A two-dimensional labile aether through homogenization

TL;DR

This paper demonstrates that a specific two-dimensional two-phase elastic problem can be homogenized despite lacking strong ellipticity, revealing novel wave propagation properties akin to a two-dimensional aether.

Contribution

It shows that homogenization is possible in a two-phase elastic setting without strong ellipticity, challenging traditional assumptions and revealing new wave phenomena.

Findings

Homogenization can be performed without strong ellipticity in certain 2D elastic problems.

Some two-phase laminates support plane wave propagation in the lamination direction.

Material blocks longitudinal waves, acting as a 2D aether.

Abstract

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.