Homogenization of Friction in a 2D Linearly Elastic Contact Problem

Contact problems with Coulomb friction in linear elasticity are notoriously difficult, and their mathematical analysis is still largely incomplete. In this paper, a model problem with heterogeneous friction coefficient is considered in two-dimensional elasticity. For this model problem, an existence and uniqueness result is proved, relying heavily on harmonic analysis. A complete and rigorous homogenization analysis can be performed in the case of a highly oscillating friction coefficient, being the first result in that direction. The Coulomb law is found to hold in the limit, and an explicit formula is provided to calculate the effective friction coefficient. This effective friction coefficient is found to differ from the spatial average, showing an influence of the coupling between friction and elasticity on the homogenized limit.