Viscoelastic fluids II: Linear viscoelasticity

Mechanical analogues

• “Viscoelastic” suggests that constitutive law should depend on strain and strain rate

• Two principal tests to probe creep and relaxation behaviour– Step increase in strain rate with

stress measured (relaxation of stress)

– Step increase in stress with strain measured (creep)

• Starting point is simple linear mechanical models– Maxwell element– Kelvin-Voigt element– Jeffreys element

• Fluids terminology:– Stress: σ↔ τ– Moduli: E ↔ G0– Viscous parameter c ↔η0– Strain ε ↔ γ, etc

Irgens 2008

Linear Maxwell model

• Spring & dashpot in series – Same stress– Displacement is additive

– λ1=η0/G0 = relaxation time; G0=elastic modulus

• Integral form: suppose the shear history is known, i.e. �̇�𝛾 𝑡𝑡 is a specified function:

γηγηττλ

ηττγγγγητ

γτ

0

Viscous

0

Elastic

1

00

212

0

10 1

==+

+=+=⇒

=

=

dtd

dtd

dtd

Gdtd

dtd

dtd

dtd

G

Differential form

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

−=−−=−== ∫∫∫∫

∞−

−−

∞−∞−∞−

−− dssedssstMdssstGdssett

/stttt

/st γληγγγ

λητ λλ 11

21

0

functionMemory ModulusRelaxation

1

0

Relaxation modulus G(t)

• Response of material to an instantaneous unit of shear• Allows us to generalize the linear Maxwell through

different G(t), e.g.– Single mode (Maxwell): G(t) = G0e-t/λ1

– Multi-modal: G(t) = G1e-t/λ1 + G2e-t/λ2+ G3e-t/λ3 +…..– Elastic solid: G(t) = G0

– Viscous fluid: G(t) = 2η0δ(t)

• In general, we might expect G(t) to decay to zero at large times (memory effect)

• Example 1: step increase in strain rate from zero

• Example 2 (step strain experiment): Apply a unit of strain, then measure τ(t):

0-T

Storage and loss moduli

• Suppose instead that we impose an oscillatory shear: γ(t) = a sin ωt – Amplitude a small enough that material response is linear – Suppose we do not know the relaxation modulus, but measure τ(t)

– This is a Fourier- type integral, defining a complex function of ω– Complex shear modulus:

( ) ( ) ( ) ( )

( ) ( )

( )

′′=

′′−′=

−=−=

′−∞

∞

∞−∞−

∫

∫

∫∫

tdetGea

tdtttGa

dssastGdssstGt

titi

tt

ωωω

ωω

ωωγτ

0

0

cos

cos

Re

( ) ( ) ( ) ( )

ModulusLoss

ModulusStorage0

ωωωω ω GiGtdetGiG ti* ′′+′=′′= ′−∞

∫

Storage and loss moduli 2

• Measured τ(t):

E.g.– Purely elastic solid: G(t) = G0 and we expect τ(t)= G0 γ(t)– Purely viscous fluid: G(t) = 2η0δ(t) and we expect τ(t)= η�̇�𝛾 𝑡𝑡

• G’ is analogous to the elastic shear modulus G0, which measures the ability to store elastic energy

• G’’ is related to the ability of the material to loseenergy through viscous dissipation– Note that: G’’/ω has dimensions of viscosity, denoted η’

( ) ( )[ ]( ) [ ][ ]( )

[ ] ( ) ( )ωγγωω

ωωωτ ω

tGtGtsinGtcosGa

GiGtsinitcosaiGeat *ti

′′+′=′+′′=

′−′′+=−= ReRe

Linear Maxwell model

• G(t) = G0e-t/λ1

– We find (exercise)

– High frequencies: ωλ1 >> 1 ⇒ G*≈ G0 an elastic solid– Low frequencies: ωλ1 << 1 ⇒ G*≈ iωλ1G0=iG’’, a viscous fluid with

viscosity λ1 G0 (=η0)

