Macroscopic stress, couple stress and flux tensors derived through energetic equivalence from microscopic continuous and discrete heterogeneous finite representative volumes

TL;DR

This work develops a rigorous, finite-size representative-volume framework to derive macroscopic fields in heterogeneous systems with independent displacements and rotations, using energetic equivalence in Cosserat continua. It furnishes two equivalent macroformulations—one based on external actions and one on internal actions—for the macroscopic stress $\bm{\upsigma}^{\mathrm{mac}}$, couple stress $\bm{\upmu}^{\mathrm{mac}}$, and flux $\mathbf{a}^{\mathrm{mac}}$, including a novel term in the couple stress that accounts for the macroscopic point location. The methodology is validated through Poisson and mechanical benchmarks in steady and transient regimes, demonstrating consistency with Love-Weber discretizations and highlighting the boundary-radius gap as a key correction in coarse discretizations. The results bridge continuous Cosserat homogenization and discrete-element representations, offering practical formulas for finite RVEs and broad applicability to various discrete and continuum microstructures.

Abstract

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.

Figures (15)

• Figure 1: Static variables of Cosserat continuum in two dimensions at macro and microscales.
• Figure 2: Components of displacement fields according to Eqs. \ref{['eq:virtkinematics_stretching']} and \ref{['eq:virtkinematics_bending']}: $\alpha$ modes account for stretching and shearing with zero rotations, $\beta$ modes describe curvatures and torsions for which the rotations are not visualized.
• Figure 3: Discrete element geometry, internal and external forces, couples and sink/sources in two dimensions.
• Figure 4: Components of virtual displacement fields in 2D: $\alpha$ modes account for stretching and shearing with zero rotations, $\beta$ modes describe local rotations and curvatures.
• Figure 5: Schematic description of two variants (with or without boundary-radius gap term) for evaluation of macroscopic flux in the discrete system: a) the control volume is dictated by a square bin depicted by dashed line, b) the control volume is a single triangle of weighted Delaunay triangulation.