  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.
