230
5 Plasticity
Obviously, the expressions in Eqs. 5.107 and 5.116 are inverse relations. Identifying ˙
hi with |˙ p | and setting σ hi = 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. 5.116 can be rephrased in terms of the postulate of
maximum dissipation (due to isotropic-hardening plasticity) that follows from the
reverse Legendre transformation
π(˙ p , ˙
hi )) = max
σ p ,σ hi
{d(σ p , σ hi ; ˙
p , ˙
hi ) − I A (σ p , σ hi )}
(5.118)
= max
{σ p ,σ hi }∈ A
{d(σ p , σ hi ; ˙
p , ˙
hi )},
whereby d(σ p , σ hi ; ˙
p , ˙
hi ) := σ p ˙
p + σ hi ˙
hi denotes the dissipation power density.
The postulate of maximum dissipation can, alternatively, be recast as a variational
inequality: For given {˙ p , ˙
hi }, find {σ p , σ hi } ∈ A as the solution of
d(σ p , σ hi ; ˙
p , ˙
hi ) ≥ d(σ
p , σ
hi ; ˙
p , ˙
hi ) ∀{σ
p , σ
hi } ∈ A,
(5.119)
whereby {σ
p , σ
hi } denote any admissible plastic and isotropic-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 y − σ hi ] by the Lagrange multiplier λ ≥ 0
(σ p , σ hi , λ; ˙
p , ˙
hi ) := −d(σ p , σ hi ; ˙
p , ˙
hi ) + λ
|σ p | − [σ y − σ hi ]
.
(5.120)
In accordance with Eq. 5.116 the stationarity conditions of this constrained optimization problem then read
˙
p = λ
σ p
|σ p |
and ˙
hi = λ,
(5.121)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format
λ ≥ 0, |σ p | ≤ [σ y − σ hi ], λ |σ p | = λ [σ y − σ hi ].
(5.122)
Note that it follows immediately from Eq. 5.121 that |˙ p | = ˙
hi = λ. Finally, the
plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of
the accumulated plastic deformation, i.e.
κ =
˙
κ dt with ˙
κ := |˙ p | = ˙
hi = λ ≥ 0.
(5.123)
5 Plasticity
Obviously, the expressions in Eqs. 5.107 and 5.116 are inverse relations. Identifying ˙
hi with |˙ p | and setting σ hi = 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. 5.116 can be rephrased in terms of the postulate of
maximum dissipation (due to isotropic-hardening plasticity) that follows from the
reverse Legendre transformation
π(˙ p , ˙
hi )) = max
σ p ,σ hi
{d(σ p , σ hi ; ˙
p , ˙
hi ) − I A (σ p , σ hi )}
(5.118)
= max
{σ p ,σ hi }∈ A
{d(σ p , σ hi ; ˙
p , ˙
hi )},
whereby d(σ p , σ hi ; ˙
p , ˙
hi ) := σ p ˙
p + σ hi ˙
hi denotes the dissipation power density.
The postulate of maximum dissipation can, alternatively, be recast as a variational
inequality: For given {˙ p , ˙
hi }, find {σ p , σ hi } ∈ A as the solution of
d(σ p , σ hi ; ˙
p , ˙
hi ) ≥ d(σ
p , σ
hi ; ˙
p , ˙
hi ) ∀{σ
p , σ
hi } ∈ A,
(5.119)
whereby {σ
p , σ
hi } denote any admissible plastic and isotropic-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 y − σ hi ] by the Lagrange multiplier λ ≥ 0
(σ p , σ hi , λ; ˙
p , ˙
hi ) := −d(σ p , σ hi ; ˙
p , ˙
hi ) + λ
|σ p | − [σ y − σ hi ]
.
(5.120)
In accordance with Eq. 5.116 the stationarity conditions of this constrained optimization problem then read
˙
p = λ
σ p
|σ p |
and ˙
hi = λ,
(5.121)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker format
λ ≥ 0, |σ p | ≤ [σ y − σ hi ], λ |σ p | = λ [σ y − σ hi ].
(5.122)
Note that it follows immediately from Eq. 5.121 that |˙ p | = ˙
hi = λ. Finally, the
plastic strain arc-length, denoted κ, may conveniently be introduced as a measure of
the accumulated plastic deformation, i.e.
κ =
˙
κ dt with ˙
κ := |˙ p | = ˙
hi = λ ≥ 0.
(5.123)