This paper presents a rigorous derivation of equations to evaluate the macroscopic stress tensor, the couple stress tensor, and the flux vector equivalent to underlying microscopic fields in continuous and discrete heterogeneous systems with independent displacements and rotations. Contrary to the classical asymptotic expansion homogenization, finite size representative volume is considered. First, the macroscopic quantities are derived for a heterogeneous Cosserat continuum. The resulting continuum equations are discretized to provide macroscopic quantities in discrete heterogeneous systems. Finally, the expressions for discrete system are derived once again, this time considering the discrete nature directly. The formulations are presented in two variants, considering either internal or external forces, couples, and fluxes. The derivation is based on the virtual work equivalence and elucidates the fundamental significance of the couple stress tensor in the context of balance equations and admissible virtual deformation modes. Notably, an additional term in the couple stress tensor formula emerges, explaining its dependence on the reference system and position of the macroscopic point. The resulting equations are verified by comparing their predictions with known analytical solutions and results of other numerical models under both steady state and transient conditions.

翻译：本文提出了严格推导方程的方法，用于评估具有独立位移和旋转的连续与离散非均匀系统中宏观应力张量、偶应力张量及与底层微观场等效的通量矢量。与经典的渐近展开均匀化方法不同，本研究考虑了有限尺寸的代表性体积。首先，针对非均匀Cosserat连续体推导了宏观物理量。将所得连续体方程离散化，以提供离散非均匀系统中的宏观物理量。最后，再次推导离散系统的表达式，此次直接考虑了系统的离散特性。公式推导呈现为两种形式，分别基于内力或外力、力偶及通量。推导基于虚功等效原理，阐明了偶应力张量在平衡方程与容许虚变形模式背景下的基本意义。值得注意的是，偶应力张量公式中出现了一个附加项，解释了其对参考系和宏观点位置的依赖性。通过将方程预测结果与已知解析解及其他数值模型在稳态和瞬态条件下的结果进行比较，验证了所得方程的正确性。