MaterialNonlinearAnalysis1MaterialNonlinearAnalysisAfundamentaldifferencebetweenelasticandplasticmaterialbehaviorsisthatnopermanentdeformationsoccurinthestructureinelasticbehavior,whereaspermanentorirreversibledeformationsoccurinthestructureinplasticbehavior.■PlasticitytheoryThecomponentsofstaticplasticstrainareconstitutedbythefollowingassumptions:�Constitutiveresponseisindependentoftherateofdeformation.�Elasticresponseisnotinfluencedbyplasticdeformation.�Additivestraindecompositionintoelasticandplasticpartsisdefinedbyεεε=+���ep(1)where,ε�:totalstrainsε�e:elasticstrainsε�p:plasticstrainsAndthefollowingbasicconceptsareusedtoformulatetheequations:�Yieldcriteriatodefinetheinitiationofplasticdeformation�Flowruletodefinetheplasticstraining�HardeningruletodefinetheevolutionoftheyieldsurfacewithplasticstrainingYieldcriteriaTheyieldfunction(orloadingfunction),F,whichdefinesthelimitfortherangeofelasticresponse,isasfollows(Fig.2.16):(,,)(,)()0ppepFσεκσσεκε=−≤����(2)ANAYSIS&DESIGN2where,σ�:currentstressesσe:equivalentoreffectivestressκ:hardeningparameterwhichisafunctionofpε�εp:equivalentplasticstrainInclassicalplasticitytheory,astateofstressatwhichthevalueoftheyieldfunctionbecomespositiveisnotadmissible.Whenyieldingoccurs,thestateofstressiscorrectedbyscalingplasticstrainsuntiltheyieldfunctionisreducedtozero.Thisprocessisknownastheplasticcorrectorphaseorreturnmapping.Fig2.16GeometricillustrationofassociatedflowruleandsingularityFlowruleTheflowruledefinestheplasticstraining,whichisexpressedasfollows(Fig2.16):ε�pdaSmoothPlasticpotentialσσ==g()F()0��Cornerσ�aσ�,bελσ∂=∂��pfdε�pdσ�dMaterialNonlinearAnalysis3ελλσ∂==∂b���pgddd(3)where,σ∂∂�g:thedirectionofplasticstrainingλd:plasticmoduluswhichidentifiesthemagnitudeofplasticstrainingThefunctiongistermedasthe‘plasticpotential’function,whichisgenerallydefinedintermsofstressinvariants.Ifg=F,itistermedas‘associatedflowrule’,andifg≠F,itisreferredtoas‘non-asso...
