6.3 Perzyna Hardening Model
401
φ = φ(σ vp , σ h ) := ϕ h (σ vp , σ h ) − σ y ≤ 0.
(6.262)
Here φ = φ(σ vp , σ h ) is the overstress function and ϕ h (σ vp , σ h ) denotes the equivalent (combined visco-plastic and hardening) stress that is compared to the initial
yield limit σ y , a material property. Then the evolution law for the visco-plastic and
hardening strains (i.e. the associated flow rules) follow alternatively to Eq. 6.261a
from the postulate of maximum dissipation (due to hardening visco-plasticity)
˜
1/η (σ vp , σ h ; ˙
p , ˙
ε h ) := −d(σ vp , σ h ; ˙
vp , ˙
ε h ) +
1
2
φ(σ vp , σ h )
2
η
→ min, (6.263)
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity
condition of this unconstrained optimization problem reads
˙
vp = λ ∂ σ vp φ and ˙
ε h = λ ∂ σ h φ with λ := =φ(σ vp , σ h )/η ≥ 0.
(6.264)
It shall be noted that collectively Eqs. 6.262 and 6.264 are entirely equivalent
statements to Eqs. 6.261a and 6.261b.
As a further interesting aspect the dissipation d = σ vp ˙
vp + σ h ˙
ε h shall next be
examined more closely. From Eqs. 6.260a and 6.260b the dissipation d is alternatively
expressed in terms of the dissipation potential π and the dual dissipation potential
π
∗ as
d = π(˙ vp , ˙
ε h ) + π
∗
(σ vp , σ h ) ≥ 0.
(6.265)
Thereby, based on the above introduction of the overstress function φ (and in view
of Eqs. 6.260a, 6.261a, 6.263 and 6.264) the dual dissipation potential is identified
as
π
∗
(σ vp , σ h ) =
⎧
⎨
⎩
0
φ(σ vp , σ h ) ≤ 0
for
1
2
φ(σ vp , σ h )
2
/η
φ(σ vp , σ h ) > 0
⎫
⎬
⎭
(6.266)
=
1
2
φ(σ vp , σ h )
2
η
.
Finally for an equivalent (combined visco-plastic and hardening) stress that
is homogeneous of degree one in the visco-plastic and hardening stresses (thus
σ vp ∂ σ vp ϕ h + σ h ∂ σ h ϕ h = ϕ h ), the dissipation d = σ vp ˙
vp + σ h ˙
ε h is exclusively given
in terms of the overstress function φ (with abbreviation λ := =φ/η ≥ 0 for the viscoplastic multiplier and equivalent stress ϕ h = φ + σ y ≥ 0), since then
d = λ
σ vp ∂ σ vp ϕ h + σ h ∂ σ h ϕ h
= λ ϕ h = =φ [φ + σ y ]/η.
(6.267)
The generic Perzyna hardening model is summarized in Table 6.13.
401
φ = φ(σ vp , σ h ) := ϕ h (σ vp , σ h ) − σ y ≤ 0.
(6.262)
Here φ = φ(σ vp , σ h ) is the overstress function and ϕ h (σ vp , σ h ) denotes the equivalent (combined visco-plastic and hardening) stress that is compared to the initial
yield limit σ y , a material property. Then the evolution law for the visco-plastic and
hardening strains (i.e. the associated flow rules) follow alternatively to Eq. 6.261a
from the postulate of maximum dissipation (due to hardening visco-plasticity)
˜
1/η (σ vp , σ h ; ˙
p , ˙
ε h ) := −d(σ vp , σ h ; ˙
vp , ˙
ε h ) +
1
2
φ(σ vp , σ h )
2
η
→ min, (6.263)
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity
condition of this unconstrained optimization problem reads
˙
vp = λ ∂ σ vp φ and ˙
ε h = λ ∂ σ h φ with λ := =φ(σ vp , σ h )/η ≥ 0.
(6.264)
It shall be noted that collectively Eqs. 6.262 and 6.264 are entirely equivalent
statements to Eqs. 6.261a and 6.261b.
As a further interesting aspect the dissipation d = σ vp ˙
vp + σ h ˙
ε h shall next be
examined more closely. From Eqs. 6.260a and 6.260b the dissipation d is alternatively
expressed in terms of the dissipation potential π and the dual dissipation potential
π
∗ as
d = π(˙ vp , ˙
ε h ) + π
∗
(σ vp , σ h ) ≥ 0.
(6.265)
Thereby, based on the above introduction of the overstress function φ (and in view
of Eqs. 6.260a, 6.261a, 6.263 and 6.264) the dual dissipation potential is identified
as
π
∗
(σ vp , σ h ) =
⎧
⎨
⎩
0
φ(σ vp , σ h ) ≤ 0
for
1
2
φ(σ vp , σ h )
2
/η
φ(σ vp , σ h ) > 0
⎫
⎬
⎭
(6.266)
=
1
2
φ(σ vp , σ h )
2
η
.
Finally for an equivalent (combined visco-plastic and hardening) stress that
is homogeneous of degree one in the visco-plastic and hardening stresses (thus
σ vp ∂ σ vp ϕ h + σ h ∂ σ h ϕ h = ϕ h ), the dissipation d = σ vp ˙
vp + σ h ˙
ε h is exclusively given
in terms of the overstress function φ (with abbreviation λ := =φ/η ≥ 0 for the viscoplastic multiplier and equivalent stress ϕ h = φ + σ y ≥ 0), since then
d = λ
σ vp ∂ σ vp ϕ h + σ h ∂ σ h ϕ h
= λ ϕ h = =φ [φ + σ y ]/η.
(6.267)
The generic Perzyna hardening model is summarized in Table 6.13.