6.2 Perzyna Model
331
π ( ˙
vp ) = max
σ vp
{σ vp ˙
vp − π
∗
(σ vp )},
(6.110a)
π
∗
(σ vp ) = max
˙
vp
{σ vp ˙
vp − π ( ˙
vp )}.
(6.110b)
Then the stationarity conditions corresponding to Eqs. 6.110a and 6.110b are the
constitutive relations
˙
vp (σ vp ) ∈ d σ vp π
∗
(σ vp ),
(6.111a)
σ vp ( ˙
vp ) ∈ d ˙
vp π ( ˙
vp ).
(6.111b)
Obviously the relations in Eqs. 6.111a and 6.111b determine entirely the dissipative behavior of the generic Perzyna model, thus the formulation would be completed
at this stage.
To be more explicit, however, alternatively to Eq. 6.111b the closed and convex
admissible domain A in the σ vp -space is introduced. It is characterized by the convex
yield condition
φ = φ(σ vp ) := ϕ(σ vp ) − σ y ≤ 0.
(6.112)
Here φ = φ(σ vp ) is the overstress function and ϕ(σ vp ) denotes the equivalent
(visco-plastic) stress that is compared to the yield limit σ y , a material property. Then
the evolution law for the visco-plastic strain (i.e. the associated flow rule) follows
alternatively to Eq. 6.111a from the postulate of maximum dissipation (due to viscoplasticity)
˜
1/η (σ vp ; ˙
vp ) := −d(σ vp ; ˙
vp ) +
1
2
φ(σ vp )
2
η
→ min,
(6.113)
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 with λ := =φ(σ vp )/η ≥ 0.
(6.114)
It shall be noted that collectively Eqs. 6.112 and 6.114 are entirely equivalent
statements to Eqs. 6.111a and 6.111b.
As a further interesting aspect the dissipation d = σ vp ˙
vp shall next be examined
more closely. From Eqs. 6.110a and 6.110b the dissipation d is alternatively expressed
in terms of the dissipation potential π and the dual dissipation potential π
∗ as
d = π(˙ vp ) + π
∗
(σ vp ) ≥ 0.
(6.115)
Thereby, based on the above introduction of the overstress function φ (and in view
of Eqs. 6.110a, 6.111a, 6.113 and 6.114) the dual dissipation potential is identified
331
π ( ˙
vp ) = max
σ vp
{σ vp ˙
vp − π
∗
(σ vp )},
(6.110a)
π
∗
(σ vp ) = max
˙
vp
{σ vp ˙
vp − π ( ˙
vp )}.
(6.110b)
Then the stationarity conditions corresponding to Eqs. 6.110a and 6.110b are the
constitutive relations
˙
vp (σ vp ) ∈ d σ vp π
∗
(σ vp ),
(6.111a)
σ vp ( ˙
vp ) ∈ d ˙
vp π ( ˙
vp ).
(6.111b)
Obviously the relations in Eqs. 6.111a and 6.111b determine entirely the dissipative behavior of the generic Perzyna model, thus the formulation would be completed
at this stage.
To be more explicit, however, alternatively to Eq. 6.111b the closed and convex
admissible domain A in the σ vp -space is introduced. It is characterized by the convex
yield condition
φ = φ(σ vp ) := ϕ(σ vp ) − σ y ≤ 0.
(6.112)
Here φ = φ(σ vp ) is the overstress function and ϕ(σ vp ) denotes the equivalent
(visco-plastic) stress that is compared to the yield limit σ y , a material property. Then
the evolution law for the visco-plastic strain (i.e. the associated flow rule) follows
alternatively to Eq. 6.111a from the postulate of maximum dissipation (due to viscoplasticity)
˜
1/η (σ vp ; ˙
vp ) := −d(σ vp ; ˙
vp ) +
1
2
φ(σ vp )
2
η
→ min,
(6.113)
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 with λ := =φ(σ vp )/η ≥ 0.
(6.114)
It shall be noted that collectively Eqs. 6.112 and 6.114 are entirely equivalent
statements to Eqs. 6.111a and 6.111b.
As a further interesting aspect the dissipation d = σ vp ˙
vp shall next be examined
more closely. From Eqs. 6.110a and 6.110b the dissipation d is alternatively expressed
in terms of the dissipation potential π and the dual dissipation potential π
∗ as
d = π(˙ vp ) + π
∗
(σ vp ) ≥ 0.
(6.115)
Thereby, based on the above introduction of the overstress function φ (and in view
of Eqs. 6.110a, 6.111a, 6.113 and 6.114) the dual dissipation potential is identified