296
6 Visco-Plasticity
6.1.2 Specific Bingham Model: Algorithmic Update
The integration algorithm for the specific Bingham model is based on the augmented
Lagrange multiplier format of the postulate of maximum (visco-plastic) dissipation
that allows to incorporate the kinematic constraint = vp (in rate form). Thereby
the evolution law for the visco-plastic strain vp is integrated by the implicit Euler
backwards method to render
n
vp :=
n
vp −
n−1
vp = λ
σ
n
|σ n |
,
(6.40)
whereby the incremental visco-plastic multiplier λ is defined as
λ := t
n
λ
n
:= t
n |σ
n
| − σ y
η
≥ 0.
(6.41)
Consequently, and based on the assumption that the kinematic constraint is satisfied at the end of the previous time step
n−1
vp −
n−1
= 0, the total stress σ (versus
visco-plastic stress σ vp and kinematic constraint violation e relation) is updated at
the end of the time step by
σ
n
= σ
k−1
vp − E [
n
vp −
n
] =: σ
− E
n
vp .
(6.42)
Here σ
k−1
vp denotes the visco-plastic stress, i.e. the Lagrange multiplier enforcing
the kinematic constraint within an Usawa iteration. Moreover E := E//t is a penalty
parameter of dimension stress. In contrast to the somewhat naive penalty format, the
augmented Lagrange multiplier format allows for arbitrary small penalty parameters
that do not compromise the condition number of the equation (system) to be solved.
The trial total stress σ
is computable exclusively from known quantities at the
beginning of the time step and the previous Usawa update, it follows as
σ
:= σ
k−1
vp − E [
n−1
vp −
n
].
(6.43)
Incorporating the discretized evolution law for the visco-plastic strain then renders
σ
n
= σ
− E λ
σ
n
|σ n |
.
(6.44)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress
|σ
n
| + E λ
σ
n
|σ n |
= σ
.
(6.45)
6 Visco-Plasticity
6.1.2 Specific Bingham Model: Algorithmic Update
The integration algorithm for the specific Bingham model is based on the augmented
Lagrange multiplier format of the postulate of maximum (visco-plastic) dissipation
that allows to incorporate the kinematic constraint = vp (in rate form). Thereby
the evolution law for the visco-plastic strain vp is integrated by the implicit Euler
backwards method to render
n
vp :=
n
vp −
n−1
vp = λ
σ
n
|σ n |
,
(6.40)
whereby the incremental visco-plastic multiplier λ is defined as
λ := t
n
λ
n
:= t
n |σ
n
| − σ y
η
≥ 0.
(6.41)
Consequently, and based on the assumption that the kinematic constraint is satisfied at the end of the previous time step
n−1
vp −
n−1
= 0, the total stress σ (versus
visco-plastic stress σ vp and kinematic constraint violation e relation) is updated at
the end of the time step by
σ
n
= σ
k−1
vp − E [
n
vp −
n
] =: σ
− E
n
vp .
(6.42)
Here σ
k−1
vp denotes the visco-plastic stress, i.e. the Lagrange multiplier enforcing
the kinematic constraint within an Usawa iteration. Moreover E := E//t is a penalty
parameter of dimension stress. In contrast to the somewhat naive penalty format, the
augmented Lagrange multiplier format allows for arbitrary small penalty parameters
that do not compromise the condition number of the equation (system) to be solved.
The trial total stress σ
is computable exclusively from known quantities at the
beginning of the time step and the previous Usawa update, it follows as
σ
:= σ
k−1
vp − E [
n−1
vp −
n
].
(6.43)
Incorporating the discretized evolution law for the visco-plastic strain then renders
σ
n
= σ
− E λ
σ
n
|σ n |
.
(6.44)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress
|σ
n
| + E λ
σ
n
|σ n |
= σ
.
(6.45)