Loss of ellipticity by homogenization in 2D elasticity and application to elastodynamics

In this work in collaboration with G. Francfort (University Paris 13) we obtain a homogenization result in 2D linear elasticity for the L^{^2}-weak convergence of the displacements, for a two-phase material a phase of which is not very strongly elliptic. In the particular case of the two-phase laminate studied by Gutiérrez (1990), the homogenized tensor turns out to be not strongly elliptic in the direction perpendicular to the lamination. As a consequence the associated homogenized elastodynamics equation is showed to allow transverse wave planes while inhibiting longitudinal waves

• Briane's seminar (1 MB)