A membrane in adhesive contact

A model for the process of quasi-static evolution of an elastic membrane in adhesive contact with a rigid obstacle is developed, analyzed, and numerically simulated. The model consists of an elliptic variational inequality for the membrane displacements and a nonlinear ordinary differential equation for the evolution of the adhesion field. By using regularity results from the theory of elliptic variational inequalities and a fixed point argument, the system is shown to have a unique weak solution. A fully discrete algorithm is described and shown to converge, and its error estimates are derived. In this process we make critical use of the regularity properties of the solution. Finally, the results of numerical simulations, based on the fully discrete algorithm, are presented.