The out-of-plane behaviour of dielectric membranes: Description of wrinkling and pull-in instabilities

Voltage controlled dielectric membranes exhibit two fundamental types of instability, strongly affecting their performances: the occurrence of wrinkling, which is due to membranal compressive stresses, and the onset of pull-in, a catastrophic thinning localisation that preludes electrical breakdown. In this manuscript we provide a unifying energetic description of both instabilities for large, out-of-plane and inhomogeneous deformations. By using the ideas of relaxation and regularisation of the energy, originally proposed by Pipkin (1986) and Hilgers and Pipkin (1992) for purely elastic membranes, we show that the onset and development of wrinkling can be effectively described by the relaxed electroelastic energy. For axially symmetric membranes and neo-Hookean materials, we show that pull-in corresponds to failure of the strong ellipticity condition of the regularised electroelastic energy, thus extending to out-of-plane deformations the validity of a previous estimate for planar systems (Zurlo et al., 2017). In agreement with ubiquitous experimental evidence, we also show that wrinkled states are always stable below the pull-in voltage. Our theoretical findings are assessed by the comparison with experiments on out-of-plane, voltage-actuated annular membranes, showing good agreement both in terms of description of wrinkled states, and for the prediction of the pull-in instability.