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1、Chapter4.ConstitutiveEquationsofLinearElasticMaterials;FormulationofProblemsofElasticityJ.Q.TarnOctober14,20151IntroductionTheconstitutiveequationsofthematerialaremathematicalmodelingofthematerialbehaviorsundervariousagents.Theyarephenomenologicaldescriptionofthem
2、aterialpropertiesbasedonreproduciblemeasurementsintestingthematerialsamples.Ingeneraltermstheconstitutiveequationsdescribetherelation:·σ=f(ε,ε(strainrate),t(time),T(temperature),moisture,crystallinestructure,chemicaleffect,electromagneticeffect,size(?),shape(?),geom
3、e-try(?),loading(?),···)Elasticity:σ=f(ε)(thestress—strainrelations).Thermalstressanalysis:σ=f(ε,T).Thermoelasticity:σ=f(ε,T,t).·Viscoelasticity:σ=f(ε,ε,T,t).Note:1.Thematerialpropertiesmustbeindependentoftheexternalexcitations,suchastheloadingconditionsandgeometr
4、y(shapeandsize)ofthebody.2.Thematerialparametersintheconstitutiveequationsaredeterminedindirectlyfromthemeasurementsintestingthematerialspecimens.1TensiletestTorsiontestl:effectivelengthsimplepuretensionshearFigure1:3.Toderivereliableformulasfortheparameteridentifi
5、cation,thetestdesignshouldbesimpleandeasytocarryout.Tensiletestandtorsiontestofstan-dardspecimensaretwosimpletestsfordeterminingthematerialproperties.2Observationsinsimpletests1.Tensiletest(producingsimpletensilestressinthespecimen)MeasuringPand∆l,P∆lσ=,ε=.AlWithi
6、ntheelasticlimit:ε∼10−3order,σ∼25GPa(≈36,000psi)foryymildsteel.⇒Linear(proportional)relationshipbetweenσandε:σσ=Eε,ε=,E:Young’smodulus.E2Figure2:3Observationinsimpletensiletestshowsthatextensioncauseslateralcontraction.Poisson’sratioνisintroducedtodefinethecontract
7、ionsinx2andx3directionsduetoextensioninx1directionsuchthatσ11ννε11=,ε22=−νε11=−σ11,ε33=−νε11=−σ11.EEE2.Torsiontest(producingpureshearstressstateinthecircularcylindricalspecimen)MeasuringMtand∆θ,Mtr∆θτ=,γ=.Jl⇒Linear(proportional)relationshipbetweenτandγ:τ=Gγ,G:shea
8、rmodulus.3.Tri-axialtestinsoilmechanicsApplycompressionandhydrostaticpressuretoasoilsampletoproduceatriaxialstressstate,thenmeasurethestraininthesample.
