弹性力学第3章

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弹性力学第3章

4MaterialBehavior—LinearElasticSolidsTheprevioustwochaptersestablishelasticity?eldequationsrelatedtothekinematicsof

smalldeformationtheoryandtheequilibriumoftheassociatedinternalstress?eld.Based

onthesephysicalconcepts,threestrain-displacementrelations(2.2.5),sixcompatibility

equations(2.6.2),andthreeequilibriumequations(3.6.5)weredevelopedforthegeneral

three-dimensionalcase.Becausemomentequilibriumsimplyresultsinsymmetryofthestress

tensor,itisnotnormallyincludedasaseparate?eldequationset.Also,recallthatthe

compatibilityequationsactuallyrepresentonlythreeindependentrelations,andtheseequa-

tionsareneededonlytoensurethatagivenstrain?eldwillproducesingle-valuedcontinuous

displacements.Becausethedisplacementsareincludedinthegeneralproblemformulation,the

solutionautomaticallygivescontinuousdisplacements,andthecompatibilityequationsarenot

formallyneededforthegeneralsystem.Thus,excludingthecompatibilityrelations,itisfound

thatwehavenowdevelopednine?eldequations.Theunknownsintheseequationsinclude3

displacementcomponents,6componentsofstrain,and6stresscomponents,yieldingatotalof

15unknowns.Thus,the9equationsarenotsuf?cienttosolveforthe15unknowns,and

additional?eldequationsareneeded.Thisresultshouldnotbesurprisingsinceuptothispoint

inourdevelopmentwehavenotconsideredthematerialresponse.Wenowwishtocomplete

ourgeneralformulationbyspecializingtoaparticularmaterialmodelthatprovidesreasonable

characterizationofmaterialsundersmalldeformations.Themodelwewilluseisthatofa

linearelasticmaterial,anamethatcategorizestheentiretheory.Thischapterpresentsthe

basicsoftheelasticmodelspecializingtheformulationforisotropicmaterials.Relatedtheory

foranisotropicmediaisdevelopedinChapter11.Thermoelasticrelationsarealsobrie?y

presentedforlateruseinChapter12.

4.1MaterialCharacterization

Relationsthatcharacterizethephysicalpropertiesofmaterialsarecalledconstitutiveequa-

tions.Becauseoftheendlessvarietyofmaterialsandloadings,thestudyanddevelopmentof

constitutiveequationsisperhapsoneofthemostinterestingandchallenging?eldsinmechan-

ics.Althoughcontinuummechanicstheoryhasestablishedsomeprinciplesforsystematic

developmentofconstitutiveequations(Malvern1969),manyconstitutivelawshavebeen

developedthroughempiricalrelationsbasedonexperimentalevidence.Ourinteresthereis

69

limitedtoaspecialclassofsolidmaterialswithloadingsresultingfrommechanicalorthermal

effects.Themechanicalbehaviorofsolidsisnormallyde?nedbyconstitutivestress-strain

http://./doc/4b8f853243323968011c9279.htmlmonly,theserelationsexpressthestressasafunctionofthestrain,strainrate,

strainhistory,temperature,andmaterialproperties.Wechoosearathersimplematerialmodel

calledtheelasticsolidthatdoesnotincluderateorhistoryeffects.Themodelmaybe

describedasadeformablecontinuumthatrecoversitsoriginalcon?gurationwhentheloadings

causingthedeformationareremoved.Furthermore,werestricttheconstitutivestress-strain

lawtobelinear,thusleadingtoalinearelasticsolid.Althoughtheseassumptionsgreatly

simplifythemodel,linearelasticitypredictionshaveshowngoodagreementwithexperimental

dataandhaveprovidedusefulmethodstoconductstressanalysis.Manystructuralmaterials

includingmetals,plastics,ceramics,wood,rock,concrete,andsoforthexhibitlinearelastic

behaviorundersmalldeformations.

