Transition from Continuous to Jumping Solutions in 2D Quasi-static Elastic Contact Problems with Coulomb Friction: the Mathematics Underlying the Onset of Brake Squeal

TL;DR

This paper analyzes the mathematical conditions under which solutions to 2D elastic contact problems with Coulomb friction transition from continuous to jump discontinuities, indicating the onset of brake squeal vibrations.

Contribution

It establishes an optimal friction coefficient condition ensuring solution continuity and links solution jumps to the physical phenomenon of brake squeal onset.

Findings

Existence of absolutely continuous solutions under certain friction conditions

Spontaneous jumps occur when friction exceeds a critical threshold

Jumps indicate a transition from quasi-static to dynamic behavior

Abstract

We formulate the quasi-static elastic contact problem with Coulomb friction in a very general setting, with possible jumps in time for both the load and the solution. Exploiting ideas originating in our recent paper [4], we exhibit an optimal condition on the magnitude of the friction coefficient under which we prove the existence of an absolutely continuous solution for arbitrary absolutely continuous loads in the case of the most general 2D problem. We provide examples showing that, when the condition is violated, spontaneous jumps in time of the solution may occur, even when the load varies absolutely continuously in time. We argue that these spontaneous jumps in time of the solution in the quasi-static problem reveal a transition of the process from a quasi-static nature to a dynamic nature, interpreted as the mathematical signature of the onset of friction-induced vibrations in the elastodynamic contact problem with dry friction.