We now consider a multi-crack problem. Namely, we compute crack opening displacements and the stresses within the body having many cracks in its interior. Analyses of this type have applications in the fracture mechanics of rock-like materials.

Specifically, we consider an infinite elastic body which contains an
array of 8
8
8 (=512) penny-shaped cracks of the
same radius *a*_{0}. The centres of these cracks are located regularly
at the interval of 4 *a*_{0} in all the coordinate directions. The
directions of these cracks are taken random. The whole system is
subjected to a uniform remote tension of the magnitude *p*_{0}. Hence
the function
which appears in
(3.100) is written as
where

(3.113) |

Again, the material constants are chosen as .

Fig.3.34 shows the mesh for 512 cracks (total
DOF=241,152), where each crack is discretised with the 471 DOF mesh
shown in Fig.3.29. In Fig.3.35 we display the
computed crack opening displacements by superposing the non-dimensional
crack opening displacement
on the
non-dimensional position vector
.
The required CPU time for
computing the crack opening displacements was 20,854 (sec), and the
number of iterations needed for the convergence of GMRES was
10. We next show the results of the stress computation. We take a
surface ,
shown in Fig.3.36, which has no intersection with
the cracks. Observation points are taken to be the 200200
lattice points located on
at an equal interval. Fig.3.37
shows the distribution of the second invariant of the stress deviator
defined by

computed on these observation points. The additional CPU time required for the calculation of stress components at these 40,000 points was 477 (sec) after the computation of the crack opening displacements. Finally, Fig.3.38 shows the crack opening displacements for a case similar to the one in Fig.3.35 except that the radius of each crack is also varied randomly between