5.3 Prandtl Hardening Model
231
Table 5.7 Summary of the specific Prandtl isotropic hardening model
(1) Strain
= e + p
(2) Energy ψ =
1
2 E [ − p ] 2 +
1
2 H 2
hi
(3) Stress
σ = E [ − p ] ≡ σ = −σ
p
(4) Stress
σ hi = −H hi
(5) Potential π = σ hi
y |˙ p | − H hi ˙
hi with σ hi
y := σ y + H hi
(6) Stress
σ p = σ hi
y
˙
p
|˙ p |
≡ σ
p for ˙
p = 0
(7) Stress
σ hi = −H hi
or
(5) Yield
0 ≥ |σ p | − σ hi
y
(6) Evolution ˙
p = λ
σ p
|σ p |
(7) Evolution ˙
hi = λ
(8) KKT
λ ≥ 0, |σ p | ≤ σ hi
y , λ |σ p | = λ σ hi
y
The specific Prandtl isotropic hardening model is summarized in Table 5.7.
5.3.2 Specific Prandtl Isotropic Hardening Model:
Algorithmic Update
For the specific Prandtl isotropic hardening model the evolution laws for the plastic
strain p and the isotropic-hardening strain hi are integrated by the implicit Euler
backwards method to render
n
p :=
n
p −
n−1
p
= λ
σ
n
p
|σ n
p |
and
n
hi :=
n
hi −
n−1
hi
= λ,
(5.124)
whereby λ = t
n
λ
n . Consequently, the plastic stress σ p and the isotropichardening stress σ hi are updated at the end of the time step by