Planar cracks of arbitrary shapes in elastic media with ellipsoidal anisotropy: efficient numerical solution

The elasticity problem for planar cracks in infinite homogeneous anisotropic media subjected to arbitrary external stress fields is considered. The problem is reduced to solving the integral equation for the displacement jump on the crack surface. An efficient numerical method of solution of this equation is proposed for media with ellipsoidal anisotropy. In the method, Gaussian radial functions shifted at the nodes of a regular grid on the crack surface are used to approximate the solution. The problem is reduces to a linear algebraic system for the coefficients of the approximation (the discretized problem). For Gaussian approximating functions, the elements the matrix of this system are presented in the forms of standard 1D-integrals. These integrals can be tabulated for small values of the arguments, and they have simple analytical asymptotics for large arguments. As the result, the matrix of the discretized problem is calculated fast. For regular grids of approximating nodes, this matrix has Toeplitz’ structure, and fast Fourier transform algorithm can be used for calculating matrix-vector products by iterative solution of the discretized problem. It substantially accelerates the construction of the numerical solutions. Examples of the solutions for circular, annular and square cracks of various orientations in a strongly anisotropic elastic medium are presented.