Contents:Textbook:Example:Topic16Use of ElasticConstitutiveRelationsinUpdatedLagrangianFormulation•UseofupdatedLagrangian(U.L.)formulation•DetailedcomparisonofexpressionsusedintotalLagrangian(T.L.)andV.L.formulations;strains,stresses,andconstitutiverelations•StudyofconditionstoobtaininageneralincrementalanalysisthesameresultsasintheT.L.formulation,andviceversa•Thespecialcaseofelasticity•TheAlmansistraintensor•One-dimensionalexampleinvolvinglargestrains•Analysisoflargedisplacement/smallstrainproblems•Exampleanalysis:LargedisplacementsolutionofframeusingupdatedandtotalLagrangianformulations6.4, 6.4.16.19SO FAR THE USE OFTHE T.L. FORMULATIONWAS IMPLIEDNow suppose that we wishtouse theU.L.formulationinthe analysis.Weask•Isit possibletoobtain, using theU.L.formulation, identically the samenumerical results (for each iteration)asare obtained using theT.L.formulation?Inother words, the situationisProgram 1TopicSixteen16·3Transparency16-1Transparency16-2• Only T.L. formulationis implemented- Constitutive relations areJSij-= function of displacementsdoSij-=oCijrsdoErsInformation obtained from physicallaboratory experiments.P~~Program 1 results16-4Elastic Constitutive Relations in u'L.F.Transparency16-3Program 2 Question:1-------------1• Only U.L. formulation Howcanweobtainisimplemented with program 2- Constitutive relations are identically the samet'Tij.=~CDresultsasaredtSij.=~® obtained from'-------'program1?Transparency16-4Toanswer, we consider the linearizedequations of motion:~vOCijrSoers80ei~°dV +~vJSijo80'T)i~OdV]~T.L._t+~tt 0- m-0vOSi~ 80ei~dVTermsusedinthe formulations:T.L.U.L.formulation formulation Transformationtfc°dV1tdV°dV =oPtdV°vtvPt tOeij,OT)i}teij'tTli}Oei}=OXr,iOXs,}terst tOT)i}=OXr,iOXs,}tT)rsOOeij,OOT)i}Oteij,OtT)i}OOei}=JXr,iJXS,}OtersOOT)i}=JXr,iJXS,}OtT)rsDerivationofthesekinematicrelationships:A fundamental property ofJCi}isthatJCi}dOXidOx}=~(CdS)2-(OdS)2)Similarly,t+dJci}dOXidOx}=~((t+dtds)2-(OdS)2)andtCrsdXrdtxs=~((t+dtds)2-CdS)2)TopicSixteen16-5Transparency16-5Transparency16-616-6ElasticConstitutiveRelationsinu'L.F.Transparency16-7timeDTransparency16-8FiberdO~of lengthodsmoves tobecomedt~of lengthtds.Hence, by subtraction,weobtainOCi}doXidOx}=tCrsdtxrdtxsSince this relationship holds forarbitrary material fibers,wehavet tOCi}=OXr,iOXs,pCrsNowwesee thatt t t toe~+o'T)~=OXr,iOXs,}ters+OXr,iOXS,}t'T)rsSince the factors6Xr,i6Xs,~do notcontain the incremental displacementsUi,wehavettl".oe~=OXr,ioXs,pers~InearInUit t dt""oTJ~=OXr,ioXsJtTJrs~quaraICInUiInaddition,wehaveoOei}=6Xr,i6Xs,}otersOOTJi}=6Xr,i6Xs,}OtTJrsThese follow because the variationistakenontheconfi~urationt+Lltandhence the factorsOXr,i6Xs,}are takenasconstant during the variation.TopicSixteen16-7Transparency16-9Transparency16-1016-8ElasticConstitutiveRelationsinU.L.F.Transparency16-11Transparency16-12Wealso haveT.L. U.L.Transformationformulation formulation0JSi}ttsPOt01'i}o i} = t t Xi,m1'mnt Xj,nP0oCijrstC~rsCPo0Coooits= t t Xi,a t Xj,b t abpq t Xr,p t Xs,qP(Tobederived below)Consider the tangent constitutivetensorsoC~rsandtCiys:Recall thatdOSi}=oCijrSdoErs~d'ff'1dtS"=tG·dtEI erentlaIiIrsrsincrements.s--Nowwenote thatodOSii· =~PXiaPXibdtSabp , "doErs=JXp,rJxq,sdtEpqHence(~PXi,.?Xj.bd,Sab)=DC".