Cluster method in the theory of fibrous elastic composites

Consider a 2D multi-phase random composite with different circular inclusions. A finite number $n$ of inclusions on the infinite plane forms a cluster. The corresponding boundary value problem for Muskhelishvili's potentials is reduced to a system of functional equations. Solution to the functional equations can be obtained by a method of sucessive approximations or by the Taylor expansion of the unknown analytic functions. Next, the local stress-strain fields are calculated and the averaged elastic constants are obtained in symbolic form. Extensions of Maxwell's approach and other various self-consisting methods are discussed. An uncertainty when the number of elements $n$ in a cluster tends to infinity is analyzed by means of the conditionally convergent series. Basing on the fields around a finite cluster without clusters interactions one can deduce formulae for the effective constants only for dilute clusters. The Eisenstein summation yields new analytical formulae for the effective constants for random 2D composites with high concentration of inclusions.