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1、9LinearViscoelasticity9.1IntroductionThepreviouschaptershaveconsideredconstitutiveequationsthatdealpri-marilywithtwodifferenttypesofmaterialbehavior:elasticresponseofsolidsandviscousflowoffluids.Examplesofmaterialsthatbehaveelasti-callyundermodestloading,andatmoderatetempera
2、tures,aremetalssuchassteel,aluminum,andcopper,certainpolymers,andevencorticalbone.Examplesofviscousflowmayinvolveavarietyoffluidsrangingfromwatertopolymersundercertainconditionsoftemperatureandloading.Polymersareespeciallyinterestingbecausetheymaybehave(respond)ineitherelast
3、ic,viscous,orcombinedmanners.Atarelativemoderatetemper-atureandloading,apolymersuchaspolymethalmethacrylite(PMMA,plexi-glass),maybeeffectivelymodeledbyalinearelasticconstitutiveequation.However,atasomewhatelevatedtemperature,thesamematerialmayhavetobemodeledasaviscousfluid.
4、Polymersarebynomeanstheonlymaterialsthatexhibitdifferentbehaviorunderalteredtemperature/frequencyconditions.Steel,aswellasaluminum,copper,andothermetals,becomesmoltenathightemperaturesandcanbepouredintomoldstoformingots.Additionally,atahighenoughdeforma-tionrate,forexample
5、,atthe48km/hrrateofavehiclecrash,steelwillexhibitconsiderablyalteredstiffnessproperties.Justascontinuummechanicsisthebasisforconstitutivemodelsasdistinctaselasticsolids(stress/strainlaws)andviscousfluids(stress/strain-ratelaws),italsoservesasthebasisforconstitutiverelations
6、thatdescribematerialbehavioroverarangeoftemperature/frequencyandtime.Oneofthesimplestmodelsforthiscombinedbehavioristhatoflinearviscoelasticity.9.2ViscoelasticConstitutiveEquationsinLinearDifferentialOperatorFormOneoftheprincipalfeaturesofelasticbehavioristhecapacityformat
7、erialstostoremechanicalenergywhendeformedbyloading,andtoreleasethisenergytotallyuponremovaloftheloads.Conversely,inviscousflow,mechanicalenergyiscontinuouslydissipatedwithnonestored.Anumberofimportantengineeringmaterialssimultaneouslystoreanddissipatemechanicalenergywhensub
8、jectedtoappliedforces.Infact,allactualmate-rialsstoreanddissipateenergyinvaryingdegreesdu
