【课程】弹性力学-课件

【课程】弹性力学-课件0TheoryofElasticityIntroductionElasticityofSolidsFieldEquationsofElasticity-DifferentialFormulationPrismaticRodsPlaneProblems–TheoryandSolutions

PlaneProblems–ApplicationsVariationalFormulationofElasticityThree-dimensionalProblemsIndex0TheoryofElasticityIntroductElasticityofSolids

DefinitionofElasticity

TwoPhysicalOriginsofElasticity

TensorDescriptionofElasticity

PhysicalFoundationofElasticSymmetry

1Chapter2ElasticityofSolidsDefinitioReferencesJ.H.Weiner，Statisticalmechanicsofelasticity,Wiley,1981Green&Zerna，Theoreticalelasticity,1968Ashby&Jones，Engineeringmaterials2Chapter2ReferencesJ.H.Weiner，StatisticDefinitionofElasticityDifferencebetweensolidsandfluids MechanicsofSolids,TheNewEncyclopediaofBritannica,15thedition,Vol.23,pp.734-747,2002,“Amaterialiscalledsolidratherthanfluidifitcanalsosupportasubstantialshearingforceoverthetimescaleofsomenaturalprocessortechnologicalapplicationofinterest.”

J.R.Rice3Chapter2.1DefinitionofElasticityDifferDefinitionofElasticityElasticity

Where

ExplicitdependenceonXcanbeeliminatedforhomogeneousmaterial4Chapter2.1DefinitionofElasticityElastiDefinitionofElasticityRemarksStressisirrelevanttothestrainrate,aswellastothehistoryofdeformation.Nohysteresis:theoriginalconfigurationisrecoveredafterunload. infinitesimaldeformation homogeneousmaterial linearelasticity6Chapter2.1DefinitionofElasticityRemarkDefinitionofElasticityHyperelasticityTwoassumptions:Theresponseoftheelasticbodyonlydependsonitscurrentstate.Thecurrentstateofanelasticbodycanbedescribedbyatensor.

PathindependentconditionbygreatmathematicianGreen7Chapter2.1DefinitionofElasticityHypereDefinitionofElasticityHyperelasticitylinearelastic:

generalizedHooke’slaw:8Chapter2.1DefinitionofElasticityHypereTwoPhysicalOriginsofElasticity

EnergeticandEntropicStresses

Helmholtzfreeenergy:

Maxwellrelation:9Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityEnergeticandEntropicStresses10Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystalsTheprimarybonding

ionic,covalentormetallic

meltat1000-5000K

Thesecondarybonding

VanderWaalsandhydrogenbonds

meltat100-500K

11Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystals12Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystals

13Stiffness：Maximuminter-atomicforceoccursat

ElasticityoflinearHooke’slaw

Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystals,crystallatticesites

acompositecell

14Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystals,15Chapter2.2siliconTwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityCrystals,16Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymers

Characteristic:highlyelasticintermsoflargeextensibilitylowelasticmodulusnearlyincompressibleGough-Jouleeffect

17Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymers18Chapter2.2backbone:taunt,covalentsidegroups:muchweaker

TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityGaussiandistribution

19Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymersGaussiandistribution

bdenotestheeffectivelength

20Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymersconfigurationentropy

Theentropystress

Stressiscausedbythereductionofconfigurationentropyduetoelongation!vibrationentropy

Sphardlychanges

21Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymers

Neo-HookeianlawJamesandGuth(1947)

:thestretchingratioN:thenumberoflinksperunitvolume

22Chapter2.2TwoPhysicalOriginsofElastiTwoPhysicalOriginsofElasticityLongChainPolymerstwonoticeabledeviations23Chapter2.2TwoPhysicalOriginsofElastiTensorDescriptionofElasticity

VoigtSymmetryForathree-dimensionalproblem,allindicesi,j,kandlmayhave3possiblenumbers.Thatgivesthemaximumpossiblecombinationsofindicesas34=81.

24Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityVoigtSymmetry25Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityVoigtSymmetry26Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityVoigtSymmetryVoigtrelationoftheelasticitytensor27Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityMatrixRepresentation21elasticconstants

28Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityCoordinateTransform,29Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityReflectiveSymmetry30Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityReflectiveSymmetryOnly13elasticconstants31Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityReflectiveSymmetryorthotropic:9

independentelasticconstants

32Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityRotationSymmetry2-folds:3-folds:4-folds:6-folds:

33Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityTransverseIsotropy

5independentelasticconstants

34Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityIsotropyarbitraryrotation

VoigtSymmetryGeneralizedHooke’slaw

35Chapter2.3LameConstantsTensorDescriptionofElasticiTensorDescriptionofElasticitySimpleShearshearmodulus36Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityDilation37Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityDilationGeneralcase

GeneralizedHooke’slaw

Strainenergy

38Chapter2.3TensorDescriptionofElasticiTensorDescriptionofElasticityUniaxialTension39Chapter2.3TensorDescriptionofElasticiPhysicalFoundationofElasticSymmetry

CrystalsandBravaisLattices40Chapter2.4PhysicalFoundationofElasticCauchy’ssymmetryadds

(j,l)(l,j)

PhysicalFoundationofElasticSymmetryCauchy’sRelationVoigtsymmetry:(I,j)(j,i),(k,l)(l,k),(i,j)(k,l)

6additionalrelations:

only15independentelasticconstants

41Chapter2.4Cauchy’ssymmetryaddsPhysicalPhysicalFoundationofElasticSymmetryCentralForceAssumptionofCauchy

inter-atomicpotential:42Chapter2.4PhysicalFoundationofElasticPhysicalFoundationofElasticSymmetryCentralForceAssumptionofCauchyDerivativesoftheinter-atomicpotential:Independentcombinations:(1111,2222,3333),(1112,1113,2221,2223,3331,3332),(1122,1133,2233)and(1123,2213,3312).

only15independentelasticconstants

43Chapter2.4PhysicalFoundationofElasticPhysicalFoundationofElasticSymmetryCentralForceAssumptionofCauchySpecialcaseofisotropicmaterials:

44Chapter2.4PhysicalFoundationofElasticPhysicalFoundationofElasticSymmetryGreenElasticityversusCauchyElasticityGreenelasticitytheorybasedontheexistenceofanelasticpotentialthatisindependentofthedeformationpathCauchyElasticitybasedonthecentralforceassumptionofinter-atomicpotential

Formaterialswithionicandcovalentbonds,thePoisson’sratioiscloseto1/4.45Chapter2.4PhysicalFoundationofElasticTheoryofElasticityIntroductionElasticityofSolidsFieldEquationsofElasticity-DifferentialFormulationPrismaticRodsPlaneProblems–TheoryandSolutions

PlaneProblems–ApplicationsVariationalFormulationofElasticityThree-dimensionalProblemsIndex0TheoryofElasticityIntroductiFieldEquationsofElasticity-DifferentialFormulationBalanceLawsofMomentumandMoment

CompatibilityEquationFieldEquationsofDynamicElasticityQuasi-staticFieldEquationsConstraintsBoundaryConditionsFormulationofElasticity

1chapter3FieldEquationsofElasticityBalanceLawsofMomentumandMoment

FieldVariables3components6components6components15fieldvariables2chapter3.1BalanceLawsofMomentumandMBalanceLawsofMomentumandMomentBalanceLawofLinearMomentumQuasi-staticcase3chapter3.1BalanceLawsofMomentumandMBalanceLawsofMomentumandMomentBalanceofAngularMomentumTheoryofCosseratMindlincouplestressStraingradienttheory

Reciprocaltheoremofshearstress

4chapter3.1BalanceLawsofMomentumandMCompatibilityEquation

Kinematics5chapter3.2CompatibilityEquationKinematCompatibilityEquationCompatibilityConditionsProperty1:

6chapter3.2CompatibilityEquationCompatibCompatibilityEquationCompatibilityConditionsProperty2:Planarcase7chapter3.2CompatibilityEquationCompatibCompatibilityEquationSimplyandMultiplyConnectedRegionSimply-connectedregions

8chapter3.2CompatibilityEquationSimplyaCompatibilityEquationSimplyandMultiplyConnectedRegionMultiply-connectedregions

9chapter3.2CompatibilityEquationSimplyaCompatibilityEquationGlobalCompatibilityIsthedisplacementunique?

