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

We consider a spectral homogenization problem for the linear elasticity system posed in a domain \(\varOmega \) of the upper half-space \(\mathbb{R}^{3+}\), a part of its boundary \(\varSigma \) being in contact with the plane \(\{x_{3}=0\}\). We assume that the surface \(\varSigma \) is traction-free out of small regions \(T^{\varepsilon }\), 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 \(\beta (\varepsilon )\) that can be very large when \(\varepsilon \to 0\). The size of the regions \(T^{\varepsilon }\) is \(O(r_{\varepsilon })\), where \(r_{\varepsilon }\ll \varepsilon \), and they are placed at a distance \(\varepsilon \) between them. We provide all the possible spectral homogenized problems depending on the relations between \(\varepsilon \), \(r_{\varepsilon }\) and \(\beta (\varepsilon )\), while we address the convergence, as \(\varepsilon \to 0\), of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on \(\varSigma \). New capacity matrices are introduced to define these strange terms.