Asmentioned,experimentaltestingiscommonlyemployedinordertocharacterizethemechanicalbehaviorofrealmaterials.Onesuchtechniqueisthesimpletensiontestinwhicha

speciallypreparedcylindricalor?atstocksampleisloadedaxiallyinatestingmachine.Strain

isdeterminedbythechangeinlengthbetweenprescribedreferencemarksonthesampleandis

usuallymeasuredbyaclipgage.Loaddatacollectedfromaloadcellisdividedbythecross-

sectionalareainthetestsectiontocalculatethestress.Axialstress-straindataisrecordedand

plottedusingstandardexperimentaltechniques.Typicalqualitativedataforthreetypesof

structuralmetals(mildsteel,aluminum,castiron)areshowninFigure4-1.Itisobservedthat

eachmaterialexhibitsaninitialstress-strainresponseforsmalldeformationthatisapproxi-

matelylinear.Thisisfollowedbyachangetononlinearbehaviorthatcanleadtolarge

deformation,?nallyendingwithsamplefailure.

Foreachmaterialtheinitiallinearresponseendsatapointnormallyreferredtoastheproportionallimit.Anotherobservationinthisinitialregionisthatiftheloadingisremoved,

thesamplereturnstoitsoriginalshapeandthestraindisappears.Thischaracteristicisthe

primarydescriptorofelasticbehavior.However,atsomepointonthestress-straincurve

unloadingdoesnotbringthesamplebacktozerostrainandsomepermanentplasticdeform-

ationresults.Thepointatwhichthisnonelasticbehaviorbeginsiscalledtheelasticlimit.

Althoughsomematerialsexhibitdifferentelasticandproportionallimits,manytimes

thesevaluesaretakentobeapproximatelythesame.Anotherdemarcationonthestress-strain

curveisreferredtoastheyieldpoint,de?nedbythelocationwherelargeplasticdeformation

begins.

s

70FOUNDATIONSANDELEMENTARYAPPLICATIONS

Becausemildsteelandaluminumareductilematerials,theirstress-strainresponseindicatesextensiveplasticdeformation,andduringthisperiodthesampledimensionswillbechanging.Inparticularthesample’scross-sectionalareaundergoessigni?cantreduction,andthestresscalculationusingdivisionbytheoriginalareawillnowbeinerror.Thisaccountsforthereductioninthestressatlargestrain.Ifweweretocalculatetheloaddividedbythetruearea,thetruestresswouldcontinuetoincreaseuntilfailure.Ontheotherhand,castironisknowntobeabrittlematerial,andthusitsstress-strainresponsedoesnotshowlargeplasticdeformation.Forthismaterial,verylittlenonelasticornonlinearbehaviorisobserved.Itisthereforeconcludedfromthisandmanyotherstudiesthatalargevarietyofrealmaterialsexhibitslinearelasticbehaviorundersmalldeformations.Thiswouldleadtoalinearconstitutivemodelfortheone-dimensionalaxialloadingcasegivenbytherelations?Ee,whereEistheslopeoftheuniaxialstress-straincurve.Wenowusethissimpleconcepttodevelopthegeneralthree-dimensionalformsofthelinearelasticconstitutivemodel.

4.2LinearElasticMaterials—Hooke’sLaw

Basedonobservationsfromtheprevioussection,inordertoconstructageneralthree-dimensionalconstitutivelawforlinearelasticmaterials,weassumethateachstresscomponentislinearlyrelatedtoeachstraincomponent

sx?C11extC12eytC13ezt2C14exyt2C15eyzt2C16ezx

sy?C21extC22eytC23ezt2C24exyt2C25eyzt2C26ezx

sz?C31extC32eytC33ezt2C34exyt2C35eyzt2C36ezx

txy?C41extC42eytC43ezt2C44exyt2C45eyzt2C46ezx

tyz?C51extC52eytC53ezt2C54exyt2C55eyzt2C56ezx

tzx?C61extC62eytC63ezt2C64exyt2C65eyzt2C66ezx(4:2:1)

wherethecoef?cientsCijarematerialparametersandthefactorsof2arisebecauseofthesymmetryofthestrain.Notethatthisrelationcouldalsobeexpressedbywritingthestrainsasalinearfunctionofthestresscomponents.Theserelationscanbecastintoamatrixformatas

sxsysztxytyztzx2666666437777775?C11C12áááC16C21áááááááááááááááááá

áááááC61ááááC662666666437777775exeyez2exy2eyz2ezx2666666437777775(4:2:2)

