Using Eq. (5.4b) for the rate-independent case or Eq. (5.8) for the rate-dependent
case results in the following nonlinear scalar equation for the consistency parameter
γ which can be solved by a local Newton method:
g Δγ
ð Þ ¼ S n À X
Φ
n
2 þ 1 À Φ
ð
Þ2μΔe nþ1 þ b nþ1 ΔγX
Φ
n
2
n
þ2 S n À X
Φ
n
À
Á : 1 À Φ
ð
Þ2μΔe nþ1 þ b nþ1 ΔγX
Φ
n
Â
Ã
'
:
1=2
À 1 À Φ
ð
Þ
ffiffi ffi
2
3
r
K α n þ
ffiffi ffi
2
3
r
Δγ
!
À Δγ 1 À Φ
ð
Þ2μ þ a nþ1
½
Š À Θ
Δγη
Δt
ð5:26Þ
Once Eq. (5.26) is solved for Δγ,the following updating scheme can be used
α nþ1 ¼ α n þ
ffiffiffiffiffi
2 = 3
p Δγ
ð5:27Þ
ε
vp
nþ1 ¼ ε
vp
n þ Δγ
B n
B n
k k
ð5:28Þ
X
Φ
nþ1 ¼ X
Φ
n þ a nþ1 Δγ
S nþ1 À X
Φ
nþ1
S nþ1 À X
Φ
nþ1
À
c
0
2
c 1
X
Φ
n
!
ð5:29Þ
ξ
Φ
nþ1 ¼ 1 À Φ
ð
ÞK α nþ1
ð
Þ
B n
B n
k k
ð5:30Þ
S nþ1 ¼ ξ
Φ
nþ1 þ X
Φ
nþ1
ð5:31Þ
σ nþ1 ¼ ҡ 1 À Φ
ð
Þtr ε nþ1
ð
ÞI þ 2μ 1 À Φ
ð
Þ
 e nþ1 À ε
vp
n À γ nþ1
B n
B n
k k
À e
θ
nþ1
ð5:32Þ
5.3.1 Linearization (Consistent Jacobian)
Differentiating Eq. (5.32) with respect to the total strain but not entropy at the end of
the step yields
5.3 Return Mapping Algorithm
211