Interfacial homogenization of a periodically corrugated surface in linear elasticity

Abstract

This work presents a homogenization framework for modeling the mechanical behavior of three-dimensional linear elastic bodies with a periodically corrugated surface subjected to Dirichlet boundary conditions. The surface microstructure is assumed to be invariant along one spatial direction and periodic along the other. By combining asymptotic homogenization with matched asymptotic expansions near the surface corrugations, we derive an effective interface constitutive model that replaces the corrugated surface and the Dirichlet boundary condition with a flat boundary governed by a mixed (Robin-type) boundary condition. This boundary condition involves a second-order effective tensor, computed from elementary problems set on a representative periodic unit cell, hence allowing to account for the effect of the microstructure on the macroscopic response. We prove the symmetry and positive definiteness of the effective tensor and establish a uniqueness result of the effective problem. The model is assessed by comparison with 2D and 3D full-field simulations, demonstrating excellent agreement in both global and local responses. In particular, a cost-efficient post-processing strategy is proposed to reconstruct the local fields near the corrugations by use of a simple periodic unit cell, providing access to fine-scale information without the need for full-resolution computations.