectors (Problems1 to 20 ) 3

Chapter2 . Fields (Problems21 to 25 ) 10

Chapter3. MatterTensorsand Coordinate Transformations

(Problems26 to 75) 13

Chapter4. Stress (Problems76 to 97) 44

Chapter5. Infinitesimal Strain (Problems98 to 115) 56

Chapter6. Finite Strain (Problems116 to124) 72

Chapter7. Effectsof Stress (Problems125 to135) 85

Chapter8. StrainHistoryand Polar D ecom position (Problem 136) 94

Answers 97S u m m a r yofFormulae 2 70

AppendixA . Elastic Param etersfor Isotropic Bodies 2 86

AppendixB . Unitsof Stress 287

References 2 88

Index 2 89


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S Y M B O L SA N DNOTATION

0 nu l l vec to ra i co m ponen t , o r ig ina l pos i tion

vector

A ij c o f a c t o rA ij e le m e n t, t r a n s f o r m a t i o n

m a t r i x

a ij d i rec t ion cos ineA, B vectors

A, B m a t r i c e sa i j d i rec t ion angle

c dis tance , or ig in to M ohrc i rc lec e n t e r

Sij componen t , Lagrang ian

Cauchy's tensorC i j c o m p o n e n t , E u l e r ia n

Cauchy's tensor

c ij e l e m e n t ,stiffness m a t r i xC i j k l c o m p o n e n t ,stiffnesstensorC i i m p l i c i tspa t i a l de r iv a t ive o f

C

D dip

Dij. component, i n f in i te s i m a ld i s t o r t i o n

Dij com ponen t , Green ' sd is tor t ion tensor

Di j component, A l m a n s i ' s

d i s to r t iont ensordS i nc rem en t o f f ina l l eng thds i n c r e m e n tof in i t i a l l eng thdV i n c r e m e n tof final v o l u m edv i n c r e m e n tof in i t i a l v o l u m e

A d i l a t a t i o n

A l i nea r d i l a t a t ionA ij com ponent , devia torof

stress

< 5 K ronecker del ta ( e lem en t ,

i d e n t i ty m a t ri x )

( com ponent , Green ' s tensor

e . . component, Almansi's

tensor

i ? . . c o m p o n e n t , d i sp l ac e m e n tg r a d i e n t

. . c o m p o n e n t , i n f i n i te s i m a l

s t r a i n

e-- f , e lem en t , a l t e rna t in gm a t r i x

: com ponen t , s t r a in r a te

F a s y m m e t ri c tr a n s fo r -

m a t i o nm a t r i xF i c o m p o n e n t ,body force

/() f u n c t i o n< P angle of ro ta t ion

< p ( . . . ) cons t r a in tf u n c t i o n

G i c o m p o n e n t ,body to rqueg j com ponen t , acc e lera tion

due to gravi tyY eng inee r ing shears t ra in

Y engineer ing shear s t ra in

rateH Hubble cons tant

H z her tz (u ni t of f requenc y)i, j , k unit vectors paral le lto

coordinate axes


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S Y M B O L SA N DN O T AT I O N

1 I ,2 I ,3 I invariants, second-rank

tensor

J joule (unit of energy)K kelvin (unit of temperature)

K thermal conductivity

K component, dielectric

p e r m i t ti v i t ytensorKi component, thermal

conductivity tensor

/. component, unit direction-vec to r

A elongation

A Lagrange multiplier

A Lame's constant

A principal tensor magnitude,

g e n e r a l

'A particular principale l o n g a t i o n

m mass

m meter (unit of length)

[ i coefficient of "internal

f r i c t i o n "p. coefficient of viscosity

/ J Lame's constant, shearmodulus

N magnitude of normal

component

N J component, normal force

(traction)

N newton (unit of force)

1 angular velocity< U - component, rotation vector

P confining pressure

P plunge

f pressure

Pa pascal ( unit of stress)

G J . . component, infinitesimal

rotation tensor

G S j j component, rotation ratetensor

q t component, heat flow

R rotation matrix, polar

decomposition

/?,. generator of rotations

R,a, b axis-subscripts, modulo 3

R, R first and second timederivat ivesof R

r radius of Mohr circle

p density

' p particular principal

a n g u l a rdensity of axes' p particular principal

angular density of polesS strike

S a g component, two-

dimensional tensor

Si component, stretch tensor

S ij. component, symmetric

tensor

sf- element, compliancematrix

Sj- f r i component, compliance

tensor

s second (unit of time)

a magnitude of normal

traction

c r. . component, stress tensorO~j,cr2,c r3 principal stresses

T trend

T magnitude of tangential

component

T temperature


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x i i STRESSAND DEFORMATION

Ti com ponent , second-ranktensor

T j c om pone nt , shear ( tangential)force ( t rac t ion)

\ i componen t , t empera tu re( t h e r m a l )grad ien t

T m a g n i t u d eof shear( tangential) t ract ion

T octahedral shear

(tangential)stressT(. . co m ponent , shear ( tangential)

stressQ angleof shear

6 angleof coordinate rotat ionU stretch tensor,left-polar

d e c o m p o s i t i o n

u i disp lacement componen tV v o l u m e

V stretch tensor, r ight-polard e c o m p o s i t i o n

V ; vector component

vv v2 , v^ cartesian vectorcomponen t s

X1, X2,X 3 car tes ian c oordinatesXR,XaXb rotation axis

fo l lowingaxes

x i general cartesian coordi-nate,or component ,

f ina lposi t ion vector$?,}, 2Ecoordinatesin Mohr space( . . . ) ordered arrayof m a t r i x

e l e m e n t s

(ttjj) ro ta t ion m at r ix(M{) m a t r i x

[ . . . ] ordered arrayof tensor

componen t s[A ] ant isymmetr ic second-rank

tensor[ S j j ] symmetr ic second-rank

tensor[ 7). ] second-rank tensor[ UVW] crys ta l lographic d i rec t ion

[ v i] vector[Mij] de term inant (of a m atr ix

or tensor)I SI absolute va lue(of ascalar)X vector cross-

m u l t i p l i c a t i o n vec to r do t -m u l tip l i ca t ion

V del (ornabla)


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S T R E S SA N DD E F O R M AT I O N


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1VECTORS

The reader, evenif fami l ia rwith vectors, willfind ituseful to work th roughthis chapter because it introduces notation that will be used throughout thisbook.W e will take vec torsto beentities that possess magnitude, orientation,and senseinthree-dim ensional space. Grap hic ally, we will represent them as arrows wi th sense from tailto head, magni tude propor t ionalto the length,and orientat ionindicated b y the ang les they form w ith a given set of referenc e direc tions . T

different kind s of sym b ol wi l l b e used to des ignate vec tors a lgeb raic al lyboldface letters(and the boldfac e nu m b er zero for a vec tor of zero m ag nitu de)and subscr ipted le t tersto be in t roduced la ter.The firstproblems deal wi thsim ple vector geom etry and i ts algebraic representat ion. M u lt iply ing a vectoa scalaraffectsonly its magn i tude( length)without changingi ts direct ion.

Problem1. State the necessaryand sufficientcondit ionsfor thethree vectorsA ,B ,and C to forma t r iangle. (Problems1-9, 12-14, 19- 23, and 25 fromSokolnikoff& Redheffer, 1958.)

Problem2 . Giventhe sum S = A + B and thedifferenceD = AB, f ind A andB intermsof S and D (a)graphicallyand (b)algebraical ly.

Prob lem 3. (a) State the un it vector a with the sam e direction as a nonzero v eA . (b) Let twononzero vectorsA and Bissue fromth e same point , forming

an angle between them; usingthe resultof (a), find avector that bisects thisangle.

Problem4. Using vector methods, showthat a l ine fromone of theverticesof aparallelogramto themidpo in tof one of thenonadjacent sides trisectsone ofthe diagonals.

Two vectorsare said to form with each othertwo dis tinct produ cts :a scalar,

the dotproduct, and avector,the crossproduct. The dotproductD of thevectorsAand B,form ing with each otheran angle 6, is calculatedas:

D = A . B s ABcos8= E.A = BAcos6 . (1.1)

Hencethe dotproduc tis commutativeand is A t imesthe projectionof B on A or

3


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4 S T R E S SANDD E F O R M A T I O N

B t imesthe pro jec t ionof A on B . Twovectorsare perpendicu la rif and only iftheirdot p roduc tis zero. The cross produ ctC of A and B is:

where u is the u nit vector perp endic u lar to the A B plane, posi t ive in the scrr u l e d i r ec t i onfor the angle 6 f rom A to B . Thecross p roduc tis notc o m m u t a t i v e ,and twovectorsare parallelor antiparal lel(parallelwith opposi te

sense)if andonly if their cross produc tis a nu l lvector.

Problem5. (a)Which co ndition m akes the otherwise arbitrary vec tors A , B , andC coplanar?(b ) Givean example.

Problem6. Canboth equat ionsA X B = 0 a nd A B = 0 b et rue without ei therA or B beinga nu ll vec tor?

Problem7. Findan exampleof three unequal vectors such thatthe cross productof any two ofthem fo rmsa r ight ang le withthe third.

L et three m u tu al ly or thogonal u ni t vectorsi, j, and khave their tailsat theoriginof a r ight-handed cartesian coordinate system.This allowsus to defineasthe position vector for everypoint in thecoordinate systema vectorthat is the sumof the produ cts of the three u nit vectors, each m u lt ipl ied by an a ppro priate scaNote that all position vectors havethe i r tailsat thecoordinate or ig in .

