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1、Numer.Math.53,513-538(1988)NumedscheMathematik~Springer-Verlag1988AFamilyofMixedFiniteElementsfortheElasticityProblemRolfStenbergHelsinkiUniversityofTechnology,InstituteofMathematics,SF-02150Espoo,FinlandSummary.Anewmixedfiniteelementformulationfortheequationsoflinearelasti
2、cityisconsidered.Intheformulationthevariablesapproximatedarethedisplacement,theunsymmetricstresstensorandtherotation.Therota-tionactasaLagrangemultiplierintroducedinordertoenforcethesymmetryofthestresstensor.Basedonthisformulationanewfamilyofbothtwo-andthree-dimensionalmixe
3、dmethodsisdefined.Optimalerrorestimates,whicharevaliduniformlywithrespecttothePoissonratio,arederived.Finally,anewpostprocessingschemeforimprovingthedisplacementisintro-ducedandanalyzed.SubjectClass!fications:AMS(MOS):65N30;CR:Gt.8.1.IntroductionThepurposeofthispaperistocon
4、tinueourdiscussion1-22,24]onmixedfiniteelementmethodsinelasticity.BymixedmethodsweheremeanmethodswhereboththedisplacementvectorandthestresstensorareapproximatedsimultaneouslyusingtheHellinger-Reissnerprinciple.Thisapproachhasanumberofadvantagescomparedwithtraditionaldisplac
5、ementmethods.First,itallowsadirectapproximationofthestresseswhichusuallyarethevariablesofprimaryinterest.Second,acorrectlydesignedmixedmethod(i.e.satisfyingtheequilibriumcondition,seebelow)givesagoodmodellingofincompressibleandnearlyincompressiblematerialsforwhichstandarddi
6、splacementmethodsbreakdown.Third,theapproachhasbeenshown[4,16]tobeefficientforplasticityproblems,forwhichtheeliminationofthestressesfromtheequilibriumandconstitutiveequationsisdifficult.Ithas,however,turnedouttobedifficulttofindgoodmixedmethods.Adirectapproachofusinge.g.con
7、tinuouspiecewisepolynomialsforboththestressesandthedisplacementdoesnotusuallywork.Forthisthereistwodifferentreasons.Themaindifficultyisthatusuallythesemethodsareunstable.Inpracticalcalculationsthisisseenintheappearanceofdisplacementmodeswhicharenearlyunphysical"mechanisms"(
8、oftenalsoreferredtoas"zeroenergy514R.Stenbergmodes").Infact,mechanismsareonlyavoid
