Sensitivity analysis of a scalar mechanical contact problem with perturbation of the Tresca's friction law

TL;DR

This paper performs a sensitivity analysis of a scalar mechanical contact problem with Tresca's friction law, focusing on perturbations of the friction threshold and establishing differentiability of solutions with numerical validation.

Contribution

It introduces a parameterized Tresca friction problem, proves the solution's differentiability with respect to perturbations, and characterizes the derivative via a boundary value problem.

Findings

Differentiability of the solution with respect to perturbations

Characterization of the derivative through a boundary value problem

Numerical simulations confirming theoretical results

Abstract

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca's friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular the friction threshold involved in the Tresca's friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally numerical simulations are provided in order to illustrate our main result.