Isotropic conductivity of two-dimensional three- and four-phase symmetric composites: duality and universal bounds

TL;DR

This paper derives universal algebraic bounds for the isotropic effective conductivity of 2D three- and four-phase symmetric composites, demonstrating their consistency with known results and superiority over existing variational bounds.

Contribution

It introduces universal, structure-independent bounds for effective conductivity that incorporate physical symmetries and properties, advancing the theoretical understanding of composite materials.

Findings

Bounds agree with numerical and exact results

Bounds are stronger than existing variational bounds

Bounds satisfy physical properties like homogeneity and duality

Abstract

We consider the problem of isotropic effective conductivity $\sigma_e(\sigma_1,\ldots,\sigma_n)$ in two-dimensional three- and four-phase symmetric composites with a partial isotropic conductivity $\sigma_j$ of the $j$-th phase. The upper $\Omega(\sigma_1,\ldots,\sigma_n)$ and lower $\omega(\sigma_1,\ldots,\sigma_n)$, $n=3,4$, bounds for effective conductivity, found by the algebraic approach, are universal (independent of the composite micro-structure) and possess all algebraic properties of $\sigma_e(\sigma_1,\ldots,\sigma_n)$ that follow from physics: first-order homogeneity, full permutation invariance, Keller's self-duality, positivity, and monotony. The bounds are compatible with the trivial solution $\sigma_e(\sigma,\ldots,\sigma)=\sigma$ and satisfy Dykhne's ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for isotropic effective conductivity $\sigma_e(\sigma_1,\ldots,\sigma_n)$ of two-dimensional three- and four-phase composites showed complete agreement. The bounds $\Omega(\sigma_1,\ldots,\sigma_n)$ and $\omega(\sigma_1,\ldots,\sigma_n)$ in both cases $n=3,4$ are stronger than the currently known variational bounds.