A singularly perturbed non-ideal transmission problem and application to the effective conductivity of a periodic composite

TL;DR

This paper studies how the effective thermal conductivity of a periodic composite material changes as the size of inclusions varies, using advanced mathematical analysis to understand the behavior near the point where inclusions become infinitesimally small.

Contribution

The paper introduces a mathematical framework to analytically continue the effective conductivity as the inclusion size approaches zero, providing new insights into the composite's behavior at this limit.

Findings

Effective conductivity can be analytically continued around zero inclusion size.

The analysis reveals the behavior of the composite as inclusions collapse to points.

Mathematical methods applied to singular perturbation problems in composite materials.

Abstract

We investigate the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. The diameter of each inclusion is assumed to be proportional to a positive real parameter \epsilon. Under suitable assumptions, we show that the effective 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.