214
5 Plasticity
where I A denotes the indicator function of the admissible domain A in the σ p -space.
The evolution law (the associated flow rule) for the plastic strain then follows as
some sub-derivative of the dual dissipation potential with respect to its conjugated
variable
˙
p (σ p ) ∈ d σ p π
∗
(σ p ) = d σ p I A (σ p ) =
⎧
⎪ ⎨
⎪ ⎩
0
|σ p | < σ y
for
λ
σ p
|σ p |
|σ p | = σ y
⎫
⎪ ⎬
⎪ ⎭
,
(5.69)
whereby d σ p π
∗ denotes the set of sub-derivatives, i.e. the sub-differential of π
∗ with
respect to σ p and λ is a positive Lagrange (or rather plastic) multiplier.
Obviously, the expressions in Eqs. 5.61 and 5.69 are inverse relations. The
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. 5.69 can be rephrased in terms of the postulate
of maximum dissipation (due to perfect plasticity) that follows from the reverse
Legendre transformation
π(˙ p ) = max
σ p
{d(σ p ; ˙
p ) − I A (σ p )} = max
σ p ∈ A
{d(σ p ; ˙
p )},
(5.70)
whereby d(σ p ; ˙
p ) := σ p ˙
p denotes the dissipation power density. The postulate of
maximum dissipation can, alternatively, be recast as a variational inequality: For
given ˙
p , find σ p ∈ A as the solution of
d(σ p ; ˙
p ) ≥ d(σ
p ; ˙
p ) ∀σ
p ∈ A,
(5.71)
whereby σ
p denotes any admissible plastic 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 y by the Lagrange multiplier λ ≥ 0
(σ p , λ; ˙
p ) := −d(σ p ; ˙
p ) + λ
|σ p | − σ y
.
(5.72)
In accordance with Eq. 5.69 the stationarity condition of this constrained optimization
problem then reads
˙
p = λ
σ p
|σ p |
,
(5.73)