362
6 Visco-Plasticity
formation
π(˙ vp , ˙
hk ) = max
σ vp ,σ hk
d(σ vp , σ hk ; ˙
vp , ˙
hk ) −
1
2
|σ vp + σ hk | − σ y
2
η
, (6.189)
Table 6.9 Summary of the specific Perzyna kinematic hardening model
(1) Strain
= e + vp
(2) Energy ψ =
1
2 E [ − vp ] 2 +
1
2 K 2
hk
(3) Stress
σ = E [ − vp ] ≡ σ ≡ −σ
vp
(4) Stress
σ hk = −K hk
(5) Potential π v =
1
2 η |˙ vp | 2
(6) Potential π p = σ y |˙ vp | + K hk [˙ vp − ˙
hk ]
(7) Stress
σ vp = η ˙
vp + σ y
˙
vp
|˙ vp |
+ K hk ≡ σ
vp
(8) Stress
σ hk = −K hk
↑for ˙
vp = 0
or
(5) Potential π ∗
v =
1
2 |σ v | 2 /η
(6) Yield
0 ≥ |σ hk
p | − σ y with σ hk
p := σ p − K hk
(7) Evolution ˙
vp = λ
σ hk
p
|σ hk
p |
=
σ v
η
(8) Evolution ˙
hk = λ
σ hk
p
|σ hk
p |
=
σ v
η
(9) KKT
λ ≥ 0, |σ hk
p | ≤ σ y , λ |σ hk
p | = λ σ y
or
(5) Potential π ∗ =
1
2 |σ hk
vp | − σ y 2 /η with σ hk
vp := σ vp − K hk
(6) Evolution ˙
vp =
|σ hk
vp | − σ y
η
σ hk
vp
|σ hk
vp |
(7) Evolution ˙
hk =
|σ hk
vp | − σ y
η
σ hk
vp
|σ hk
vp |