Prob lem 8. Let the ve rt ices of a t r iang le b e defin ed b y the posit ion vec toi + j + k, 2j + k, and 2 i + j . F orm this triangle b y vectors, head against tail , verifythat th e vectorsum is thenu ll vector0 .

With the help of the axis-paralleluni t vec torsthe cross productcan be

redef ined:

where the ver t ica l bars enclos ing the three-by- three array des ignate thdeterminant of that array. Explici t ly, that determ inantis:


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V E C T O R S 5

Angular brackets{ ) arechosen in eq. (1.4) becauseth e square bracketsare reservedfor arrays designating vectors (andother tenso rs), as in eq. (1.6) below. If we letthe traditional cartesian coordinate systemx, y, and z coincide withthat def inedby the three unit vectorsof Problem 8, concepts of analyticaland vector geometrycan be combined.

Problem 9. Which vector is perpendicularto the plane ax+ by+ cz+ d= 0?

Problem 10. T h r e e plane mirrorsare silveredon both sides and arranged so thateach intersects the others at r ight angles, forming eight concave corners.Show, by vector methods, thata beam of l ight enteringth e system from anydirection will emerge travelingin exactly the opposite direction. (This isthe principleof radar targetsand of many highwayreflectors.)

Problem 11. Find A B, A X B, and B X A if A = i + 2j + 3k andB = 5i + 7j + l l k .

Problem 12. (a) Show thatthe cross productof any two of thefollowing vectorsis paral lel to the th i rd: i + j + k, i - k, and i 2j + k. (b) What are theimplications for the three vectors?

V ector a lgebracan be s impl i f ied by in t roducing the subscr ip t no ta t ion .Replace the let ters x, y, and z for cartesian coordinatesby the subscripted variablesx\, X 2 > and x$. T o characterizea vector, first decompose it in to componen t sparallel to the coordinates:

and then give th e three scalarfactors a, b,and c the new designations v\, v^, andv$. Let further a letter subscript,such as i, j, etc., stand collectivelyfor the integernumber subscr ipts1, 2, or 3. Then th e vector of eq. (1.5) can be characterizedbyth e ordered array of scalars [ v j, v2, v-$}. T he vector natureof the array as awhole is indicated by the brackets. T o simplify the writ ing, the commas can beeliminated, and the vector in subscript notation is:


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6 S T R E S SA N DD E F O R M A T I O N

Because the brackets indicate a complete array, brackets around one term of anequa t ion require brackets around every other termof the same equa t ion . It isun de rstoo d that a let ter sub script , if repeated in the same term of an exp ression,impl ies summat ion overthe numer i c a l r angeof the subsc r i p tby a conven t ionin t roduced by and namedafter A lbert Ein stein . Becau se the let ter repe ated in anyone term of an equat ion can be replaced by another let ter, i t iscalled a dumm')subscript. A different le t ter subscr ip tis used for every separate term thatis to bes u m m e d .A n example fo l lowsof wha t the Eins te in sum m at ion conven t ion thusi m p l i e s :

N o t e tha t th is convent ion doesnot hold fo r numer ic subscr ip tand thus doesnotaffect the r ight-hand side of eq. (1.7) .

In subscript notat ionthe sum S of the twovectorsA and B can bewrit tenas:

or more commonly byspecifyingan arbi t rary com po nen t of the sum vector as :

In this usage,th e subscr ip t z is calleda free subscript, and every termof an equat ionm u s t necessarily possessthe same free subscr ip t . In subscr ip t no ta t ion ,the scalardot product D of the two vectors of eq.(1.8) is w rit ten as:

where /' is a dummy subscr ip tand should be in te rp re tedas in eq.(1.7) .

P rob l em 13. (a) Using subsc r ip t no t a t i on , de t e rm ine whe the rthe vec to r s1 v = i + j + k,2 v = i k, and 3 v= i 2j + k arem utua l l y o r t hogona l.N o t e :T h r o u g h o u t th is b o o k , lef t supersc r ip t s a re t aken to se rve on ly fo rident i f ica t ionand not tohave any fur ther func t ion , (b) Select scalarsx, y,andz so as to m a k e th e vec tors 1 w = i + j + 2 k , 2 w = j + ^)i] a nd3 w = 2i + j y j+ zk mutua l l y o r t hogona l .

Prob lem 14. (a) Let A be anonzero vector ; doesthe equat ion A B = A Censure that B = C? If not , p rov idean e x a m p l e to the con t r a ry, (b) IfA B = A C forevery vec torA ,mus t C equa l B ? (c) Usesubscript notat ionto prove your answer.


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VECTORS 7

T o take full advan tageof the subsc r ip t no ta t ion ,w e m u s t i n t r o d u c ethealternating matrix e :k . Each of itsthree subscr ip tsis an integer modulo3 . Bythat we unders tandthat th e seriesof integern u m b e r srestarts ov erand overat 1everyt i m eafter the n u m b e r3 hasbeen reached, thu s2 > 1 and 3 > 2, bu t 1 > 3.Setsof three integers modulo3 have cyclicpermutations, calculatedb y addingto themthe integers1 and 2,thus the cyc li c pe rm u ta t ionsof theascending series1, 2, 3 are2 , 3, 1 and 3,1,2; thoseof thedescending series3, 2 , 1 are 1, 3, 2 and 2 ,1 ,3 .These two sets of series containall possible complete setsof all three posi t iveintegers1, 2 , and 3. The 2 7elementsof thea l t e rna t ing m at rixare d e t e r m i n e db ythe fo l lowing ru les :

e; :k = 1 if /',j, k = 1, 2 , 3 orcycl ic permutat ions,et ,k = 1 if /',j, k 3, 2, 1 orcycl ic permutat ions,e i j k ~ 0 if ne i the rof these. (1 .11)

Prob lem15 . F i n d the componen t sof thevector in1 + jn2 + kn3 that is n o r m a lto theplane con ta in ingthe twounit vectors fromthe or ig in ,iv1 + jv2 + kz3

and iw

1+ ]w

2+ kw

3.

F i n dan expressionfor thestrikeand dip of theplaneif i points north,j east,and k down.

Problem16 . Suppose thata forceF actson thecenterof onefaceof ac u b eof edge2 a, and forces Fon thecenterof theopposi teface,P on that of a third, andQ on tha t of its oppos i t eface, (a) M a k e a diagramand (b)expla in therelat ionshipsbetween P, Q, and Fsuch tha tthe cube wi l lbe in e q u i l i b r i u m(i.e.,so as toavoidanyaccelerat ion) .

To specifythe direc t ionof a vector with respectto a coordinate system,it issuff ic ientand most convenientto find thecom ponents , calledthe direction cosinesof theu n i t , or direc t ion , vec tor poin t ingin this direction; thesec o m p o n e n t sarethe direc t ion vector ' s pro jec t ionsonto the three coordinate axesand hence thecosinesof its direction angles.This is an al ternat iveto the geological conventionof statingthe a t t i tudeof a l ine in t e r m sof itst rend, the a z i m u t hof itshorizontalprojec t ion ,and itsp l u n g e ,the angle betweenthe t rend and thedownward branchof the l ine. We m a ywant to extend the conceptof plungeto vectorsif they arereferred to geological coordinates .The p l u n g eof a vector witha downwardcomponen t( X3, posi t ive) wil lb e regardedas posi t iveand that of avector withanupward component (x3 negative) as negative.

Problem17. In thegeological coordinatesystem ( xi ) = ( north, east,down) ,twovectorsare givenb y [ui] = [ 3 4 1 2 ] and [v;] = [ 5 12 84 ] .Find (a) the


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trend T andplunge P of ui and itsm a g n i t u d eand (b) theangle includedbetween ui and v;. Angles needb e stated onlyto the nearest degree,(c ) Dete rminethe direction cosinesof [ui] and [vi\. (d ) What is theangle< pbetweenui and x1?

A n alternativeway ofcalculatingthe trend and plunge of avector stated ingeological coordinates usesthe correspondenceof the trend T and theangle(90 - P) to thespherical coordinates 6 and (patconstantp = 1 inwhich 9 and q>are relatedto ( xi ) = ( north,east,down) accordingto thescheme:

Cartesian direction cosines li, have the spherical counterparts:

wherethe secondexpressionfor 6 servesto breakth e s ign ambigui tyof the first,

and thethird is usefulfor greater accuracyat smal l(p (s teep plunge) .The trendin these coordinatesis T = 9 and theplungeP = (90 < p). Conversely, given9and (p 90 P of aline, its direc tion c osines are:

Vectorscan bedifferentiatedwith respect to tim e; their tim e deriva tives arealso vectors.

Problem18 . Let thepositionof apart icleat t i m et b e definedby thevectorfromthe origin:R = ia cos(2nt/T) + ja sin (2nt/T). (a) Interpretthe equationand find the physical signific anc e of the c onstantsa and T . (b) Find the


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V E C T O R S 9

velocityV and (c) theaccelerationA of thepart icle,and thei r magni tudesand direct ionsas func t ionsof t ime .

Problem1 9 . Show that Rx R =O if R is atwice di fferent iab le fu nc t iono ft i m e t as follows:

R = A+ f(t) B,

whereA and B areconstant.

Prob lem 20 .When a particle of constant m assm is accelerated by the force F and

has an instantaneous velocityV, show thatits kinetic energy increasesat therate of:

Hint: Newton's law is: F = d(mV}/dt.


