Dispersion tensor and its unique minimizer in Hashin-Shtrikman micro-structures

TL;DR

This paper introduces the dispersion tensor in Hashin-Shtrikman micro-structures and proves the existence of a unique minimizer, called Apollonian-Hashin-Shtrikman micro-structure, for two-phase periodic materials.

Contribution

It defines the dispersion tensor in generalized Hashin-Shtrikman structures and establishes the uniqueness of its minimizer in two-phase periodic cases.

Findings

Dispersion tensor introduced for Hashin-Shtrikman micro-structures.

Unique minimizer identified as Apollonian-Hashin-Shtrikman micro-structure.

Proves minimizer's uniqueness in two-phase periodic materials.

Abstract

In this paper, we introduce the macroscopic quantity, namely the dispersion tensor or the \textit{Burnett coefficient}s in the class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}. In the case of two-phase materials associated with the periodic Hashin-Shtrikman structures, we settle the issue that the dispersion tensor has an unique minimizer, which is so called Apollonian-Hashin-Shtrikman micro-structure.