5.3 Prandtl Hardening Model
283
Here φ = φ(σ p , σ h ) is the yield function and ϕ h (σ p , σ h ) denotes the equivalent (combined plastic and hardening) stress that is compared to the initial yield limit σ y , a
material property. Then the evolution laws for the plastic and hardening strains (i.e.
the associated flow rules) follow alternatively to Eq. 5.215a from the postulate of
maximum dissipation (due to hardening plasticity) with a Lagrange functional
incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0
(σ p , σ h , λ; ˙
p , ˙
ε h ) := −d(σ p , σ h ; ˙
p , ˙
ε h ) + λ φ(σ p , σ h ).
(5.217)
Consequently, the stationarity conditions of this constrained optimization problem
read
˙
p = λ ∂ σ p φ and ˙
ε h = λ ∂ σ h φ,
(5.218)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form
λ ≥ 0, φ ≤ 0, λ φ = 0.
(5.219)
It shall be noted that collectively Eqs. 5.216, 5.218 and 5.219 are entirely equivalent
statements to Eqs. 5.215a and 5.215b.
As a further interesting aspect the dissipation d = σ p ˙
p + σ h ˙
ε h shall next be
examined more closely. From Eqs. 5.214a and 5.214b the dissipation d is alternatively
expressed in terms of the dissipation potential π and the dual dissipation potential
π
∗ as
d = π(˙ p , ˙
ε h ) + π
∗
(σ p , σ h ) ≥ 0.
(5.220)
However, based on the above introduction of the yield condition φ ≤ 0 the dual
dissipation potential is identified as the indicator function I A of the admissible domain
A
π
∗
(σ p , σ h ) = I A (σ p , σ h ) :=
⎧
⎨
⎩
0
φ(σ p , σ h ) ≤ 0
for
∞
φ(σ p , σ h ) > 0
.
(5.221)
Thus for the generic Prandtl hardening model the dual dissipation potential equals
zero in the admissible domain A. Consequently, provided the plastic and hardening
stresses are admissible, the dissipation is indeed expressed in terms of the dissipation
potential only
d = π(˙ p , ˙
ε h ) ≥ 0.
(5.222)
Finally for an equivalent (combined plastic and hardening) stress that is homogeneous
of degree one in the plastic and hardening stresses (thus σ p ∂ σ p ϕ h + σ h ∂ σ h ϕ h =
283
Here φ = φ(σ p , σ h ) is the yield function and ϕ h (σ p , σ h ) denotes the equivalent (combined plastic and hardening) stress that is compared to the initial yield limit σ y , a
material property. Then the evolution laws for the plastic and hardening strains (i.e.
the associated flow rules) follow alternatively to Eq. 5.215a from the postulate of
maximum dissipation (due to hardening plasticity) with a Lagrange functional
incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0
(σ p , σ h , λ; ˙
p , ˙
ε h ) := −d(σ p , σ h ; ˙
p , ˙
ε h ) + λ φ(σ p , σ h ).
(5.217)
Consequently, the stationarity conditions of this constrained optimization problem
read
˙
p = λ ∂ σ p φ and ˙
ε h = λ ∂ σ h φ,
(5.218)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form
λ ≥ 0, φ ≤ 0, λ φ = 0.
(5.219)
It shall be noted that collectively Eqs. 5.216, 5.218 and 5.219 are entirely equivalent
statements to Eqs. 5.215a and 5.215b.
As a further interesting aspect the dissipation d = σ p ˙
p + σ h ˙
ε h shall next be
examined more closely. From Eqs. 5.214a and 5.214b the dissipation d is alternatively
expressed in terms of the dissipation potential π and the dual dissipation potential
π
∗ as
d = π(˙ p , ˙
ε h ) + π
∗
(σ p , σ h ) ≥ 0.
(5.220)
However, based on the above introduction of the yield condition φ ≤ 0 the dual
dissipation potential is identified as the indicator function I A of the admissible domain
A
π
∗
(σ p , σ h ) = I A (σ p , σ h ) :=
⎧
⎨
⎩
0
φ(σ p , σ h ) ≤ 0
for
∞
φ(σ p , σ h ) > 0
.
(5.221)
Thus for the generic Prandtl hardening model the dual dissipation potential equals
zero in the admissible domain A. Consequently, provided the plastic and hardening
stresses are admissible, the dissipation is indeed expressed in terms of the dissipation
potential only
d = π(˙ p , ˙
ε h ) ≥ 0.
(5.222)
Finally for an equivalent (combined plastic and hardening) stress that is homogeneous
of degree one in the plastic and hardening stresses (thus σ p ∂ σ p ϕ h + σ h ∂ σ h ϕ h =