Effective elastic properties of random two-dimensional composites

Consider 2D two-phase random composites with circular inclusions of concentration f. New analytical formulae for the effective constants are deduced up to O(f4) for macroscopically isotropic composites. It is shown that the second order terms O(f2) do not depend on the location of inclusions whilst the third order terms do. This implies that the previous analytical methods (effective medium approximation, differential scheme, Mori-Tanaka approach and so forth) can be valid at most up to O(f3) for macroscopically isotropic composites. First, the local elastic field for a finite number n of inclusions arbitrarily located on the plane are found by a method of functional equations. Further, the limit n → ∞ yields conditionally convergent series defined by the Eisenstein summation method. One of the series for periodic composites is the famous lattice sum S2=π deduced by Rayleigh for a conductivity problem.