Analysis and numerical simulations of a dynamic contact problem with adhesion

The dynamic process of frictionless contact between a viscoelastic body and a reactive foundation is modelled, analyzed, and simulated. The contact is adhesive and it is described by introducing an internal variable, the bonding field β, which measures the fractional density of active bonds. The evolution of β is described by an ordinary differential equation that depends on the process history, taking into account possible adhesive degradation during cycles of debonding and rebonding. The existence of the unique weak solution of the model is proved by using arguments of nonlinear evolutionary equations with monotone operators and a fixed-point theorem. A fully discrete numerical scheme is proposed for the model and implemented in a computer code. Numerical simulations of one- and two-dimensional examples are presented.