Relations(4.2.1)canalsobeexpressedinstandardtensornotationbywriting

sij?Cijklekl(4:2:3)

whereCijklisafourth-orderelasticitytensorwhosecomponentsincludeallthematerialparametersnecessarytocharacterizethematerial.Basedonthesymmetryofthestressandstraintensors,theelasticitytensormusthavethefollowingproperties(seeExercise4-1):

MaterialBehavior—LinearElasticSolids71

Cijkl?Cjikl

(4:2:4)

Cijkl?Cijlk

Ingeneral,thefourth-ordertensorCijklhas81components.However,relations(4.2.4)

reducethenumberofindependentcomponentsto36,andthisprovidestherequiredmatch

withform(4.2.1)or(4.2.2).LaterinChapter6weintroducetheconceptofstrainenergy,and

thisleadstoafurtherreductionto21independentelasticcomponents.ThecomponentsofCijkl

orequivalentlyCijarecalledelasticmoduliandhaveunitsofstress(force/area).Inorderto

continuefurther,wemustaddresstheissuesofmaterialhomogeneityandisotropy.

Ifthematerialishomogenous,theelasticbehaviordoesnotvaryspatially,andthusallelasticmoduliareconstant.Forthiscase,theelasticityformulationisstraightforward,leadingtothe

developmentofmanyanalyticalsolutionstoproblemsofengineeringinterest.Ahomogenous

assumptionisanappropriatemodelformoststructuralapplications,andthusweprimarily

choosethisparticularcaseforsubsequentformulationandproblemsolution.However,thereare

acoupleofimportantnonhomogeneousapplicationsthatwarrantfurtherdiscussion.

Studiesingeomechanicshavefoundthatthematerialbehaviorofsoilandrockcommonlydependsondistancebelowtheearth’ssurface.Inordertosimulateparticulargeomechanics

problems,researchershaveusednonhomogeneouselasticmodelsappliedtosemi-in?nite

domains.Typicalapplicationshaveinvolvedmodelingtheresponseofasemi-in?nitesoil

massundersurfaceorsubsurfaceloadingswithvariationinelasticmoduliwithdepth(seethe

reviewbyPoulosandDavis1974).Anothermorerecentapplicationinvolvesthebehaviorof

functionallygradedmaterials(FGM)(seeErdogan1995andParameswaranandShukla1999,

2002).FGMsareanewclassofengineeredmaterialsdevelopedwithspatiallyvarying

propertiestosuitparticularapplications.Thegradedcompositionofsuchmaterialsiscom-

monlyestablishedandcontrolledusingpowdermetallurgy,chemicalvapordeposition,or

centrifugalcasting.Typicalanalyticalstudiesofthesematerialshaveassumedlinear,exponen-

tial,andpower-lawvariationinelasticmodulioftheform

Cij(x)?Coij(1tax)

Cij(x)?Coijeax

(4:2:5)

Cij(x)?Coijxa

whereCoijandaareprescribedconstantsandxisthespatialcoordinate.Furtherinvestigationof

formulationresultsforsuchspatiallyvaryingmoduliareincludedinExercises5-6and7-12in

subsequentchapters.

