# A frictional contact problem with wear diffusion

## TL;DR

This paper develops a mathematical model for the dynamic interaction of a viscoelastic body with a moving foundation, incorporating wear, debris diffusion on curved surfaces, and contact mechanics, with proven existence of solutions.

## Contribution

It introduces a novel model accounting for wear debris diffusion on a manifold contact surface, relevant for joint prostheses and implants, with a proof of solution existence.

## Key findings

- Model captures wear and debris diffusion on curved contact surfaces.
- Existence of weak solutions established using fixed point theorem.
- Applicable to biomechanics and mechanical joint wear analysis.

## Abstract

This paper constructs and analyzes a model for the dynamic frictional contact between a viscoelastic body and a moving foundation. The contact involves wear of the contacting surface and the diffusion of the wear debris. The relationships between the stresses and displacements on the contact boundary are modeled by the normal compliance law and a version of the Coulomb law of dry friction. The rate of wear of the contact surface is described by the differential form of the Archard law. The effects of the diffusion of the wear particles that cannot leave the contact surface on the surface are taken into account. The novelty of this work is that the contact surface is a manifold and, consequently, the diffusion of the debris takes place on a curved surface. The interest in the model is related to the wear of mechanical joints and orthopedic biomechanics where the wear debris are trapped, they diffuse and often cause the degradation of the properties of joint prosthesis and various implants. The model is in the form of a differential inclusion for the mechanical contact and the diffusion equation for the wear debris on the contacting surface. The existence of a weak solution is proved by using a truncation argument and the Kakutani--Ky Fan--Glicksberg fixed point theorem.