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1、COMPUTERMETHODSINAPPLIEDMECHANICSANDENGINEERING4X(1985)101-118NORTH-HOLLANDCONSISTENTTANGENTOPERATORSFORRATE-INDEPENDENTELASTOPLASTICITYJ.C.SIMOandR.L.TAYLORDivisionofStructuralEngineeringandStructuralMechanics,DepartmentofCivilEngineering,UniversityofCalifornia,Berke
2、ley,CA94720,U.S.A.Received1May1984Itisshownthatforproblemsinvolvingrateconstitutiveequations,suchasrate-independentelastoplasticity,thenotionofconsistencybetweenthetangent(stiffness)operatorandtheintegrationalgorithmemployedinthesolutionoftheincrementalproblem,playsac
3、rucialroleinpreservingthequadraticrateofasymptoticconvergenceofiterativesolutionschemesbaseduponNewtonsmethod.Withintheframeworkofclosest-point-projectionalgorithms,amethodologyispresentedwherebytangentoperatorsconsistentwiththisclassofalgorithmsmaybesystematicallydev
4、eloped.Towit,associativeJZflowruleswithgeneralnonlinearkinematicandisotropichardeningrules,aswellasaclassofnon-associativeflowrulesareconsidered.TheresultingiterativesolutionschemepreservestheasymptoticquadraticconvergencecharacteristicofNewtonsmethod,whereasuseofthes
5、o-calledelastoplastictangentinconjunctionwitharadialreturnintegrationalgorithm,aprocedureoftenemployed,resultsinNewtontypeofalgorithmswithsuboptimalrateofconvergence.Applicationismadetoasetofnumericalexampleswhichincludesaturationhardeninglawsofexponentialtype.1.Intro
6、ductionInthecontextofthefiniteelementmethod,theformulationandnumericalsolutionofnonlinearproblemsincontinuummechanicsreliescruciallyontheweakformofthemomen-tumbalanceequation(virtualwork).Ifthematerialiselastic,onehasdirectlyavailablearesponsefunctionforthestresstenso
7、rwhichmaybeintroducedintothevirtualworkequationtoobtainanonlinearvariationalequationinvolvingonlydisplacementsanddisplacementgradients(displacementmodel).Typically,thesolutiontothisnonlinearproblemisachievedbyNewtonsmethodinwhichasequenceoflinearproblemsissolved.Ifthe
8、linearproblemisobtainedbyconsistentlylinearizingthenonlinearproblem,onehasquadraticrateof(asymp-totic)convergence.Incontrast
