Homogenization with quasistatic Tresca's friction law: qualitative and quantitative results

TL;DR

This paper investigates the homogenization of a linear elasticity system with Tresca's friction law, providing both qualitative convergence results and quantitative error estimates, supported by numerical experiments.

Contribution

It offers new homogenization results for systems with Tresca friction law, including quantitative error estimates and numerical validation.

Findings

Weak convergence of solutions under H-convergence

Quantitative $H^1$-norm error estimates

Numerical validation of convergence rates

Abstract

Modeling of frictional contacts is crucial for investigating mechanical performances of composite materials under varying service environments. The paper considers a linear elasticity system with strongly heterogeneous coefficients and quasistatic Tresca friction law, and studies the homogenization theories under the frameworks of H-convergence and small $\epsilon$-periodicity. The qualitative result is based on H-convergence, which shows the original oscillating solutions will converge weakly to the homogenized solution, while our quantitative result provides an estimate of asymptotic errors in $H^1$-norm for the periodic homogenization. This paper also designs several numerical experiments to validate the convergence rates in the quantitative analysis.