Localized Thickening of a Compressed Elastic Band

Abstract

An infinite elastic band is compressed along its unbounded direction, giving rise to a continuous family of homogeneous configurations that is parameterized by the compression rate β < 1 (β = 1 when there is no compression). It is assumed that, for some critical value β0, the compression force as a function of β has a strict local extremum and that the linearized equation around the corresponding homogeneous configuration is strongly elliptic. Under these conditions, there are nearby localized deformations that are asymptotically homogeneous. When the compression force reaches a strict local maximum at β0, they describe localized thickening and they occur for values of β slightly smaller than β0. Since the material is supposed to be hyperelastic, homogeneous and isotropic, the localized deformations are not due to localized imperfections. The method follows the one developed by A. Mielke for an elastic band under traction: interpretation of the nonlinear elliptic system as an infinite dimensional dynamical system in which the unbounded direction plays the role of time, its reduction to a center manifold and the existence of a homoclinic solution to the reduced finite dimensional problem in [A. Mielke, Hamiltonian and Lagrangian fiows on center manifolds, Lecture Notes in Mathematics 1489. Springer, Berlin Heidelberg New York, 1991]. The main difference lies in the fact that Agmon's condition does not hold anymore and therefore the linearized problem cannot be analyzed as in Mielke's work.