The literature in the field of higher-order homogenization is mainly focused on 2-D models aimed at composite materials, while it lacks a comprehensive model targeting 3-D lattice materials (with void being the inclusion) with complex cell topologies. For that, a computational homogenization scheme based on Mindlin (type II) strain gradient elasticity theory is developed here. The model is based on variational formulation with periodic boundary conditions, implemented in the open-source software FreeFEM to fully characterize the effective classical elastic, coupling, and gradient elastic matrices in lattice materials. Rigorous mathematical derivations based on equilibrium equations and Hill–Mandel lemma are provided, resulting in the introduction of macroscopic body forces and modifications in gradient elasticity tensors which eliminate the spurious gradient effects in the homogeneous material. The obtained homogenized classical and strain gradient elasticity matrices are positive definite, leading to a positive macroscopic strain energy density value—an important criterion that sometimes is overlooked. The model is employed to study the size effects in 2-D square and 3-D cubic lattice materials. For the case of 3-D cubic material, the model is verified using full-field simulations, isogeometric analysis, and experimental three-point bending tests. The results of computational homogenization scheme implemented through isogeometric simulations show a good agreement with full-field simulations and mechanical tests. The developed model is generic and can be used to derive the effective second-grade continuum for any 3-D architectured material with arbitrary geometry. However, the identification of the proper type of generalized continua for the mechanical analysis of different cell architectures is yet an open question.

高阶均质化领域的文献主要集中在针对复合材料的二维模型，而缺乏针对具有复杂单元拓扑的3维晶格材料（包含空隙）的综合模型。为此，这里开发了一种基于 Mindlin（II 型）应变梯度弹性理论的计算均质化方案。该模型基于具有周期性边界条件的变分公式，在开源软件 FreeFEM 中实现，以充分表征晶格材料中的有效经典弹性矩阵、耦合矩阵和梯度弹性矩阵。提供了基于平衡方程和希尔-曼德尔引理的严格数学推导，导致宏观体力的引入和梯度弹性张量的修改，从而消除均质材料中的寄生梯度效应。获得的均质经典弹性矩阵和应变梯度弹性矩阵是正定的，导致宏观应变能密度值为正——这是一个有时被忽视的重要标准。该模型用于研究 2-D 方格和 3-D 立方晶格材料的尺寸效应。对于 3-D 立方材料的情况，使用全场模拟、等几何分析和实验三点弯曲测试来验证模型。通过等几何模拟实现的计算均质化方案的结果显示出与全场模拟和力学测试良好的一致性。开发的模型是通用的，可用于导出任何具有任意几何形状的 3D 建筑材料的有效二级连续体。然而，确定用于不同细胞结构的力学分析的广义连续体的正确类型仍然是一个悬而未决的问题。