10chapter3.2CompatibilityEquationGlobalCCompatibilityEquationforanyinfinitesimalloops

Forasimply-connectedregion

11chapter3.2foranyclosedcurvespassingAandBGlobalCompatibility(Cesaro'sIntegral)CompatibilityEquationforanyCompatibilityEquationGlobalCompatibilitymultiply-connectedregion

foranyinfinitesimalloops

i＝1，…，n－1

AND12chapter3.2CompatibilityEquationGlobalCchapter3.2CompatibilityEquationDisplacementsviaStrainIntegration13chapter3.2CompatibilityEquatCompatibilityEquationDisplacementsviaStrainIntegrationPathindependencerequires:

i.e.14chapter3.2CompatibilityEquationDisplaceCompatibilityEquationDisplacementsviaStrainIntegrationThepathindependencerequires:

15chapter3.2CompatibilityEquationDisplaceFieldEquationsofElasticDynamics

DisplacementEquations16chapter3.3FieldEquationsofElasticDynFieldEquationsofElasticDynamicsDisplacementEquationsMultiply,integrateoverV

integrationbyparts

17chapter3.3FieldEquationsofElasticDynFieldEquationsofElasticDynamicsDisplacementEquations18chapter3.3ClapeyronTheoremFieldEquationsofElasticDynFieldEquationsofElasticDynamicsIsotropicMaterialsandNavierEquationisotropicmaterials

19chapter3.3NeglectbodyforceFieldEquationsofElasticDynFieldEquationsofElasticDynamicsHelmholtzRepresentation andWaveEquationsOnedilatationalwaveandtwotransversewaves.

20chapter3.3FieldEquationsofElasticDynFieldEquationsofElasticDynamicsPlaneWavesandTravelingWavesonedimensionalproblem:similarly:

21chapter3.3FieldEquationsofElasticDynFieldEquationsofElasticDynamicsPlaneWavesandTravelingWavesChristoffelacoustictensor

22chapter3.3FieldEquationsofElasticDynFieldEquationsofElasticDynamicsPlaneWavesandTravelingWavesA

:eigen-vectoroftheacoustictensor;:eigen-valuePositivedefinitenessoftheacoustictensorandthepositivenatureofdensityinferthatc2isnon-negativeandreal.

23chapter3.3FieldEquationsofElasticDynQuasi-staticFieldEquations

Quasi-staticProcessesJustification:24chapter3.4Quasi-staticFieldEquationsQQuasi-staticFieldEquationsDisplacementEquationsQuasi-staticNavierequation:

25chapter3.4Quasi-staticFieldEquationsDiQuasi-staticFieldEquationsGreenFunctionsPointforce

Greenfunction

26chapter3.4Quasi-staticFieldEquationsGrQuasi-staticFieldEquationsApplicationsofGreenFunctionsSolutionsforaninfinitebodyunderarbitrarybodyforcecanbeobtainedbyintegratingtheGreenfunctionAfinitebodywitharbitrarybodyforcecanbereducedtheproblemwithoutbodyforcebutwithknownboundarytraction.

27chapter3.4Quasi-staticFieldEquationsApQuasi-staticFieldEquationsApplicationsofGreenFunctionsServesasthebasisfor“boundaryintegralequation”and“boundaryelementmethod”.ReducestoKelvinsolutionforthespecialcaseofanisotropicmaterial.28chapter3.4Quasi-staticFieldEquationsApQuasi-staticFieldEquationsFourierTransformDirectandinverseFouriertransform

Fouriertransform29chapter3.4Quasi-staticFieldEquationsFoQuasi-staticFieldEquationsGreenFunctionsIntegrationbypartstwice

30chapter3.4Quasi-staticFieldEquationsGrQuasi-staticFieldEquationsGreenFunctionsInverseFouriertransformdetailsofdeduction31chapter3.4Quasi-staticFieldEquationsGrQuasi-staticFieldEquationsdetailsofdeduction32chapter3.4Quasi-staticFieldEquationsdeQuasi-staticFieldEquationsdetailsofdeductionInasphericalcoordinatesystem

thevolumeelement

33chapter3.4Quasi-staticFieldEquationsdeQuasi-staticFieldEquationsdetailsofdeductionback34chapter3.4Quasi-staticFieldEquationsdeQuasi-staticFieldEquationsGreenFunctions

