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

Gilles Francfort 1 Antoine Gloria 2, 3
3 MEPHYSTO - Quantitative methods for stochastic models in physics
LPP - Laboratoire Paul Painlevé - UMR 8524, ULB - Université Libre de Bruxelles [Bruxelles], Inria Lille - Nord Europe
Abstract : 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 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. Résumé. Nous savons depuis l'article fondateur de Geymonat, Müller et Triantafyl-lidis qu'enélasticitéenélasticité linéaire l'homogénéisation périodique ne conserve pas nécessairement l'ellipticité forte. Ce phénomène est lié au flambage microscopique des composites. Notre contribution consiste examiner le rôle de l'isotropie dans ce type de pathologie. Le mélange de deux phases isotropes peut en effet conduirè a cette perte si l'arrangement est celui d'un laminé. Nous montrons qu'en revanche, si un arrangement de type ma-trice/inclusion produit un tenseur homogénéisé isotrope, alors la forte ellipticité est con-servée.
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Gilles Francfort, Antoine Gloria. Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity. Comptes Rendus Mathématique, Elsevier Masson, 2016, 354, pp.1139 - 1144. ⟨10.1016/j.crma.2016.09.014⟩. ⟨hal-01398518⟩

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