6.3 Perzyna Hardening Model
341
˙
vp (σ vp , σ hi ) =
|σ vp | − [σ y − σ hi ]]
η
σ vp
|σ vp |
,
(6.145a)
˙
hi (σ vp , σ hi ) =
|σ vp | − [σ y − σ hi ]]
η
.
(6.145b)
Finally, the visco-plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of the accumulated visco-plastic deformation, i.e.
Table 6.7 Summary of the specific Perzyna isotropic hardening model
(1) Strain
= e + vp
(2) Energy ψ =
1
2 E [ − vp ] 2 +
1
2 H 2
hi
(3) Stress
σ = E [ − vp ] ≡ σ ≡ −σ
vp
(4) Stress
σ hi = −H hi
(5) Potential π v =
1
2 η |˙ vp | 2
(6) Potential π p = σ hi
y |˙ vp | − H hi ˙
hi with σ hi
y := σ y + H hi
(7) Stress
σ vp = η ˙
vp + σ hi
y
˙
vp
|˙ vp |
≡ σ
vp for ˙
vp = 0
(8) Stress
σ hi = −H hi
or
(5) Potential π ∗
v =
1
2 |σ v | 2 /η
(6) Yield
0 ≥ |σ p | − σ hi
y
(7) Evolution ˙
vp = λ
σ p
|σ p |
=
σ v
η
(8) Evolution ˙
hi = λ
(9) KKT
λ ≥ 0, |σ p | ≤ σ hi
y , λ |σ p | = λ σ hi
y
or
(5) Potential π ∗ =
1
2 |σ vp | − σ hi
y 2 /η
(6) Evolution ˙
vp =
|σ vp | − σ hi
y
η
σ vp
|σ vp |
(7) Evolution ˙
hi =
|σ vp | − σ hi
y
η