  We show that a smooth giant voltage actuation of soft dielectric plates is
not easily obtained in practice. In principle one can exploit, through
pre-deformation, the snap-through behavior of their loading curve to deliver a
large stretch prior to electric breakdown. However, we demonstrate here that
even in this favorable scenario, the soft dielectric is likely to first
encounter the plate wrinkling phenomenon, as modeled by the onset of
small-amplitude sinusoidal perturbations on its faces. We provide an explicit
treatment of this incremental boundary value problem. We also derive
closed-form expressions for the two limit cases of very thin membranes (with
vanishing thickness) and of thick plates (with thickness comparable to or
greater than the wavelength of the perturbation). We treat explicitly examples
of ideal dielectric free energy functions (where the mechanical part is of the
neo-Hookean, Mooney-Rivlin or Gent form) and of dielectrics exhibiting
polarization saturation. Finally we make the link with the classical results of
the Hessian electro-mechanical instability criterion and of Euler buckling for
an elastic column.
