Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic
body and a rigid foundation, when both the effects of the
(irreversible) adhesion and of the friction are taken into account.
We describe the adhesion phenomenon in terms of a damage surface
parameter according to Frémond’s theory, and we model unilateral
contact by Signorini conditions, and friction by a nonlocal
Coulomb law. All the constraints on the internal variables as well
as the contact and the friction conditions are rendered by means
of subdifferential operators, whence the highly nonlinear character
of the resulting PDE system. Our main result states the existence
of a global-in-time solution (to a suitable variational formulation)
of the related Cauchy problem. It is proved by an approximation
procedure combined with time discretization.