Nonlinear Dynamics of a Dielectric Elastomer Membrane Under Compressive Loading

This paper investigates the large-amplitude, non-linear vibration of dielectric elastomer membrane disks. Dielectric elastomers are non-conducting materials with extreme stretchability and electro-mechanical coupling, making them compelling candidates for actuation (soft robotics, artificial muscles), sensing, and energy harvesting. A common configuration for dielectric elastomers is a circular or rectangular membrane. A stretchable conductor (for example, an electrically-conducting grease) is applied to the top and bottom surfaces. A voltage is applied across the membrane, which forces positive charge on one surface and negative charge on the other. These opposite charges attract, which results in compression of the membrane. In response, the membrane’s thickness decreases and its surface area increases. In this paper, the dielectric elastomer is modeled as an incompressible, isotropic, ideal dielectric, with mechanical stiffening at large stretches captured using the Gent hyperelastic constitutive model. The fully non-linear, coupled electro-mechanical equation of motion is derived using Hamilton's principle. The disk achieves steady equilibrium where the compressive stresses due to the applied voltage balance the tensile stresses from the applied radial loads. System equilibria are calculated numerically for a wide range of radial loads, applied voltages, and limiting stretches. It is possible for the disk to have two stable steady equilibria at given radial load and applied voltage (one at small stretch and the other at large stretch), which gives rise to an instability where extreme stretches occur for infinitesimal changes in applied voltage. The equation of motion is determined for small vibrations of the system about equilibrium. Unlike thin membrane disks, the vibrating mass of thick membrane disks depends on the steady equilibrium stretch. The natural frequency meaningfully decreases with increasing membrane thickness due to the inertia associated with dynamic changes in the membrane thickness. The amount of axial inertia depends on the ratio of the nominal disk thickness to its radius and the steady equilibrium stretch. The limitations of the small-vibration model are numerically determined in terms of the vibration amplitude, applied voltage, and membrane disk thickness. The linear model is less accurate for thick membrane disks than thin ones, where the inertia associated with changes in thickness is negligible. The linear model also suffers from inaccuracy in the vicinity of the bifurcation voltages, where large stretches occur with small changes in voltage. Large amplitude vibrations are numerically investigated for a wide range of system parameters. Finite-element simulations are in good agreement with the numerical solutions of our analytical model.