Exact Expansion Formalism for Transport Properties of Heterogeneous Materials Characterized by Arbitrary Continuous Random Fields

TL;DR

This paper introduces an exact formalism for calculating the effective conductivity of heterogeneous materials with continuous random fields, providing highly accurate approximations for both isotropic and anisotropic media.

Contribution

It generalizes existing models by deriving an exact contrast-expansion formalism applicable to arbitrary continuous random fields, with closed-form approximations and rigorous structure-property closures.

Findings

First-order approximation achieves percent-level accuracy for isotropic media.

Second-order approximation achieves sub-2% accuracy for anisotropic media.

The formalism enables precise design of materials with tailored transport properties.

Abstract

We derive an exact contrast-expansion formalism for the effective conductivity of heterogeneous materials (media) with local properties described by arbitrary continuous random fields, significantly generalizing the widely used binary-field models. The theory produces a rapidly convergent Neumann-series that, upon Gaussian closure via a Hermite expansion, yields closed-form first-, second- and third-order approximations, which achieve percent-level accuracy at first order for isotropic media. For anisotropic media, second-order approximations achieve sub-2% accuracy across a wide range of local property contrasts and correlations. Our formalism provides mathematically rigorous structure-property closures, with significant implications for the discovery and design of novel graded and architected materials with tailored transport properties.