Fast Solution of 3D-Elasticity Problem of a Planar Crack of Arbitrary Shape

A planar crack of an arbitrary shape in a homogeneous elastic medium is considered. The problem is reduced to integral equation for the crack opening vector. Its numerical solution utilizes Gaussian approximating functions that drastically simplify construction of the matrix of a linear algebraic system of the discretized problem. For regular grids of approximating nodes, this matrix turns out to have the Teoplitz structure. It allows one to use the Fast Fourier Transform algorithms for calculation of the matrix-vector products in the process of iterative solution of the discretized problem. The method is applied to a crack bounded by the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^{2p}_{1} + x^{2p}_{2}\leq 1}$$\end{document} for 0.2 ≤ p ≤ 4. The contribution of a crack to the overall effective elastic constants is calculated.