Homogenization of friction in a 2D linearly elastic contact problem

TL;DR

This paper rigorously analyzes the homogenization of a 2D elastic contact problem with heterogeneous Coulomb friction, establishing existence, uniqueness, and deriving an explicit formula for the effective friction coefficient.

Contribution

It provides the first rigorous homogenization analysis for highly oscillating friction coefficients in elastic contact problems, revealing the impact of coupling on the effective friction.

Findings

Coulomb law holds in the homogenized limit

Explicit formula for the effective friction coefficient

Effective friction differs from spatial average

Abstract

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.