北航计算固体力学课件-第三章

ComputationalSolidMechanics,Chapter31CHAPTER3TWODIMENSIONALFEMComputationalSolidMechanics,Chapter32§3-1PLANEELASTICPROBLEM1.CONSTITUTIVERELATIONSijkkijijijkkijijEGG212ijij2jiwhenijiscalledengineeringstrainkisdummyindex,rangeover1,2G,aretheLameconstants)1(2EG)21)(1(EGG21G21ComputationalSolidMechanics,Chapter332.GEOMETRICALRELATION0(,)(,)(,)(,)0(,)(,)xyxxyuxyxyxyvxyyxyyxεComputationalSolidMechanics,Chapter343EQUATIONSOFEQUILIBRIUMx1x2x3111111dxx111212dxx111313dxx112131333131dxx222323dxx222121dxx222222dxx0dddddddddddd3211123133313131212221213211111111xxxXxxxxxxxxxxxxDividingthroughbydx1dx2dx3,wehaveComputationalSolidMechanics,Chapter3531112111230Xxxx0jiijXxCyclicpermulationofsymbolsIftheredonotexistexternalmomentproportionaltoavolume,thisleadtothestresstensorissymmetric,ijjiInabovederivation,Twoimportantassumptionsarebasedon:(1)Thereisdefinitioneverywhereinthebody.(2)Thestressiscontinuous.0ddfx0ddqxQForrodForbeamComputationalSolidMechanics,Chapter364.PLANESTATEOFSTRESSAssumptions:(1)Uniformandthinplate(2)Twosurfacesarefree,allexternalloadinthexy-planeareindependentofcoordinatez(3)Elasticpropertiesareindependentofz0yzxzzyxyxE)1(2111Eεσ][Tyxσ][TyxεComputationalSolidMechanics,Chapter37)(1)(yxyxzEQuestion:doesthedirectstrainzvanish?No,becausethetwosurfacesarefree!ComputationalSolidMechanics,Chapter385.PLANESTATEOFSTRAINAssumptions:(1)Uniformandlongcolumn(2)allexternalloadparalleltothesectionareindependentofcoordinatez(3)Elasticpropertiesareindependentofz0yzxzzyxyxE)1(2211111)21)(1()1(ComputationalSolidMechanics,Chapter39Question:doesthedirectstresszvanish?)()(yxyxzNo,thedeformationalongzdirectionisconstrained.Attentions:(1)Theequilibriumandgeometricalequationsaresameforplanestressandplanestrainproblems?(2)Theuniquedistinctionistheconstitutiverelations,andtransformationisavailable.SameFEModelcanbeused2(planestress)/(1)/(1)(planestress)EEPLANESTRAINComputationalSolidMechanics,Chapter3106.TheMinimumPotentialEnergyPrinciplexfyfxpypBuB()dd()dxyAxyBUfufvxypupvsEεεεσTT2121UThestrainenergydensityfunctionisBu:fixedboundaryB:freeboundaryσd)(dd)(ByxAyxsvpupyxvfufUComputationalSolidMechanics,Chapter311xvyuyvxuUyxεEεTuσσd)()(d)(ddBBxyyyxxByxAyyxxsvnnunnsvpupyxvfxyufyx00yyxxfxyfyxyxyyxyxxpnnpnncos),cos(xnnxsin),cos(ynnyEulerequationsNaturalboundaryconditionsComputationalSolidMechanics,Chapter3127.RayleighQuotient02ddstTyxUAAyxvuT220d)d(21KineticcoefficientwithoutconcentratedmassMaximumpotentialenergywithoutconcentratedspringsTT11dddddd22AAAUxyxyxyσεεEεComputationalSolidMechanics,Chapter3138.FiniteElements(1).Rectangularelements:inCartesiancoordinates,suitableforregulardomain.(2).Triangularelements:inareacoordinates,suitableforregularandirregulardomain,ortakingastransitionelementofdifferentsizerectangularelement(3).Iso-parametricelements:inCartesiancoordinates,suitableforanydomain.ComputationalSolidMechanics,Chapter3149.RectangularElements)1,1()1,1()1,1()1,1(b1234)()(),(yuxuyxuUseonedimensionalfunctions:niiiaxu1)()(niiiayu1)()(144112212,2yyyyyxxxxxQuestions(3):(1)C0orC1?Completeness?(2)Thenumberofnodalparametersforeachnode?(3)Compatibilityatnodeandalongsideline?ComputationalSolidMechanics,Chapter3151111212122221212121233T331212(,)()()nnjjjiiijinnijijijuxyuuuuuuuuuNu41112213224121(,)iiiuxyuuuuNuBilinearelement:n=241112213224121(,)iiivxyvvvvNvsimilarlyQuestions(2):(1)why123411212212(,,,)(,,,)uuuuuuuu?(2)Howtointerprettheshapefunctiongeometrically?ComputationalSolidMechanics,Chapter316vukkkkvuvvvuuvuuAyxU21ddTTdd2122121)1(2dd221111222uvvuvabvbaubauabEyxUAThediscretizationofstrainenergygivestheelementstiffnessmatrixAttention:(1)Stiffnessmatrixdependsona/bonly.largeorsmallelementhasthesamestiffness?(2)Finedmeshisgoodornot?(3)Thenodaldisplacementareexact?ComputationalSolidMechanics,Chapter317224(1)8844833493433893884489343349388448338934338448938334888334938889348SymEk83388(1),2ab双线性矩形单元ComputationalSolidMechanics,Chapter3双线性矩形单元的质量矩阵1840420402041020401020420102040201020436称对abm][44332211TvuvuvuvudComputationalSolidMechanics,Chapter319dd41111lkjivvuuab