Similartohomogeneity,anotherfundamentalmaterialpropertyisisotropy.Thispropertyhastodowithdifferencesinmaterialmoduliwithrespecttoorientation.Forexample,many

materialsincludingcrystallineminerals,wood,and?ber-reinforcedcompositeshavedifferent

elasticmoduliindifferentdirections.Materialssuchasthesearesaidtobeanisotropic.Note

thatformostrealanisotropicmaterialsthereexistparticulardirectionswherethepropertiesare

thesame.Thesedirectionsindicatematerialsymmetries.However,formanyengineering

materials(moststructuralmetalsandmanyplastics),theorientationofcrystallineandgrain

microstructureisdistributedrandomlysothatmacroscopicelasticpropertiesarefoundtobe

essentiallythesameinalldirections.Suchmaterialswithcompletesymmetryarecalled

isotropic.Asexpected,ananisotropicmodelcomplicatestheformulationandsolutionof

problems.WethereforepostponedevelopmentofsuchsolutionsuntilChapter11andcontinue

ourcurrentdevelopmentundertheassumptionofisotropicmaterialbehavior.

72FOUNDATIONSANDELEMENTARYAPPLICATIONS

Thetensorialform(4.2.3)providesaconvenientwaytoestablishthedesiredisotropicstress-strainrelations.Ifweassumeisotropicbehavior,theelasticitytensormustbethesameunderallrotationsofthecoordinatehttp://./doc/4b8f853243323968011c9279.htmlingthebasictransformationpropertiesfromrelation(1:5:1)5,thefourth-orderelasticitytensormustsatisfy

Cijkl?QimQjnQkpQlqCmnpq

Itcanbeshown(ChandrasekharaiahandDebnath1994)thatthemostgeneralformthatsatis?esthisisotropyconditionisgivenby

Cijkl?adijdkltbdikdjltgdildjk(4:2:6)

wherea,b,andgarearbitraryconstants.Veri?cationoftheisotropypropertyofform(4.2.6)isleftashttp://./doc/4b8f853243323968011c9279.htmlingthegeneralform(4.2.6)instress-strainrelation(4.2.3)gives

sij?lekkdijt2meij(4:2:7)

wherewehaverelabeledparticularconstantsusinglandm.TheelasticconstantliscalledLame′’sconstant,andmisreferredtoastheshearmodulusormodulusofrigidity.SometextsusethenotationGfortheshearmodulus.Equation(4.2.7)canbewrittenoutinindividualscalarequationsas

sx?l(exteytez)t2mex

sy?l(exteytez)t2mey

sz?l(exteytez)t2mez

txy?2mexy

tyz?2meyz

tzx?2mezx

(4:2:8)

Relations(4.2.7)or(4.2.8)arecalledthegeneralizedHooke’slawforlinearisotropicelasticsolids.TheyarenamedafterRobertHookewhoin1678?rstproposedthatthedeformationofanelasticstructureisproportionaltotheappliedforce.Noticethesigni?cantsimplicityoftheisotropicformwhencomparedtothegeneralstress-strainlaworiginallygivenby(4.2.1).Itshouldbenotedthatonlytwoindependentelasticconstantsareneededtodescribethebehaviorofisotropicmaterials.AsshowninChapter11,additionalnumbersofelasticmoduliareneededinthecorrespondingrelationsforanisotropicmaterials.

Stress-strainrelations(4.2.7)or(4.2.8)maybeinvertedtoexpressthestrainintermsofthestress.Inordertodothisitisconvenienttousetheindexnotationform(4.2.7)andsetthetwofreeindicesthesame(contractionprocess)toget

skk?(3lt2m)ekk(4:2:9)Thisrelationcanbesolvedforekkandsubstitutedbackinto(4.2.7)toget

eij?

1

2m

sijà

l

3lt2m

skkdij

MaterialBehavior—LinearElasticSolids73

whichismorecommonlywrittenas

eij?1tn

E

sijà

n

E

skkdij(4:2:10)

whereE?m(3lt2m)=(ltm)andiscalledthemodulusofelasticityorYoung’smodulus,andn?l=[2(ltm)]isreferredtoasPoisson’sratio.Theindexnotationrelation(4.2.10)maybewrittenoutincomponent(scalar)formgivingthesixequations

ex?1

E

sxàn(sytsz)??

ey?1

E

syàn(sztsx)??

ez?1

E

szàn(sxtsy)??

exy?1tn

E

txy?

