Loss of ellipticity through homogenization in linear elasticity
Résumé
It was shown in \cite{geymonat.muller.triantafyllidis93} that, in the setting of linearized elasticity, a $\Gamma$-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in $\mathbb{R}^N$ is zero while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate {construction} of \cite{gutierrez99}. We also demonstrate that the predicted loss of strict strong ellipticity resulting from the construction in \cite{gutierrez99} is unique within a ``laminate-like" class of microstructures. It will only occur for the specific micro-geometry investigated there. Our results thus confer both a rigorous, and a canonical character to those in \cite{gutierrez99}.