PropertiesofGreenfunction:

35chapter3.4Quasi-staticFieldEquationsGrQuasi-staticFieldEquationsKelvinSolutionIsotropiccase36chapter3.4Quasi-staticFieldEquationsKeQuasi-staticFieldEquationsStressFormulationCompatibilityConditions:Equilibriumequation:37chapter3.4Quasi-staticFieldEquationsStStressFormulationIsotropicMaterialsCompliancetensor38chapter3.4StressFormulationIsotropicMaStressFormulationBeltrami-MichellEquations

39chapter3.4StressFormulationBeltrami-MicStressFormulationBeltrami-MichellEquations

40chapter3.4StressFormulationBeltrami-MicStressFormulationBeltrami-MichellEquationsAbsenceofbodyforce41chapter3.4StressFormulationBeltrami-MicStressFormulationStressFunctionsStressfunctions

Compatibilityequations

42chapter3.4StressFormulationStressFunctStressFormulationStressFunctionsMaxwellstressfunctionMorerastressfunction43chapter3.4StressFormulationStressFunctConstraints

InternalConstraintsIncompressibilityconstraintInextensibleconstraintGeneralconstraint44chapter3.5ConstraintsInternalConstrainConstraintsConstrainingStressConstrainingstresscannotdeliveranyworkConstrainingStress45chapter3.5ConstraintsConstrainingStressConstrainingStressExamplestheincompressibilityconstraint:forisotropicmaterials

46chapter3.5ConstrainingStressExamplesforConstrainingStressExamplestheinextensibleconstraint47chapter3.5ConstrainingStressExamples47cConstrainingStressExamplestheinextensibleconstraint48chapter3.5ConstrainingStressExamples48cBoundaryConditionsBoundaryComposition49chapter3.6BoundaryConditionsBoundaryCoBoundaryConditionsTypesofBoundaryConditionsAll-arounddisplacementboundarycondition.

Allaroundtractionboundarycondition.rigidbodymotioncannotbeeliminated;allaroundtractionshouldsatisfytheglobalequilibrium.Mixedboundarycondition.Partoftheboundaryisprescribedbydisplacement,andpartbytraction.50chapter3.6BoundaryConditionsTypesofBoBoundaryConditionsTypesofBoundaryConditionsHybridboundarycondition.51chapter3.6BoundaryConditionsTypesofBoBoundaryConditionsTypesofBoundaryConditionsSpring-likeboundarycondition.Anelasticinclusionisembeddedinarigidmatrixviaanelasticinterlayer.

52chapter3.6BoundaryConditionsTypesofBoBoundaryConditionsSymmetryConditionsSymmetryalongx=053chapter3.6BoundaryConditionsSymmetryCoBoundaryConditionsSymmetryConditions

Undersymmetricloading

Allanti-symmetricfieldvariablesandtheirderivativeswithrespecttoxoftheevenordersshouldvanishatx=0.Thederivativeswithrespecttoxoftheoddordersofallsymmetricfieldvariablesshouldvanishatx=0.54chapter3.6BoundaryConditionsSymmetryCoBoundaryConditionsSymmetryConditionsUnderanti-symmetricloading

Allsymmetricfieldvariables

andtheirderivativeswithrespecttoxoftheevenordersshouldvanishatx=0.Thederivativeswithrespecttoxoftheoddordersofallanti-symmetricfieldvariablesshouldvanishatx=0.55chapter3.6BoundaryConditionsSymmetryCoSymmetryConditionsexamples56chapter3.6SymmetryConditionsexamples56BoundaryConditionsConditionsforSingleValuedDisplacementFields57chapter3.6BoundaryConditionsConditionsBoundaryConditionsInitialConditions

Elasto-dynamicsproblems

Navierequations

58chapter3.6BoundaryConditionsInitialConBoundaryConditionsSaintVenantPrinciplePointforce,pointcoupleanddipleappliedalongtheboundary.59chapter3.6BoundaryConditionsSaintVenanBoundaryConditionsSaintVenantPrinciple60chapter3.6BoundaryConditionsSaintVenanFormulationofElasticity