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2

FIELDSA scalar determ ined at every poi nt in a given dom ain, analyt ical ly or otherwconst i tutes ascalar field. Vectors s im i lar ly determ ined co nst i tu te avector field.The def in inganaly t ica l express ionsof a th ree -d imens iona lfield arec o m m o n l ydifferent iablewith respect to space; hence in a cartesian coordinate system they arame na b l eto par t ia ldifferent iat ionwith respectto x\, X2, and x3. In this contextit is useful to define severaldifferential operators.The op erator V is ca lled the

"del" or the "nabla" and is defined asfollows:

It can beseen thatthe del is avector. B yconvention, however,it is notrenderedin

boldface . B efore we def ine add i t io na ldifferent ia loperators , we extend thesu b sc r i p t no t a t i on fu r t he r and l e t a sub sc r i b ed com m a ind i c a t e pa r t idi fferent ia t ion .A c o m m a p r e c e d i n ga le t ter subscript ,say i, is taken to i m p l ydi fferent ia t ionwith respectto xf. Thus, if (p(xt) is a scalar func t ionof posi t ionand thus defines a scalar f ield, i tsgradient, ano ther different ialopera tor, i sdefinedby theequat ion:

Thus the g rad i en t of a scalar is a vec to r. A vec tor field wi th su i t ab lydifferent iablef u n c t i o n sof pos i t ionhas adivergencedefinedb y:

which is short for:

B ecau se the div ergenc e is the dot prod u c t of two vectors it is a scalar.With the help of the a l t e rna t ing m at r ix ,w e def ine anotherdifferent ia l

operator,the curl of a vectorfield:

10


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F I E L D S 11

Note that in this equationi is a free subscript , occurring once and onlyonceineachof theterms;j and k, on theotherhand,bothoccur twicein thesame termandthus aredummy subscr ip t s imply ing summat ion .To understandthe impl icat ionsof eq. (2.6), the reader should write it outexplicitly,term by term, to find that itis short for:

Beinga crossproduct,the cu rl is avector.

Problem2 1. If r is aposit ion vectorof the form r = i xl + j x2 + k x y

dem onstrate that:

Problem2 2 . Demonstra te thatthe followingequations holdif a is aconstantvectorand r avariableposition vector:(a) V ( a r ) = a , ( b ) V x ( a X r ) =2 a,(c ) V ( aX r) = 0. D rawgeneral conclusionsfrom yourfindings.

If vectoror scalarfieldssatisfyc ertain conditions, theyare said to possessoneor m ore of the follow ing charac teristics. Vec tor fields aresolenoidal if:

whereV is thevectorfield and A is avector function.Irrotational vector fieldssatisfy

either:VxV = 0 , (2.10)

or:V = Vp . (2.11)

where (p is a scalarfunc t ion .A well-behavedvectoror scalar field in as implyconnected region(a doughnut-shaped regionis doubly connected)satisfiesany oneof the followingfour condi t ions:

or:


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12 S T R E S SAND D E F O R M A T I O N

whereV is avectorfield, A avector fu nction,and (p a scalarfield.

Problem23. A rigidbodyrotates witha constant angular velocityil abou tanaxis in itsinterior that passes througha poin t O with the position vector0 ;choosethis po in t as the originof a local coordinate system .Let thelocalposition vector, relativeto O, of anarbitrary poin tR in therotat ing bodyb er; then the velocityv of that point relativeto theexternalreference frameis:

wh ere v is the veloc ity ofO in the externalf rame, (a ) Show thatat 7?cu r lv = 2fl (angu lar velocity eq u als one-halfthe cu r lof thevelocityfield),(b ) Demonstrate thatv issolenoidal. Hints: The velocityv is independentof the local coordinatesin which r is given. Use one of theresultsofProblem22. (c )Characterizeth e velocityfield.

Problem24. The"red shift"of the spectraof distant galaxiesis interpretedasindicating that galaxies generally move away fromth e earth witha velocityvpropor t ionalto their distancer, thus thatv = //r, where H is theso-called

H u b b l econstant ,(a) What is the divergence of the velocity field v ? (b)Showthat observationsfrom another galaxy wou ld indicateth e same constantHandthusth e same expansionof theuniverse.

Problem2 5. If theposition vectorr = ix^+ ] x2 + k x y (a)demonstrate thatth evectorfield v =r" r is irrotational.(b) Is itsolenoidal?

Fieldscan befo rmedb y tensorsof thesecondand higher ranks,and wewill

encountera few ofthem in later chapters.


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3

MATTERT E N S O R SAND C O O R D I N AT ET R A N S F O R M A T I O N SVectors,the sub jec tof theprevioustw o chapters,m ay b eclassifiedas m e m b e r sof aclassof m athem atical entit ies cal ledtensors, insofaras theycan beexpressedin theform of ordered arrays, or m atrices, and insofar as they fu rther confo rm condit ionsto beexploredin thepresent chapter.Tensors can have variousranks,and vectorsare tensorsof the firstrank,which in three-dimensional space have31

or three components .Much of this, and later, chapters dealswith tensorsof thesecond rank whichin thesam e space have3 or nine components .Tensors ofhigher (nth) rankdo existand have 3" components ,and so do, atleast nom inally,tensors of zero rank w ith a single, or 3 , co m ponen t, w hich m akes themscalars.

Tensors of the second ran k for three dim ensions are written as three-by -thrematr ices wi th each component markedby twosub scr ipts , whichm a y b eeitherlettersor n u m b e r s .The firstsubscript indicatesthe row, the secondthe c o l u m n ,

of the com ponen t asfollows:

where the brackets indicate thatTij is a tensorand notsomeother matr ix . Amatr ixalwaysis a tensorif it relatestwo vectorsto each otherb y amatr ix product ,or tensor equation,like the following:

or shorter:

or even shorter(aswith vectors,a f u l l tensorcan beindicatedby itssubscriptedgeneralizedco m ponent ; i .e ., the brackets indicat ing the com plete tensor m ay omit ted) :

13


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14 S T R E S S A N D D E F O R M AT I O N

where in the previou s two equ ations the Einstein su m m ation conv ention im ps u m m a t i o n o v e rthe d u m m ys u b s c r i p tj. A m o r e precise c ond i t ion thatdisc r imina testensorsfrom other matrices willb e introduced later.A m o n gotheruses, tensorsof the second rank serveto descr ibe proper t iesof ma te r i a l s ,geological and other, thatdiffer from one direc t ion to anoth er and are cal ledanisotropic (somet imes a lsoheterotropic), in cont rad i s t inc t ionto isotropicmater ia ls ,in whichpropert ies are the same in al l direct ions.Tensorsof thiskind are calledmatter tensors (Nye, 1964).

Prob lem26 . Therelat ion betweenthe f low ofheat q i and thet empe ra tu r e

gradient T : in a homogeneous mediumis givenby the tensor equation:

in whichthe second-rank tensor,Kf ., is the thermal conduct iv i ty tensor.Inthe coordinate system( x i ) = ( north eastdown) the conduct iv i tyof acertainbodyof rock is given by:

(a ) Is the body therm al ly i sotropicor anisotropic?(b ) Qualitatively, whatisthe pr in c ip al therm al charac ter is t ic of th is body ofrock? This range ofvariat ionwith di rec t ionis typicalof m a n yrocks,(c ) Suppose,as anexample ,that the tem peratu re gradientis givenb y:

In which di rec t ion doesthe t em pera tu re decreaseat the m a x i m u mrate?(d) Find theflow of heat and i ts direct ion in terms of t rend and plunge,

(c )Suppose thatthe t em pe ra tu r e g r ad ien tis given insteadb y:

In which d irec t ion ( trend and plu ng e) does the tem per atu re decrease at thm a xi m u m ra tein this case? ( f ) F i n dthe flow ofheat and itsdirection cosines


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MATTERT E N S O R S 15

and trend and plunge; comparethe directionsof heat flow andt empera tu regradient , (g ) W h a t error wouldb e c o m m i t t e dif the geothermalheat flowwere basedon t em pe ra tu r e m easu rem ent sin adeep vertical welland if itwerecalculatedu p o n the assu m pt ion that the tem peratu re gradient was ver t ica l lyupwardand if theconduc t iv i tyof the rock were takento be isotropicat theaverage of 50 Jrrf1 s l 1C1?

Problem2 7. The twovectors p i and q . are related accordingto the tensorequa t ion :

in which the tensorS . is given by :

Supposethat [ < ? , ]= [ 1 /3 ~7 , - T b

] (a) If (x ;) = ( north eastd o w n ) ,whatare the m a g n i t u d eand direction (trendand plunge)of q . (b )F i n dp t, giventhis qf. (c )What i s the m agn i tu deof p?. (d) Find the angle included between/>, .and qf. Now replace[q t] by [q\] = [ 3 3^3" 6/3~ ] (e)W h a tare them a g n i t u d eand direct ion ( trendand p lunge )of q -? ( f )Whatis themagn i tudeof theresu l t ing//? (g )Findthe angleincluded between//and q 't.

