Upper bounds for multiphase composites in any dimension

TL;DR

This paper establishes a rigorous upper bound on the effective conductivity of isotropic multiphase composites in any dimension, extending classical bounds and matching Hashin-Shtrikman bounds under specific conditions.

Contribution

It introduces a new upper bound for effective conductivity in multiphase composites applicable in any dimension, generalizing existing bounds.

Findings

The upper bound matches Hashin-Shtrikman bounds when all but two phases vanish.

The bound is rigorous and applies to isotropic composites in any dimension.

Provides a theoretical limit for effective conductivity in complex composite materials.

Abstract

We prove a rigorous upper bound for the effective conductivity of an isotropic composite made of several isotropic components in any dimension. This upper bound coincides with the Hashin Shtrikman bound when the volume ratio of all phases but any two vanish.