Continuum model for converse flexoelectricity in a thin plate

Flexoelectricity - the effect where inhomogeneous polarization causes deformation - substitutes symmetry-forbidden piezoelectricity in nano-scale centrosymmetric materials. Its mathematical treatment in terms of continuum mechanics represents a challenge: correct models are subjects of debates during the past decades. In the present article we formulate a model with reasonable approximations for converse flexoelectric effect in a plate. Except for very specific boundary conditions the theory inevitably include nonclassical higher order terms. These are treated using variational calculus and are essentially reduced to a modification of the boundary conditions for the elastic variables. Analytical solutions are obtained for a circular plate with round electrodes, where deflections are proportional to the inverse square of plate thickness. For mechanically free boundary conditions at plate's edges deflection grows monotonically with the electrode radius. In contrast, a plate with clamped edge does not deform at all when fully electroded, an optimal electrode size is found rendering maximal deflection.