Second-rank tensors have a dist inctmagnitudefor each direct ion.W h e r eP i S j - q - ,the scal ar m agn i tud eS of [St] is the project ionof p onto a l ineparallel to q dividedby thevec to r m agn i tu deq , S= (p qj)/q, or subs t i tu t ingS^ q . for pf:

Because[ qj q\ is a unit vector,the conceptof m a g n i t u d ecan begeneral izedandm ade independentof theexistenceof avector [q^\. Replace[ qj q\ with the un i tvector / . for thedirect ionin whichthe magn i tudeof thetensoris sought :

Beinga m atrix, a sec ond-ran k tensor has aninversewhich can be calculated byCramer's rule. The invetse Tt ~ of thetensor T . is:


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1 6 S T R E S SA N D D E F O R M AT I O N

where D is the determinant of the tensor:

and A (. . [note th e t ransposi t ionof the subscriptson the right-hand sideof eq. 3.7]are the cofactorr.

I n other terms, an individual cofactor A . . is the determinant tha t remainsafterdelet ing the z'th row and the^ ' th c o l u m n of Tf ., m u l t i p l i e d by a sign factor of( 1)' +J , as in the pa rticu lar caseo f :

Problem 28. F indq,, given that p.= Ti q., that [ / >;] = [ 1 6 20 24 ], and that:

(a) Solve for q., t r ea t ing the tensor equat ionas a system of l inear equat ions .(b) Solve by matrix inversion, using Cramer 's rule.

Up to now, w e have considered vectors,and in the last few problems second-rank tensors w ith referen ce to a single cartes ian coo rdina te system. It is, however,com mo nly necessary torefer a tensor, of any rank, tomore than a single set ofcoordinates . I f we omit t r ans la t io ns , tha t i s , i f we le t two or mo re suchcoordinate systems share their origin, then one cartesian coordinate system isrelated to another by the set ofangles that eachof the three coordinate axesof thefirst system makes with each of those of the second. F igu re 3.1 shows the angle sthat a unit vector along one of the axes of the second system forms with the threeaxes of the first and its project ionsonto these axes.

T he complete set of these angles can be thou ght to form athree-by-threearrayof direction angles,which are conventionally orderedso that the angles formed bythe same axis of theo ld system are arranged in the three columns of the array, eachcolumn in turn being ordered so that the angles formed with each of the axes of


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Figu re3.1. Projectionsof the unit vectori1 alongthe

axis xr ' jof a second coordinate system,x f onto the axesxfof the first. Anglesa ! are the direction anglesof i'.

the new systemform the rowsof the array. The arrayof direction angles(Xi is,accord ing ly :

To arriveat the"new" coordinatesin termsof the"old,"however,we need th eproject ionsof the oldonto the newaxes;henceth e cosinesof theanglesin eq.(3.11). Thearray, orderedas in eq.(3.11),consistsof the three setsof direct ion

cosinesof the newcoordinate axes,referredto the oldsystem, fo rm ingthe threerows of the rotation matrix. The t h r ee co lumnsso f o r m e dare also setsofdirection cosines, thoseof the oldc oord inate axesreferredto the newsystem. Therotationmatrix correspondingto thearrayof direction angleseq. (3.11) is:


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Because eachrow ande a c h c o l u m nof t hi s m a t r ixis a uni t vec tor,the fo l lowingholds:

a p r o p e r t y t h a t fol lows also fromthe s ix i n d e p e n d e n t e q u a t i o n sof theorthogonality conditions that holdfor thero ta t ion matr ix :

where < S; , the Kroneckerdelta, has the fo l lowing de f in i t ion :

so that , forexample :

Note t h a t m u l t i p l y i n gthe K ronec ker de l ta m atr ix wi th another m atr ix resul t sint ha t i d e n t i c a l m a t r i x ,for which reasonit is also calledthe identity matrix.Becauseof the i m p o s i t i o nof the sixc o n d i t i o n sof eq. ( 3 . 1 4 )on the ro ta t ionm a t r i x (a.--), i t is h ig hly red u nd an t , and o nly three of i ts n in e e lem ents a reindependent .The condi t ionson a i also havethe consequence:

where the negative signaffects on ly t r ans f o rm at ion s tha t changethe coord ina tesf rom r igh t -to l e f t -handedness ,or vice versa. A set ofi n t e r r e l a t ionsof theelem ents of the rotat ion m atr ix is :

or e x p l i c i t l y :


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M AT T E RTENSORS 19

where the negative s igns apply onlyto t ransformat ions involvinga changeofhandedness,and fur thermore:

and:

Becauseof the grea t numberof poss ib le cycl ic permuta t ions ,it is preferabletostate this ruleb y ident i fy ingthe four free subscr ip tsi, j, f, and q by theorder inwhich they occuron theleft-hand sideof eq.(3.21) accordingto thescheme:

The orthogonali tyof cartesian coordinateshas the consequence:

Problem 2 9. Why can thee lementsa t of a t r a n s f o r m a t i o n m a t r i xfor the

rotat ionof coordinate axes neverb e greater than+1 orless than 1?

Prob lem 30 . (a) F ind the m atr ix (a .t ) for the tr an sfo rm ati on of a set of old axes( x ; ) = ( north eastd o w n )to new"(*; . ) =( 1 2 0 ,0 30,-60 210,-30).(Di rec t ionsare stated as t rends and plunges , negat ivei f u p w a r d , w i t haz imuthsin degrees counting from northto east,sou th ,and west.) Hint: U sethe m e t h o dof Problem17. (b)Evaluatethe d e t e r m i n a n t\ a i \. (c) A re the( x - ) axes right-or lef t -handed?(d) F i n d the m a t r i x(a'j .) to t ransform f rom

new axes ( x }) to newer ( x\ ) = (300, 0 210, 60 30, 30); thisis calledan inversionof axes, (e)Evaluatethe de te rminan t| a '^ | . (f) A re the( x " ) axesright- orlef t -handed? (g) Find the matr ix(a.'\ ) to t ransform from the old(x ; ) to thenewer axes(*".). (h) Evaluatethe de te rminan t\a" i \. (i) Find thet r a n s f o r m a t i o nm a t r i x( a i; ) for at ransformat ion f romnew axes (x^) to theo ld axes ( x1, - ) . ( j ) F i n d the d e t e r m i n a n tof ( # , - ) . (k ) F i n d the m a t r i x( # , ) to t r ans fo rm f rom the new axes(x \ ) to ref lec ted axes( * , - ) = (120,0 210 ,60 210,-30). (1 )What is themirror p lanefor


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2 0 STRESSA ND DEFORMATION

this ref lect ion? (m) Evaluate the determinant | a ,_ ,\. (n) Are the axes ( * , - )right- or left-handed?

The transformation ( a ] : ) of Problem 30 is an inversion, ( a f ) a reflection;these are typical symmetry operations.

Problem 31. Find the element a 2^ of the transformation matrix from an old

coordinate system with axes ( xf ) = (north east down) to new axes(*;.) = (225, 0 315,-30 315, 60).

Problem 32. Suppose that:

and that the old axes were ( x - )= ( north east down ). (a) Find the directionsof the new axes (x^ ) obtained f rom the old by means of the transformationmatrix ( < * , - . ) .(b) Are the newaxes orthogonal?

Problem 33. Suppose ( < 2( . . ) is the transformation matrix for the rotation of a set

of right-handed orthogonal coordinates. Express a , < z3 2, a n d a in terms

O fa u ,a l2 ,a l3 ,a 2l ,a 22 ,anda 23 .

The principal application of the rotation matrix is to transform tensors

referredto one cartesian coordinate system into the form appropriate for another.Note that the physical entity described by the tensor obviously cannot be affectedby the choice of a di ffe ren treference system. The rule for the transformationfrom "old" to "new" coordinates of a tensor of the first rank, that is, a polar

vector, is:

and that for the transformation of a second-rank tensor is:

or e x p l i c i t l y :


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MATTERT E N S O R S 2 1

Problem34. Transformthe tensor:

to thenewcoordinates definedby thet ransform at ion m atr ix :

The new coordinate axesare calledthe principal axes of the tensor [5).], theyform its principal coordinates. Verify that each of the three term s S'fi(no s u m o nz ) satisfiesthe cubic equat ion:

The secular equationof Problem34 for anysymmet r ic t ensorof the secondrank can besolved to find theprincipal values (also cal ledeigenvalues) of thattensor.

Problem 35. To show fo rm al ly tha ta diagonal tensor with equal diagonal

components ,such as:

is invariantto the t r ans format ionof axes ( a t ,) ,use theK roneck er deltain anappropriate equationfor subscripted variables.


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Problem36. Let [T . ] b e atensor. U sing su b script notation, show thatTif isinvariant ,whichis to saythat the Tfi are thesamefor allor ienta t ionsof thecoordinate axes.

Problem37. Using subscript notation, show thatthe dotproductof twovectors,say p; and q^, is invariantto the t ransformat ionof axes (af } .

Problem38. U sing sub script notat ion, show thatthe squareof themagn i tudeof avectorp., and henceth e magni tude ,is invariantto thet r ans format ionof axes


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subjectto theconstraint :

the const ra int implying thatthe x( in these tw o equationsare not independent,mu l t i p lythe const ra int equat ionby thearbi t rary Lagrange mul t ip l ierA . Thensolve:

for A together with:

W e furtherhaveto defineth e representation quadricof asymmetr ic second-ranktensor. Sucha tensorm a y b erepresentedby thequadric centeredon thecoordinateo r i g i n :

with semiaxes5(- I . B ychoosingan appropriate sign,one canavoid imaginarybranchesof thequadr ic .Let frf ] be theradius vectorto x ;; then the magn i tudeofthe tensoris:

Spat ia l di ffe ren t i a t ionallows us to f ind thetangent plane t h roughanarbitrarypointon asurfaceand anoutward normalvector through that point .Let

th e equationof thesurfaceb e:

Then at thepoint x i the equat ionof thetangent planeis:

and thenormalw; to thetangent planeis:

Note that the expression:

in eqs. (3.34)and (3.35) doesnot designatea mul t ip l i ca t i on ,and hence doesnoti m p l y s u m m a t i o n .