(6xp,rD'xq,Sd'E~...I I • TdoSikdoErsSolving fordtSabgivesd,S.b=(~6x.'i6xb.j.DCijrs6xp,rJx."s)litEpq\ 'tCabpqAndwetherefore observe that thetangent material relationshiptobeusedistC_Ptt C t tt abpq -0-=-OXa,iOXb.J-o·iJ"soXp,r oXq,sPTopic Sixteen16-9Transparency16-13Transparency16-1416-10Elastic Constitutive RelationsinD.L.F.Transparency16-15Transparency16-16Now compare each of the integrals appearingintheT.L.and U.L. equations of motion:1)favdSij.8oeij.°dV =ftvtTij.8tettdV ?, ,True, asweverifybysubstituting theestablished transformations:Lv(~PXi,m~mnPX~n)(dXr,idxs.;.8ters) °dV.IS. 30eiLo t •=rtTmn8ters(~Xi,mdXr,i)(~Xj.,ndxs,j.)~p°dV~vp3m•3ns-1-dV-=rtTmn8temntdVJtv2)fovdSt8011t °dV =JvtTt8tl1ttdV ?I ITrue,asweverifybysubstituting theestablished transformations:fav(:PPXi,mtTmnPX~n)(dXr,idxs,j.8tl1rs)°dV. P " ,. .riSt30TJij-=rtTmn8tl1rs(PXi,mdXr,i)(?X~,n dxs,~)~P°dVJovP----=---~~3m• 3nsIdV= rtTmn8tl1mntdVJtv3)fovoCijrsoers80eij°dV = ttC~rsters&e~tdV?I ITrue, as we verify by substituting theestablished transformations:fa(0 )Po 0 0 00vtptXi,atXj,btCabpqtxr,ptXs,qX\ I IOCijrs(dXk,rdxe.stekf)(dXm,idxn.j-8temn) °dV\ T I \ JProvided the establishedtransformations are used, the threeintegrals are identical. Therefore theresulting finite element discretizationswill alsobeidentical.(JKL+ JKNddU=t+~tR- JF(~KL+~KNddU=t+~tR-~FTopicSixteen16-11Transparency16-17Transparency16-18dKL=asLdKNL=asNLdE=tEThesame holds foreachequilibrium iteration.16-12Elastic Constitutive Relations in u'L.F.Transparency16-19Transparency16-20Hence, to summarize once more,program 2 gives the same results asprogram1,providedCD~The Cauchy stresses arecalculated fromtt_PttstTiJ--0:::-OXi,m 0 mn oXj,nP®~The tangent stress-strain law iscalculated fromtC_Ptt C t ttifs-upOXi,a OXj,b 0 abpq oXr,p oXs,qConversely, assume that the materialrelationships for program 2 are given,hence, from laboratory experimentalinformation,tTij-andtCijrsfor theU.L.formulation are given.Then we can show that, provided theappropriate transformationsots-POt0oit--ttXi,m Tmn tXj,nPoC-Po0Cooo ijrs --ttXi,a tXj,b t abpq tXr,p tXs,qPare usedinprogram 1 with theT.L.formulation, again the same numericalresults are generated.Hence the choice of formulation(T.L.vs.U.L.)isbased solelyonthenumerical effectiveness of the methods:• The~BLmatrix(U.L.formulation)contains less entries than thetiBLmatrix(T.L.formulation).• The matrix product BT C Bislessexpensive using theU.L.formulation.•Ifthe stress-strain lawisavailableinterms oftis,then theT.L.formulationwillbeingeneral most effective.- Mooney-Rivlin material law-Inelasticanalysis allowing for largedisplacements / large rotations, butsmall strainsTopic Sixteen16-13Transparency16-21Transparency16-2216-14Elastic Constitutive RelationsinU.L.F.Transparency16-23Transparency16-24THE SPECIAL CASEOF ELASTICITYConsider that the componentsJCi1Saregiven:tstcta i} = aijrsaErsFrom the above discussion, to obtainthe same numerical results with theU.L. formulation, we would employtt _ P t(tct)tTi} -opaXi,m a mnrsaErs aXj.ntCPtt C t tt ilS = 0 aXi,aaXj,ba abpq aXr,p aXs,qPWesee thatinthe above equation, theCauchy stresses are related to theGreen-Lagrange strains by atransformation acting onlyonthe mand n