Some remarks on the asymptotic invertibility of the linearized operator of nonlinear elasticity in the context of the displacement approach

In this article we study the invertibility of the linearized operator coming from the nonlinear elasticity in the special case of a two-dimensional thin beam of thickness 2ε in one direction and of length 2πL and periodic in the other direction. In the context of the displacement approach, we show that the linearized operator is not invertible for some small compressions of orderO(ε/L) in the direction of the thickness of the beam, and not in the direction of the length as it is usually considered. In particular, we study the kernel of an associated linear operator on an infinite strip. This linear operator depends on a parameter δ which describes the compression with respect to the thickness for δ < 0. For small enough δ > 0, we prove that the kernel is trivial; on the contrary for δ < 0, we rigorously find periodic solutions in the kernel. This last fact is related to the non-invertibility of the previous linearized operator coming from nonlinear elasticity.