308
6 Visco-Plasticity
(t 1 ) = 21. ¯
3 and (t 2 ) = 448, respectively, and a final rigid phase for σ(t) ∈ [1, 0]
with (t) = 469. ¯
3.
The resulting σ = σ() diagram is highlighted in Fig. 6.9c.
Figure 6.9d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.9e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10], thus κ max = 469. ¯
3.
6.1.4 Generic Bingham Model: Formulation
A generic formulation of the Bingham model can be obtained from generalizing
the specific Bingham model in Fig. 6.1 by assuming the viscous dashpot or/and the
frictional slider as nonlinear.
For the generic Bingham model the free energy density ψ vanishes identically
ψ() ≡ 0.
(6.55)
Thus the energetic stress σ
vanishes identically as well
σ
() ≡ 0.
(6.56)
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 Bingham 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)},
(6.57a)
π
∗
(σ) = max
˙
{σ ˙
− π ( ˙
)}.
(6.57b)
Then the stationarity conditions corresponding to Eqs. 6.57a and 6.57b are the
constitutive relations
˙
(σ) ∈ d σ π
∗
(σ),
(6.58a)
σ( ˙
) ∈ d ˙
π ( ˙
).
(6.58b)
Obviously the relations in Eqs. 6.58a and 6.58b determine entirely the dissipative
behavior of the generic Bingham model, thus the formulation would be completed
at this stage.
To be more explicit, however, alternatively to Eq. 6.58b the closed and convex
admissible domain A in the σ-space is introduced. It is characterized by the convex