Comptes Rendus
no. 11
Mathematical problems in mechanics
Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity

Presented by: Jean-Michel Coron

Gilles A. Francfort1; Antoine Gloria2, 3
1 L.A.G.A. (UMR CNRS 7539), Université Paris-Nord, Villetaneuse, France
2 Université libre de Bruxelles (ULB), Brussels, Belgium
3 Inria, Lille, France
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1139-1144.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.09.014

Gilles A. Francfort 1; Antoine Gloria 2, 3

1 L.A.G.A. (UMR CNRS 7539), Université Paris-Nord, Villetaneuse, France
2 Université libre de Bruxelles (ULB), Brussels, Belgium
3 Inria, Lille, France 
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Gilles A. Francfort; Antoine Gloria. Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1139-1144. doi : 10.1016/j.crma.2016.09.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.014/

[1] M. Briane; G.A. Francfort Loss of ellipticity through homogenization in linear elasticity, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 5, pp. 905-928

[2] G. Geymonat; S. Müller; N. Triantafyllidis Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Ration. Mech. Anal., Volume 122 (1993), pp. 231-290

[3] S. Gutiérrez Laminations in linearized elasticity: the isotropic non-very strongly elliptic case, J. Elasticity, Volume 53 (1998/1999) no. 3, pp. 215-256

[4] S. Müller Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Ration. Mech. Anal., Volume 99 (1987) no. 3, pp. 189-212

[5] J. Nečas Sur les normes équivalentes dans Wk(Ω) et sur la coercivité des formes formellement positives, Équations aux derivées partielles, Séminaire de mathématiques supérieures, vol. 19, 1966, pp. 101-108

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