Size-dependent piezoelectricity
  In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric solids. This theory shows that electric polarization can be generated as the result of coupling to the mean curvature tensor, unlike previous flexoelectric theories that postulate such couplings with other forms of curvature and more general strain gradient terms ignoring the possible couple- stresses. The present formulation represents an extension of recent work that establishes a consistent size-dependent theory for solid mechanics. Here by including scale-dependent measures in the energy equation, the general expressions for force- and couple-stresses, as well as electric displacement, are obtained. Next, the constitutive relations, displacement formulations, the uniqueness theorem and the reciprocal theorem for the corresponding linear small deformation size-dependent piezoelectricity are developed. As with existing flexoelectric formulations, one finds that the piezoelectric effect can also exist in isotropic materials, although in the present theory the coupling is strictly through the skew-symmetric mean curvature tensor. In the last portion of the paper, this isotropic case is considered in detail by developing the corresponding boundary value problem for two dimensional analyses and obtaining a closed form solution for an isotropic dielectric cylinder. 
Ali R. Hadjesfandiari
physics.gen-ph