Effective conductivity of a singularly perturbed periodic two-phase composite with imperfect thermal contact at the two-phase interface

TL;DR

This paper analyzes how the effective thermal conductivity of a periodic two-phase composite behaves asymptotically as inclusions shrink to points, considering imperfect thermal contact at interfaces, using functional analysis and potential theory.

Contribution

It provides a new analytical continuation of the effective conductivity around the degenerate case where inclusions collapse, extending understanding of composite materials with imperfect interfaces.

Findings

Effective conductivity can be analytically continued around the degenerate point

As inclusions shrink, the composite's thermal properties are characterized precisely

The approach combines functional analysis and potential theory for rigorous results

Abstract

We consider the asymptotic behaviour of the effective thermal conductivity of a two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material and of size proportional to a positive parameter \epsilon. We are interested in the case of imperfect thermal contact at the two-phase interface. Under suitable assumptions, we show that the effective thermal conductivity can be continued real analytically in the parameter \epsilon around the degenerate value \epsilon=0, in correspondence of which the inclusions collapse to points. The results presented here are obtained by means of an approach based on functional analysis and potential theory and are also part of a forthcoming paper by the authors.