Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagram

TL;DR

This paper investigates how strong ellipticity can be lost during homogenization in 2D linear elasticity, identifying laminate microstructures as the primary cause and extending the analysis to ergodic settings.

Contribution

It proves that laminate microstructures are essentially the only configurations causing loss of strong ellipticity in homogenized elastic materials.

Findings

Laminate microstructures lead to loss of strong ellipticity.

Loss of ellipticity is not generic; it is primarily due to specific microstructures.

The analysis extends to stationary, ergodic random settings.

Abstract

Since the seminal contribution of Geymonat, M\"uller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Guti\'errez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.