Solution Method
By using a first order finite difference approximation wit step size
6.41 get the form
 |
(141) |
and
 |
(142) |
where
is the stress at the precious time step. With
 |
(143) |
we have
 |
(144) |
where
with  |
(145) |
The upper bound
makes sure that yield condtion 6.47 holds. With this setting the eqaution 6.53 takes the form
 |
(146) |
After inserting 6.57 into 6.48 we get
 |
(147) |
Combining this with the incomressibilty condition 6.40 we need to solve a
Stokes problem as discussed in section 6.1.1 in each time step.
If we set
 |
(148) |
we need to solve the nonlinear problem
 |
(149) |
We use the Newton-Raphson Scheme to solve this problem
 |
(150) |
where
denotes the derivative of
with respect of
and
.
Looking at the evaluation of
in 6.59 it makes sense formulate
the iteration 6.61 using
.
In fact we have
with  |
(151) |
As
 |
(152) |
we have
with  |
(153) |
which leads to
 |
(154) |
esys@esscc.uq.edu.au