On the elastic moduli of three-dimensional assemblies of spheres: characterization and modeling of fluctuations in the particle displacement and rotation

TL;DR

This study investigates how the elastic properties of random sphere packings depend on preparation methods, analyzing particle displacement and rotation fluctuations to improve macroscopic modeling of their elastic moduli.

Contribution

It introduces a detailed analysis of displacement and rotation fluctuations in sphere packings, linking microscopic contact behavior to macroscopic elastic properties.

Findings

Elastic moduli vary with preparation history and coordination number.

Displacement fluctuations counteract normal strain, rotations influence tangential response.

Fluctuations with short correlation lengths significantly affect macroscopic behavior.

Abstract

The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. The four samples are assembled with different preparation procedures, two of which aim to reproduce experimental compaction by vibration and lubrication. The mechanical properties of the samples are found to change with the preparation history, and to depend much more on coordination number than on density. Secondly, the fluctuations in the particle displacements from the average strain are analysed, and the way they affect the macroscopic behavior analyzed. It is found that only the average over equally oriented contacts of the relative displacement these fluctuations induce is relevant at the macroscopic scale. This average depends on coordination number, average geometry of the contact network and average contact stiffness. As far as the separate contributions from particle displacements and rotations are concerned, the former is found to counteract the average strain along the contact normal, while the latter do in the tangential plane. Conversely, the tangential components of the center displacements mainly arise to enforce local equilibrium, and have a small, and generally stiffening effect at the macro-scale. Finally, the fluctuations and the shear modulus that result from two approaches available in the literature are estimated numerically. These approaches are both based on the equilibrium of a small-sized representative assembly. The improvement of these estimate with respect to the average strain assumption indicates that the fluctuations relevant to the macroscopic behavior occur with short correlation length.