Local fields and effective conductivity of composites with anisotropic components

Abstract

A homogeneous anisotropic conductive medium with a set of anisotropic heterogeneities of arbitrary shapes is considered. Calculation of local fields in the medium subjected to arbitrary external fields is reduced to systems of volume integral equations. For numerical solution, these equations are discretized using Gaussian approximating functions concentrated at the nodes of a regular grid. The elements of the matrices of the discretized problems have form of 1D-integrals that can be tabulated. For regular node grids, these matrices have Teoplitz’ structures, and fast Fourier transform algorithms can be used for iterative solution of the discretized problems. The method is applied to calculation of fields around isolated anisotropic spherical and cylindrical inclusions in an anisotropic homogeneous host medium. The results are used for calculation of the tensor of effective conductivity of the medium containing random sets of cylindrical inclusions. The self-consistent effective field method is used for solution of the homogenization problem. Dependencies of the components of the tensor of the effective conductivity on the volume fraction and orientations of the inclusions are presented.