248
5 Plasticity
whereby d σ p π
∗ and d σ hk π
∗ denote the sets of sub-derivatives, i.e. the sub-differentials
of π
∗ with respect to σ p and σ hk , respectively, and λ is a positive Lagrange (or rather
plastic) multiplier.
Obviously, the expressions in Eqs. 5.143 and 5.152 are inverse relations. Identifying ˙
hk with ˙
p and setting σ hk = 0, the remaining non-smooth dissipation and
dual dissipation potentials π = π(˙ p ) and π
∗
= π
∗
(σ p ) together with the resulting
non-smooth constitutive relations σ p = σ p (˙ p ) and ˙
p = ˙
p (σ p ) are similar to those
displayed in Fig. 5.3.
Interestingly, the result in Eq. 5.152 can be rephrased in terms of the postulate of
maximum dissipation (due to kinematic-hardening plasticity) that follows from the
reverse Legendre transformation
π(˙ p , ˙
hk )) = max
σ p ,σ hk
{d(σ p , σ hk ; ˙
p , ˙
hk ) − I A (σ p , σ hk )}
(5.154)
= max
(σ p ,σ hk )∈ A
{d(σ p , σ hk ; ˙
p , ˙
hk )},
whereby d(σ p , σ hk ; ˙
p , ˙
hk ) := σ p ˙
p + σ hk ˙
hk denotes the dissipation power density.
The postulate of maximum dissipation can, alternatively, be recast as a variational
inequality: For given {˙ p , ˙
hk }, find {σ p , σ hk } ∈ A as the solution of
d(σ p , σ hk ; ˙
p , ˙
hk ) ≥ d(σ
p , σ
hk ; ˙
p , ˙
hk ) ∀{σ
p , σ
hk } ∈ A,
(5.155)
whereby {σ
p , σ
hk } denote any admissible plastic and kinematic-hardening stress. As
yet another alternative, the postulate of maximum dissipation may be reformulated
as constrained optimization problem with a Lagrange functional incorporating the
admissibility constraint |σ p + σ hk | ≤ σ y by the Lagrange multiplier λ ≥ 0
(σ p , σ hk , λ; ˙
p , ˙
hk ) := −d(σ p , σ hk ; ˙
p , ˙
hk ) + λ
|σ p + σ hk | − σ y
.
(5.156)
In accordance with Eq. 5.152 the stationarity conditions of this constrained optimization problem then read
˙
p = λ
σ p + σ hk
|σ p + σ hk |
and ˙
hk = λ
σ p + σ hk
|σ p + σ hk |
,
(5.157)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format
λ ≥ 0, |σ p + σ hk | ≤ σ y , λ |σ p + σ hk | = λ σ y .
(5.158)
Note that it follows immediately from Eq. 5.157 that |˙ p | = |˙ hk | = λ. Finally, the
plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of
the accumulated plastic deformation, i.e.
5 Plasticity
whereby d σ p π
∗ and d σ hk π
∗ denote the sets of sub-derivatives, i.e. the sub-differentials
of π
∗ with respect to σ p and σ hk , respectively, and λ is a positive Lagrange (or rather
plastic) multiplier.
Obviously, the expressions in Eqs. 5.143 and 5.152 are inverse relations. Identifying ˙
hk with ˙
p and setting σ hk = 0, the remaining non-smooth dissipation and
dual dissipation potentials π = π(˙ p ) and π
∗
= π
∗
(σ p ) together with the resulting
non-smooth constitutive relations σ p = σ p (˙ p ) and ˙
p = ˙
p (σ p ) are similar to those
displayed in Fig. 5.3.
Interestingly, the result in Eq. 5.152 can be rephrased in terms of the postulate of
maximum dissipation (due to kinematic-hardening plasticity) that follows from the
reverse Legendre transformation
π(˙ p , ˙
hk )) = max
σ p ,σ hk
{d(σ p , σ hk ; ˙
p , ˙
hk ) − I A (σ p , σ hk )}
(5.154)
= max
(σ p ,σ hk )∈ A
{d(σ p , σ hk ; ˙
p , ˙
hk )},
whereby d(σ p , σ hk ; ˙
p , ˙
hk ) := σ p ˙
p + σ hk ˙
hk denotes the dissipation power density.
The postulate of maximum dissipation can, alternatively, be recast as a variational
inequality: For given {˙ p , ˙
hk }, find {σ p , σ hk } ∈ A as the solution of
d(σ p , σ hk ; ˙
p , ˙
hk ) ≥ d(σ
p , σ
hk ; ˙
p , ˙
hk ) ∀{σ
p , σ
hk } ∈ A,
(5.155)
whereby {σ
p , σ
hk } denote any admissible plastic and kinematic-hardening stress. As
yet another alternative, the postulate of maximum dissipation may be reformulated
as constrained optimization problem with a Lagrange functional incorporating the
admissibility constraint |σ p + σ hk | ≤ σ y by the Lagrange multiplier λ ≥ 0
(σ p , σ hk , λ; ˙
p , ˙
hk ) := −d(σ p , σ hk ; ˙
p , ˙
hk ) + λ
|σ p + σ hk | − σ y
.
(5.156)
In accordance with Eq. 5.152 the stationarity conditions of this constrained optimization problem then read
˙
p = λ
σ p + σ hk
|σ p + σ hk |
and ˙
hk = λ
σ p + σ hk
|σ p + σ hk |
,
(5.157)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format
λ ≥ 0, |σ p + σ hk | ≤ σ y , λ |σ p + σ hk | = λ σ y .
(5.158)
Note that it follows immediately from Eq. 5.157 that |˙ p | = |˙ hk | = λ. Finally, the
plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of
the accumulated plastic deformation, i.e.