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24 S T R E S S A N DD E F O R M A T I O N

A m o n gother usefulapplicat ions,the representat ionqu adr ic of asecond-ranktensor helpswith the visualizationof therelative orientationsof twovectorsthatare interrelatedb y a tensor equation.The relat ionshipis called the radius-normalproperty of therepresentat ion quadric.If />(.= 5. q , then thedirectionofp for agivenq isfoundb y drawinga radius vectorof therepresentat ion quadricfor S i parallelto q (Fig. 3.2),then drawingthe tangentto thequadr icsurfacewhereit touchesthe radiu s vector.The outward norm alto that tangent planethenpara l le ls p . F i g u r e3 .2 i sdrawn for an el l ipsoidal quadric,but the s amerule-ho lds i f the qu adric is a parab oloid or hy perb oloid.

In the pr incipal d i rec t ionsof a symmetr ic tensor,the vectorsp and q are

parallel . This fact can beused to determine pr incipal d i rec t ionsof a tensoranalyticallyb y amethod describedat the end ofthis chapter.

Problem41 . Supposep t and q i are relatedby thetensor equationp f= S t q -,where:

and where S j > 5 2 > S y (a) Is [ S i ] referredto itspr inc ipa laxes? (b) If q t isa variable-direct ion unit vector, which direct ionfor q i will give the greatestm agn i tudep for / > . ? (c ) What is the direct ionof p i in that case? (d) What isthe equa t ionof therepresenta t ion qu adr icfor 5 ; . ? (e)Find the radius vectorx i from the or ig into the surfaceof the quadr ic ,in the direct ionof a givenuni t vectoru f . ( f ) F i n dthe unit vectorw; n o r m a lto thequadr icat the poin twh ere the qu adric is touc hed byxf (g) F indpf given q ^= (.. (h )What is theanglebetweenp i and w(?

F i g u r e3.2 . The radius-normal proper tyofth e r ep r e sen t a t i on e l l i p so id . C en t r a lsect ion of the el l ipsoid conta in ingb o t hpand q and therefore perpendicu larto the

tangent.


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M A T T E RT E N S O R S 2 5

Where the context prevents confusion with a vector, it is common practice todesignate principal values of symmetric second-rank tensors by symbols with a

singlesubscript. Use of this convention permits, for example, the s im pl i fi ca t ionof eq. (3.6) for the magnitude of a tensor referredto its principal coordinates:

where S- are the principal values of the symmetric second-rank tensor 5(- and / ;the direction cosines for the direction in which the magnitude S is sought.

Beware, however, of applying the transformation rule for vectors to a singly

subscripted variables if, as in eq. (3.36), it does not represent a vector.Second-rank tensors can be asymmetric, and in that case they can be divided

into summands, one of which is symmetric [eq. (3.25)] and the otherantisymmetric. An antisymmetric second-rank tensor A (. . has the property:

which i m p l i e s :

Problem 42 . Show that any tensor 2 can be expressed as the sum of as y m m e t r i c a ltensor Sf and an antisymmetrical tensor A (. .; that is, show thatthere exist S and A such that:

where S = Sjf and A = Ajf Find expressions for S and A . .in terms ofT

a-

Prob lem43. Find a symmetric tensor S and an antisymmetrical tensor A r suchthat:

where:

Tensors of the second and of higher ranks are used to describe physicalproper t iesof anisotropic materials, especially of crystals (Nye, 1964, pp. 20-21).


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2 6 S T R E S SAND D E F O R M A T I O N

Physical properties of crystals are subject toNeumann'sprinciple: The symmetryelements of any physical propertyof a crystal must include, but may exceed, thesymmetry elementsof thepoint group of the crystal. An exam ple is the opticalisotropyof cubic crys ta ls ;the symmetry e lementsof c om plete isotropy exceedthose of even the class wi th the highest point grou p in the c u b ic system ,z3m. A san empirical fact , noneof theknowncrystal propert ies requiresan asymmet r i csecond-rank tensor for their description;hencem atter tensors ofthat rank wil lhere be assum ed to be sym m etric.

Problem44. (a)Writethe t r ans form at ion m at rix/( , - . - )for a rotat ionof axesthroughan arb i t rary angle9 abou tthe x2 axis (senseof rotat ion posit iveinthe direct ion fromx^ to X j ) . (b )Writethe othertwo t ransform at ion m atr icesper fo rmingrotat ions about coordinateaxes, (c )Supposenow that 9 = 180;f ind Ig()o2( < z, - . - ) (d) Trans fo rm a s y m m e t r i c t e n s o r[ 5;- ] u s i n g thet ransformat ion matr ixIg0o ( # , ) to producethe newtensorcalled i g o [r]-(e) If the tensor Oo[ , - ]represents som e tensor prop ertyof a m o n o c l i n i ccrystal,then what does Neumann ' spr inc ip lesay a b o u t i8 0D[ ^ , ] >as sumingthat the dyad axisof the monocl inic crys ta lis parallelto x2? ( f ) From this ,deducethe general formof o [ S ..] for m onoc l inic crys ta ls .

Problem45. Usingthe methodof Problem44, find thegeneral formof a tensorSj representing some tensor propertyof a crystalin the t r igonal sys tem ,incoordinates withx3 parallelto the threefold axis.

F igu re 3 .3 . U ni t ce l lof d i m e n s i o n sa, b, and cm easured a long cry s ta llographic axesx , y,and z that

form interax ial anglesC C ,j3 ,and y.


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M A T T E RT E N S O R S 27

B y convention, the generally unequal dimensions of the unit cell of a crystal,designated as a, b, and c, are measured along the generally nonorthogonalcrystallographic axes x, y, and z, which form interaxial angles Of, [3 , and J(Fig. 3.3). The lengths a, b, and c are treated as the respective unit lengths for the

axes x,y, and z. Crystallographic directions [UVW] are vectors, referredto thissystem of crystallographic axes, drawn from an origin at a corner of a unit cell toanother corner of the same or of another unit cell. They have integer componentsU, V,and W, chosen so as not topossess a common denominator (multiplicationwith a common factor does not change the direction of a vector). In cartesiancoordinates with the same origin as the crystallographic coordinates (and with a

conventional relative orientation of the two coordinate systems), the vectors[UVW] become t/a+ K b + We if a, b, and c are vectors of lengths a, b, and cin the crystallographic x, y, and z directions, but referred to the cartesiancoordinates.

Problem 46. Find an expression for the unit vector [ (.] in the crystallographicdirection [UVW] in the cubic system. Follow the usual conventions for theorientation of crystallographic axes, for example, those used by Nye (1964,

pp.276-88). Use[UVW] = [ l 1 1 ] as anexample.

Problem 47. Find an expression for theunit vector [ .] in thecrystallographicd i r e c t i o n [UVW] in the monoclinic system. Follow conventions as inProblem 46.

The case of the monoclinic system is still too highly symmetric to allow thederivation of a general rule for f indingunit vectors in given crystallographicdirections. That rule (stated without derivation) is: Cartesian components of aunit vector in terms of crystallographiccomponents are.

where:

Note that in evaluating the quantity in brackets it is not permissible to cancel theapparent square represented by the product of the two identical quantities inparentheses against the exponent outside the brackets. The reason is the impliedsummation of the nine products with repeated subscript that results f rom the ful lm u l t i p l i c a t i o nof the two parentheses. An expression of t h i s type must beevaluated f rom the inside out. To evaluate the squared normalization constant k 1

for the monoclinic system in terms of the general rule, we write:


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where:

The f ramed termsin eq.(3.41) vanish, because whenevera i ^0, then bf = 0 , andwheneverbf ^ 0, then cf = 0 . Hence:

which agreeswith eq . (3.40). In the t r icl inic system,eq . (3.41)represents2 7explicitproducts,and noneof them vanishes although9 areduplicated becauseofthe commuta t ive p roper tyof mul t ip l ica t ion .

Problem 48. The re la t ive perm it t iv i ty tensor[A!*, . ] (the tensor formof thedielectricconstant) of a single crystal has been expe rim entally determ ined tbe:

at an applied frequencyof 200kHz. What can bededucedfrom this findingconcerning the symmetry of the crystal? Explain your reasoning.

Prob lem 49. (a) Of thefour threefold symmetry axes arrangedl ike the bodydiagonalsof a cube thatare requiredfor m e m b e r s h i pin the cubic sys tem,choose one, give it a sense, and find its direction cosines in a conventionaoriented coordinatesys tem, (b ) Wri te out the t r ans fo rma t ion ma t r ixeffect ingthe symmetry operat ion consis t ingof a 120 rotat ion aboutthethreefoldaxis specifiedby theanswerto (a), (c )Verifywhetherthe symmet ryoperat ions determinedin the answer to (b)ensurethe second-rank tensorpropert ies requiredb y Neumann'sPrinciple for cubic crystals.


