Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity

TL;DR

This paper investigates how isotropy in composite materials influences the preservation of strong ellipticity during homogenization in linear elasticity, showing that macroscopic isotropy prevents loss of ellipticity even in laminate arrangements.

Contribution

It demonstrates that macroscopic isotropy in mixtures of isotropic phases ensures the preservation of strong ellipticity during homogenization.

Findings

Macroscopic isotropy prevents loss of strong ellipticity.

Laminate arrangements of isotropic phases can cause loss of ellipticity.

Isotropy at the macroscopic level guarantees ellipticity preservation.

Abstract

Since the seminal contribution of Geymonat, M{\"u}ller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved by homogenization in linear elasticity. This phenomenon is typically related to microscopic buckling of the composite material. The present contribution is concerned with the interplay between isotropy and strong ellipticity in the framework of periodic homogenization in linear elasticity. Mixtures of two isotropic phases may indeed lead to loss of strong ellipticity when arranged in a laminate manner. We show that if a matrix/inclusion type mixture of isotropic phases produces macroscopic isotropy, then strong ellipticity cannot be lost. R{\'e}sum{\'e}. Nous savons depuis l'article fondateur de Geymonat, M{\"u}ller et Triantafyl-lidis qu'en{\'e}lasticit{\'e}en{\'e}lasticit{\'e} lin{\'e}aire l'homog{\'e}n{\'e}isation p{\'e}riodique ne conserve pas n{\'e}cessairement l'ellipticit{\'e} forte. Ce ph{\'e}nom{\`e}ne est li{\'e} au flambage microscopique des composites. Notre contribution consiste examiner le r{\^o}le de l'isotropie dans ce type de pathologie. Le m{\'e}lange de deux phases isotropes peut en effet conduir{\`e} a cette perte si l'arrangement est celui d'un lamin{\'e}. Nous montrons qu'en revanche, si un arrangement de type ma-trice/inclusion produit un tenseur homog{\'e}n{\'e}is{\'e} isotrope, alors la forte ellipticit{\'e} est con-serv{\'e}e.