6.3 Perzyna Hardening Model
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Table 6.11 Summary of the specific Perzyna mixed hardening model
(1) Strain
= e + vp
(2) Energy ψ =
1
2 E [ − vp ] 2 +
1
2 H 2
hi +
1
2 K 2
hk
(3) Stress
σ = E [ − vp ] ≡ σ ≡ −σ
vp
(4) Stress
σ hi = −H hi
(5) Stress
σ hk = −K hk
(6) Potential π v =
1
2 η |˙ vp | 2
↓ σ hi
y := σ y + H hi
(7) Potential π p = σ hi
y |˙ vp | − H hi ˙
hi + K hk [˙ vp − ˙
hk ]
(8) Stress
σ vp = η ˙
vp + σ hi
y
˙
vp
|˙ vp |
+ K hk ≡ σ
vp
(9) Stress
σ hi = −H hi
↑for ˙
vp = 0
(10) Stress
σ hk = −K hk
or
(6) Potential π ∗
v =
1
2 |σ v | 2 /η
(7) Yield
0 ≥ |σ hk
p | − σ hi
y with σ hk
p := σ p − K hk
(8) Evolution ˙
vp = λ
σ hk
p
|σ hk
p |
=
σ v
η
(9) Evolution ˙
hi = λ
(10) Evolution ˙
hk = λ
σ hk
p
|σ hk
p |
=
σ v
η
(11) KKT
λ ≥ 0, |σ hk
p | ≤ σ hi
y , λ |σ hk
p | = λ σ hi
y
or
(6) Potential π ∗ =
1
2 |σ hk
vp | − σ hi
y 2 /η with σ hk
vp := σ vp − K hk
(7) Evolution ˙
vp =
|σ hk
vp | − σ hi
y
η
σ hk
vp
|σ hk
vp |
(8) Evolution ˙
hi =
|σ hk
vp | − σ hi
y
η
(9) Evolution ˙
hk =
|σ hk
vp | − σ hi
y
η
σ hk
vp
|σ hk
vp |