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Aniso t rop ic phys ica l p roper t i esare not r e s t r i c t edto s ingle crys ta ls .C rystall ine m aterials in which thecrystallographicaxes of consti tuent grains arenot oriented at random b u t are stat is t ical ly al igned are anisotrop ic, and so layered rockbodies ,if consideredas a whole and if the i r inhomogene i t i esareneglected.

Note that the physical measurement of an anisotropic material property thatcan bedefinedb y a tensor of the second rank involvesthe appl icat ionof anindepen dent vec torial variable to a sam ple of the m aterial in aspecificdirect ionand them e a s u r e m e n tof the dependent vectorial variablein the same direction,repeatedin a suff ic ientn u m b e rof different directions. This amounts to the

de t e rm ina tionof a series of tensor m agni tu desfor the m a t t e r tensor to bedetermined.

Problem 50 . The tem peratu re gradient in a certainbody of rock is given b y:

and the correspondingflow of heat as:

F i n d the the rm a l condu c t iv i t y [ sho r tfo r m a g n i t u d eof the t h e r m a lconduct iv i tytensor,see eqs. (3.5)and (3.6)]in thedirectionof [T (. ]. Notethat the sametw ovectors playa role in Problem26 andhereb u t that theyarenot nec essarily relatedto eachother by thesam e tensor[ K - ] .

Matter tensorsc an also describe m anu fac tu red m aterials .

Problem51. In a specimenof insula t ing m ater ialthe thermal conductivi tyK invariousdirect ionswas foundin a set ofexperiments:

Tempera ture g rad ien t(1 0 K m ) M agni tude of conduc t iv i ty in thedirectionof thegradient(J nT 1 s~l K "1)

[ 1.1 0.0 0.0 ] 8[ 0.0 1.7 0.0 ] 16[ 0.0 0.0 1.0 ] 8[ 0.8 0.8 0.0 ] 24[ 0.0 0.7 0.7 ] 20[ 0.7 0.0 0.7 ] 12

F indthe thermal conduct iv i ty[ Kf ] for this m aterial .


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There follow several exercises dealing with general features of second-ranktensors.

Problem 52. Two vector quanti t iesp - and q t are related b y the tensor prop erty:

accordingto the equat ion,pf = Sf q ^. Findthe magn i tudeof the propertySf

in the direct ion (a) of thevector f 1 /3~ 2/3~ ] and (b) of thevector[ 1 /3~ 2-/3~ ]. Hint: C o m p a r ewith Problem34.

Problem53. Supposea tensoris:

(a) Find the m a g n i t u d eT in the direct ionof the xaxis. (b ) F i n d themagn i tudeT' in the direc tion of the vector[1 1 1 ].

Problem 54 . Show that the tensor:

is invariant to rotation of coordinate axes aboutx 3.

D eterm inants of m atrices (and thus of tensors) have uses beyond those we haalready encountered; theirusefulnessis enhancedby thefact that de te rminan tsareunaffectedb y cer ta in m anipu la t ions .

Problem 55. (a) Show that the determinant of a matrix ( A j j ) i s :

(b ) Show that the determ inant der ivedfrom \Apq\ b y in terchangingtw oc o l u m n sor tworows is equalto \A f> c,\ , and therefore that,if two co lumns


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or two rows of | A \ are identical, necessarily\A \ = 0. (c) From theseresults,show that:

(d) Also show that,if IA . I and IB.\ are de te rminan tsand C. = A . , B - , ,I] I] lj I K JKthen:

P rob lem 5 6 . Using one of theresults of the prev ious p rob lemand theorthogonal i ty re la t ionsof a ro ta tion m at r ixfor cartesian coordinatesa.^.,showthat | a i | =1.

The representa t ion quadr ic [eq.(3.3 1)] is not theonly geometr icf iguredescriptive of a second-rank ten sor.

Problem57. Let asym m etr ic second- rank tensorS f . have x i as itsp r i n c i p a l

co ordinates. Prove thatthe ovaloidsurface:

has in every direction a radius vectorr with magni tuder equal to the tensorm a g n i t u d eS in that direct ion.

This surfac e represents a second-rank tensorless conven ien t lythan therepresentat ionquadr icof eq.(3.31)and Figure 3 .2 .

Let asym m etric tensor relateth e vector[pt ] to thevector[ q .]:

Principaldirections of this tensor are those inwhich the radius-norm al proper tyof the representat ion qu adric ensuresthat [ f>f ] parallels[^ . ] andthus that:

where C is a prop or t iona l i ty co nstant . B ydef in i t ion ,the m agn i tud e of ourtensoris:

Choosingas an instanceof [q i ] the u nit vector[ /;.], eq. (3.44) becomes:


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32 S T R E S S A N DD E F O R M A T I O N

andeq .(3.46):

Principal coordinates x i of the tensor 5 r are paral le l to its principaldirections; thus, in each of the three principal directionsx i , both [p^ and [ /(. ]have onlyone component ,and this component equals their magnitudes,p and 1.A cc ordingly, in pr inc ipal c oordinates the vector m agni tu deslp in the threep r inc ipa l d i r ec t ions mus tb e iden t i ca lwi th th e three pr inc ipa l t ensormagni tudes'5 of [5. .] and thus wi ththe three solutions'A of the secular

equationand also withthe proport ionali ty c onstantsC in eq.(3.45). Therefore,in each of theprincipal directions:

This findingcannot dependon thechoiceof coordinates,and it canthereforeb egeneral izedfor pr inc ipal d i re c t ion s of a sym m etr ic tensor referred to anarbitrarycoordinate system;for this pu rpose,we subst i tu tefor thecomponentsp -of a vector with the m agn itu dep in eq,(3.47)those of a vector of m ag nitu deA , orA/,:

where /(. are the direction cosinesof one of theprincipal directionsof [5(. ].Eq u ation (3.50) represents three hom ogeneou sl inearequationsin thevariables/ ; .They havea solution onlyif the following con dit ionis met :

whichis thesecular equationwe have seenbefore.

Problem58. F ind (a) thecoefficientsof thesecu lar equ ationand (b) thethreeprincipalvalues of the tensor:

(c ) F i n d the direct ion cosinesof the shortest principal axisof therepresentat ion quadric of [ . . ] and (d) those of the other twoaxes, (e)Writethe secular equationfor thesame tensorin princ ipal c oordinates.


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General rule: Invarianceto rotation of the coefficients of the secular eq uationisa necessaryproperty of a second-ranktensor. They are, therefore,commonly cal ledthe first,second,and third invariantof asecond-rank tensorTf and aredefinedasfo l lows:

In general, i t is less convenient to solve for the principal directions of sym m etric tensor than in Problem 58. A un iversally applicab le m ethod is neU n i tvectors/ ; in the principalor eigendirectionsof a symmetr ic tensor beingsubjectto eq. (3.50), in which A is any one of the prin cip al values'/I of the tensor,the ind ividu al equ ations for the three eigendirections can b e restated as:

where the ' X are vectorsof arb i t ra ry m agni tu de para l le lto the pr inc ipa ldirect ionsof the tensor ;it is general ly impossibleto solve for these vectorsdirectly becau se of their u nc ertain m agn itud e. Instead, one solves for the ratiotwo of thecomponentsof each' X . Define:

and arbi trar i lyset:

F i n d the d i r ec t ion cos ines (com ponen t sof a uni t vec tor )for the z'the igendi rec t ion b y norm al iza t ion :

Note that a n u m b e rof arb i t ra r i ly accura te nu m er ica lmethodsexist for thesim u ltaneous determ ination of eigenvalues and eigendirections.


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34 S T R E S SA N D D E F O R M AT I O N

The orthogonality conditions of the rotation matrix [eq. (3.14)] and the(5 rule [eq. (3.21)] commonly introduce the Kronecker delta, or identity matrix,into tensor equations. The Kronecker delta has a useful attribute, the substitutionproperty.

Where a subscripted variable shares a dummy subscript with the Kronecker delta,the second subscript of the Kronecker delta can be substituted for the dummysubsc r ip t of the variable, while the Kronecker delta itself is eliminated. Thisproperty can easily be verif ied by executing eq. (3.57) explicitly. Note that

numer ica lsubscripts cannot be dummy subscripts and do not, therefore, convey thesubstitution property to the Kronecker delta; the Kronecker delta with numericalsubscripts simply has the value of 0 or of 1.

Problem 59. Eliminate the Kronecker deltas f rom the following expressions bymeans of the substitution property and characterize them as either scalars (inthe i r simplest form they have no free subscripts), vectors (one subscript), orsecond-rank tensors (two subscripts): (a) 5. TV; (b) 8pq8 r S ;

M S tJ 5 mn A jm B np C M ; (d) S p9 S ir S f, T st ; (e) Sfj S j{ .

Problem 60. Transform the following expressions by rotation, s imp l i fy the

results where possible using the orthogonality relations of rotation matricesand the substitution property of the Kronecker delta, and characterize theexpressions as scalars, vectors, or tensors: (a) AB mn ; (b) B tu C vw ; (c) p k q k\($P m A m k->(3A mp B nf .

Problem 61. Eliminate the Kronecker deltas f rom the following expressions:S f9 S fl T f,; S 23 T 13 ;(c)(S ii )

2 ;( S ii2 .

The most reliable criterion that distinguishes tensors of any rank f rom othersubscripted variables (matrices) is the conservation of the tensor invariants whenit is rotated according to the rule appropriate for its rank. Equations (3.23) and(3.24) are the rules for tensors of the first and second rank, and tensors of higher

ranks are rotated analogously. Thus, for a tensor of the third rank with threesubscripts, the rule is :

for one of the fourth rank:

and so on.


