chap3_Field Equations of Elasticity - Differential

3．FieldEquationsofElasticity-DifferentialFormulationThischapterwillexplorethefieldequationsofelasticityviewedasdifferentialequations.Thematerialspresentedherehavepartialoverlapwiththegeneralformulationincontinuummechanics.Thefocuswillbeonthesimplestcase:anelasticbodyunderinfinitesimaldeformation.Westartwiththebalancelaws.3.1BalanceLawsofMomentumandMomentFieldVariablesConsiderageneralthree-dimensionalbody(intheshapeofapotato)occupyingadomainVunderlinearelasticandinfinitesimaldeformation,asshowninthefigurebelow.Thebasicfieldvariablesincludeadisplacementvectoriu(3components),asymmetricsecondrankstraintensorij(6components),andanothersymmetricsecondrankstresstensorij(6components).Thatgivesatotalnumberoffieldvariablesof15.Ingeneral,theyareallfunctionsofthespatialcoordinatesixandtemporalvariablet.tSuSV3X2X1XFigure3.1Athree-dimensionalbodyObviously,oneneedstoformulate15equationstosolvethesefieldvariables.Toshortentheequationstobederived,thetemporalandspatialderivativesareabbreviatedas:tandiix,inthesequel.BalanceLawofLinearMomentumConsiderthebalanceofthelinearmomentumfirst.Fromthepreviousknowledgeincontinuummechanics,onehasiijijuf,(3.1)wheredenotesthedensity(massperunitvolume)ofthebody,andifthebodyforceperunitvolume.Forthespecialcaseofquasi-staticmotionwheretheparticlevelocityismuchlessthanthestresswavespeedsthatwillbespecifiedafter(3.20),thebalanceequation(3.1)reducestotheequilibriumone:0,ijijf.(3.2)BalanceofAngularMomentumNextconsiderthebalanceofangularmomentum.Iftheeffectofbodycouplecanberegardedashigherorderinfinitesimal,theequilibriumofaninfinitesimalcubewillleadtothefollowingreciprocaltheoremofshearstress:jiij.(3.3)Abasicassumptiontoderive(3.3)isthatthebodymoment(Vm)ismuchlessthanthemomentVintroducebythesurfacetractiononthecube.Iwouldliketoremindyouthattheassumption(3.3)onasymmetricstresstensorsometimesbreaksdown.Thecaseofstrongmagneticfieldisoneexception,alongwiththeothers.ThetheoryofCossarabrotherswereproposedmorethanacenturyagotodealwiththesituation,where9independentcomponents(insteadofsix)ofstresstensorshouldbeengaged.Inmorerecentliteratures,theasymmetricshearstressesaretermedthe“couplestresses”.Mindlinwaslargelyresponsibleindevelopingthecouplestressconceptinelasticity.Thecurrentdevelopmentsofstraingradienttheory(suchasthoseundertakenbyK.C.Hwangandhiscollaborators)somewhatborrowtheideaofasymmetricstressundertheframeworkofcouplestresselasticity.3.2CompatibilityEquationKinematicsUnderinfinitesimaldeformation,thestraintensorcanbeeasilyderivedasthesymmetricgradientofthedisplacementvector:ijjiijuu,,21.(3.4)The6symmetriccomponentsofthestraintensorcanbeexpressedthrough3displacementcomponents.Accordingly,thestraintensorcannotbearbitrarilyprescribed.Otherwise,gaporoverlapwouldoccurinthedeformationmanifold.Theconditionstoavoidgaporoverlapindeformationarecalledthecompatibilityequations.gapoverlapcompatibilityyyyFigure3.2CompatibilityofdeformationCompatibilityConditionsWeexaminethecompatibilityconditioninthegeneralthree-dimensionalcase.From(3.4),itisstraightforwardtoverifythatthesymmetricdoublecurloperationonthestraintensorwouldvanish0,mnklijnilmjkLeeε(3.5)wheredenotesthegradientoperator,thevectorproduct,andmjkethealternationsymbol.Theyareallintroducedinthecourseofcontinuummechanics.Fromthefirstglance,equations(3.5)seemstorepresent9conditionsforthe“incompatibletensor”mnL.However,alargeportionofequationsin(3.5)areredundant.ListedbelowaretwousefulpropertiesformnLtoremovetheredundancy:Property1SymmetryofmnLnmmnLL(3.6)ThispropertycanbeprovedbymeansoftheidentityklkiknjljijnmlmimnnilmjkeebetweenthealternationsymbolmjkeandtheKroneckerdeltamj.Property2Bianchiidentity0,nmnL(3.7)Namely,the“incompatibletensor”mnLisdivergencefree.Conditions(3.6)and(3.7)give6constraintsontheincompatibletensormnL.Therefore,among9compatibilityconditionsin(3.5),onlythreeofthemareindependent.Forthespecialcaseofplanarelasticity,thestraincompatibilitydegeneratestoascalarcondition,writtenas,,(3.8)wheretheGreekindicesonlyhavetherangefrom1to2.SimplyandMultiplyConnectedRegionFigure3.3Simply-connectedregionsThecompatibilityconditions(3.5)or(3.8)areexpressedinalocalsense,namelyinthevicinityofagivenpoint.Thestraincompatibilityalsorequiresglobalimpact,intermsofthesinglevalueconditionofthedisplacementfields.Toillustratethispoint,onehastodistinguishasimply-connectedregionandamultiply-connectedone.TheformerisshowninFigure3.3.Anyclosedcurvesinasimply-connectedregioncancontinuouslyshrinktoapoint.Severalcasesformultiply-connectedregionsaredelineatedinFig.3.4,whereatleastsomeclosedcurvescannotbeshrunkontoapoint.Figure3.4Multiply-connectedregions.GlobalCompatibilityConsidertwopoints,AandB,asshowninFig.3.5.Onemayaskthequestion:ifthedisplacementofpointAisknown,canweuniquelycomputethedisplacementatpointBfromtheinformationofthestrainfield?SABFigure3.5FromthedisplacementatpointAtothedisplacementatpointBThereareinfinitelymanypathstojointpointAandpointB.Takeanarbitrarypath,markbySinFigure3.5.Ifthedispl