366
6 Visco-Plasticity
E
n
a = E − H 0 ((λ)
E
2
E + K + η//t n .
(6.209)
Note that, consequently, the algorithmic tangent degenerates to the plastic case
for η → 0 and λ > 0, likewise it degenerates to E a = E for t
n
→ 0. In one
dimension the algorithmic tangent trivially coincides with its continuous counterpart.
It shall be noted, however, that this is at variance with the corresponding result in
two and three dimensions.
The algorithmic step-by-step update for the specific Perzyna hardening model
capturing kinematic hardening is summarized in Table 6.10.
Table 6.10 Algorithmic update for the specific Perzyna kinematic hardening model
Input
n n−1
vp
n−1
hk
Trial Strain
vp = n−1
vp
hk =
n−1
hk
Trial Stress
σ
vp = −E [
vp − n ]
σ
hk = −K
hk
Trial Yield
φ = |σ
vp + σ
hk | − σ y
Loading Check IF φ < 0 THEN
λ = 0
ELSE
λ =
φ
E + K + η//t n
ENDIF
Update Strain n
vp =
vp + λ
σ
vp + σ
hk
|σ
vp + σ
hk |
n
hk =
hk + λ
σ
vp + σ
hk
|σ
vp + σ
hk |
Update Stress σ n = E [ n − n
vp ]
Tangent
E n
a = E − H 0 ((λ)
E 2
E + K + η//t n
Output
σ n n
vp n
hk E n
a