Abstract <p>The effect of the energy dissipation mechanisms, including the viscoelasticity and the charge leakage, on the electromechanical coupling behavior of circular dielectric elastomer membranes is the focus of this study. Based on the principle of non-equilibrium thermodynamics and the theory of nonlinear dissipative dielectrics, this study develops a physical model to describe the influence of the viscoelasticity and the charge leakage on the electromechanical coupling behavior, allowing for the examination of both their independent and combined effects. Through numerical simulations, we first analyze the effect of the viscoelasticity on membrane stress and deformation while isolating the charge leakage. Next, we investigate the radial distribution characteristics of the leakage current in a fixed membrane state. Finally, we examine the combined effects of the viscoelasticity and the charge leakage under varying voltage slopes. The results indicate that both the viscoelasticity and the charge leakage significantly affect the electromechanical response rate of the membrane, while the voltage slope primarily influences the growth rates of various physical quantities within the membrane. This effect is particularly evident in the responses of the radial stretch and the leakage current. This study elucidates the complex effects of the viscoelasticity and the charge leakage on the electromechanical performance of dielectric elastomer membranes, offering valuable reference data for optimizing and designing their applications.</p>