268
5 Plasticity
Table 5.11 Summary of the specific Prandtl mixed hardening model
(1) Strain
= e + p
(2) Energy ψ =
1
2 E [ − p ] 2 +
1
2 H 2
hi +
1
2 K 2
hk
(3) Stress
σ = E [ − p ] ≡ σ ≡ σ
p
(4) Stress
σ hi = −H hi
(5) Stress
σ hk = −K hk
(6) Potential π = [σ y + H hi ] |˙ p | − H hi ˙
hi + K hk [˙ p − ˙
hk ]
(7) Stress
σ p = [σ y + H hi ]
˙
p
|˙ p |
+ K hk ≡ σ
p for ˙
p = 0
(8) Stress
σ hi = −H hi
(9) Stress
σ hk = −K hk
or
(6) Yield
0 ≥ |σ hk
p | − σ hi
y with σ hi
y := [σ y + H hi ]
(7) Evolution ˙
p = λ
σ hk
p
|σ hk
p |
with σ
hk
p := [σ p − K hk ]
(8) Evolution ˙
hi = λ
(9) Evolution ˙
hk = λ
σ hk
p
|σ hk
p |
(10) KKT
λ ≥ 0, |σ hk
p | ≤ σ hi
y , λ |σ hk
p | = λ σ hi
y
and
n
hi :=
n
hi −
n−1
hi
= λ,
(5.196)
whereby λ = t
n
λ
n . Consequently, the plastic stress σ p , the kinematic-hardening
stress σ hk and the isotropic-hardening stress σ hi are updated at the end of the time
step by
σ
n
p = −E [
n
p −
n
] =: σ
p − E
n
p ,
(5.197)
σ
n
hk = −K
n
hk
=: σ
hk − K
n
hk ,
σ
n
hi = −H
n
hi
=: σ
hi − H
n
hi .