The effective conductivity of a periodic lattice of circular inclusions

TL;DR

This paper derives an exact analytic formula for the effective electrical conductivity tensor of a 2D composite material with a periodic array of circular inclusions, including its volume fraction expansion.

Contribution

It provides a novel exact analytic expression for the effective conductivity tensor of a periodic composite with circular inclusions.

Findings

Exact formula for effective conductivity tensor

Expansion in terms of volume fraction

Illustrative examples of results

Abstract

We determine the effective conductivity of a two-dimensional composite consisting of a doubly periodic array of identical circular cylinders within a homogeneous matrix. We obtain an exact analytic expression for the effective conductivity tensor as well as its expansion in terms of volume fraction of the cylinders. Results are illustrated by examples.