Computing Critical Indices and Critical Thresholds of Effective Conductivity in Polydisperse Random Composites

This paper is devoted to reviewing constructive methods for computing the effective conductivity (permeability and permittivity) of random, densely packed dispersed composites. The review is based on the formulation of boundary value problems within the framework of homogenization theory. The resulting problems can be solved using both numerical methods and analytical techniques realized as asymptotic methods. Particular attention is given to aRVE (analytical Representative Volume Element) theory based on the efficient evaluation of structural sums that characterize the geometric arrangement of inclusions, as these quantities play a crucial role in determining the critical parameters governing the effective properties of random dispersed composites. The resulting analytical expressions for the effective conductivity components λij(e) can be obtained as polynomial functions of the inclusion concentration f valid from the dilute regime f=0 up to the critical concentration fcp corresponding to touching inclusions, or to the transport percolation threshold fc. As f approaches fcp, the composite becomes densely packed, and percolation chains may form. For materials with high contrast in their physical properties, the bounded polynomial behavior of λij(e), as f→fc, may transition to a singular one, a phenomenon first identified by J. B. Keller in 1963. Beginning in 2001, L. Berlyand and A.G. Kolpakov extended Keller’s result to random composites and developed a theory describing pairwise interactions of inclusions of various shapes. These asymptotic developments were eventually unified into what is now known as the Network Method within the broader framework of Structural Approximation (SA). To maintain consistency with homogenization principles, we extend the SA framework by formulating a periodic boundary value problem on the corresponding Delaunay graph. This modification enables the application of SA to macroscopically anisotropic conducting composites. Special attention is given to elliptic inclusions governed by different probabilistic distributions. The analytical formulas obtained in this setting are then asymptotically transformed via renormalization techniques into expressions applicable to densely packed systems of ellipses. The method is extended to polydisperse composites containing circular inclusions of varying sizes.