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

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