6.1 Bingham Model
297
As an immediate consequence the equivalent stress and its trial value are related
via
|σ
n
| = |σ
| − E λ.
(6.46)
A direct further consequence that alleviates the computation of the flow direction
at the end of the time step in terms of trial values is then obviously
σ
n
|σ n |
≡
σ
|σ |
.
(6.47)
Eventually, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
| − σ y = φ
− E λ.
(6.48)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
| − σ y .
(6.49)
Next for visco-plastic loading with λ > 0 the definition for the incremental
visco-plastic multiplier is regrouped to render
φ
n
− λ
η
t n = 0.
(6.50)
Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form
from
λ =
φ
E + η//t n ≥ 0.
(6.51)
Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t
n
→ 0. Once λ is computed
all other variables may be updated. In particular, the elastic strain (that represents
the kinematic constraint) at the end of the time step reads as
n
e =
n
−
n
vp with
n
vp =
n−1
vp + λ
σ
|σ |
.
(6.52)
Then, if
n
e exceeds a given tolerance, the visco-plastic stress is reset according to
an Usawa update scheme as
σ
k
vp = σ
k−1
vp − E [
n
vp −
n
]
(6.53)
297
As an immediate consequence the equivalent stress and its trial value are related
via
|σ
n
| = |σ
| − E λ.
(6.46)
A direct further consequence that alleviates the computation of the flow direction
at the end of the time step in terms of trial values is then obviously
σ
n
|σ n |
≡
σ
|σ |
.
(6.47)
Eventually, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
| − σ y = φ
− E λ.
(6.48)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
| − σ y .
(6.49)
Next for visco-plastic loading with λ > 0 the definition for the incremental
visco-plastic multiplier is regrouped to render
φ
n
− λ
η
t n = 0.
(6.50)
Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form
from
λ =
φ
E + η//t n ≥ 0.
(6.51)
Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t
n
→ 0. Once λ is computed
all other variables may be updated. In particular, the elastic strain (that represents
the kinematic constraint) at the end of the time step reads as
n
e =
n
−
n
vp with
n
vp =
n−1
vp + λ
σ
|σ |
.
(6.52)
Then, if
n
e exceeds a given tolerance, the visco-plastic stress is reset according to
an Usawa update scheme as
σ
k
vp = σ
k−1
vp − E [
n
vp −
n
]
(6.53)