  We investigate the theoretical nonlinear response, Hessian stability, and
possible wrinkling behaviour of a voltage-activated dielectric plate immersed
in a tank filled with silicone oil. Fixed rigid electrodes are placed on the
top and bottom of the tank, and an electric field is generated by a potential
difference between the electrodes. We solve the associated incremental boundary
value problem of superimposed, inhomogeneous small-amplitude wrinkles,
signalling the onset of instability. We decouple the resulting bifurcation
equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric
plate, we show that a potential difference between the electrodes can induce a
thinning of the plate and thus an increase of its planar area, similar to the
scenarios encountered when there is no silicone oil. However, we also find
that, depending on the material and geometric parameters, an increasing applied
voltage can also lead to a thickening of the plate, and thus a shrinking of its
area. In that scenario, Hessian instability and wrinkling bifurcation may then
occur spontaneously once some critical voltages are reached.
