3 Linear viscoelasticity - UCL

1 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)?

2 Oneway wouldbe to carryoutastepstrainexperiment: forshear ow thismeansyouwouldsetupyourmaterialbetwee ntwo platesandleave it to settle,so it losesthememoryof the ow thatputit hashadtimeto relax,youshearit throughoneshearunit( distancethesameas thedistancebetweentheplates). maintaintheplatesin positionas this:11-6-6 ShearrateTimeStressTimeWhatis therealrelationshipbetweenthestressfunct ion (t) andtherelaxationmodulusG(t) in thiscase?Supposethelengthof timethematerialis shearedforisT. Thentheshearrateduringthattimemustbe 1=T(togeta totalshearof 1):_ (t0) =8<:0t0< T1=T T t0 00t0>0 Thenthestressfunctionbecomes (t) =Zt 1G(t t0) _ (t0) dt0=1TZ0 TG(t t0) dt0:Now supposethatwe dotheshearveryfastso thatTis TG(t t0) dt0 T G(t)andso thestressis (t) G(t):Thus therelaxationmodulusis actuallytheresponseof thesystemto verydi cultto achieve in ,workingin industry,arefarmorelikelyto placedin aCouettedevice, which is essentiallya pairof con-centriccylinders,oneof which is putin thenarrow gapbetweenthecylinders,andthefreecylinde ris ow setupbetweenthetwo cylinders12is almostsimpleshear ow (aslongas thegapis narrow relative to theradiusof thecylinders,andnoinstabilitiesoccur).

3 Therotationis setupto be a simpleharmonicmotion,withsheardisplaceme nt: (t) = sin(!t)where is theshear,!thefrequencyand reality mustbe keptsmallto ensurethematerialis keptwithinitslinearr egime;butif we aredealingwithanideallinearviscoelasticm aterial, canbe any the uid,_ (t), willbe_ (t) = !cos(!t):Supposethismotionstarteda longtimeago(t! 1). Thenat timet, thestresspro lefora generallinearviscoelasticmaterialbecomes (t) =Zt 1G(t t0) _ (t0) dt0=Zt 1G(t t0) !cos(!t0) dt0We cantransformthisintegralby changingvariablesusings=t t0: (t) =Zt 1G(t t0) !cos(!t0) dt0= !Z10G(s) cos(!(t s)) dsandnow if we writecos(!(t s)) =<[exp[i!(t s)]] we get: (t) = !Z10G(s)<[exp[i!(t s)]] ds= !< Z10G(s) exp[i!(t s)] ds = !< exp[i!t]Z10G(s) exp[ i!s] ds andtheintegralontheright is now a hasnodependenceont, it is justa de nethecomplexshearmodulus,G , as:G =i!Z10G(s) exp[ i!s] ds;anditsrealandimaginaryparts:G =G0+ (t) = <[exp[i!t]( iG )] = <[(cos(!t) +isin(!t))(G00 iG0)]= [G0sin(!t) +G00cos(!t)] =G0 (t) +G00!

4 _ (t):13 Now a purelyviscous uidwouldgive a response (t) = _ (t) = !cos(!t)anda purelyelasticsolidwouldgive (t) =G0 (t) =G0 sin(!t):We canseethatifG00= 0 thenG0takes theplaceof theordinaryelasticshearmodulusG0: henceit is calledthestoragemodulus,becauseit measuresthematerial'sability tostoreelasticenergy. Similarly, themodulusG00is relatedto theviscosity ordissipationof energy:in otherwords,theenergywhich is ^oleof theusualNewtonianviscosity is taken byG00=!, it is alsocommontode ne 0=G00!as thee ective viscosity; however,thestorageandlossmoduliG0andG00a rethemostcommonmeasuresof check a fewvalues:G(t) = (t)G (!) =i! G(t) =G0G (!) =G0NB:ByobservationnotintegrationG(t) =G0exp[ t= ]G (!) =i!G0 (1 +i! )=G0(!2 2+i! )1 +!2 2 Letus lookmoreat slow speeds! 1 we haveG iG0! so themateriallookslike a viscous uidwithviscosityG0 . At highspeeds! 1, wehave insteadG G0andthemateriallookslike we plotthetwo moduli,G0andG00against!thegraphlookslike this:G0G00 Frequency,!14 Thepoint wherethetwo graphscrossis givenby:G0=G00G0!

5 2 21 +!2 2=G0! 1 +!2 2! =1:A real uidhasmorecomplexdynamics:Typicaldynamic dataforanelasticliquid-6 Frequency,!buta rstengineeringapproximationoftenusedis to take thevalueof!cat which thetwo curves cross,anduse =! 1cas therelaxationtimeof the , aseriesof relaxationmodesis usedto t thecurves as wellas