Analytical solution for doubly-periodic harmonic problems with circular inhomogeneities and superconducting or membrane-type interfaces

A novel analytical solution for two-dimensional harmonic problems involving doubly-periodic arrays of circular inhomogeneities with superconducting or membrane type interfaces is derived. The complex potential inside each inhomogeneity is sought in the form of power series, while its counterpart inside the matrix is represented by the series in terms of Weierstrass ζ-function and its derivatives. Compliance with the interface conditions results in an infinite system of linear algebraic equations for unknown series coefficients. A rigorous theoretical study of the system properties is performed. The solution is used for evaluating the local fields and overall properties of composites. For the case of square and hexagonal unit cells, accurate formulas for the effective properties are provided. Numerical examples are presented and comparison with the results reported in the literature is performed.