Inflation of rectangular hyperelastic membranes with dielectric properties

This paper investigates the electromechanically driven inflation of pre-stretched rectangular hyperelastic membranes, taking into account dielectric properties of the material. The problem, which is relevant to several emerging engineering applications, is typically addressed numerically due to its mathematical complexity. Here, to accurately capture the inflated configuration of the membrane while retaining a compact formulation, the membrane deformation is represented through suitable kinematic assumptions. Further, a novel variational formulation is developed, based on the stationarity of the potential energy of the system by a Ritz method. To this aim, an incompressible Mooney-Rivlin material model is adopted assuming an ideal dielectric elastomer behavior. The resulting formulation reduces the problem to a small set of nonlinear algebraic equations, enabling efficient prediction of the membrane profile and pressure-deflection response. Notably, for long rectangular membranes, a simplified analytical pressure-deflection expression is also derived, providing direct insight into the role of mechanical pre-stretch, material parameters, geometry, and electrical actuation. The accuracy of the proposed formulation is assessed through comparisons with finite element simulations. Overall, the results offer a practical modeling framework for the analysis and design of rectangular dielectric elastomer membranes in soft actuators, sensors, and related devices.