Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

Abstract

We consider a spectral homogenization problem for the linear elasticity system posed in a domain of the upper half-space R3+, a part of its boundary being in contact with the plane {x3=0}. We assume that the surface is traction-free out of small regions T, where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function M(x) and a reaction parameter () that can be very large when 0. The size of the regions T is O(r), where r, and they are placed at a distance between them. We provide all the possible spectral homogenized problems depending on the relations between , r and (), while we address the convergence, as 0, of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on . New capacity matrices are introduced to define these strange terms.