6.1 Bingham Model
293
π(˙ ) = max
σ
d(σ; ˙
) −
1
2
|σ| − σ y
2
η
,
(6.24)
whereby d(σ; ˙
) := σ ˙
denotes the dissipation power density. Interestingly, the
reverse Legendre transformation in Eq. 6.24 embodies the unconstrained optimization problem
˜
1/η (σ; ˙
) := −d(σ; ˙
) +
1
2
|σ| − σ y
2
η
→ min
σ
,
(6.25)
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility constraint |σ| ≤ σ y penalized by the penalty parameter 1/η. In accordance with Eq. 6.20
the stationarity condition of this unconstrained optimization problem then reads
˙
(σ) =
|σ| − σ y
η
σ
|σ|
.
(6.26)
Finally, the total strain arc-length, denoted κ, may conveniently be introduced as
a measure of the accumulated total deformation, i.e.
κ =
˙
κ dt with ˙
κ := |˙ | =
|σ| − σ y
η
≥ 0.
(6.27)
The specific Bingham model is summarized in Table 6.1.
Table 6.1 Summary of the specific Bingham model
(1) Strain
= vp
(2) Potential π v =
1
2 η |˙ | 2
(3) Potential π p = σ y |˙ |
(4) Stress
σ = η ˙
+ σ y
˙
|˙ |
≡ σ
for ˙
= 0
or
(2) Potential π ∗
v =
1
2 |σ v | 2 /η
(3) Yield
0 ≥ |σ p | − σ y
(4) Evolution ˙
= λ
σ p
|σ p |
=
σ v
η
(5) KKT
λ ≥ 0, |σ p | ≤ σ y , λ |σ p | = λ σ y
or
(2) Potential π ∗ =
1
2 |σ| − σ y 2 /η
(3) Evolution ˙
=
|σ| − σ y
η
σ
|σ|