1

2m

txy

eyz?1tn

tyz?

1

tyz

ezx?1tn

E

tzx?

1

2m

tzx

(4:2:11)

Constitutiveform(4.2.10)or(4.2.11)againillustratesthatonlytwoelasticconstantsare

neededtoformulateHooke’slawforisotropicmaterials.Byusinganyoftheisotropicforms

ofHooke’slaw,itcanbeshownthattheprincipalaxesofstresscoincidewiththeprincipal

axesofstrain(seeExercise4-4).Thisresultalsoholdsforsomebutnotallanisotropic

materials.

4.3PhysicalMeaningofElasticModuli

Fortheisotropiccase,thepreviouslyde?nedelasticmodulihavesimplephysicalmeaning.

Thesecanbedeterminedthroughinvestigationofparticularstatesofstresscommonlyusedin

laboratorymaterialstestingasshowninFigure4-2.

4.3.1SimpleTension

Considerthesimpletensiontestasdiscussedpreviouslywithasamplesubjectedtotension

inthexdirection(seeFigure4-2).Thestateofstressiscloselyrepresentedbytheone-

dimensional?eld

sij?

s000000002

4

3

5

Usingthisinrelations(4.2.10)givesacorrespondingstrain?eld74FOUNDATIONSANDELEMENTARYAPPLICATIONS

eij?s

E000ànEs000ànE

s2666437775Therefore,E?s=exandissimplytheslopeofthestress-straincurve,whilen?àey=ex?àez=existheratioofthetransversestraintotheaxialstrain.Standardmeasure-mentsystemscaneasilycollectaxialstressandtransverseandaxialstraindata,andthusthroughthisonetypeoftestbothelasticconstantscanbedeterminedformaterialsofinterest.

4.3.2PureShear

Ifathin-walledcylinderissubjectedtotorsionalloading(asshowninFigure4-2),thestateofstressonthesurfaceofthecylindricalsampleisgivenby

sij?0t0t0

0000

2435Again,byusingHooke’slaw,thecorrespondingstrain?eldbecomes

eij?0t=2m0

t=2m

00000

2

435andthustheshearmodulusisgivenbym?t=2exy?t=gxy,andthismodulusissimplytheslopeoftheshearstress-shearstrain

curve.(SimpleTension)(PureShear)(HydrostaticCompression)

FIGURE4-2Specialcharacterizationstatesofstress.

MaterialBehavior—LinearElasticSolids75

4.3.3HydrostaticCompression(orTension)

The?nalexampleisassociatedwiththeuniformcompression(ortension)loadingofacubicalspecimen,asshowninFigure4-2.Thistypeoftestwouldberealizableifthesamplewasplacedinahigh-pressurecompressionchamber.Thestateofstressforthiscaseisgivenby

sij?

àp00

0àp0

00àp

2

4

3

5?àpdij

ThisisanisotropicstateofstressandthestrainsfollowfromHooke’slaw

eij?

à

1à2n

E

p00

0à

1à2n

E

p0

00à

1à2n

E

p2

66

66

4

3

77

77

5

Thedilatationthatrepresentsthechangeinmaterialvolume(seeExercise2-11)isthusgiven

byW?ekk?à3(1à2n)p=E,whichcanbewrittenas

p?àkW(4:3:1)wherek?E=[3(1à2n)]iscalledthebulkmodulusofelasticity.Thisadditionalelastic

constantrepresentstheratioofpressuretothedilatation,whichcouldbereferredtoasthe

volumetricstiffnessofthematerial.NoticethatasPoisson’sratioapproaches0.5,thebulk

modulusbecomesunboundedandthematerialdoesnotundergoanyvolumetricdeformation

andhenceisincompressible.

