Dynamic Adhesive Contact of a Membrane

Abstract

This work presents a dynamic model for adhesive contact between a stretched viscoelastic membrane and a reactive obstacle that lies beneath it. The adhesion is described by a bonding field, and the model allows for failure, that is complete debonding in finite time. It is two-dimensional, but retains the essential mathematical structure of the full three-dimensional model. It is set as a hyperbolic equation for the vibrations of the membrane coupled with a nonlinear ordinary differential equation for the evolution of the bonding field. Existence and uniqueness of regular solutions are established in the case of positive viscosity, and in the case of no viscosity, existence of weak solutions is obtained, while the uniqueness of the solutions remains unresolved.