  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.
