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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1139-1144 (novembre 2016)
Doi : 10.1016/j.crma.2016.09.014
Received : 6 September 2016 ;  accepted : 29 September 2016
Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity
L'isotropie interdit la perte d'ellipticité forte par homogénéisation en élasticité linéaire

Gilles A. Francfort a , Antoine Gloria b, c
a L.A.G.A. (UMR CNRS 7539), Université Paris-Nord, Villetaneuse, France 
b Université libre de Bruxelles (ULB), Brussels, Belgium 
c Inria, Lille, France 


Since the seminal contribution of Geymonat, Müller, 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 concerns 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.

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Nous savons, depuis l'article fondateur de Geymonat, Müller et Triantafyllidis, qu'en élasticité linéaire l'homogénéisation périodique ne conserve pas nécessairement l'ellipticité forte. Ce phénomène est lié au flambage microscopique des composites. Notre contribution consiste à examiner le rôle de l'isotropie dans ce type de pathologie. Le mélange de deux phases isotropes peut en effet conduire à cette perte si l'arrangement est celui d'un laminé. Nous montrons en revanche que, si un arrangement de type matrice/inclusion produit un tenseur homogénéisé isotrope, alors la forte ellipticité est conservée.

The full text of this article is available in PDF format.

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