A new mixed formulation for elasticity 1988 Arnold Falk NM

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1、NumerischeNumer.Math.53,13-30(1988)Mathematik9Springer-Verlag1988ANewMixedFormulationforElasticityDouglasN.Arnold2,.andRichardS.Falk2,it.tDepartmentofMathematics,UniversityofMaryland,CollegePark,MD20742,USA2DepartmentofMathematics,RutgersUniversity,NewBrunswick,NJ08903,USADedicatedtoProfesso

2、rIvoBabugkaontheoccasionofhissixtiethbirthdaySummary.Weproposeanewmixedvariationalformulationfortheequationsoflinearelasticity.Itdoesnotrequiresymmetrictensorsandconsequentlyiseasytodiscretizebyadaptingmixedfiniteelementsdevelopedforscalarsecondorderellipticequations.SubjectClassifications:AM

3、S(MOS):65N30;73C35;73K25;CR:G1.8.1.IntroductionInthispaperwepresentanewmixedvariationalformulationfortheproblemoflinearelastostatics.OurformulationisverysimilartotheclassicalHellinger-Reissnerformulation,butappearssuperiorforfiniteelementdiscretization.TomakeplaintherelationbetweentheHellinge

4、r-Reissnerformulationandthepresentone,weconsiderfirstanelasticbodyoccupyingaregiong?inEuclideann-space(n=2or3)subjecttogivenbodyforcesfandwhosedisplacementgonF-0Risknown.TheHellinger-Reissnerprincipleseeksasaddle-pointofthequadraticfunctionalJ(~,v)=~[
89~g-~n.(1.1)g?0~Thevariables~andvrangeov

5、erspacesofsuitablysmoothfunctionsonQwithvaluesin~,+,thespaceofsymmetricnxntensors,and~=IR",respective-ly.ThefourthordertensorAisthecompliancetensor,whichcharacterizestheelasticproperitiesofthematerial.FurthernotationsareexplainedinSect.2.Underreasonableassumptionsthereisauniquesaddle-point(if

6、,u)of(l.1)and,moreover,~isthestressfieldanduthedisplacementfield.TheEuler-Lagrangeequationsassociatedwith(1.1)formanellipticsystemoforder2n.Thepresentformulationalsoseeksasaddle-pointofaquadraticfunctionaloftheform(1.1).Thefunctionaldiffersonlyinthatthecompliancetensoris*SupportedbyNSFGrantDM

7、S-8313247**SupportedbyNSFGrantDMS-840261614D.N.ArnoldandR.S.Falkreplacedbyadifferentfourthordertensor,whichdependsonAinasimplefashion.AmoreessentialdifferenceisthatinourformulationthevariablerangesoverallsuitablysmoothfunctionswithvaluesinIR(