Ourdiscussionofelasticmoduliforisotropicmaterialshasledtothede?nitionof?veconstantsl,m,E,n,andk.However,keepinmindthatonlytwooftheseareneededto

characterizethematerial.Althoughwehavedevelopedafewrelationshipsbetweenvarious

moduli,manyothersuchrelationscanalsobefound.Infact,itcanbeshownthatall?veelastic

constantsareinterrelated,andifanytwoaregiven,theremainingthreecanbedeterminedby

usingsimpleformulae.ResultsoftheserelationsareconvenientlysummarizedinTable4-1.

Thistableshouldbemarkedforfuturereference,becauseitwillprovetobeusefulfor

calculationsthroughoutthetext.

TypicalnominalvaluesofelasticconstantsforparticularengineeringmaterialsaregiveninTable4-2.Thesemodulirepresentaveragevalues,andsomevariationwilloccurforspeci?c

materials.Furtherinformationandrestrictionsonelasticmodulirequirestrainenergycon-

cepts,whicharedevelopedinChapter6.

Beforeconcludingthissection,wewishtodiscusstheformsofHooke’slawincurvilinearcoordinates.Previouschaptershavementionedthatcylindricalandsphericalcoordinates(see

Figures1-4and1-5)areusedinmanyapplicationsforproblemsolution.Figures3-9and3-10

de?nedthestresscomponentsineachcurvilinearsystem.Inregardstothese?gures,itfollows

thattheorthogonalcurvilinearcoordinatedirectionscanbeobtainedfromabaseCartesian

systemthroughasimplerotationofthecoordinateframe.Forisotropicmaterials,theelasticity

tensorCijklisthesameinallcoordinateframes,andthusthestructureofHooke’slawremains

thesameinanyorthogonalcurvilinearsystem.Therefore,form(4.2.8)canbeexpressedin

cylindricalandsphericalcoordinatesas

76FOUNDATIONSANDELEMENTARYAPPLICATIONS

sr?l(erteytez)t2mer

sR?l(eRteftey)t2meRsy?l(erteytez)t2mey

sf?l(eRteftey)t2mefsz?l(erteytez)t2mez

sy?l(eRteftey)t2meytry?2mery

tRf?2meRftyz?2meyz

tfy?2mefytzr?2mezrtyR?2meyR(4:3:2)

Thecompletesetofelasticity?eldequationsineachofthesecoordinatesystemsisgiveninAppendixA.

4.4ThermoelasticConstitutiveRelations

Itiswellknownthatatemperaturechangeinanunrestrainedelasticsolidproducesdeform-ation.Thus,ageneralstrain?eldresultsfrombothmechanicalandthermaleffects.Withinthecontextoflinearsmalldeformationtheory,thetotalstraincanbedecomposedintothesumofmechanicalandthermalcomponentsas

TABLE4-1RelationsAmongElasticConstants

En

kmlE,nEn

EEEnE,kE3kàE

6kk3kE9kàE3k(3kàE)9kàEE,mEEà2m

mEmm(Eà2m)E,lE2l

EtltR

Et3ltR6Eà3ltR4ln,k3k(1à2n)nk3k(1à2n)2(1tn)3kn1tnn,m2m(1tn)n

2m(1tn)m2mnn,ll(1tn)(1à2n)nn

l(1tn)3nl(1à2n)2nlk,m9km3kà2m

kmkà2mk,l9k(kàl)3kàll

3kàlk32(kàl)lm,l

m(3lt2m)ltml

2(ltm)3lt2m3mlR?????????????????????????????????E2t9l2t2ElpMaterialBehavior—LinearElasticSolids77

eij?e(M)ijte(T)ij(4:4:1)

IfToistakenasthereferencetemperatureandTasanarbitrarytemperature,thethermalstrainsinanunrestrainedsolidcanbewritteninthelinearconstitutiveform

e(T)ij?aij(TàTo)(4:4:2)

whereaijisthecoef?cientofthermalexpansiontensor.Noticethatitisthetemperaturedifferencethatcreatesthermalstrain.Ifthematerialistakenasisotropic,thenaijmustbeanisotropicsecond-ordertensor,and(4.4.2)simpli?esto

e(T)ij?a(TàTo)dij(4:4:3)

whereaisamaterialconstantcalledthecoef?cientofthermalexpansion.Table4-2providestypicalvaluesofthisconstantforsomecommonmaterials.Noticethatforisotropicmaterials,noshearstrainsarecreatedbytemperaturechange.Byusing(4.4.1),thisresultcanbecombinedwiththemechanicalrelation(4.2.10)togive

eij?1tnsijànskkdijta(TàTo)dij(4:4:4)

