Frictionless contact problem with adhesion for nonlinear elastic materials

We consider a quasistatic frictionless contact problem for a nonlinear elastic body. The contact is modelled with Signorini's conditions. In this problem we take into account of the adhesion which is modelled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We derive a variational formulation of the mechanical problem and we establish an existence and uniqueness result by using arguments of time-dependent variational inequalities, differential equations and Banach fixed point. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solution of a penalized problem as the penalization parameter converges to 0