In this article, we develop a phenomenological model for the contact mechanics and the dissipation mechanism at an oscillatory contact with adhesion. We consider an inflated spherical membrane held between two large rigid static/oscillating parallel flat plates. For the contact problem, the critical energy release rate is used to determine the detachment of the membrane from the plate surface. An analytical relation between the adhesion energy and the adhesive bond force is established. For the geometry considered, we obtain a relation between the adhesive bond force (and hence also the energy release rate) and the displacement of the plate at equilibrium without any external force. An interesting and counterintuitive stretch-induced softening phenomenon (in terms of the equilibrium displacement) is observed in certain material models. In the oscillatory contact problem, the plates are assumed to provide a kinematic oscillating boundary condition. The effective energy dissipated through hysteresis during a period of oscillation of the plate against the membrane is determined. The effect of initial inflation on energy dissipation is studied.