PositiveDefinitenessofStrainEnergy

61chapter3.7FormulationofElasticityPosiFormulationofElasticityPositiveDefinitenessofElasticityFormulationTotalenergyUandconsequentlytheelasticityformulationarepositivedefinite.62chapter3.7FormulationofElasticityPositFormulationofElasticityPrincipleofSuperpositionUniquenessTheorem63chapter3.7FormulationofElasticityPrincSummaryofChapter3DifferentialFormulationDifferentialequations(15)equilibrium(3),kinematics(6),constitutiverelation(6)Boundaryconditions(3)(orreplacedbysymmetryconditions)Initialconditions(6)Constraintconditionsandexpressionsforconstrainingstress(nconstraintsleadtonunknownconstrainingparametersdeterminedbyequilibrium).64chapter3SummaryofChapter3DifferentiSummaryofChapter3SimplifiedFormulationsDisplacementformulation-

3Navierequationsand3displacementboundaryconditions.Stressformulation-6Beltrami－Michellequationsor6equationsorstressfunctionequationsand3tractionboundaryconditions.65chapter3SummaryofChapter3SimplifiedSummaryofChapter3BasicPrinciplesPrincipleofsuperpositionPositivedefinitestrainenergy(ellipticity)Uniquenesstheorem SaintVenantprinciple.BasicSolutionsTravelingwavesolutionPlanewavesolutionPointforcesolution(Greenfunction).

66chapter3SummaryofChapter3BasicPrinTheoryofElasticityIntroductionElasticityofSolidsFieldEquationsofElasticity-DifferentialFormulationPrismaticRodsPlaneProblems–TheoryandSolutions

PlaneProblems–ApplicationsVariationalFormulationofElasticityThree-dimensionalProblemsIndex0TheoryofElasticityIntroductiPrismaticRods

FormulationforPrismaticRods

UniaxialTensionandPureBending

CorrectiveSolution(SaintVenantDecay)FreeTorsionofaPrismaticRod

InverseandSemi-inverseSolutions

FormulationofAnti-planeProblems

Chapter40PrismaticRodsFormulationforGeometry

PrismaticRodsChapter4.11Theproblems:Three-dimensionalquasi- staticelasticityequationis observedinVoccupiedbytheprismaticrod.Traction

independentofzisprescribedalongtheside surface.Twoendsoftherodare subjectedtoarbitrarybutself-equilibratedtraction.GeometryPrismaticRodsChapterPrismaticRodsDecompositionChapter4.12PrismaticrodwithatractionfreesidesurfaceAnti-planeproblemPlanestrainproblemPrismaticRodsDecompositionChaPlanestrainproblemPrismaticRodsLateraltractioncomponentsFieldvariablesChapter4.13PlanestrainproblemPrismaticPrismaticRodsPlanestrainproblemRemainingfieldvariablesBCsChapter4.14PrismaticRodsPlanestrainproAnti-planeproblemPrismaticRodsTractionFieldvariablesTheproblemwillbeformulatedandsolvedinSection4.6Chapter4.15Anti-planeproblemPrismaticRoPrismaticrodwithatractionfreesidesurface

PrismaticRodsChapter4.16TheendtractionPrismaticrodwithatractionFurtherDecompositionforEnd-loadedPrismaticRodsPrismaticRodsChapter4.17FurtherDecompositionforEnd-UniaxialTensionandPureBendingUniaxialTensionChapter4.28Aspecialcase:UniaxialTensionandPureBendUniaxialTensionUniaxialTensionandPureBendingHooke’slawDisplacementsChapter4.29UniaxialTensionUniaxialTensiUniaxialTensionUniaxialTensionandPureBendingThegeneralformofthedisplacements:Remarksifthetranslationandrotationatx＝0arefixed,thesolutiongivenherecanbeappliedtoprismaticrodsoflengthLofanycross-section.thetotalelongationoftherodisChapter4.210UniaxialTensionUniaxialTensiUniaxialTensionandPureBendingPureBendingTheendtractionThecorrespondingmomentsattheendsChapter4.211UniaxialTensionandPureBendPureBendingUniaxialTensionandPureBendingStressesStrainThedisplacementgradientChapter4.212PureBendingUniaxialTensionaPureBendingUniaxialTensionandPureBendingIntegratedthedisplacementgradient:Chapter4.213PureBendingUniaxialTensionaPureBendingUniaxialTensionandPureBendingThevanishingvaluesofallshearstraincomponentsleadto:Chapter4.214PureBendingUniaxialTensionaPureBendingUniaxialTensionandPureBendingTheaboveexpressionsleadtoCombiningthisexpressionwiththepreviousonesChapter4.215PureBendingUniaxialTensionaPureBendingUniaxialTensionandPureBendingSubstitutingintothelasttwoexpressionsof:Therefore,Chapter4.216PureBendingUniaxialTensionaPureBendingUniaxialTensionandPureBendingThedisplacementfieldsChapter4.217PureBendingUniaxialTensionaCorrectiveSolution

