  This paper develops the equilibrium equations describing the flexoelectric
effect in soft dielectrics under large deformations. Previous works have
developed related theories using a flexoelectric coupling tensor of mixed
material-spatial character. Here, we formulate the model in terms of a
flexoelectric tensor completely defined in the material frame, with the same
symmetries of the small-strain flexocoupling tensor and leading naturally to
objective flexoelectric polarization fields. The energy potential and
equilibrium equations are first expressed in terms of deformation and
polarization, and then rewritten in terms of deformation and electric
potential, yielding an unconstrained system of fourth order partial
differential equations (PDEs). We further develop a theory of geometrically
nonlinear extensible flexoelectric rods under open and closed circuit
conditions, with which we examine analytically cantilever bending and buckling
under mechanical and electrical actuation. Besides being a simple and explicit
model pertinent to slender structures, this rod theory also allows us to test
our general theory and its numerical implementation using B-splines. This
numerical implementation is robust as it handles the electromechanical
instabilities in soft flexoelectric materials.
