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1、CONTINUUMMECHANICS-ImplicitConstitutiveRelations-K.R.RajagopalIMPLICITCONSTITUTIVERELATIONSK.R.RajagopalTexasA&MUniversity,USAKeywords:ConstitutiveEquations,Implicitmodels,ImplicitTheories,Solid,FluidsContents1.Introduction2.ASimpleImplicitModel3.MoreGeneralImpli
2、citTheoriesforFluids4.ImplicitTheoriesforElasticSolids5.MoreGeneralImplicitTheories6.ConcludingRemarksGlossaryBibliographyBiographicalSketchSummaryClassicalConstitutiveequationsliketheNavier-Stokesmodelforfluidsandthelinearizedelasticmodelforsolidsareexplicitmode
3、lsinthattheyprovideexplicitexpressionsforthestressintermsofkinematicalquantities.Similarly,theconstitutiveequationforaSimpleMaterialisalsoanexplicitequationforthestressintermsofthehistoriesofthedeformationgradientandthedensity.However,manyratetypemodelsthathavebe
4、endevelopedtodescribeviscoelasticandinelasticmaterialsareimplicitinthataequationisprovidedforthestress,anditstimerates,aswellasappropriatekinematicalquantitiesandtheirtimerates.Evenwhentimeratesarenotinvolved,suchimplicitmodelsareveryusefulindescribingalargeclass
5、ofmaterials,especiallythosewhereinthematerialmodulidependontheLagrangemultiplierthatisassociatedwithaconstraint.Thus,forexample,suchimplicitequationscandescribeincompressiblematerialsinwhichthematerialmodulidependonthepressure(meannormalstress).Inthischapterwedis
6、cussverybrieflytheroleofimplicitconstitutiverelationsinUNESCO–EOLSSmechanics.1.IntroductionSAMPLECHAPTERSClassicalconstitutiverelationssuchasthosethatareusedtodescribethelinearizedelasticandlinearviscoelastic(Tobeprecise,themodelshouldalsobereferredtoasthelineari
7、zedviscoelasticmodelasthestrainthatisusedintheconstitutiverepresentationisthelinearizedstrainratherthanafullyandpropernon-linearstrainmeasure.)responseofsolids,namelyHooke’slaw(seeHooke’sPotentiadeResistuvainGunther(1931))andthelinearviscoelasticsolidmodel(Boltzm
8、ann(1874)),aswellastheEulerfluidmodel(Euler(1752),(1755)andtheNavier-Stokes(Navier(1823),Poisson(1831),St.Venant(1843),Stokes(1845))modelsforcompressibleandinc
