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