6.1 Bingham Model
309
yield condition
φ = φ(σ) := ϕ(σ) − σ y ≤ 0.
(6.59)
Here φ = φ(σ) is the overstress 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. 6.58a
from the postulate of maximum dissipation (due to visco-plasticity)
˜
1/η (σ; ˙
) := −d(σ; ˙
) +
1
2
φ(σ)
2
η
→ min,
(6.60)
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility constraint φ ≤ 0 penalized by the penalty parameter 1/η. Consequently, the stationarity
condition of this unconstrained optimization problem reads
˙
= λ ∂ σ φ with λ := =φ(σ)/η ≥ 0.
(6.61)
It shall be noted that collectively Eqs. 6.59 and 6.61 are entirely equivalent statements to Eqs. 6.58a and 6.58b.
As a further interesting aspect the dissipation d = σ ˙
shall next be examined
more closely. From Eqs. 6.57a and 6.57b 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.
(6.62)
Thereby, based on the above introduction of the overstress function φ (and in view
of Eqs. 6.57a, 6.58a, 6.60 and 6.61) the dual dissipation potential is identified as
π
∗
(σ) =
⎧
⎨
⎩
0
φ(σ) ≤ 0
for
1
2
φ(σ)
2
/η
φ(σ) > 0
⎫
⎬
⎭
=
1
2
φ(σ)
2
η
.
(6.63)
Finally for an equivalent stress that is homogeneous of degree one in the stress
(thus σ ∂ σ ϕ = ϕ), the dissipation d = σ ˙
is exclusively given in terms of the overstress function φ (with abbreviation λ := =φ/η ≥ 0 for the visco-plastic multiplier
and equivalent stress ϕ = φ + σ y ≥ 0), since then
d = λ σ ∂ σ ϕ = λ ϕ = =φ [φ + σ y ]/η.
(6.64)
The generic Bingham model is summarized in Table 6.3.