The mechanical behavior of a circular dielectric elastomer membrane (DEM) under the contact of a rigid ball is studied in this paper. The out-of-plane deformation of the DEM is unfolded to an equivalent in-plane deformation, and the mechanical behavior is further studied through the Helmholtz free energy theory and the Gent model. The theoretical results obtained from the proposed analysis approach are validated through the out-of-plane deformation experiments, and the influence of the DEM's thickness on the mechanical behavior is revealed and explained. Furthermore, the influences of some key dimensional, dynamical and electrical parameters on the DEM's mechanical behavior are investigated and discussed. The research results are helpful for the system analysis of dielectric elastomer actuators and dielectric elastomer generators with out-of-plane deformations.