Justifying Linearization for Nonlinear Boundary Homogenization on a Grill-Type Winkler Foundation

Abstract

We address a nonlinear boundary homogenization problem associated to the deformations of a block of an elastic material with small reaction regions periodically distributed along a plane. We assume a nonlinear Winkler-Robin law which implies that a strong reaction takes place in these reaction regions. Outside, on the plane, the surface is traction-free while the rest of the surface is clamped to an absolutely rigid profile. When dealing with critical sizes of the reaction regions, we show that, asymptotically, they behave as stuck regions, the homogenized boundary condition being a linear one with a new reaction term which contains a capacity matrix depending on the macroscopic variable. This matrix is defined through the solution of a parametric family of microscopic problems, the macroscopic variable being its parameter. Among others, to show the convergence of the solutions, we develop techniques that extend those both for nonlinear scalar problems and linear vector problems in the literature. We also address the extreme cases.