On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.