  We investigate the finite bending and the associated bending instability of
an incompressible dielectric slab subject to a combination of applied voltage
and axial compression, using nonlinear electro-elasticity theory and its
incremental version. We first study the static finite bending deformation of
the slab. We then derive the three-dimensional equations for the onset of
small-amplitude wrinkles superimposed upon the finite bending. We use the
surface impedance matrix method to build a robust numerical procedure for
solving the resulting dispersion equations and determining the wrinkled shape
of the slab at the onset of buckling. Our analysis is valid for dielectrics
modeled by a general free energy function. We then present illustrative
numerical calculations for ideal neo-Hookean dielectrics. In that case, we
provide an explicit treatment of the boundary value problem of the finite
bending and derive closed-form expressions for the stresses and electric field
in the body. For the incremental deformations, we validate our analysis by
recovering existing results in more specialized contexts. We show that the
applied voltage has a destabilizing effect on the bending instability of the
slab, while the effect of the axial load is more complex: when the voltage is
applied, changing the axial loading will influence the true electric field in
the body, and induce competitive effects between the circumferential
instability due to the voltage and the axial instability due to the axial
compression. We even find circumstances where both instabilities cohabit to
create two-dimensional patterns on the inner face of the bent sector.
