A rigorous approach to the field recursion method for two-component composites with isotropic phases

TL;DR

This paper rigorously derives the field recursion method for two-component isotropic composites, providing new solvability conditions, explicit operator representations, and a continued fraction approach to estimate effective material properties.

Contribution

It offers a rigorous mathematical foundation for the field recursion method, including solvability conditions and explicit operator formulas, enhancing the analysis of composite materials.

Findings

Provided new solvability conditions for Z- and Y-problems

Derived explicit representations of Z- and Y-operators

Developed a continued fraction representation for the effective tensor

Abstract

In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we give a rigorous derivation of the field equation recursion method in the abstract theory of composites to two-component composites with isotropic phases. This method is of great interest since it has proven to be a powerful tool in developing sharp bounds for the effective tensor of a composite material. The reason is that the effective tensor $\bf L_*$ can be interpreted in the general framework of the abstract theory of composites as the $Z$-operator on a certain orthogonal $Z(2)$ subspace collection. The base case of the recursion starts with an orthogonal $Z(2)$ subspace collection on a Hilbert space $\cal H$, the $Z$-problem, and the associated $Y$-problem. We provide some new conditions for the solvability of both the $Z$-problem and the associated $Y$-problem. We also give explicit representations of the associated $Z$-operator and $Y$-operator and study their analytical properties. An iteration method is then developed from a hierarchy of subspace collections and their associated operators which leads to a continued fraction representation of the initial effective tensor $\bf L_*$.