Internal resonance and nonlinear dynamics of a dielectric elastomer circular membrane

Dielectric elastomers (DE) are widely used in several applications such as artificial muscles, sensors or energy harvesters. Their hyperelastic nature complicates their design because of the obtained rich dynamics. The electrical field applied in the material, couples the dynamics of these devices to the applied voltage through compliant electrodes. In this work, a circular DE membrane under mechanical and electrical excitations is analyzed in order to seek internal resonances capable of increasing its performance for future applications. The hyperelastic properties of the material and the Maxwell stresses are taken into account to derive the nonlinear governing equation. Analytical and numerical approaches are used to obtain the dynamic solution of the device under different prestretch and electrical conditions. The effect of the prestretch on the first transverse and radial natural frequencies is first calculated analytically and validated by a finite element software. One-to-one, two-to-one and three-to-one internal resonances are possible as the prestretch applied to the membrane is varied. Using a Galerkin procedure and a multiple scale technique, the frequency response curves are obtained for different applied voltages. The results are validated using a Runge–Kutta time discretization technique. Finally, the effect of the applied voltage is analyzed for the frequency response curves of the radial and transverse displacements.