  In this paper we analyze a new temperature-dependent model for adhesive
contact that encompasses nonlocal adhesive forces and damage effects, as well
as nonlocal heat flux contributions on the contact surface. The related PDE
system combines heat equations, in the bulk domain and on the contact surface,
with mechanical force balances, including micro-forces, that result in the
equation for the displacements and in the flow rule for the damage-type
internal variable describing the state of the adhesive bonds. Nonlocal effects
are accounted for by terms featuring integral operators on the contact surface.
The analysis of this system poses several difficulties due to its overall
highly nonlinear character, and in particular to the presence of quadratic
terms, in the rates of the strain tensor and of the internal variable, that
feature in the bulk and surface heat equations. Another major challenge is
related to proving strict positivity for the bulk and surface temperatures. We
tackle these issues by very careful estimates that enable us to prove the
existence of global-in-time solutions and could be useful in other contexts.
All calculations are rigorously rendered on an accurately devised time
discretization scheme in which the limit passage is carried out via variational
techniques.
