Anatoly Markov (Center for Design, Manufacturing and Materials, Skolkovo Institute of Science and Technology, Moscow, Russia). A novel numerical method for the solution of the problems of inhomogeneous cracked media based on Gaussian approximation.

An efficient numerical method for the solution of elasticity problems for an infinite homogeneous medium containing inhomogeneities (cracks and inclusions) is developed. Cracks and inclusions of interest occupy a finite region of the medium that is subjected to arbitrary external forces. The problem is reduced to a system of surface integral equations for crack opening vectors and volume integral equations for stress tensors inside the inclusions. The method is mesh free. Stress fields inside the inclusions and crack opening vectors are approximated by Gaussian functions centered at a system of nodes. The elements of this matrix are calculated in closed analytical forms (for inclusions) or expressed in terms of five standard 1D-integrals (for cracks) that can be tabulated. For regular node grids, the matrix of discretized system has Toeplitz’s structure, and a Fast Fourier Transform technique can be used for calculation of matrix-vector products with such matrices. A technique of calculation of stress intensity factors at the crack edges in the framework of the method is proposed. An iterative method for the construction of the equilibrium shape of a crack (shape for which a stress intensity factor is equal to the fracture toughness of the medium at any point of the crack edge) is developed. This method was applied to the hydraulic fracture process simulation; calculation of the equilibrium crack is an essential part of the simulation. It is possible to apply the method to solve different problems involving integral equations with only some minor changes.

Examples of numerical solutions of some well-known problems for planar cracks and spherical inclusions are presented and compared with analytical and numerical solutions available in the literature. The developed method is applied to some previously unsolved problems.