Nonlinear Problems - The Boundary Element Method with Programming - page 422

whereas in the second case

' t t c t tan M

(15.77)

p s n

We now propose the following solution procedure:

1. The system is solved in the normal way using the interface compatibility and

equilibrium conditions.

2. The yield conditions F 1 and F 2 are computed at each interface node. If both are zero

then the analysis is finished.

3. If one of the yield conditions is greater than zero residual tractions are computed

according to equations (15.76) or (15.77).

4. The interface matrix K is re-assembled taking into consideration the relaxed

continuity conditions for interface points which are separating or slipping.

5. The system is solved with the residual tractions applied as loading in the opposite

direction.

6. Points 2 to 5 are repeated until the yield conditions are satisfied at all interface

nodes.

The extension of the method to three dimensions is straightforward. In 3-D we have
two instead of one shear traction (

s t ) and when we check the yield condition we
have to work with a resultant shear traction. This is given by

and

1 s 1

2
2

2

(15.78)

t t t

s s 1 s

15.4.3 Example of application

As an example of application we present an analysis of the delamination of a cantilever
beam. The beam consists of two finite regions described by quadratic boundary
elements, as shown in Figure 15.14. At the interface the tensile strength of the material
was assumed to be zero. Shear loading is applied to the bottom half of the beam as
shown. Figure 15.15 shows the distribution of normal stress at the interface after the
linear analysis. It can be clearly seen that the yield condition for tension is violated.
Figure 15.16 shows that after iteration step 1, delamination starts as a consequence.
Further examples of the application of the method to the modelling of faulted rock can
be found in [12,13]. The method can also be applied to the simulation of dynamic crack
propagation 14 .