• Plot G’ and G’’– Curves cross when ωλ1 =1

• For more complex fluids– Different behaviour, but typically both increase– Construct G’ and G’’ from rheometer– Physical interpretation same:

Elastic dominates at high frequencies Viscous behaviour at low frequencies

– Intercept taken as an indicative relaxation time

( ) ( )( )21

12

10

1

01

11 ωλωλωλ

ωλωλω

++

=+

=iG

iGiG*

Example: CTAB, wormlike micellar solution

Hegelson et al., J. Rheol. 2009

Toy model for linear Maxwell model:

• Simple unsteady shear flow – Linear elasticity, no pressure gradient

– If λ1 =0, leads to parabolic problems for u(y,t) Start up flow in channel, Stokes 1st problem, etc

• Combining 2 equations: – Note G0=η0 /λ1 and so as λ1 →∞

With c2 = G0/ρ, i.e. c is an elastic wavespeed

– At intermediate λ1 2nd term provides “damping” from elastic relaxation

ytu

yu

t

∂∂

=∂∂

∂∂

=+∂∂

τρ

ηττλ 01

2

2

1

0

12

2 1yu

tu

tu

∂∂

=∂∂

+∂∂

ρλη

λ

2

22

2

2

yuc

tu

∂∂

=∂∂

E

c σE

c σKelvin-Voigt

Maxwell

Kelvin-Voigt element

• Step stress (creep) experiment– Maxwell has strain increasing with t

• Kelvin-Voigt: spring-dashpot in parallel– Same displacement, stress is additive

– λ2=η2/G0 = retardation time

• Creep experiments: – Apply constant τ0 at t=0– Release at t=t1– Elastic element retards γ to zero

• Note however that Kelvin-Voigt is not good for step strain rate

+=+=+=⇒

=

=

dtdG

dtdG

dtd

G γλγγηγτττγητ

γτ202021

22

01

( ) ( )210

0 λτγ /teG

t −−=

( ) ( ) ( ) 211

λγγ /ttett −−=

Jeffreys element

• Replaces spring of Maxwell with a Kelvin-Voigt element– 4 unknowns– 3 equations

– Eliminate γ1 & γ2

– Jeffreys model: λ1 relaxation time λ2 retardation time Can explore limits of zero relaxation & retardation times η1 is same as zero-shear rate viscosity η0 from Maxwell

dtd

dtd

dtd 21 γγγ

+=

dtd 1

1γητ =

dtdG 2

220γηγτ +=

+=+

+=+

+

2

2

211

2

2

0

21

0

21

dtd

dtd

dtd

dtd

Gdtd

dtd

G

γλγηττλ

γηγηττηη

Jeffreys model:

• Assume finite strain rate and stress as t=-∞– Solve the DE (exercise):

• Relaxation modulus form:

• Memory function form:

+=

+=+

dtd

dtd

dtd

dtd γλγηγλγηττλ

212

2

211

( ) ( ) ( ) ( )ttdtett

/tt γλληγ

λλ

λητ λ

2

11

2

1

1

1 11 +′′

−= ∫

∞−

′−−

( ) ( ) ( )

( )

( ) tdtttett

ttG

/tt ′′

′−+

−= ∫

∞−

′−

′−− γδλλη

λλ

λητ λ

:modulus Relaxation

2

11

2

1

1

1 21 1

( ) ( ) ( )

( )

( ) tdttttd

dett

ttM

/tt ′′

′−

′+

−−= ∫

∞−

′−

′−− γδλλη

λλ

λητ λ

:functionMemory

2

11

2

121

1 21 1

Linear elastic models

• Motivations: – Materials scientists &

chemists have developed experiments that relate structure to linear mechanical responses

– Linear responses provide a common comparative language useful for quality control

– Background for nonlinear elastic responses

• Caveat: we have been rather sloppy in this lecture– Fluid (viscous) elements:

stress ∝ strain rate at t

– Elastic (spring) elements: stress ∝ strain, but strain is relative to an isotropic reference state (at t0)

• Large times & strains?– Nonlinearity in strains– Time derivatives of γ and τ?

