Size-dependent theories of piezoelectricity: Comparisons and further   developments for centrosymmetric dielectrics
  Here the recently developed size-dependent piezoelectricity and the strain gradient theory of flexoelectricity are compared. In the course of this investigation, the strain gradient theory of flexoelectricity is shown to violate fundamental rules of mathematics, continuum mechanics and electromagnetism. The major difficulties are associated with ill-posed boundary conditions, the missing angular (moment) equilibrium equation and the appearance of a non-physical extraneous vectorial electrostatic law. Therefore, the strain gradient flexoelectricity must be classified as an inconsistent theory. The present investigation further reveals that the new size-dependent piezoelectricity is the more consistent theory to describe linear electromechanical coupling in dielectrics. Some new aspects of this theory are presented for isotropic and centrosymmetric cubic dielectric materials, whose coupling effect is described by only one parameter. 
Ali R. Hadjesfandiari
physics.class-ph