Do Discrete Fine-Scale Mechanical Models with Rotational Degrees of Freedom Homogenize Into a Cosserat or a Cauchy Continuum?

Abstract

This article answers the question of whether homogenization of discrete fine-scale mechanical models, such as particle or lattice models, gives rise to an equivalent continuum that is of Cauchy-type or Cosserat-type. The study employs the machinery of asymptotic expansion homogenization to analyze discrete mechanical models with rotational degrees of freedom commonly used to simulate the mechanical behavior of heterogeneous solids. The proposed derivation has general validity in both stationary (steady-state) and transient conditions (assuming wavelength much larger that particle size) and for arbitrary nonlinear, inelastic fine-scale constitutive equations. The results show that the unit cell problem is always stationary, and the only inertia term appears in the linear momentum balance equation at the coarse scale. Depending on the magnitude of the local bending stiffness, mathematical homogenization rigorously identifies two limiting conditions that correspond to the Cauchy continuum and the Cosserat continuum. A heuristic combination of these two limiting conditions provides very accurate results also in the transition from one limiting case to the other. Finally, the study demonstrates that cases for which the Cosserat character of the homogenized response is significant are associated with non-physically high fine-scale bending stiffness and, as such, are of no interest in practice.

Figures (8)

• Figure 1: Two rigid bodies $I$ and $J$ in contact; local reference system and vectors $\mathbf{r}_{C\!I}$ and $\mathbf{r}_{C\!J}$; tetrahedron for evaluation of volumetric strain.
• Figure 2: Example of a fine-scale model of size $d_c=100$ mm with periodic internal structure.
• Figure 3: Macroscopic elastic parameters computed from 150 fine-scale periodic models with different internal structure for each size. The upper row reports estimation of the mean value, the bottom row shows the standard deviation.
• Figure 4: Mean value estimates of coarse-scale elastic parameters computed from 150 fine-scale periodic models with different internal structure. Only size $d_c=0.2$ m and limiting case LC2 is shown.
• Figure 5: Structural bending stiffness of the full discrete model and homogenized models with different homogenization schemes. The averages are taken over 6 samples (full model) or 150 samples (homogenized model), respectively.