6.3 Perzyna Hardening Model
389
with abbreviations ¯
σ
vp := σ
vp + σ
hk and ¯
E := E + K . As a conclusion the algorithmic tangent E a is thus finally expressed as
E
n
a = E − H 0 ((λ)
E
2
E + H + K + η//t n .
(6.257)
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
Table 6.12 Algorithmic update for the specific Perzyna mixed (isotropic and kinematic) hardening
model
Input
n n−1
vp
n−1
hi
n−1
hk
Trial Strain
vp = n−1
vp
hi =
n−1
hi
hk =
n−1
hk
Trial Stress
σ
vp = −E [
vp − n ]
σ
hi = −H
hi
σ
hk = −K
hk
Trial Yield
φ = |σ
vp + σ
hk | − σ y + σ
hi
Loading Check IF φ < 0 THEN
λ = 0
ELSE
λ =
φ
E + H + K + η//t n
ENDIF
Update Strain n
vp =
vp + λ
σ
vp + σ
hk
|σ
vp + σ
hk |
n
hi =
hi + λ
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 + H + K + η//t n
Output
σ n n
vp n
hi n
hk E n
a