Elastic Analyses of Planar Cracks of Arbitrary Shape

A numerical method is presented for planar cracks of arbitrary shape. The fundamental solution for a dislocation segment is obtained from the point force solution and used to derive three coupled surface integral equations in which the crack-face tractions are expressed in terms of the gradients of the relative crack-surface displacements. Because the singularity of the kernel in the integral equations is one order less for fundamental solutions based on dislocation segments than for those based on dislocation loops or the body force method, no special numerical techniques are required. Most of the integrations over elements are evaluated analytically. The integral equations are solved numerically by covering the crack surface with triangular elements, and taking the relative displacements to vary linearly over the elements. The mesh is generated by optimizing the local aspect ratio, which is related to the difference in the principal stretches of the mapping of a square reference mesh onto the fracture surface. This mesh generator allows cracks of a wide variety of shapes to be analyzed with good accuracy. Comparison with known solutions indicate that accurate numerical solutions are obtained with a relatively coarse mesh.