A two-dimensional labile aether through homogenization

Marc Briane (1) , Gilles A. Francfort (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Homogenization in linear elliptic problems usually assumes coercivity of the accompanying Dirichlet form. In linear elasticity, coercivity is not ensured through mere (strong) ellipticity so that the usual estimates that render homogenization meaningful break down unless stronger assumptions, like very strong ellipticity, are put into place. Here, we demonstrate that a L^2-type homogenization process can still be performed, very strong ellipticity notwithstanding, for a specific two-phase two dimensional problem whose significance derives from prior work establishing that one can lose strong ellipticity in such a setting, provided that homogenization turns out to be meaningful.A striking consequence is that, in an elasto-dynamic setting, some two-phase homogenized laminate may support plane wave propagation in the direction of lamination on a bounded domain with Dirichlet boundary conditions, a possibility which does not exist for the associated two-phase microstructure at a fixed scale. Also, that material blocks longitudinal waves in the direction of lamination, thereby acting as a two-dimensional aether in the sense of e.g. Cauchy.

Mots clés: lamination; homogenization; Linear elasticity; wave propagation; ellipticity

Domaines: Science des matériaux [cond-mat.mtrl-sci]; Equations aux dérivées partielles [math.AP]; Mécanique des matériaux [physics.class-ph]

Marc Briane, Gilles A. Francfort. A two-dimensional labile aether through homogenization. Communications in Mathematical Physics, 2019, 367 (2), pp.599-628. ⟨10.1007/s00220-019-03333-7⟩. ⟨hal-01696161v2⟩