高级工程材料-作业答案-第二章晶体级别的应力

1.Showthat“9isfororthorhombiccrystals”.Solution:Constitutiveequation:111111121314151622222122232425263333313233343536414243444546232351525354555613136162636465661212CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC(1)Cij:36independentstiffnessconstants.Becauseofmechanicalequilibriums,Cij=Cji,independentstiffnessconstantschangeto21.Fororthotropicmaterial,nointeractionexistsbetweennormalstressesσ11,σ22,σ33andshearstrainsγ23,γ13,γ12;i.e.normalstressesactingalongprincipalmaterialdirectionsproduceonlynormalstrains.Nointeractionexistsbetweenshearstressesτ23,τ13,τ12andnormalstrainsε11,ε22,ε33;i.e.shearstressesactingonprincipalmaterialplanesproduceonlyshearstrains.Nointeractionexistsbetweenshearstressesandshearstrainsondifferentplanes;i.e.ashearstressactingonaprincipalplaneproducesashearstrainonlyonthatplane.Then,Eq.(1)changeto:111111121322222122233333313233442323551313661212000000000000000000000000CCCCCCCCCCCCThenumberofindependentconstantsisreducedto9.2.3Planestressandplanestrainareimportantconcepts,particularlyintherealmoffracture.Planestressisdefinedbytwofiniteprincipalstresscomponents,oneprincipalstresscomponentbeingzero.Planestrainisdefinedanalogously.UseEq.(2.3)anditsanalogstoshowthatplanestressconditionsleadtothreeprincipalstraincomponentsandthatplanestrainconditionsresultinthreeprincipalstresscomponents.Solution:FromEq.(2.3)312312132132123312123EEEEEEEEEEEEThen112322313312EEEEEEEEEApplicationofauniaxialstressleadstolongitudinalextensionalongthetensileaxis(1=L1/L1)andtotransversecontractionsalongthetwoperpendicularaxes(2=30).Forlinearelasticdeformation,thelongitudinalandlateralstrainsarerelativethroughPoisson’sratio,2=3=-1Ontheotherhand,whenapplyauniaxialstressontwoperpendicularaxis1leadtotransversecontractionsalongtheaxis1,thustotalstainonaxis1is1-2-3;=-1/2=-1/3132-2,-31E1112.9Foriron,C11=237GN/m2,C12=141GN/m2andC44=116GN/m2.aDetermineE[100],E[110]andE[111]ofFe.bSupposeapolycrystallineFewireiscomposedofgrainswitheither[100],[110],or[111]crystaldirectionslyingalongthewireaxis.Ifallthethreeorientationsarepresentinequalamounts,whatisEforthewire?cThepolycrystallineelasticmodulusofFeis209GN/m2.Howdoesthiscomparewiththemodulusestimatedinpart(b)?dBychangingtheprocessingconditions,adifferentkindoftexturecanbeproducedinFewire;70%ofthegrainsareorientedsothata100directionisalignedwiththewireaxisandtheremaining30%areevenlysplitbetween110and111directionslyingalongthisaxis.Whatisthemodulusalongtheaxisforthiscase.eIsEtransversetothewireaxisdifferentfromthemodulusalongtheaxisforthesituationspertainingtoparts(b)and(d)?Explainyourreasoning.Solution:a:111211111211121212111211124444()(2)()(2)1SSCSSSSSCSSSSCS321211111112111232121211121112324444()7.587*10/(2)()2.823*10/(2)()118.62*10/116CCSGNmCCCCCSGNmCCCCSGNmCThemodulusalonganarbitrary[hkl]directionisobtainedas[100][110][001][111][010]22222211111244[]112()()2hklSSSSEFor[100],222222=0,E[100]=131.81GN/m2For[110],222222=0.25,E[110]=220.57GN/m2For[111],222222=0.33,E[111]=284.43GN/m2b:Case1σ1σ2σ1*A=σ2*A=E*ε=(E[100]+E[110]+E[111])*ε/3E1=212.27GN/m2Case2σσσ=Eε=E[100]ε[100]=E[110]ε[110]=E[111]ε[111]ε=ε[100]+ε[110]+ε[111]1/E=(1/3E[100]+1/3E[110]+1/3E[111])E2=191.86GN/m2c:ThepolycrystallineelasticmodulusofFeis209GN/m2.Itisintherangeof191.86~212.27GN/m2.closeto212.27GN/m2.d:E=70%E[100]+30%(E[110]+E[111])/2=70%*131.81+15%(220.57+284.43)=168.02GN/m2.e:underdifferentcondition,thevalueofEisdifferent.Eistransversetothewireaxisdifferentfromthemodulusalongtheaxisforthesituationspertainingtoparts(b)and(d).Itwillbechangedintherangeof168.02isintherangeof131.81~284.43GN/m2.E[100]E[110]E[111]E[100]E[110]E[111]2.18aTheadditionalspringanddashpotinserieswiththevoigtmodelofFig.2.14constituteastandardlinearsolid(Fig.2.16).ThespringanddashpotinseriesalonearereferredtoasaMaxwellmodel.Ifastressisappliedattimet=0andheldconstant,sketchthestrain-timeresponseexpectedforaMaxwellmodel.bReleasetheloadinpart(a)sometimeafteritsapplication.Sketchstrainvs.timefollowingthisunloading.cShowthattheadditionofthestrainsdrawnin(a)and(b)tothoseschematizedinFig.2.15leadstotheresponseillustratedinFig.2.17.Solution:2.20RefertoFig.2.20b.Schematicallyplotmaxmax/asafunctionoftheappliedfrequency.(maxisthemaximumstressobservedduringthehysteresiscycleandmaxisthemaximumstrain;notemaxtypicallyisnotfoundwhenmax,cf.Fig.2.20b).Solution:Fig.2.20bshowsthestress-straincurve.Weseethatitispath-dependent,inthatthestress-straincurvesonloadingandunloadingdiffer.Theaveratemodulusasdefined,forexample,bytheratiomaxmax/inFig.2.20bisbetweenErandEusincesome,butnotall,ofthepotentialviscoelasticstrainismanifestedatthisintermediatefrequency.Theareabetweentheunloadingtheloadingcurves(thecross-hatchedregioninFig.2.20b)representsanirreversibleorhyste