208
5 Plasticity
The resulting σ = σ() diagram is highlighted in Fig. 5.6c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = 5 obviously tend to ∞ with t → 0.
Figure 5.6d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint p (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 5.6e follows linear-constant-linear in time
from integrating ˙
κ(t) = |˙ (t)| = {5, 0, 5} over the time interval t ∈ [0, t max = 10],
thus κ max = 10.
5.1.4 Generic St. Venant Model: Formulation
A generic formulation of the St. Venant model can be obtained from generalizing the
specific St. Venant model in Fig. 5.1 by assuming the frictional slider as nonlinear.
For the generic St. Venant model the free energy density ψ vanishes identically
ψ() ≡ 0.
(5.45)
Thus the energetic stress σ
vanishes identically as well
σ
() ≡ 0.
(5.46)
Recall that the energetic and the dissipative stresses are constitutively related to
the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
, thus (with
σ
≡ 0) the total stress σ ≡ σ
will exclusively be used in the sequel.
Furthermore, for the generic St. Venant model the convex but non-smooth dissipation and dual dissipation potentials introduced as π = π(˙ ) and π
∗
= π
∗
(σ),
respectively, are related via corresponding Legendre transformations
π ( ˙
) = max
σ
{σ ˙
− π
∗
(σ)},
(5.47a)
π
∗
(σ) = max
˙
{σ ˙
− π ( ˙
)}.
(5.47b)
Then the stationarity conditions corresponding to Eqs. 5.47a and 5.47b are the constitutive relations
˙
(σ) ∈ d σ π
∗
(σ),
(5.48a)
σ( ˙
) ∈ d ˙
π ( ˙
).
(5.48b)