Structural sums on the complex plane and their application to composite materials

Consider a two-dimensional, two-component composite consisting of non-overlapping, identical circular disks embedded in a uniform host. The conductivity of the considered composites is modeled by the R -linear conjugation problem for functions analytic in the domains occupied by the components of composites and H¨older continuous in their closures. The effective conductivity of these dispersed random composites can be determined using constructive homogenization theory, which assumes strictly stationary fields. The Eisenstein summation method is applied to analyze conditionally convergent sums that arise during the homogenization process. Additionally, the Clausius-Mossotti approximation, which is valid for macroscopically isotropic two-dimensional composites up to $O(f3)$, is justified for random composites. A new analytical formula for the effective conductivity tensor of macroscopically anisotropic composites is derived up to $O(f3)$. This formula contains a conditionally convergent sum $S'_2$ coinciding with the Rayleigh lattice sum for the square array of disks calculated by the Eisenstein summation method. Moreover, $S'_2$ depends on a curve connecting a finite point with infinity. This approach provides a robust framework for understanding the behavior of such materials and offers insights into their effective conductivity properties.