Approximating the effective tensor as a function of the component tensors in two-dimensional composites of two anisotropic phases

Abstract: A conducting two-dimensional periodic composite of two anisotropic phases with anisotropic, not necessarily symmetric, conductivity tensors is considered. By finding approximate representations for the relevant operators, an approximation formula is derived for the effective matrix valued conductivity as a function of the two matrix valued conductivity tensors of the phases. This approximation should converge to the exact effective conductivity function as the number of basis fields tends to infinity. Using the approximations for the relevant operators one can also directly obtain approximations, with the same geometry, for the effective tensors of coupled field problems, including elasticity, piezoelectricity, and thermoelectricity. To avoid technical complications we assume that the phase geometry is symmetric under reflection about one of the centerlines of the unit cell.