Homogenization of high-contrast Mumford-Shah energies

TL;DR

This paper establishes a homogenization result for high-contrast Mumford-Shah energies in brittle composites with periodic weak inclusions, revealing complex limit behaviors influenced by the contrast in toughness moduli.

Contribution

It introduces a novel homogenization analysis for Mumford-Shah energies with high contrast, showing unique limit interactions between volume and surface energies.

Findings

Limit volume energy is not quadratic in the critical scaling

High contrast leads to interaction effects in the limit

Different from classical quadratic energy limits

Abstract

We prove a homogenization result for Mumford-Shah-type energies associated to a brittle composite material with weak inclusions distributed periodically at a scale ${\varepsilon}>0$. The matrix and the inclusions in the material have the same elastic moduli but very different toughness moduli, with the ratio of the toughness modulus in the matrix and in the inclusions being $1/\beta_{\varepsilon}$, with $\beta_{\varepsilon}>0$ small. We show that the high-contrast behaviour of the composite leads to the emergence of interesting effects in the limit: The volume and surface energy densities interact by $\Gamma$-convergence, and the limit volume energy is not a quadratic form in the critical scaling $\beta_{\varepsilon} = {\varepsilon}$, unlike the ${\varepsilon}$-energies, and unlike the extremal limit cases.