Generalized continuum mechanical theories such as second gradient elasticity can consider size and localization effects, which motivates their use for multiscale modeling of periodic lattice structures and metamaterials. For this purpose, a numerical homogenization method for computing the effective second gradient constitutive models of cubic lattice metamaterials in the infinitesimal deformation regime is introduced here. Based on the modeling of lattice unit cells as shear-deformable 3D beam structures, the relationship between effective macroscopic strain and stress measures and the microscopic boundary deformations and rotations is derived. From this Hill–Mandel condition, admissible kinematic boundary conditions for the homogenization are concluded. The method is numerically verified and applied to various lattice unit cell types, where the influence of cell type, cell size and aspect ratio on the effective constitutive parameters of the linear elastic second gradient model is investigated and discussed. To facilitate their use in multiscale simulations with second gradient linear elasticity, these effective constitutive coefficients are parameterized in terms of the aspect ratio of the lattices structures.

诸如第二梯度弹性之类的广义连续力学理论可以考虑尺寸和局部化效应，这促使它们用于周期性晶格结构和超材料的多尺度建模。为此，本文介绍了一种数值均化方法，用于计算无限小变形状态下立方晶格超材料的有效第二梯度本构模型。基于晶格晶胞作为可剪切变形的3D梁结构的建模，得出有效的宏观应变和应力测度与微观边界变形和旋转之间的关系。根据这种Hill-Mandel条件，可以得出均质化的运动学边界条件。该方法经过了数值验证，并适用于各种晶格单位晶格类型，研究并讨论了像元类型，像元大小和纵横比对线性弹性第二梯度模型的有效本构参数的影响。为了便于将其用于具有第二梯度线性弹性的多尺度模拟中，这些有效的本构系数根据晶格结构的纵横比进行参数设置。