Flexoelectricity - the generation of electric field in response to a strain gradient - is a universal electromechanical coupling, dominant only at small scales due to its requirement of high strain gradients. This phenomenon is governed by a set of coupled fourth-order partial differential equations (PDEs), which require $C^1$ continuity of the basis in finite element methods for the numerical solution. While Isogeometric analysis (IGA) has been proven to meet this continuity requirement due to its higher-order B-spline basis functions, it is limited to simple geometries that can be discretized with a single IGA patch. For the domains, e.g., architected materials, requiring more than one patch for discretization IGA faces the challenge of $C^0$ continuity across the patch boundaries. Here we present a discontinuous Galerkin method-based isogeometric analysis framework, capable of solving fourth-order PDEs of flexoelectricity in the domain of truss-based architected materials. An interior penalty-based stabilization is implemented to ensure the stability of the solution. The present formulation is advantageous over the analogous finite element methods since it only requires the computation of interior boundary contributions on the boundaries of patches. As each strut can be modeled with only two trapezoid patches, the number of $C^0$ continuous boundaries is largely reduced. Further, we consider four unique unit cells to construct the truss lattices and analyze their flexoelectric response. The truss lattices show a higher magnitude of flexoelectricity compared to the solid beam, as well as retain this superior electromechanical response with the increasing size of the structure. These results indicate the potential of architected materials to scale up the flexoelectricity to larger scales, towards achieving universal electromechanical response in meso/macro scale dielectric materials.

翻译：挠曲电效应——由应变梯度产生电场的现象——是一种普遍的机电耦合效应，仅在小尺度下占主导地位，因其需要高应变梯度。该现象由一组耦合的四阶偏微分方程（PDEs）描述，其在有限元数值求解中要求基函数具有$C^1$连续性。虽然等几何分析（IGA）因其高阶B样条基函数已被证明能满足此连续性要求，但其仅限于可用单一IGA贴片离散化的简单几何结构。对于需要多个贴片进行离散化的域（例如结构材料），IGA在贴片边界处面临$C^0$连续性的挑战。本文提出了一种基于间断伽辽金法的等几何分析框架，能够求解基于桁架的结构材料域中的挠曲电四阶偏微分方程。通过实施基于内部惩罚的稳定化方法以确保解的稳定性。相较于类似的有限元方法，本公式具有优势，因其仅需计算贴片边界上的内部边界贡献。由于每个杆件仅需两个梯形贴片即可建模，$C^0$连续边界的数量得以大幅减少。此外，我们考虑了四种独特的单胞来构建桁架晶格，并分析其挠曲电响应。与实心梁相比，桁架晶格表现出更高强度的挠曲电效应，并且随着结构尺寸增大，这种优异的机电响应得以保持。这些结果表明，结构材料具有将挠曲电效应提升至更大尺度的潜力，从而有望在介观/宏观尺度介电材料中实现普适的机电响应。