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MATTERTENSORS 35

Problem62. Using the Kronecker del taand the orthogonali ty relat ions, showthat if A fj and B,j are tensors, the nC,-y:

is alsoa tensor.

Problem63. Using the Kronecker del taand the or thogonal i ty re la t ions , showthat if A jj , BJJ, and C,-yare tensors, then/),-.:

is alsoa tensor.

Problem64. Usingthe Kronecker de l t aand the or thogonal i ty re la t ions , showthat if :

then the fol low ing is t rue :

Problem65. Transformthe vector:

by m eans o f the t r ansform at ion m at rix :

What geom etric re la t ionship does th is vec tor bearto the t r ans format ionof

axes represen tedby the mat r i x( a t . )?

Antisymmetric tensors Vji Vij (none describ es m aterial prope rt ies) havediagonal com pon ents that are necessarily zero, and an antisy m m etric tensor caexpressedin t e r m sof an axial vector. In con t rad i s t inc t ionto the ord inary,orpolar, vectors , axial vec tors t ransf orm acc ording to the ru le:


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36 STRESSANDDEFORMATION

A s a consequenceof this rule, axial vectorsdo not change signupon invers ion ,whereas polar vectorsdo. Inversionis thet rans fo rm at ionb y m e a n sof thenegativeident i ty m atr ix [eq.(A.30.4)]. A llvectors that describea rotat ionareaxial.

Prob lem 6 6. Given two arb i trary vectors[/>;]and [q^] and the relationship:

show (a ) that [ V jj ] is a n t i s y m m e t r i cand (b)that it can bestatedin t e rmsof anaxial vector [r, - ] . (c) D em onstrate that [V j\ is a tensor by showing that i ts

secular equat ionis invar ian tto rotat ion, (d ) Does [ V;.- ] have rea l pr inc ipalvalues?

G eneral rule: N o real solutions exist for the secular eq uations of antisym m etrictensors.

The a n t is y m m e t ri c t en s or[ Vjj] can benormal izedb y d i v i d i n gthe axialvector [r;] by i t s m agn i tu de . The resu l t ing an t i sym m et r i c t ensor i s ca l led thgenerator of rotations [Rfj]:

This m a k e s[ r ; ] a u ni t vec torand hencethe generatorof rotat ionsis:

It represents a rotationof+90 relative to a fixedreference frame. The rotation isa b o u ta l ine wi ththe direct ion cosines/; and clockwise,seen in the di rec t ionof

[/,'].T h e r o t a ti o n m a t r i x( ^ , - y ) and the genera to ro f ro ta t ions [/?,/] a re

necessari ly interrelated. Sinc e i t t rans form s thereferencef rame, the compar i sonrota tion m atr ix m ust ro ta teth e reference frame itself counterclockwiseabout [ /,].This is theeffect of a m atr ix with the elem ents:

for a rotat ion through the angle6 . In term s of the generator of rotat ions the sam erota t ionis:


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Conversely,sin 9 and cos 9, and also /; and thus RJJ by eq .(3.62),can becalculated froma known rotat ion matrix( a , - y ) :

Problem67. Show thatan ant isym m etric tensorhas noprincipal directions; thatis, show that, ifpi and q ^ are real, nonzero, finite vectors, if furtherA;.- is a

tensor such thatA / .= Ajj, and ifpj = A jj q;, then the vectorsp; and q cannot b e parallel.

Problem 68 . Byapplyingall pertinent tests,show thatS\\ is a pr inc ipa lcomponentand x \ a principalaxis in the tensor:

General rule: A nonzero diagonaltensor component accompanied byzeroes in theremainder of the same column andof the same row is a principal componentof thattensor.

Rotationof a sym m etr ic second-rank tensor abou tone of itsprincipal axeshas a geometrical analog, called theMohr circle after its discoverer(1882 ,1914).

Let thecoordinate axes coincide withth e principal tensor axes,and let thetensorthusbe:

The m atrix that rotates this tensor ab ou t thex-$axis, expressed as a fu nc tion of thepositive angleof coordinate rotation6, is:


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Figu re3.4. M o h rcircleconstruction.InM o h r space , d iagona l c om ponentsareplot ted a longthe 3? axis ; nondiagonalcomponents a longthe ^ axis. A circlewith the radius r is the locus for thesec om pone nts at various angles of clockwiserotat ion6 about thex^ axis in real space.

The d o u b l e a n g l e 29 is p o s i t i v ecounte rc lockwise ,and the poin t P hascoordinates- $ 2 2and 512 (solid vert icalline) for a particular angle2 9. The

dashed lines serveto localizeS\\ .

rotat ion axis and the axes wi th su b sc ripts fol low ingR in ascending orderm o d u l o3 (if R \, then a 2 and b= 3, or in thepresent case,if R = 3, thena= 1 and b = 2), and letS}j be theresultof a rotation abou tX Rof Sj.-, then erectin M o h rspacea positiveS'a /, in the %direction fromthe point c = S'f,/,on theabscissa (a negativeS' ah is droppedfrom the sam e poin t). InM o h rspace, theordinatesfor 5 at 3? = S'a(, have th e opposite signof that in the tensor itself;they are therefore drawn dashed throu gh ou t this b ook.

Use of th is convent ion enforcesa def in i te assoc ia t ionof subsc r ip tassignmentsand senseof rotation betweenMohr and real space.A positiveM o h rangle 29 (counterclockwisein thedrawing plane becauseth e third Mohr axis 2Epoints towardthe viewer) correspondsto apositive (clockwise seenin thepositivedirect ionof the rotationaxis) rotation of the two moving coordinate axes inrealspace. Thismeans that theaxis x 'a movesfrom i ts original orientat iontowardthe original or ienta t ionof x f, (Fig. 3.5) . (Som e au thorsu se conventionsforM o h r spacethat dependon the order of the tensor's principal values.This isinconvenient , because within a coherent tensor f ield principal values commchange their sequential position alongone and thesam e trajectory.)


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Figure3.5. Rotationin real space(a) andM ohr space (b).A positivereal rota t ion throughan arbitrary 15,and ofthe corresponding 30 inM o h r space, appears counter-clockwisein bo th becausethe rotation axesX R in real

space andS in M o h rspace point towardthe viewer.

M o h r circ le c onstru c t ions could b e used to obta in , ra ther labo r iouslyreasonablyaccurate graphic solut ions; thei rreal usefulness,however,is the

possibilityto sketch a co nstruc tion and then f ind the appropriate tr igon om etrrelat ionshipsbetween the desired solutions for the unknowns of a problem anthe data. Inspection of F igu re 3.4 yields, am ong others:

where:

and:

Problem69. (a)Show withthe help of a M o h rcirc le construct ion thatthedete rminant :

is invariantto rotat ion aboutxj,. (b) Istherea geometric interpretat ion(inM o h rspace)for your f inding?

The rota t ion matr ices for rota t ions through6 of coordinate axes about thex \ and X2axes are:


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Find these rotationsby theM o h rcircle method.

Prob lem 72. (a) U sing theM o h rcircle construction, find the tensor5,- ,-after acoordinate rotat ionof 45 a b o u t x \ ( rota t ionof xi f rom i ts or ig ina lorientation towardX T ,is considered positive), where:

(b ) By thesame const ruct ion, t ransformS;j to its principal coordinates .What is the or ien ta t ionof the principal axes relat iveto the or ig ina lcoordinates(beforethe 45rotation)?

Problem73. (a)Using M ohr c i rc le const ruct ion,find thetensors U\i and V\;after a coordinate rotationof 30abou tx ^ and #3 , respectively (rotationof*3 from its original orientat ion towardx\ is considered positive,and so isrotat ionof x\ from i ts original orientation towardx^):

(b) Bythe sam e constru ction,transformU jj and V jj to principal coordinatesand find thenecessary rotation s.

Problem74. UsingM o h rcircle construction, t ransformthe following tensorsto the principal coordinates with axes nearest those givenand find thenecessary rotations.


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Problem75 . Rotatethe tensor:

abou tx\ by thesmallest angle that makes,$22= 6 . Find the direct ionandangle ofrotationand therotated tensor[ S } j].

The Mohr circleapproachis restr ictedto two-dimensional problems.It is,however, validfor rota t ionsof three-dim ensional second-rank tensorsa b o u toneof thecoordinateaxeseven if thisaxis is not aprincipalaxisof the tensor. Sucharota t iondoes m o d i f ythose nondiag onal c om pone nts of the tensor that share asub script with the rotation axis . No info rm ation abou t these changes is ob taif rom the Mohr c i r c l e co ns t ruc t i on . "P r inc ipa l va lues"an d " p r i n c i p a ldirect ions" foundin that caseare notthose of thetensoras awholebut of atwo-

dim ensional sect ion throu gh i t . Rotat ion of the stage of a m icroscope wcrossed polarizers to "extinc tion" o rientat io n in a thin sec t iont h r o u g hs o m egrainof a biref r ingentminera lis an analogous operat ion.


