Viscoelastic Material Models

Outline

• Introduction（引言） • Creep compliance and relaxation modulus（蠕变柔度与松弛刚

度） • Retarded elasticity（滞/粘弹性） • Steady-state creep（稳态蠕变） • Elastic vs. viscoelastic materials（弹性与粘弹性材料） • Maxwell materials（Maxwell模型） • Kelvin-voigt materials（KV模型） • Prony series representation（Prony级数模型） • WLF time/temperature equivalence（WLF时间/温度相当） • Calibrating viscoelastic models（粘弹性模型校准） • More sophisticated viscoelastic models（复杂粘弹性模型） • Generalization to 3D（向三维模型推广） • Response to harmonic loading（周期载荷的响应）

2

• Amorphous polymers show complex time-dependent behavior when subjected to

a history of stress or strain.

• Viscoelasticity theory was developed to approximate this behavior in polymers

that are subjected to modest strains (less than 0.5%).

• Polymers strongly resist volume changes at all temperatures. The bulk modulus is

comparable to that of metals or covalently bonded solids.

• The shear response of a polymer is strongly temperature dependent.

Introduction

3

Shear modulus (N/m2)

Introduction

4

Shear modulus (N/m2)

• At temperatures near the glass transition, the shear modulus is strongly time

dependent. The time dependent shear response can be measured by applying (1) a

step load or (2) a harmonic (sinusoidal) load to the specimen.

• The time-dependent modulus of polymers is also temperature dependent.

Reducing the temperature is qualitatively equivalent to increasing the strain rate.

• Most amorphous polymers are isotropic.

• Creep compliance:

the strain response to

a unit constant stress

• Relaxation modulus:

the stress response to a

unit constant strain.

Response to Step Loading

5

• The results of such a test depend on the degree of cross-linking in

the polymer.

• Heavily cross-linked materials show “retarded elastic” behavior,

whereas un-cross-linked materials show steady-state creep.

• There is always an instantaneous strain in response to a step change in stress.

• At temperatures significantly below Tg, the solid is essentially elastic.

• At temperatures significantly above Tg, the solid is very compliant.

• For a range of temperatures near Tg, the solid shows a slow transient response.

• The deformation is reversible, although this may take a very long time.

Retarded Elasticity for Crosslinked Polymers

6

• There is always an instantaneous strain in response to a step change

in stress, exactly as in crosslinked polymers.

• At temperatures well below Tg, the solid is essentially elastic and

has a very low compliance, comparable with Jg.

• At temperatures above Tg, the solid is very compliant. For most

practical ranges of loading, the compliance will increase more or

less linearly with time. The slope of compliance is strongly

temperature dependent.

Steady-state Creep for uncrosslinked polymers

7

• Above the glass transition

temperature Tg, the deformation

is irreversible.

• For a constant loading acting

from time 0 to T,

• The strain responses of elastic

materials:

Elastic vs. Viscoelastic Materials

8

EE

• The strain responses of viscoelastic materials:

• Maxwell model (steady-state creep of uncrosslinked

polymers)

Viscoelastic Models

9

EE

0

0 0

0

const ln ln

0

Et

Et Et

E EtC t Ce

E

tE C t E e G t Ee

• Creep compliance due to a constant loading σ0 acting

from time 0 to T

• Relaxation modulus due to a constant strain ε0

0 0

00

0

, 0

0

,

t t TE

t

T t T

0

1 1tJ t t

E

Kelvin-Voigt Model (Retarded Elasticity of Crosslinked Polymers)

E E

• Creep compliance due to a

constant loading σ0 acting from

time 0 to T

00

0 0

0 0 0

0 0

0 :

0 0 1

: 0

1 1

Et Et

Et Et Et

Et

Et

EtET ET ET Et

Et T E e e

de e t Ce

dt E

C C t eE E E

Et T E t De

eT De e t e e J t

E E

1ETeE

• Spring + Kelvin

Standard Linear Solid Model

11

1 2

2 2 2

2

1 1

1 2 1 2 1

E E

E

EE E

E E E E E

• Strain response to a constant

loading σ0 acting from time 0 to

T

• Spring // Maxwell

Three-parameter Model

12

2E

1E

2 1

1 1

1

1

2 2

1 2 1

1

2

E E

E E

E

E E E E E

E

1 1

1 1 1 1 1

1

1

1 2 010 1 1 2 0

1 2 02 0 2 0

2 0 1 0 2 0 1 0 2 0 1 0

2 1

0

const

0

E t E t

E t E t E t E t E t

E t

E t

E EEE E E e e

E Ede e e e E C t E Ce

dt

E C E E C E t E E e

tG t E E e

• Relaxation modulus due to a constant strain ε0

• Spring // lots of Maxwell elements

Prony Series for the Relaxation Modulus

13

G

1G2G nG2

0

1

, i

nt t

i i i i

i

G t t G G e t G

• Relaxation modulus due to a constant strain ε0

• These parameters are used directly as the properties of

the material.

• The sum of exponentials is known as the Prony series.

• At temperature T1, subject a specimen to a step change in shear

strain and measure the relaxation modulus.

• Repeat the experiment at several higher temperatures.

• The double log plot demonstrates that, if you simply shift the

modulus curves for the higher temperatures to the right, the data

collapse onto a single master curve.

Williams–Landell–Ferry Time/Temperature Equivalence

14

1log log log ,G f t A T T • WLF shift function:

1 1

1

2 1

log ,C T T

A T TC T T

• More convenient to use Tg as

the reference temperature

1

2

log ,

g

g

g g

g

C T TA T T

C T T

Calibrating Viscoelastic Models

15

Experimental and fitted WLF shift

function for polyisobutylene

Experimental and fitted relaxation

modulus (seven-term Prony series)

of polyisobutylene at Tg

• General mathematical

structure of viscoelastic

models:

More Sophisticated Models

16

• 1D linear elastic

Generalization to 3D

17

• 1D linear viscoelastic

E

• 3D linear elastic

3 ;

2 .

kk kk

ij ij

K

G

P Q E

3 ;

2 .

kk kk

ij ij

P Q K

P Q G

• 3D linear viscoelastic

• Elasticity:

Response to Harmonic Loading

18

• Viscoelasticity:

• Storage compliance: 0 0cos

• Loss compliance: 0 0sin