Analytical homogenization method for periodic composite materials

We present an easy-to-implement technique for determining the effective properties of composite materials with periodic microstructures, as well as the field distributions in them. Our method is based on the transformation tensor of Eshelby and the Fourier treatment of Nemat-Nasser of this tensor, but relies on fewer limiting assumptions as compared to prior approaches in the literature. The final system of linear equations, with the unknowns being the Fourier coefficients for the potential, can be assembled easily without a priori knowledge of the concepts or techniques used in the derivation. The solutions to these equations are exact to a given order, and converge quickly for inclusion volume fractions up to 70%. The method is not only theoretically rigorous but also offers flexibilities for numerical evaluations.