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

• A model for adhesive contact including thermal and frictional effects is analyzed.
• Entropy balance equations govern the evolution of the thermal variables.
• The PDE system is highly nonlinear: subdifferentials render contact and friction.
• The main result of the paper states the existence of global-in-time solutions.
• Existence is proved by an approximation–a priori estimates–limit passage argument.

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 χχ. The related equations are the (quasistatic) momentum balance for the vector of displacements, and parabolic-type evolution equations for χχ, 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.