A semi-analytic method for the computation of the effective properties of composites of two isotropic constituent materials

We develop a computational framework for the semi-analytic representation of the effective transport properties of two-component composite materials, in the spirit of Bergman’s spectral representation. The components of the effective permittivity (or conductivity) tensor are expanded as power series in a contrast parameter between the two phases. The coefficients (the moments of a spectral measure) of these expansions are determined recursively through the numerical evaluations of integrals defined solely on the boundary between the constituent phases, and discretized by high-order (possibly exponentially converging) quadrature schemes. The high precision reached by these coefficients allows the series representation to be recast into Padé approximants (or continued fractions) to provide analytical expressions valid over large (possibly infinite) regions of the parameter complex plane. Examples, thus far limited to two-dimensional rhombic unit cells, demonstrate that the method is reliable for microgeometries and material parameters for which other methods are far less dependable. The analytical representations are particularly useful to study the optical or electrostatic resonant behaviour of some composite materials.