6.1 Bingham Model
289
States in the interior int A of the admissible domain with |σ p | < σ y are rigid,
whereas states on the boundary ∂ A of the admissible domain with |σ p | = σ y are
visco-plastic. Recall that σ p is constitutively not determined in the rigid domain.
The corresponding dual viscous and plastic dissipation potentials π
∗
v and π
∗
p , as
determined from the Legendre transformations
π
∗
v (σ v ) = max
˙
σ v ˙
−
1
2
η |˙ |
2
(6.15a)
π
∗
p (σ p ) = max
˙
{σ p ˙
− σ y |˙ | }
(6.15b)
then read
π
∗
v (σ v ) =
1
2
1
η
|σ v |
2
(6.16a)
π
∗
p (σ p ) = I A (σ p ) :=
⎧
⎨
⎩
0
|σ p | ≤ σ y
for
∞
|σ p | > σ y
⎫
⎬
⎭
,
(6.16b)
where I A denotes the indicator function of the admissible domain A in the σ p -space.
The evolution law (the associated flow rule) for the total strain then follows either
as partial derivative of the dual viscous dissipation potential or likewise as some
sub-derivative of the dual plastic dissipation potential, in either case with respect to
its conjugated variable
˙
(σ v ) = ∂ σ v π
∗
v (σ v )
=
1
η
σ v
(6.17a)
˙
(σ p ) ∈ d σ p π
∗
p (σ p ) = d σ p I A (σ p ) =
⎧
⎪ ⎨
⎪ ⎩
0
|σ p | < σ y
for
λ
σ p
|σ p |
|σ p | = σ y
⎫
⎪ ⎬
⎪ ⎭
,
(6.17b)
whereby d σ p π
∗
p denotes the set of sub-derivatives, i.e. the sub-differential of π
∗
p with
respect to σ p and λ is a positive Lagrange (or rather plastic) multiplier.
Obviously, the expressions in Eqs. 6.7, 6.8 and 6.17a, 6.17b are inverse relations.
The smooth viscous dissipation and dual viscous dissipation potentials π v = π v (˙ )
and π
∗
v = π
∗
v (σ v ) together with the resulting smooth constitutive relations σ v = σ v (˙ )
and ˙
= ˙
(σ v ) are similar to those displayed in Fig. 4.2. The non-smooth plastic dissipation and dual plastic dissipation potentials π p = π p (˙ ) and π
∗
p = π
∗
p (σ p ) together
with the resulting non-smooth constitutive relations σ p = σ p (˙ ) and ˙
= ˙
(σ p ) are
similar to those displayed in Fig. 5.3.