Nonlinear quasi-static finite element formulations

COMMUNICATIONSINNUMERICALMETHODSINENGINEERINGCommun.Numer.Meth.Engng(2009)PublishedonlineinWileyInterScience().DOI:10.1002/cnm.1262Nonlinearquasi-staticfiniteelementformulationsforviscoelasticEuler–BernoulliandTimoshenkobeamsG.S.PayetteandJ.N.Reddy∗,†,‡AdvancedComputationalMechanicsLaboratory,DepartmentofMechanicalEngineering,TexasA&MUniversity,CollegeStation,TX77843-3123,U.S.A.SUMMARYWeakformfiniteelementmodelsforthenonlinearquasi-staticbendingandextensionofinitiallystraightviscoelasticEuler–BernoulliandTimoshenkobeamsaredevelopedusingtheprincipleofvirtualwork.Themechanicalpropertiesofthebeamsareconsideredtobelinearviscoelastic.However,largetransversedisplacements,moderaterotationsandsmallstrainsareallowedbyretainingthevonK´arm´anstraincomponentsoftheGreen–Lagrangestraintensorintheformulation.Thefullydiscretizedfiniteelementequationsaredevelopedusingthetrapezoidalruleinconjunctionwithatwo-pointrecurrencerelation.Theresultingfiniteelementequations,therefore,necessitatedatastoragefromtheprevioustimesteponly,andnottheentiredeformationhistory.MembranelockingiseliminatedfromtheEuler–BernoulliformulationthroughtheuseofselectivereducedGauss–Legendrequadrature.MembraneandshearlockingarebothcircumventedintheTimoshenkobeamfiniteelementbyemployingafamilyofhigh-orderLagrangepolynomials.ANewton–Raphsoniterativeschemeisusedtosolvethenonlinearfiniteelementequations.Copyrightq2009JohnWiley&Sons,Ltd.Received21November2008;Revised2March2009;Accepted27March2009KEYWORDS:beams;viscoelasticity;finiteelementanalysis;locking-freeelements;sheardeformablebeams1.INTRODUCTIONTherearemanyengineeringmaterialsthatcannotbeadequatelymodeledusingtheclassicalelas-ticityformulation.Onecategoryofsuchmaterialsisthesetofviscoelasticmaterials,examplesofwhichincludepolymers,concretestructuresandmetalsatelevatedtemperatures.Thetheoreticalfoundationsforviscoelasticityarewellestablished[1,2].Analyticalmethodshavebeenemployedsuccessfullyinthestudyofthemechanicalresponseofviscoelasticcontinua.TheLaplacetransform∗Correspondenceto:J.N.Reddy,AdvancedComputationalMechanicsLaboratory,DepartmentofMechanicalEngineering,TexasA&MUniversity,CollegeStation,TX77843-3123,U.S.A.†E-mail:[e-mail]‡Professor.Copyrightq2009JohnWiley&Sons,Ltd.G.S.PAYETTEANDJ.N.REDDYmethodwasemployedbyFl¨ugge[1]intheanalysisofviscoelasticbeams.ThecorrespondenceprinciplehasalsobeenusedbyChristensen[2]andFindleyetal.[3]toconvertlinearelasticitysolutionsintoviscoelasticitysolutionsthroughtheuseofintegraltransformations[4].AnalyticalsolutionsbasedontheLaplacetransformmethodorcorrespondenceprinciple,however,aretypi-callylimitedtoverysimplegeometricconfigurations,boundaryconditionsandmaterialmodels.Numericalmethodsprovideapowerfulframeworkforobtainingapproximatesolutionstoviscoelasticityproblems.Inparticular,thefiniteelementmethodhasbeenemployedsuccessfullyintheanalysisofviscoelasticbodiesbymanyresearchers.Tayloretal.[5]usedthefiniteelementmethodinconjunctionwitharecurrencerelationtosolveviscoelasticityproblemssuchthatdatafromonlytheprevioustimestep(asopposedtotheentiredeformationhistory)isneededindeterminingabody’sconfigurationatthecurrenttimestep.OdenandArmstrong[6]developedafiniteelementframeworkforthermoviscoelasticityandpresentednumericalsolutionstothick-walledcylinderproblemswithtime-dependentboundaryconditions.Intheirwork,theyextendedtherecurrenceformulationtononlinearproblems.Additionalgeneralfiniteelementformulationsforviscoelasticcontinuacanbefoundin[7–10].Althoughthree-dimensionalfiniteelementformulationsareapplicabletocontinuaingeneral,itisoftencomputationallyadvantageoustospecializethesemodelstostructuralelementssuchasbeams,platesandshells.Thereareavarietyoffiniteelementmodelsintheliteratureforviscoelasticbeams.MostofthesemodelsemploysomeformofeithertheEuler–Bernoulli(EBT)orTimoshenkobeamtheories(TBT).Themajorchallengesencounteredinanyviscoelasticfiniteelementformulationareduetotheviscoelasticconstitutiveequations,oftenexpressedinconvolutionform.Rencisetal.[11]presentedasimpleEuler–Bernoullibeamfiniteelementmodelusinganincrementalapproach.Intheiranalysis,theviscoelasticconvolutionintegralsarereplacedbycreepstrainincrementsintheformoffictitiousbodyforces.TheLaplacetransformapproachhasbeenemployedbyseveralresearchers[12–14]inconjunc-tionwiththefiniteelementmethod.Chen[12]successfullyanalyzedviscoelasticTimoshenkobeamsbyconvertingthetimedependentandconvolutionformofthefiniteelementequationsintoasetofalgebraicequationsinsspace.ThesolutioninthetimedomainwasdeterminedthroughanumericalinversionoftheLaplacetransform.Inhisanalysis,ChenassumedthatthePoissonratioisconstant.ThisassumptionisconsistentwiththefindingsofZheng-youetal.[15],whopresentedanalyticalsolutionsforTimoshenkobeamswithtime-dependentandtime-independentPoisson’sratios.Ak¨ozandKadioglu[13]presentedtwoTimoshenkobeamfiniteelementsusingtheLaplace–Carsonmethodandamixedformulation.AsintheworkbyChen[12],thefiniteelementformulationsrequirenumericalinversionfromtheLaplace–Carsondomainbacktothetimedomain.Temeletal.[14]studied