386
6 Visco-Plasticity
dance with Eqs. 6.229, 6.230, 6.231 the stationarity conditions of this unconstrained
optimization problem then read
˙
vp (σ vp , σ hi , σ hk ) =
|σ vp + σ hk | − [σ y − σ hi ]]
η
σ vp + σ hk
|σ vp + σ hk |
, (6.237a)
˙
hi (σ vp , σ hi , σ hk ) =
|σ vp + σ hk | − [σ y − σ hi ]]
η
,
(6.237b)
˙
hk (σ vp , σ hi , σ hk ) =
|σ vp + σ hk | − [σ y − σ hi ]]
η
σ vp + σ hk
|σ vp + σ hk |
. (6.237c)
Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.
κ =
˙
κ dt
(6.238)
with ˙
κ := |˙ vp | = ˙
hi = |˙ hk | =
|σ vp + σ hk | − [σ y − σ hi ]]
η
≥ 0.
The specific Perzyna mixed hardening model is summarized in Table 6.11.
6.3.8 Specific Perzyna Mixed Hardening Model: Algorithmic
Update
For the specific Perzyna mixed (isotropic and kinematic) hardening model the evolution laws for the visco-plastic strain vp , the kinematic-hardening strain hk 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
hk
|σ n
vp + σ
n
hk |
=
n
hk −
n−1
hk =:
n
hk ,
(6.239)
and
n
hi :=
n
hi −
n−1
hi
= λ,
(6.240)
whereby the incremental visco-plastic multiplier λ is defined as
λ := t
n
λ
n
:= t
n
|σ
n
vp + σ
n
hk | − [σ y − σ
n
hi ]]
η
≥ 0.
(6.241)
Consequently, the visco-plastic stress σ vp , the kinematic-hardening stress σ hk and
the isotropic-hardening stress σ hi are updated at the end of the time step by