Lecture 2: Constitutive Relations

1 Lecture2: Constitutive RelationsE. J. Hinch1 IntroductionThislecturediscussesequation sof motionfornon-Newtonian uidmustsatisfyconservationof momentum DuDt= rp+r + g(1)where is thedensity of the uid,uis thevelocity eld,pis thepressureand is thedeviatoricstresstensor(thetrace-freec omponent of thestress).1We canabsorbthebodyforce ginto a modi edpressure,andin turnwe canabsorbthemodi edpressureinto thestressgiving DuDt=r . Much of themodelingin non-Newtonian uidsconcentrateson ndinga Constitutive relationbetween andthe ow velocity uidswe useareincompressibleunlessstatedotherwis e,so we have assumedr u= 0:In many practicalapplicationsof non-Newtonian uidsinertiais willoftenusetheStokes equations.

2 R =0 a simplematerialthestress dependsonthedeformationandtherateof understandthisrelationshipwe beginby consideringhow the uiddescriptionwe follow a uidparticlein the owumapsamaterialelement to a newpositionxthatdependsonitsinitialposit ionXX!x(X; t):If we follow a materiallineelement, X, it is stretchedandrotatedin the ow accordingto X! x=A X;AiJ=@xi@XJ:1In thesenotes is usedforeitherthestressor is assumethatthesystemis , thestressat a materialpointdependsonlyonthehistoryof thatmaterialpoint, a functionalforstress (t) = fA( )g t.

3 (2)We now adopttheassumptionofmaterialframeindi erencewhich statesthatourconstitutive equationshouldnotdependonthetranslation, rotationor accelerationof theframeof extremecasesthisshouldbe a good ,we shouldgetthesameresultif we calculateourstressbeforeor afterrotatingtheframeof changeof frameof referencegivenbyx0=Q(t)x+a(t);(3)withQ(t ) a rotationmatrixandaa thenewframeofreferenceis givenby 0= fQ( )A( )QT(0)g t Q(t) fA( )g tQT(t):We require fAgto obey thisidentity forallQ(t). perfectlyelasticmaterialrespondsinstanta neouslyto thepresent strainwhich dependsonlyonthepresent how it arrivedinto itscurrent positiondoes fA(t)gbecomesa function (A).

4 We candecomposethedeformationtensorAintoa rotationtensorRanda stretchtensorUsuch thatA=R UwithRTR=IandU2=ATA:(4)ThensettingQ RTinmaterialframeindi erencegives fAg=RT(t)f(U(t))R(t);(5)thus reducingtheproblemto determiningtheunknownfunctionf(U). It is convenient toexpresstheconstitutive law in ermsof thepotentialenergyw(U) insteadof frameindi erenceleadsto theconstitutive anelasticmaterialthatis isotropicin itsreststate,andhaspotentialenergywforel asticdeformations,thisgives =1 @w@ AAT 1 @w@ A TA 1(6)where =12( 21+ 22+ 23) , =12( 21+ 22+ 23) and = 21 22 23(=1if incompressible),and problemIn a newreferenceframethestressis givenby 0=Q QTandso_ 0=Q_ QT+_Q QT+Q _QT.

5 Thus thetransformationfrom_ to _ 0does notfollow thesamerelationas thetransformationfrom to 0. We willtryto ndsomeotherderivative ow velocity isu0=Qu+_Qx+_aandthevelocity gradient is@u0@x0=Q@u@xQT+_QQT:(7)Thevelocity gradient canbe separatedintosymmetric(strainrate)andant isymmetric(vorticity) partsThetransformedstrainrateisE0=QEQT andthetransformedvorticityis 0=Q QT+ togetherwe canshow thattheco-rotational(Jaumann[1,2])timede rivative D Dt + (8)hastransformation 0 0=Q QT;(where 0 0denotestheco-roationalderivative of thestressin thenewframeof reference,)as does theco-deformational(Oldroyd[3]or upper convected)derivative5 D Dt ruT ru:(9)Note I=0but5I6= is therateof changeas observedwhilerotatingandtranslatingwitht he is therateof changeas observedwhiledeformingandtranslatingwith the is validin thelimitwhereATA I.

6 Themostgeneralformof thehistory-dependent linearconstitutive law is (t) =R(t)Z10G(s) ATA (t s)dsRT(t):(10)22 Thisis a (s) represents theelasticmemory. It is theFouriertransformof thefrequencydependent elasticmodulusG (w) de nedin Lecture1. For a Newtonian uidG(s) = (s) andforanelasticsolid,G(s) = simpleshearwithshearrate_ (t) =Z10G(s) _ (t s)ds(11)andsinceforsteadyshear_ is constant, thesteadyshearviscosity is givenbyR10G(s) uidThesecondorder uidis derivedthrougha retardedmotionexpansionandis validforslow, weak usedbecausethismodelcanhave instabilitiesin regimeswhereit doesn'tapply( , highfrequency), Newtonianwithsmalltermsadded: = pI+ 2 E 2 15E+ 2E E(12) =Z10G(s)ds 1=Z10s G(s)ds.

7 In simpleshearthesecondorder uidhasconstant viscosity =R10G(s)dsandnormalstressdi erencesN1= yy xx= 2 1_ 2,N2= zz yy= 14 2_ 2, wherexdenotesthe ow direction,yis thevelocity gradient uniaxialextensional ow theviscosity is ext= + ( 1+14 2) _ ;(13)where_ is uidsaretoo non-linearto be describedby thelinearviscoelasticor slightlynon-linearsecondordermodelsdiscu ssedabove. For these uidstherearenoexactsolutionsorexactappro ximationsandothermodelsmustbe uidfollowsthesameequationsastheNewtonian uidbuttheviscosity dependsontheshearrate_ =p2E:E.

8 AsforNewtonian uids,thestressdependsonlyontheinstantane ous ow andnotthe ow history. Theconstitutive law is = pI+ 2 ( _ )E:(14)ThegeneralizedNewtonianmodelswere developedto t experimentaldataandtheformof ( _ ) is usuallyderived empirically. Somecommonexpressionsusedto t dataare:23 Power Law [4] ( _ ) =k_ n 1;(15)wherekandnare t parameters. Carreau,Yasuda,Cross[5, 6, 7] ( _ ) = 1+ ( 0 1)[1+ ( _ )a]n 1a(16)where 0and 1aretheviscositiesatthelimitsof zeroandin niteshearrate,respectively;a,n, and are t parameters.

9 Yield uids:the uid owsonlyabove somecriticalstress y.{Bingham[8] =8<:1ifj j< y 0+ y_ ifj j> y{HerschelBulkley =8<:1ifj j< yk_ n 1+ y_ ifj j> [3]is oneof thesimplestmodelsthatincludesthehistoryo f the usethefollowingequationfortheevolutionof thedeviatoricstress, + 15 = 2 (E+ 25E);(17)where 1is therelaxationtimeand 2is a givenpressurep, theOldroyd-Bmodeloftenappearsin anequivalent formforthetotalstress: = p I+ 2 E+G A(18)5A= 1 (A I)(19)wherep =p+ 2(1 2= 1) = 1,G= 2(1 2= 1) , = 1and = 2 = 1. TheOldroyd-Bmodelreducesto Upper Convective Maxwell(UCM)when 2= 0 andviscousNewtonianwhen 2= simpleshearanOldroyd-B uidhasconstant viscosity =G=2 + andthenormalstressesareN1= 2 ( 1 2) _ 2;N2 0:(20)Theuniaxialextensionalviscosity is ext= (1 2_ 2 1 2_ 2)(1 2_ )(1+ 1_ ):(21) 10 8 6 4 20246810 Elongationrate:_ ExtensionalViscosity: extFigure1: Theextensionalviscosity for 1= 2, 2= 1.}}

10 Theextensionalviscosity is negativeforelongationratesslightlyabove 1=2 negative viscositiesat someelongationrates( gure1) which is derivedusingHooksLaw springswhich arein- reformulatedto eliminatethenegative viscosity. Assumingthatthemicrostructureis notin nitelyextensible,we get5A+f (A I) = 0(22)forsomefunctionf(A). Thestress is then = pI+ 2 sE+Gf (A I):(23)Occasionallyfappearsonlyin Equation(22)andnotin (23).TheFENE( nitelyextensiblenonlinearlyelastic)modi cationkeepsAfromgrowingtoo fastby settingf=L2L2 traceA;(24)whereLrepresents a nondimensionallengthscaleforthestretchin gof ,thesti erit EquationsBelow area fewof themany othercostitutive equations,which arederived to match exper-imentaldata.