Thecorrespondingresultsforthestressintermsofstraincanbewrittenas

sij?Cijkleklàbij(TàTo)(4:4:5)

wherebijisasecond-ordertensorcontainingthermoelasticmoduli.ThisresultissometimesreferredtoastheDuhamel-Neumannthermoelasticconstitutivelaw.Theisotropiccasecanbefoundbysimplyinvertingrelation(4.4.4)toget

sij?lekkdijt2meijà(3lt2m)a(TàTo)dij(4:4:6)

ThermoelasticsolutionsaredevelopedinChapter12,andthecurrentstudywillnowcontinueundertheassumptionofisothermalconditions.

Havingdevelopedthenecessarysixconstitutiverelations,theelasticity?eldequationsystemisnowcompletewith15equations(strain-displacement,equilibrium,Hooke’slaw)for15unknowns(displacements,strains,stresses).Obviously,furthersimpli?cationisneces-

TABLE4-2TypicalValuesofElasticModuliforCommonEngineeringMaterials

E(GPa)n

m(GPa)l(GPa)k(GPa)a(10à6=8C)Aluminum68.90.34

25.754.671.825.5Concrete27.60.20

11.57.715.311Copper89.60.34

33.47193.318Glass68.90.25

27.627.645.98.8Nylon28.30.40

10.14.0447.2102Rubber0.00190.499

0:654?10à30.3260.326200Steel207

0.2980.211116413.578FOUNDATIONSANDELEMENTARYAPPLICATIONS

saryinordertosolvespeci?cproblemsofengineeringinterest,andtheseprocessesarethe

subjectofthenextchapter.

References

ChandrasekharaiahDS,andDebnathL:ContinuumMechanics,AcademicPress,Boston,1994.

ErdoganF:Fracturemechanicsoffunctionallygradedmaterials,CompositesEngng,vol5,pp.753-770,

1995.

MalvernLE:IntroductiontotheMechanicsofaContinuousMedium,PrenticeHall,EnglewoodCliffs,

NJ,1969.

ParameswaranV,andShuklaA:Crack-tipstress?eldsfordynamicfractureinfunctionallygradient

materials,Mech.ofMaterials,vol31,pp.579-596,1999.

ParameswaranV,andShuklaA:Asymptoticstress?eldsforstationarycracksalongthegradientin

functionallygradedmaterials,J.Appl.Mech.,vol69,pp.240-243,2002.

PoulosHG,andDavisEH:ElasticSolutionsforSoilandRockMechanics,JohnWiley,NewYork,1974.Exercises

4-1.Explicitlyjustifythesymmetryrelations(4.2.4).Notethatthe?rstrelationfollowsdirectlyfromthesymmetryofthestress,whilethesecondconditionrequiresasimple

(CijkltCijlk)elktoarriveattherequiredconclusion.

expansionintotheformsij?1

2

4-2.Substitutingthegeneralisotropicfourth-orderform(4.2.6)into(4.2.3),explicitlydevelopthestress-strainrelation(4.2.7).

4-3.Followingthestepsoutlinedinthetext,inverttheformofHooke’slawgivenby(4.2.7)anddevelopform(4.2.10).ExplicitlyshowthatE?m(3lt2m)=(ltm)andn?l=[2(ltm)].

http://./doc/4b8f853243323968011c9279.htmlingtheresultsofExercise4-3,showthatm?E=[2(1tn)]andl?En=[(1tn)(1à2n)].

4-5.Forisotropicmaterialsshowthattheprincipalaxesofstraincoincidewiththeprincipalaxeso