Analysis of a temperature-dependent model for adhesive contact with friction

We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond's approach, adhesion is described in terms of a surface damage parameter χ. The related equations are the (quasistatic) momentum balance for the vector of displacements, and parabolic-type evolution equations for χ, and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character. After introducing a suitable regularization of the Coulomb law for dry friction, we address the analysis of the resulting PDE system. The main result of the paper states the existence of global-in-time solutions to the associated Cauchy problem. It is proved by passing to the limit in a carefully tailored approximate problem, via variational techniques.