( ) ( )tt γητ =

( ) ( )t,tGt 00γτ =

General linear viscoelastic models

• Expressed in integral form, with respect to relaxation modulus and memory functions

• Notes – The reference state taken for γ is that at the current time, i.e.

for fluids we have no natural reference state– Both relaxation modulus and memory function are assumed to

be positive & decrease monotonically to zero in time Physical features of fading memory, relaxation M is the derivative of G

– Our previous integral expressions for Maxwell & Jeffrey models should have γ replaced by γ(t,t’)

( ) ( ) ( )

( ) ( ) ( ) tdt,tttMt

tdtttGt

t

t

′′′−−=

′′′−=

∫

∫

∞−

∞−

γτ

γτ

Viscoelastic fluids III: Constitutive Models

Constitutive modelling

• General principles:– Constitutive equations

independent of the observer and of material frame used Constitutive equations should be

form invariant under Euclidean transformations

Frame indifference is referred to as objectivity

The same stresses should be given at time t by constitutive equations in two different reference frames, provided that the coordinates of a point coincide at time t.

– Stress tensor at t should only depend on the past t’ < t

– Constitutive equations should be local in space

• Modelling steps: – Change 1D toy models to

objective tensor quantities– Derivatives of the strain and

strain rate?– Derivatives of the stress and

what do they mean?– Nonlinear models

• Here we will work with differential forms

– For some models it is also possible to derive them from the integral form

– Ideally also we can relate the continuum model to a microscopic description

See e.g. Bird et al. 1987, Joseph 1990, Irgens 2008

Higher order fluids I

• Systematic treatment focusing on �̇�𝛾(𝑡𝑡): – Perturbation of viscous fluids to account for elastic effects – Assumes that �̇�𝛾 and its derivatives are small

• n-th rate of strain tensor:

– Example 1 simple shear: u=(u(y,t),0,0), with u(y,t) = y�̇�𝛾(𝑡𝑡), then

– Find that (exercise):

( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ,...,n,DtD

nnT

nn 321

1

=∇⋅+⋅∇−=

=

+ uγγuγγ

γγ

( ) ( ) ( ) ( )t,t γγ

=∇

==

000001000

000001010

1 uγγ

( ) ( ) ( ) ( ) ( ) ( )tdtdt

dtd,tt

dtd 2

2

2

32

2

000000001

3000001010

000000001

2000001010

γγγγ

−

=

−

= γγ

Higher order fluids II

• Assume an expansion of stress in terms of the n-th rate of strain tensors

– Ordered fluids: n-th order fluid truncates expansion at order n– Example 1 continued, steady simple shear:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )( ) ( ) ...bbbbbb : ++⋅+⋅++⋅++=

sorder term Third

11111112211233

sorder term Second

11112211 γγ:γγγγγγγγγγτ

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] 31221

211

211

221

2000001010

000010001

2000000001

2000001010

γγ

γγγ

−=⋅+⋅

=⋅

=

−=

=

γγγγγγ

γ:γγγ

,

,,

Higher order fluids III

• Finally:

• Can identify the visco-metric functions:

– Second order fluid has constant viscosity and normal stress coefficients We’d expect b2<0, b11<0 for polymeric liquids Further restrictions arise in different flows

– Third order fluid can be shear-thinning, but in that case strain rate must be limited

[ ]( ) 211

22

3111121

000010001

000000001

22000001010

γγγγ

+

−−−

= bbbbb :τ

[ ]

1122

112

212

21

2111121

22

2

bbNbbN

bbb :

=⇒=

−=⇒−=

−−=

ψγ

ψγ

γη

See chapter 6 of Bird, Armstrong & Hassager for more

Objective time derivatives

• Although the stress is objective (frame indifferent), it can be shown that its time derivative is not

