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.

翻译：本文提出了一种高阶广义周期性边界条件的构建方法，应用于包含挠曲电效应、应变梯度弹性与梯度介电性的高阶电-力学理论框架，旨在研究周期性超构材料。该理论产生四阶控制偏微分方程组，其周期性条件要求原始场（位移和电势）及其法向导数在周期边界上的连续性，以及相应广义对偶力（牵引力、双牵引力、面电荷密度与双面电荷密度）的连续性。我们并未将此类条件作为显式约束进行数值施加，而是构造了一个通过设计自动满足广义周期性的近似空间。该方法能够自然地在任意方向施加宏观场（应变/应力及电场/电位移），从而表征材料各向异性。为验证该方法的有效性，我们将其应用于具有表观压电性的周期性超构材料研究：首先通过与大型周期性结构的直接比较进行验证，其次评估几何极化二维晶格的各向异性表观压电性，最后展示该方法在三维超构材料中的应用。