Nonlinear vibration of dielectric elastomer membranes with axial inertia effects

Dielectric elastomers are a compelling class of electro-active materials that show great promise for large -deformation actuation, sensing, and energy generation applications. In many of these applications, the applied voltages are transient, leading to meaningful dynamic stretches with rich and atypical nonlinear behavior. Existing works either jettison dynamic thickness changes at the outset or do not examine their overall effects on dynamic behavior. In this paper, a Newtonian framework accounting for dynamic thickness changes during planar motion is used to model the nonlinear dynamics of circular dielectric elastomer membranes. A distinguishing feature of the resulting equation of motion is the presence of additional nonlinear terms that we call axial inertia. A small-vibration model is derived, and its agreement with the full nonlinear model is quantified in terms of the vibration amplitude, applied voltage, and membrane thickness. The natural frequency is found to be sensitive to membrane thickness when vibrations occur about small-stretch equilibria, but not large-stretch equilibria. The impact of axial inertia on the membrane's frequency response is examined for sinusoidal voltage fluctuations. At sufficiently low voltages where the membrane has only one equilibrium, the frequency response transitions from softening behavior at low amplitudes to hardening behavior at moderate-to-high amplitudes. Axial inertia results in atypical frequency response, with regions where three steady-state dynamic stretches are possible and additional jumps occur during frequency sweeps through resonance. Secondary subharmonic resonances are possible for thick membranes at low voltages, which have frequency ranges where four steady-state dynamic stretches are possible, the first known demonstration of this phenomenon. Electric fields needed to access this and other novel nonlinear dynamic response are shown to be below stretch-dependent dielectric strengths from the literature. Our results show that the effects of axial inertia on nonlinear dynamics can be significant even for thin membranes undergoing large dynamic stretches.