5.1 St. Venant Model
209
Obviously the relations in Eqs. 5.48a and 5.48b determine entirely the dissipative
behavior of the generic St. Venant model, thus the formulation would be completed
at this stage.
To be more explicit, however, alternatively to Eq. 5.48b the closed and convex
admissible domain A in the σ-space is introduced. It is characterized by the convex
yield condition
φ = φ(σ) := ϕ(σ) − σ y ≤ 0.
(5.49)
Here φ = φ(σ) is the yield function and ϕ(σ) denotes the equivalent stress that is
compared to the yield limit σ y , a material property. Then the evolution law for the
(total) strain (i.e. the associated flow rule) follows alternatively to Eq. 5.48a from the
postulate of maximum dissipation (due to plasticity) with a Lagrange functional
incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0
(σ, λ; ˙
) := −d(σ; ˙
) + λ φ(σ).
(5.50)
Consequently, the stationarity condition of this constrained optimization problem
reads
˙
= λ ∂ σ φ,
(5.51)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form
λ ≥ 0, φ ≤ 0, λ φ = 0.
(5.52)
It shall be noted that collectively Eqs. 5.49, 5.51 and 5.52 are entirely equivalent
statements to Eqs. 5.48a and 5.48b.
As a further interesting aspect the dissipation d = σ ˙
shall next be examined
more closely. From Eqs. 5.47a and 5.47b the dissipation d is alternatively expressed
in terms of the dissipation potential π and the dual dissipation potential π
∗ as
d = π(˙ ) + π
∗
(σ) ≥ 0.
(5.53)
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) = I A (σ) :=
⎧
⎨
⎩
0
φ(σ) ≤ 0
for
∞
φ(σ) > 0
.
(5.54)
Thus for the generic St. Venant model the dual dissipation potential equals zero
in the admissible domain A. Consequently, provided the stress is admissible, the
dissipation is indeed expressed in terms of the dissipation potential only