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Effective conductivity of a composite material with non-ideal contact conditions

The effective conductivity of 2D doubly periodic composite materials with circular disjoint inclusions under non-ideal contact conditions on the boundary between material components is found. The obtained explicit formula for the effective conductivity contains all parameters of the considered model, such as the conductivities of matrix and inclusions, resistance coefficients, radii and centres of the inclusions and also the values of special Eisenstein functions. The method of functional equations is used to analyse the conjugation problem for analytic functions which is equivalently derived from the initial problem. Existence and uniqueness for the solution of the problem is obtained by using a reduction to a certain mixed boundary value problem for analytic functions in special functional spaces.