344
6 Visco-Plasticity
depends implicitly on
n
∂ σ
n
vp = E − E λ ∂
σ
vp
|σ
vp |
− E
σ
vp
|σ
vp |
∂ ((λ).
(6.161)
The first derivative term on the right-hand-side computes to zero since
∂
σ
vp
|σ
vp |
=
1
|σ
vp |
E −
σ
vp
|σ
vp | 2
σ
vp
|σ
vp |
E ≡ 0.
(6.162)
It shall be noted that the corresponding tangent modulus (tensor) in more than
one dimension is different from zero. The second derivative term on the right-handside computes from requiring satisfaction of ∂ [φ
n
− λ η//t
n
] = 0 for ongoing
visco-plastic flow at the end of the time step, i.e.
Table 6.8 Algorithmic update for the specific Perzyna isotropic hardening model
Input
n n−1
vp
n−1
hi
Trial Strain
vp = n−1
vp
hi =
n−1
hi
Trial Stress
σ
vp = −E [
vp − n ]
σ
hi = −H
hi
Trial Yield
φ = |σ
vp | − σ y + σ
hi
Loading Check IF φ < 0 THEN
λ = 0
ELSE
λ =
φ
E + H + η//t n
ENDIF
Update Strain n
vp =
vp + λ
σ
vp
|σ
vp |
n
hi =
hi + λ
Update Stress σ n = E [ n − n
vp ]
Tangent
E n
a = E − H 0 ((λ)
E 2
E + H + η//t n
Output
σ n n
vp n
hi E n
a