Quantitative analysis of field concentration in presence of closely located inclusions of high contrast

Abstract

In composites consisting of inclusions and a matrix of different materials, some inclusions are located closely to each other. If the material properties of inclusions are of high contrast with that of the matrix, field concentration occurs in the narrow region between closely located inclusions. Understanding the field concentration quantitatively is important in the theory of composites and imaging since it represents stress or field enhancement. The last 30 years or so have witnessed significant progress in analyzing this phenomena of field concentration: optimal estimates and asymptotic characterization capturing the field concentration have been derived in the contexts of the conductivity equation (or antiplane elasticity), the Lamé system of linear elasticity, and the Stokes system. The purpose of this paper is to review some of them in a coherent manner.