• Instead new forms of derivative are needed to describe time derivatives of tensors used in constitutive models

– The derivative we have used for order fluids is called the upper convected derivative:

• These are not unique, e.g.– Lower convected derivative:

– Corotational (Jaumann) derivative:

( ) ( )[ ]uττuττ ∇⋅+⋅∇−=∇

T

DtD

( ) ( )[ ]uττuττ ∇⋅+⋅∇+= T

DtDΔ

( ) ( )[ ]T

DtD

uuW

WττWττ

∇−∇=

⋅+⋅−=

21

o

Maxwell like models

• Maxwell element:• Upper convected

Maxwell (UCM) fluid – Upper convected derivative– Frequently used

• Lower convectedMaxwell (LCM) fluid – Lower convected derivative

• Johnson-Segalman fluid– Linear combination of upper

convected & lower convectedderivatives a = slip parameter

– Non-monotone flow curves in simple shear

– Used for shear-banding

γηττλ 11 =+dtd

γττ 11 ηλ =+∇

γττ 1

Δ

1 ηλ =+

γτττ 1

Δ

1 21

21 ηλ =+

−

++ ∇ aa

Jeffreys framework:

• Jeffreys element: – Tensorial form– Use upper convected derivative– Oldroyd-B fluid

(Upper convected Jeffreys model) Popular model: simplest model that

contains relaxation and retardation For certain parameters, imposed step

extensional strain rate does not lead to steady extensional stress

• Special cases:– λ2=0: UCM fluid – λ1=0: Second order fluid with

zero 2nd normal stress– λ1=λ2: Newtonian fluid

+=+

dtd

dtd γλγηττλ

211

+=+

∇∇

γγττ 211 ληλ

γττ 11 ηλ =+∇

+=

∇

γγτ 21 λη

See example 7.2-1 & 7.2-2 of Bird et al 1987

Oldroyd 8-constant model

• Systematic approach to include all possible quadratic terms, within upper convectedframework

– Numerous constraints on the constants – Model contains many simpler models – Material functions for this model have been calculated

( ) ( ) ( )

( )

+⋅++=

++⋅+⋅++

∇

∇

δγγγγγγ

δγτγτγττγττ

:2

:2

tr22

74

BOldroyd

20

653

BOldroyd

1

λλλη

λλλλ

From Bird et al 1987

Calculate response in steady shear, extension etc

From Bird et al 1987

Oldroyd 4-constant modelλ3=λ 4= λ 6= λ 7= 0

From Bird et al 1987

Nonlinear models

• Oldroyd B, Maxwell, J-S are linear models, made nonlinear through the objective derivatives used– Referred to as quasi-linear models– Viscous terms linear in UCM, LCM, J-S

• Instead, can approach nonlinear modeling directly, e.g.– White-Metzner:

Generalised Newtonian form of UCM

– Gieskus: Quadratic stress terms in UCM

– Phan-Thien-Tanner (PTT) f(τ) linear or exponential increasing function of tr(τ)

( ) ( )γττ

γηγη=+

∇

0G

γττττ 11

11 η

ηαλλ =⋅++

∇

( ) γτττ 11 ηλ =+∇

f

FENE-type models

• Developed in response to problems with Oldroyd B in extensional flows

• FENE = Finite-Extensibility-Nonlinear-Elastic, e.g.– FENE-CR (Chilcott & Rallison)

Note: upper convected derivative applies to entire bracketed term

– FENE-P (Peterlin)

– Function f(τ) ensures finite extensibility L is dimensionless and relates to maximum

extension of polymer chains

( ) γτττ

11 ηλ =+

∇

f

( ) ( ) ( ) Iτ

γτ

τττ

−=+

∇

fDtD

ff11

1 ηλ

( ) ( )ττ tr1 21

1

Lf

ηλ

+=