清华弹性力学课件_variational formulation

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1、TheoryofElasticityIntroductionElasticityofSolidsFieldEquationsofElasticity-DifferentialFormulationPrismaticRodsPlaneProblems–TheoryandSolutionsPlaneProblems–ApplicationsVariationalFormulationofElasticityThree-dimensionalProblemsIndex0VariationalFormul

2、ationBasicConceptsofVariationWeakSolutionsPrincipleofVirtualWorkVariationalPrinciplesNumericalMethodsBasedonEnergyPrinciples1Chapter7BasicConceptsofVariationConceptofVariationConstraintsBasicVariationalOperationsEulerEquations2Chapter7.1WeakSolutionsS

3、trongSolutionsThestrongsolutionsofelasticityrefertothecasewherethesolutionssatisfyingthecompletesetoffieldequationsofelasticityinapoint-wisemanner.3Chapter7.2equilibriumequation:thekinematicalrelation:theconstitutiverelation:theboundaryconditions:Fiel

4、dEquationsWeakSolutionsTheweaksolutions,ontheotherhand,arebasedontheenergyprinciples.keycharacteristics:Theproblemisformulatedfromacompatiblestateinthesenseofeitherakinematicalorstaticallyadmissible.Possibleinclusionofconstraintsonthecompatiblestate.4

5、Chapter7.2keycharacteristics:Asuitableenergyprincipleisselectedastheguidingprinciple.Thecontinuity(ordifferentiability)requirementforthesolutionisrelaxedtogivethenameofweaksolution.Theproblemisapproximatedinacertainsenseoftruncation.Avarietyofapproxim

6、ationmethodsexist,includingFEM,Rayleigh-Ritzmethod,themethodofweightedresiduals,etc.WeakSolutions5Chapter7.2PrincipleofVirtualWorkACompatibleField6Chapter7.3CompatibleFieldsstaticallycompatiblefieldsIfinVandon,thenthestaticfield()iscalledstaticallycom

7、patiblefields.kinematicalcompatiblefieldsifinVandon,thenthekinematicalfield()iscalledkinematicalcompatiblefields.7Chapter7.3CompatibleFieldscompletecontinuumfield.Thecombinationofthestaticfield()andthekinematicalfield()formsthecompletecontinuumfield.t

8、otallycompatiblecontinuumfieldIf,where8Chapter7.3TheoremforTotallyCompatibleFieldsTheorem:Ifacontinuumfieldisnotonlykinematicalcompatible(KC)butalsostaticallycompatible(SC),thenitmustbetotallycompatible.Proof:9Chapter7.3DeductionTheorem1Ifacon