Analysis of an adhesive contact problem for elastic materials

The goal of this paper is to study a mathematical model which describes the adhesive frictionless contact between a deformable body and a foundation. The body consists of an elastic material and the process is assumed to be quasistatic. The adhesive contact condition on the normal direction is modeled by a version of normal compliance condition with unilateral constraint. The adhesion is modeled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We present a variational formulation of the mechanical problem and prove the existence and uniqueness of a weak solution. Also, we study a penalized contact problem which admits a unique solution. We prove that when the penalization parameter converges to zero, the solution converges to the solution of the original model. The technique of the proof is based on time-dependent variational inequalities, differential equations and the Banach fixed-point theorem.