time increments are used for isothermal cases at different temperatures, and a
suitable time increment is chosen based on convergence study. Objectivity (frame
indifference) was ensured by using stress components and their conjugate rate
components in reference configuration (in material coordinates). Final forms of
constitutive equations are obtained after a transformation of evolution equations to
reference configuration. Evolution of elastic and plastic deformation gradients is
approximated by exponential operator (Weber and Anand 1990). When exponential
operator is used in combination with backward Euler scheme, plastic
incompressibility and symmetry of state variable tensors are conserved with sufficient accuracy by only including the first two terms of series representation of
exponential function. Rate dependency of some material properties (E, μ M , θ g ) are
implemented in numerical algorithm such that property values are calculated based
on strain rates in previous increment and strain rates are updated based on initial and
final deformation gradients at the end of time increment. Therefore, equilibrium is
satisfied at discrete time increments assuring thermodynamic consistency.
Table 7.1 Material parameters for PMMA constitutive model (Gunel 2010)
Parameters
Parameters
Parameters
θ
ref
g K
ð Þ
373
ϕ
Ã
g
0.001
g r
9.55
c
g
1 K
ð Þ
32.58
ϕ
Ã
g
0
Δθ
g (K)
5
c
g
2
87.5
θ ϕ (K)
0
θ
g (K)
5
c 1 (K)
9
Δ ϕ (K)
10
X
g
g K
À1
À
Á
0.07
c 2
116
X
g
ϕ 1=K
ð
Þ
24.5 3 10
26
b 1 (MPa)
2.72 3 10
10
ν
ref
(s
À1
)
1.73 3 10
22
X
r
ϕ 1=K
ð
Þ
0
b 2 (1/K)
24.58 3 10
22
E g (MPa)
500
I
g
M
7.04
b 3
4.04 3 10
22
E r (MPa)
1.5
I
r
M
4.6
ν
p
ref s
À1
ð Þ
0.001
θ E (K)
28
θ M (K)
220
ν
o
I s
À1
ð Þ
2.43 3 10
12
Δ E (K)
15
Δ M (K)
12
Q I (J/K)
1.56 3 10
219
s E
0.06
X
g
M 1=K
ð
Þ
0
V(m
3
)
1.39 3 10
227
X
g
E MPa=K
ð
Þ 215.05
X
r
M 1=K
ð
Þ
0.001
α p
0.21
X
r
E MPa=K
ð
Þ 0
S
g
M MPa
ð
Þ
35
n I
2.17
ν g
0.31
S
r
M MPa
ð
Þ
0.2
h I
40.42
ν r
0.49
θ S (K)
5
γ(MPa)
60
μ
g
M MPa
ð
Þ
9
Δ S (K)
5
ν
o
M s
À1
ð Þ
4.33 3 10
9
μ
r
M MPa
ð
Þ
0.5
X
g
S MPa=K
ð
Þ
20.01
Q M (J/K)
1.30 3 10
219
θ μ (K)
5
X
r
S MPa=K
ð
Þ
0
h M
14.43
Δ μ (K)
14
B g (MPa)
10
n M
5
s μ
0.03
X
g
B MPa=K
ð
Þ 2.15
m s (g/mol)
100.13
X
g
μ MPa=K
ð
Þ
20.4
Δθ B (K)
5
X
r
μ MPa=K
ð
Þ
0
g g
7.97
7.7 Numerical Implementation of Dual-Mechanism Viscoplastic Model
383
suitable time increment is chosen based on convergence study. Objectivity (frame
indifference) was ensured by using stress components and their conjugate rate
components in reference configuration (in material coordinates). Final forms of
constitutive equations are obtained after a transformation of evolution equations to
reference configuration. Evolution of elastic and plastic deformation gradients is
approximated by exponential operator (Weber and Anand 1990). When exponential
operator is used in combination with backward Euler scheme, plastic
incompressibility and symmetry of state variable tensors are conserved with sufficient accuracy by only including the first two terms of series representation of
exponential function. Rate dependency of some material properties (E, μ M , θ g ) are
implemented in numerical algorithm such that property values are calculated based
on strain rates in previous increment and strain rates are updated based on initial and
final deformation gradients at the end of time increment. Therefore, equilibrium is
satisfied at discrete time increments assuring thermodynamic consistency.
Table 7.1 Material parameters for PMMA constitutive model (Gunel 2010)
Parameters
Parameters
Parameters
θ
ref
g K
ð Þ
373
ϕ
Ã
g
0.001
g r
9.55
c
g
1 K
ð Þ
32.58
ϕ
Ã
g
0
Δθ
g (K)
5
c
g
2
87.5
θ ϕ (K)
0
θ
g (K)
5
c 1 (K)
9
Δ ϕ (K)
10
X
g
g K
À1
À
Á
0.07
c 2
116
X
g
ϕ 1=K
ð
Þ
24.5 3 10
26
b 1 (MPa)
2.72 3 10
10
ν
ref
(s
À1
)
1.73 3 10
22
X
r
ϕ 1=K
ð
Þ
0
b 2 (1/K)
24.58 3 10
22
E g (MPa)
500
I
g
M
7.04
b 3
4.04 3 10
22
E r (MPa)
1.5
I
r
M
4.6
ν
p
ref s
À1
ð Þ
0.001
θ E (K)
28
θ M (K)
220
ν
o
I s
À1
ð Þ
2.43 3 10
12
Δ E (K)
15
Δ M (K)
12
Q I (J/K)
1.56 3 10
219
s E
0.06
X
g
M 1=K
ð
Þ
0
V(m
3
)
1.39 3 10
227
X
g
E MPa=K
ð
Þ 215.05
X
r
M 1=K
ð
Þ
0.001
α p
0.21
X
r
E MPa=K
ð
Þ 0
S
g
M MPa
ð
Þ
35
n I
2.17
ν g
0.31
S
r
M MPa
ð
Þ
0.2
h I
40.42
ν r
0.49
θ S (K)
5
γ(MPa)
60
μ
g
M MPa
ð
Þ
9
Δ S (K)
5
ν
o
M s
À1
ð Þ
4.33 3 10
9
μ
r
M MPa
ð
Þ
0.5
X
g
S MPa=K
ð
Þ
20.01
Q M (J/K)
1.30 3 10
219
θ μ (K)
5
X
r
S MPa=K
ð
Þ
0
h M
14.43
Δ μ (K)
14
B g (MPa)
10
n M
5
s μ
0.03
X
g
B MPa=K
ð
Þ 2.15
m s (g/mol)
100.13
X
g
μ MPa=K
ð
Þ
20.4
Δθ B (K)
5
X
r
μ MPa=K
ð
Þ
0
g g
7.97
7.7 Numerical Implementation of Dual-Mechanism Viscoplastic Model
383