342
6 Visco-Plasticity
κ =
˙
κ dt with ˙
κ := |˙ vp | = ˙
hi =
|σ vp | − [σ y − σ hi ]]
η
≥ 0.
(6.146)
The specific Perzyna isotropic hardening model is summarized in Table 6.7.
6.3.2 Specific Perzyna Isotropic Hardening Model:
Algorithmic Update
For the specific Perzyna isotropic hardening model the evolution laws for the viscoplastic strain vp and the isotropic-hardening strain hi are integrated by the implicit
Euler backwards method to render
n
vp :=
n
vp −
n−1
vp = λ
σ
n
vp
|σ n
vp |
and
n
hi :=
n
hi −
n−1
hi
= λ,
(6.147)
whereby the incremental visco-plastic multiplier λ is defined as
λ := t
n
λ
n
:= t
n
|σ
n
vp | − [σ y − σ
n
hi ]]
η
≥ 0.
(6.148)
Consequently, the visco-plastic stress σ vp and the isotropic-hardening stress σ hi
are updated at the end of the time step by
σ
n
vp = −E [
n
vp −
n
] =: σ
vp − E
n
vp ,
(6.149)
σ
n
hi = −H
n
hi
=: σ
hi − H
n
hi .
Here the trial visco-plastic stress σ
vp and the trial isotropic-hardening stress σ
hi
are computable exclusively from known quantities at the beginning of the time step
and follow as
σ
vp := −E [
n−1
vp −
n
],
(6.150)
σ
hi := −H
n−1
hi
.
Incorporating the discretized evolution law for the visco-plastic strain then renders
σ
n
vp = σ
vp − E λ
σ
n
vp
|σ n
vp |
.
(6.151)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress