On calculation of effective conductivity of inhomogeneous metals

TL;DR

This paper derives an exact perturbation series-based formula for calculating the effective conductivity of three-dimensional inhomogeneous metals, applicable when the medium's statistical properties are known, with examples including Gaussian-distributed and polycrystalline metals.

Contribution

It provides a formal, exact perturbation series approach for computing the effective conductivity of inhomogeneous metals, extending previous methods.

Findings

Derived a formal expression for effective conductivity using perturbation theory

Applicable to media with known statistical properties, allowing arbitrary series terms

Illustrated method with examples of Gaussian-distributed and polycrystalline metals

Abstract

In the framework of the perturbation theory an expression suitable for calculation of the effective conductivity of 3-D inhomogeneous metals is derived. Formally, the final expression is an exact result, however, a function written as a perturbation series enters the answer. More accurately, when statistical properties of the given inhomogeneous medium are known, our result provides the regular algorithm for calculation of the effective conductivity up to an arbitrary term of the perturbation series. As examples, we examine (i) an isotropic metal whose local conductivity is a Gaussianly distributed random function, (ii) the effective conductivity of polycrystalline metals.