We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. The relaxed micromorphic model is a generalized continuum that utilizes the $\Curl$ of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature...

翻译：我们通过最小二乘拟合方法，确定了给定周期性微结构对应的松弛微形态广义连续介质模型的材料参数。具体而言，我们将松弛微形态均质连续体的总能量与经典线弹性理论控制的完全解析非均质微结构的总能量进行拟合。松弛微形态模型是一种广义连续介质模型，其采用微畸变场的$\Curl$算子而非经典微形态理论中的全梯度算子，这带来了若干优势与差异。其中最关键的优点在于，该模型在精确界定的两个尺度之间运行，这两个尺度由微观和宏观弹性张量的线弹性决定，分别从上下界约束松弛微形态连续体的刚度。宏观弹性张量可通过标准周期一阶均匀化方法预先确定，但微观弹性张量仍需求解。此外，与曲率测量相关的特征长度参数控制着微-宏观尺度间的过渡。本文通过基于能量最小二乘拟合的计算方法，同时确定了微观弹性张量和特征长度参数。该过程需考虑足够数量的二次变形模态和不同尺寸的试件。我们对松弛微形态模型、斜对称微畸变场（Cosserat-微极模型）以及采用两种不同曲率公式的经典微形态模型的拟合结果进行了比较分析……