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

## TL;DR

This paper develops an approximation formula for the effective conductivity tensor of two-dimensional composites with anisotropic phases, applicable to various coupled field problems, based on operator approximations.

## Contribution

It introduces a new approximation method for the effective tensor in anisotropic composites, extending to coupled field problems, with convergence properties.

## Key findings

- Provides an approximation formula for effective conductivity tensor.
- Applicable to elasticity, piezoelectricity, and thermoelectricity.
- Convergence to exact effective properties as basis size increases.

## 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.