We consider a spectral homogenization problem for the linear elasticity system posed in a domain Ω of the upper half-space R^{3 +}, a part of its boundary Σ being in contact with the plane { x₃= 0 }. We assume that the surface Σ is traction-free out of small regions T^ε, 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 β(ε) that can be very large when ε→ 0. The size of the regions T^ε is O(r_ε) , where r_ε≪ ε, and they are placed at a distance ε between them. We provide all the possible spectral homogenized problems depending on the relations between ε, r_ε and β(ε) , while we address the convergence, as ε→ 0 , of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on Σ. New capacity matrices are introduced to define these strange terms.