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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
+=