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4

STRESSStress is a tensor quantity that describesthe mechanical force density (forceperunit area) on the completesurfaceof a dom ain inside a m aterial b ody. A stressexistswhereverone part of a body exertsa force on neighboring par ts .Itsorientation is not t ied to any part ic u lar directions that are intrins ic to tmateriallike, say, crystallographic axes.It isthus dist inctfrom the m atter tensorsthat were discussed in the preceding chapter, all ofwhich have def ini t ive

orientations within a crystal orother aniso tropi c m aterial; it is called afieldtensor (and so is strain ). The de fin itio n of stress depen ds on the c onc ept ofcontinuum.Let f(xi) be asingle-valued function definedfor every po intxi in aregion. This funct ionis said to becont inuousat thepoint xi if the followingholds for allpathsof approachof x i to xi:

Equivalently,for any numbere, no m atter how sm all, there exists a neighb orhoodof nonzero radiu s aroun dthe pointxi in which:

for all pointsxi in that neighborhood. A c ontinu u m is an idealized m aterialwhose phys ica l a t t r ibu tesare c o n t i n u o u s f u n c t i o n sof pos i t ion . Thusneighboring points remain neighbors , and a cont inuum cannot have gapsj u m p s(discontinuities)in its propert ies . Surfaces b ou ndin g gapsor def in ingdiscontinuit iesm u st be specially treated incontinuum mechanics. Examples aresurfacesbetweentw o fluidsof differingdensityor viscosity,or between solidswith different thermal conduct iv i tyor elastic pro perties. Real m aterialsarenevercontinua; theyare discont inuousat the atomic scale,and oftenat largerscalesas well. The notion of a c o n t i n u u mis, therefore, onlya macroscopicapproximat ion,but it allowsusefulm athem atical approaches to the treatm ent ofreal phenomena .

The definit ionof a bodyforce m ayserveas anexample:

44


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S T R E S S 45

Figure4.1. Euler's stress principle.Thecont inuous regionR containsa domainsur roundedby theclosed surfaceS. A ne lementA S on that surfacehas the out-ward normaln, and u p o n it the sur-rounding mater ia l exer tsthe force AF.

(AfterFung,1965.)

whereAf i is the total gravitational(or magnet ic ,or other) forceon thematerialin the volumeA V. Thisdefinit ion impliesthe assumption thatthe materialis ac o n t i n u u m .In a real m aterial,the vo lumeA V,if chosenat the scaleof, say,1 m ~2 0, willat most points enclose nothingb u t empty space,but at a fewpointsextrem ely dense su b atom ic m atter. Evidently, some averagingis required; thatis,A Vm u s tnot beallowedto shrinkto zerob u t only to avolume, small comparedto macroscopic dimensions,yet large comparedto atoms. In treatingpolycrystals,tu rbulen tfluids,suspensions,or emulsions,or fluid-saturated rocksas continua,A Vm u s tb e large comparedto thevolumesof constituent crystals, microcracks,eddies, suspended grains, or pores.

With these provisosin mind , the idea (if not thereality)of stress can be

defined by way ofEuler'sprinciple. Consider a material regionR to be acontinuum and, inside this region,a domainboundedby theclosed surfaceS(Fig.4.1). Considerfurthera small portionof that surface,A S,with th e outwardnormal n. Designateas A F theforce exertedby theportionsof R neighbor ingthe positive sideof n upon the m aterial insideS adjacentto A S neighboringthenegative sideof n. Thisforceis a func t ionof both the sizeand theorientationofA S. Now assumethat as A S goesto zero, the ratio A F/A S tends to a definitel i m i t dF/dS, the force density,and that any momentof force (or torque) acting


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aboutanypointon thesurfaceA S vanishin thelimit (the second conditionm aybe violated if a rare state exists that is caused by abody couple or, according to Nye,1964, a body torque;in thisbook a state that encompasseseffectsof abody torqueshallb e calleda stress in the wider sense). The lim iting vector, calleda traction orstress vector,can then be statedas:

where th e superscriptn refers to thespecif ic outward normaln of A S. It can b e

seen that tractionhas thedimensions offereeper unit area,or [m T1

t~2

]. Stressatsome point completely definesthe t rac t ionsfor allorientat ionsof n at thatpoin t .

F or s imp l i c i t y,the d o m a i n on the surfaceof which the forces act iscommonly takento be acube with edges parallelto the axes of the referencecoord ina tesystem (Fig. 4.2) .The i l lus t ra t ionm ay either b e regardedas amagnif ica t ionto finite size of a point-sized cube or, with lessdiff icul tyfor theimag ina t ion ,as the representat ionof an ac tua l lyfinite cube ins idea region

subject to ahomogeneousstress (a stress that is independent of position).Figure4.2 showsa set offorce densities,or tractions, actingon thesurfacesof

a cubical domain insidea region subjectto a homogeneous stress(in the narrowsense). The tractions thatact on theconcealed facesare antiparallelto those shown(they appearon Fig. 4 .3) ,and the d o m a i n is thereforenot s u b j e c tto an

Figure4.2 . Componentsof force density(forceper u n i t area) ac t ingon faces of ahomogeneously stressed cube.The ninecomponentscr {. constitutethe stress tensor .


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S T R E S S 47

acceleration. Sinceb o t h the outward normalsof the backwardfaces and thet ract ionson them have the opposite signsof those in front , t ract ionsand stresscomponents havethe same sign backand f ront . The stress tensor(in thenarrowsense) is sym m etric by d efin ition (because m om ents of force are taken to vanithe l imi t ) :

In the contextof geologyit is c o m m o n l ypermiss ibleto neglectall fields,exceptthe gravi ta t ionalfieldcausedby theEarth's mass,and also the dynamictransient statesof stress oc cu rring du ring acc elerat ion, say, du ringan ear thquake.The rem ainin g s tressesare said to be ins ta t i c equ i l ib r ium,a condit ion thatc anbe expressed precisely as:

where p is the local densityand g the accelerat iondue togravity. The threeequations (4.6)are thestaticequationsof equilibrium,and pg f is calleda body force.Although other bodyforcesdo exist,the one due togravityis most signif icantingeology.

It isc o m m o npracticeto use thet e rms normaland tangential stress for thediagonaland thenondiagonal com ponentsof thestress tensor. (Som e au thorsu setw o differentsymbolsfor thesame tensor,c r^ , ij, for thenormal ,and T , i&j,

Figu re4.3. Tractionson the facesof atetrahedron.The front face has theout-ward no rma l1 of uni t length ,and the

t ract ionon it is P. (AfterNye, 1964.)


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for the tangent ia l components . )A traction, too,can bedecomposedin to itsn o r m a land tangent ia l components ,and so can theforceb e integrated overadefined area.

How one can f ind thet rac t ion P. u p o n an arbi t rar i ly or iented surfaceelem ent wi th the u ni tary outward norm al/;. at a point where the stress iscr ; . canmost easily be dem onstrated for abody at staticequi l ib r ium sub jec tto ahom ogeneou s stress. Let there b e a tetrahedron- shaped elem ent of this b oboundedb y threesurfaces,each parallelto twocoordinate axes,and afou r th wi than outward normal unit vector/;. as in Fig. 4.3. Since the stress is hom og eneou sand the sizedifferencesof the four bounding surfaces are i r re levant because

tractions are normalized to unit area, we can vector-add the contributions toP ; asfo l lows :

In subscr iptnotat ion,thissimplifiesto Cauchy's equation:

The existence of this form u la, whic h relates one vec tor to a noth er l inearlalso proves thatcr; . is atensorand that stressis a tensor qu anti ty.When the stressisnot homogeneous ,or body forcesare acting,or when the body is not instatice q u i l i b r i u m ,Cauchy ' sf o r m u l astill holds because the contr ibut ions of thesedepar turesfrom the sim ple case becom e negl ig ibleas thesize of the tetrahedronof Fig.4.3 becom es van ish ing ly sm al l.

Prob lem 76. Let a b ody b e hom oge neou sly stressed, and let the stress b e:

F ind (a) thenormalforce N i and (b) thetangential forceT., in newtons, thatare exer ted across1 m of t he p l a n e p e r p e n d i c u l a rto the vec to r[ v.] = [ 1, 1, 1 ] by themater ia lon the +i>; side upon thaton the z / .side.(One pascal, Pa, is the trac tion exerted by aforceof one newton, N, acting onone m 2 of surface.)


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S T R E S S 49

Severalspecial form sof the stress tensorhave specific nam es. Referredtotheir principal axes, theyare the following:

(i ) Uniaxial stress, such as the stress in a vert icalrod orwire loadedb y aweight suspendedon itslower end:

(ii) Biaxial stress, suchas thestressin a thin plate loadedat the edges:

(iii) Triaxial stress is an al ternative nam efor ageneral stateof stress,(iv) Hydrostatic pressure,suchas thestressin a fluid atrest:

where p is the pressure. N o t e that pressure is always stated positive forcompression, whereasin physicsand geophysics,and inthis book,extensile stress

and t ract ion are convent ional ly posi t ive .(In the fields of engineer ingandgeology,th e opposite conventionis c o m m o n ,so that a com pressive stressis takento bepositive.)

(v ) Pure shear stress, a special formof biaxial stress:

which, upona 45rotat ion abou tthe x, axis,th e axis of shear, becomes:

henceits name.


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Problem77. (a)Determineth e kindof stress thatis representedby thetensor:

Hints: A determinantis multiplied by a given factor by multiplying one row orone column bythat factor, and a determinant is unaltered if a multiple of onerow (or column) is added to another row (or column), (b ) Find the tensor 'so r ien ta t ion .

A nystressm ay bedivided intotw o components ,a mean stress and adeviator ofstres

0195095030_stress and deformation etc

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