Global existence for a contact problem with adhesion

Abstract

SUMMARYIn this paper, we analyze a contact problem with irreversible adhesion between a viscoelastic body and a rigid support. On the basis of Frémond's theory, we detail the derivation of the model and of the resulting partial differential equation system. Hence, we prove the existence of global in time solutions (to a suitable variational formulation) of the related Cauchy problem by means of an approximation procedure, combined with monotonicity and compactness tools, and with a prolongation argument. In fact the approximate problem (for which we prove a local well-posedness result) models a contact phenomenon in which the occurrence of repulsive dynamics is allowed for. We also show local uniqueness of the solutions, and a continuous dependence result under some additional assumptions.

Global existence for a nonlocal model for adhesive contact

“…To obtain a contradiction, it is sufficient to extend the approximable solution (u * , χ * ), defined on the interval [0, T * ], to an approximable solution on the interval [0, T * + η] for some η > 0. The argument for this is completely analogous to the one developed in [1,Sec. 5] for the adhesive contact system without nonlocal effects.…”

“…on Γ C × (0, T ). The operators ρ and β will generalize the subdifferentials ∂I (−∞,0] and ∂I [0,1] in (1.20e). On the one hand, the requirement dom( β) ⊂ [0, +∞) guarantees χ ≥ 0 a.e.…”

“…We point out that (4.1e) and (4.1g) lead to lim sup approximate system, by repeating the very same calculations in the proof of Proposition 4.1. We refer the reader to [1,Sec. 5] for all the details of the argument.…”

“…5] for the adhesive contact system without nonlocal effects. We once again refer to [1] for all details.…”

“…The type of model investigated in the present paper was first rigorously derived and analyzed in [1] (cf. also [2]).…”