Static flexoelectric effect in a finite sample of a solid is addressed in
terms of phenomenological theory for the case of a thin plate subjected to
bending. It has been shown that despite an explicit asymmetry inherent to the
bulk constitutive electromechanical equations which take into account the
flexoelectric coupling, the electromechanical response for a finite sample is
"symmetric". "Symmetric" means that if a sensor and an actuator are made of a
flexoelectric element, performance of such devices can be characterized by the
same effective piezoelectric coefficient. This behavior is consistent with the
thermodynamic arguments offered earlier, being in conflict with the current
point of view on the matter in literature. This result was obtained using
standard mechanical boundary conditions valid for the case where the
polarization vanishes at the surface. It was shown that, for the case where
there is the polarization is nonzero at the surface, the aforementioned
symmetry of electromechanical response may be violated if standard mechanical
boundary conditions are used, leading to a conflict with the thermodynamic
arguments. It was argued that this conflict may be resolved when using modified
mechanical boundary conditions. It was also shown that the contribution of
surface piezoelectricity to the flexoelectric response of a finite sample is
expected to be comparable to that of the static bulk contribution (including
the material with high values of the dielectric constant) and to scale as the
bulk value of the dielectric constant (similar to the bulk contribution). This
finding implies that if the experimentally measured flexoelectric coefficient
scales as the dielectric constant of the material, this does not imply that the
measured flexoelectric response is controlled by the static bulk contribution
to the flexoelectric effect