(SaintVenantDecay)

SaintVenantdecay

Chapter4.318ThedecayinglengthCorrectiveSolution

(SaintVeSaintVenantdecayCorrectiveSolution

(SaintVenantDecay)

SymmetricpartAnti-symmetricpartChapter4.319SaintVenantdecayCorrectiveSSaintVenantdecayCorrectiveSolution

(SaintVenantDecay)

Airystressfunction:BCs:Chapter4.320SaintVenantdecayCorrectiveSSaintVenantdecayCorrectiveSolution

(SaintVenantDecay)

TheSaintVenantdecayingsolution:EquilibriumEquation:Chapter4.321SaintVenantdecayCorrectiveSSaintVenantdecayCorrectiveSolution

(SaintVenantDecay)

Forthesymmetricself-equilibratedtractionBCs:Anecessaryconditionforexistinganon-trivialsolutionisthevanishingofthecoefficientdeterminantChapter4.322SaintVenantdecayCorrectiveSFreeTorsionof

aPrismaticRod

NavierSolutionchapter4.41FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodNavierSolution

Contradictingboundaryconditionchapter4.42FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodNavierSolutionForacircularcross-section

chapter4.43FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolutionwarpingfunctionSaintVenantdisplacementfieldisincompressiblechapter4.44FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolutionEquilibriumequationchapter4.45FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolutionEquilibriumequationBoundarycondition

chapter4.46FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolution

Lemma:IfinAandon then.Proof:Gausstheoremchapter4.47FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolution.NullresultantforceovertheendplaneTorsionrigidityisreduced!chapter4.48FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolution(summary)SolutionprocedureunderSaintVenantmethod:

SolvethewarpingfunctionwfromFindthetorsionrigidityfromCalculatethetwistangleperunitlengthfromchapter4.49FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodSaintVenantSolutionCircularcross-sectionchapter4.410FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodPrandtlSolutionEquilibriumequation

Prandtlstressfunction

chapter4.4FatherofModernFluidDynamics11FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodPrandtlSolutionContoursofPrandtlstressfunctiondenotethestresstrajectories

chapter4.412FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodPrandtlSolutionforsimplyconnectedregion

chapter4.413FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodIftheratioofischosenas2,theprofileofthethinfilmwouldresemblethatofchapter4.4Prandtl-Taylor-GriffithMembraneAnalogyofTorsion(SimplyConnectedRegions)

14FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodMembraneAnalogyofTorsion(SimplyConnectedRegions)

chapter4.415FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodCross-sectionwithHolesGlobalcompatibilityconditions

chapter4.416FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodCross-sectionwithHoleschapter4.417FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodThinFilmAnalogy(MultiplyConnectedRegions)chapter4.418FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodResultantschapter4.4SimplyconnectedMultiplyconnected19FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodThinFilmAnalogy(MultiplyConnectedRegions)chapter4.420FreeTorsionof

aPrismaticRFreeTorsionof

aPrismaticRodThinFilmAnalogy(MultiplyConnectedRegions)Torqueisproportionaltothevolumecoveredbythinfilmandplates

chapter4.421FreeTorsionof

aPrismaticRInverseandSemi-inverseSolutions

SeveralexamplesEllipticalCross-sectionThinWallTubesThinWallTubesofOpenCross-sectionRectangularCross-sectionchapter4.51InverseandSemi-inverseSolutInverseandSemi-inverseSolutionsEllipticalCross-sectionchapter4.51InverseandSemi-inverseSolutI