  Thin dielectric elastomers with compliant electrodes exhibit various types of
instability under the action of electromechanical loading. Guided by the
thermodynamically-based formulation of Fosdick and Tang (J. Elasticity 88,
255-297, 2007), here we provide an energetic perspective on the stability of
dielectric elastomers and we highlight the fundamental energetic divide between
voltage control and charge control. By using the concept of energy relaxation,
we describe wrinkling for neo-Hookean ideal elastomers, and we show that in
voltage control wrinkling is stable as long as the tension-extension inequality
holds, whereas wrinkling is always stable in charge control. We finally
illustrate some examples involving both homogeneous and inhomogeneous
deformations, showing that the type and hierarchy of instabilities taking place
in dielectric membranes can be tuned by suitable choices of the boundary
conditions.
