Quasistatic thermoviscoelastic problem with normal compliance, multivalued friction and wear diffusion

Abstract

This paper presents and analyzes a model for quasistatic frictional contact between a thermoviscoelastic body and a moving foundation that involves wear of the contacting surface and the diffusion of the wear debris. The constitutive law includes temperature effects and the evolution of the temperature is described by a parabolic equation with a subdifferential heat exchange boundary condition. Contact is modeled with normal compliance together with a subdifferential frictional law. The rate of wear of the contact surface is described by the differential form of the Archard condition. The effects of the diffusion of the wear particles on the contact surface are taken into account. Such situations arise in mechanical joints and in orthopedic biomechanics where the wear debris is trapped, diffuses and influences the properties of joint prosthesis and implants. The variational formulation of the problem leads to a system with a time-dependent hemivariational inequality for the displacement, a parabolic hemivariational inequality for the temperature and a parabolic equation on the contact boundary for the wear diffusion. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational diffusion equation, and a fixed point argument.