  We present a formulation for high-order generalized periodicity conditions in
the context of a high-order electromechanical theory including
flexoelectricity, strain gradient elasticity and gradient dielectricity, with
the goal of studying periodic architected metamaterials. Such theory results in
fourth-order governing partial differential equations, and the periodicity
conditions involve continuity across the periodic boundary of primal fields
(displacement and electric potential) and their normal derivatives, continuity
of the corresponding dual generalized forces (tractions, double tractions,
surface charge density and double surface charge density). Rather than imposing
these conditions numerically as explicit constraints, we develop an
approximation space which fulfils generalized periodicity by construction. Our
method naturally allows us to impose general macroscopic fields
(strains/stresses and electric fields/electric displacements) along arbitrary
directions, enabling the characterization of the material anisotropy. We apply
the proposed method to study periodic architected metamaterials with apparent
piezoelectricity. We first verify the method by directly comparing the results
with a large periodic structure, then apply it to evaluate the anisotropic
apparently piezoelectricity of a geometrically polarized 2D lattice, and
finally demonstrate the application of the method in a 3D architected
metamaterial.
