Transcription of 3 Linear viscoelasticity - UCL
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3 Linear viscoelasticity - UCL
3 LinearviscoelasticityAlinearviscoelastic uidis a uidwhich hasa linearrelationshipbetweenitsstrainhistor yanditscurrent valueof stress: (t) =Zt 1G(t t0) _ (t0) dt0 ThefunctionG(t) is therelaxationmodulusof the uidcanneverremember timesin thefuture,G(t) = 0 ift < , you wouldalsoexpectthatmorerecent strainswouldbe moreimportant thanthosefromlongerago,so int >0,G(t) shouldbe a 'treallyany otherconstraints fewoften-usedformsforG(t) wheret 0 are:SingleexponentialG0exp[ t= ]Multi-modeexponentialG1exp[ t= 1] +G2exp[ t= 2] + Viscous uid (t)LinearlyelasticsolidG0 Let'sjustcheck thelasttwo. For theviscousformwe have (t) =Zt 1G(t t0) _ (t0) dt0=Zt 1 (t t0) _ (t0) dt0= _ (t)as we wouldexpectfora Newtonianviscous theelasticsolidwe have (t) =Zt 1G(t t0) _ (t0) dt0=Zt 1G0_ (t0) dt0=G0Zt 1_ (t0) dt0:Theintegralontheright is thetotalstrain(orshear)thematerialhasund ergone:sothis,too, gives theformwe sampleof material,how wouldyougoaboutmodellingit?Evenif youstartbyassumingit is a linearmaterial(andtheyallareforsmallenou ghstrains),how wouldyoucalculateG(t)?
3 Linear viscoelasticity A linear viscoelastic °uid is a °uid which has a linear relationship between its strain history and its current value of stress:
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