Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium

A planar crack of arbitrary shape in a 3D-anisotropic elastic medium subjected to an arbitrary external stress field is considered. An efficient numerical method of the solution of the problem is proposed. The problem is reduced to an integral equation for the crack opening vector on the crack surface. For discretization of this equation, Gaussian (radial) approximation functions centered at a system of nodes that covers the crack surface are used. For such functions, the elements of the matrix of the discretized problem are calculated in a quasi analytical form that involves standard non-singular integrals. If the node grid is regular, the matrix of the discretized system has Teoplitz’s structure, and the Fast Fourier Transform algorithm may be used for the calculation of matrix-vector products with such a matrix. It accelerate substantially the process of the iterative solution of the discretized system. Examples of the solutions for a circular crack in a transversally isotropic elastic medium are presented.