Effective conductivity of anisotropic media with crack-like inclusions and cracks of arbitrary shapes

Abstract

The effective field method is applied to solution of the homogenization problem for anisotropic media containing random sets of thin inclusions of low conductivity (crack-like inclusions) or cracks. The derived expression for the tensor of effective conductivity of cracked media on the one hand, takes into account peculiarities of shapes and conductivity of thin inclusions, and on the other hand, reflect statistical properties of the inclusion distributions in the host medium. The crucial part of realization of the method is solution of the so-called one-particle problem that is the conductivity problem for an isolated inclusion embedded into an anisotropic host medium and subjected to a constant external field. This problem is reduced to solution of the integral equation for the potential jump on the middle surface of a thin inclusion. An efficient numerical method of solution of this equation is proposed. The integral equation is discretized by Gaussian approximating functions and reduced to a linear algebraic system for the coefficients of the approximation (the discretized problem). For Gaussian functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms (for isotropic host media) or they reduce to a standard 1D-integral (for arbitrary anisotropic host media) that can be tabulated. As a result, the matrix of the discretized problem is calculated fast. For inclusions with planar middle surfaces and regular grids of approximated nodes, this matrix has Toeplitz’ structure, and fast Fourier transform algorithm can be used for iterative solution of the discretized problem. The cases of a strongly anisotropic medium with circular, annular and square cracks are considered. Examples of solution of the homogenization problems for the medium containing cracks of various shapes are presented.