Optimal anisotropic three-phase conducting composites: Plane problem

TL;DR

This paper derives tight bounds for the effective conductivity tensor of anisotropic three-phase composites with one material infinitely conductive, and identifies optimal microstructures that achieve these bounds.

Contribution

It extends Hashin-Shtrikman and translation bounds to anisotropic multiphase composites using localized polyconvexity techniques.

Findings

Derived explicit bounds for anisotropic three-phase composites.

Identified optimal laminate microstructures matching bounds in most cases.

Numerically estimated the gap where bounds are not achieved.

Abstract

The paper establishes tight lower bound for effective conductivity tensor $K_*$ of two-dimensional three-phase conducting anisotropic composites and defines optimal microstructures. It is assumed that three materials are mixed with fixed volume fractions and that the conductivity of one of the materials is infinite. The bound expands the Hashin-Shtrikman and Translation bounds to multiphase structures, it is derived using the technique of {\em localized polyconvexity} that is a combination of Translation method and additional inequalities on the fields in the materials; similar technique was used by Nesi (1995) and Cherkaev (2009) for isotropic multiphase composites. This paper expands the bounds to the anisotropic composites. The lower bound of conductivity (G-closure) is a piece-wise analytic function of eigenvalues of $K_*$, that depends only on conductivities of components and their volume fractions. Also, we find optimal microstructures that realize the bounds, developing the technique suggested earlier by Albin Cherkaev and Nesi (2007) and Cherkaev (2009). The optimal microstructures are laminates of some rank for all regions. The found structures match the bounds in all but one region of parameters; we discuss the reason for the gap and numerically estimate it.