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Fundamental structural characteristics of planar granular assemblies: Self-organization and scaling away friction and initial state

Takashi Matsushima1,* and Raphael Blumenfeld2,3,4,†

  • 1Department of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan
  • 2Imperial College London, London SW7 2AZ, United Kingdom
  • 3National University of Defense Technology, Changsha 410073, Hunan, China
  • 4Cavendish Laboratory, Cambridge University, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom

  • *tmatsu@kz.tsukuba.ac.jp
  • †rbb11@cam.ac.uk
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Phys. Rev. E 95, 032905 – Published 20 March, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.032905

Abstract

The microstructural organization of a granular system is the most important determinant of its macroscopic behavior. Here we identify the fundamental factors that determine the statistics of such microstructures, using numerical experiments to gain a general understanding. The experiments consist of preparing and compacting isotropically two-dimensional granular assemblies of polydisperse frictional disks and analyzing the emergent statistical properties of quadrons—the basic structural elements of granular solids. The focus on quadrons is because the statistics of their volumes have been found to display intriguing universal-like features [T. Matsushima and R. Blumenfeld, Phys. Rev. Lett. 112, 098003 (2014)]. The dependence of the structures and of the packing fraction on the intergranular friction and the initial state is analyzed, and a number of significant results are found. (i) An analytical formula is derived for the mean quadron volume in terms of three macroscopic quantities: the mean coordination number, the packing fraction, and the rattlers fraction. (ii) We derive a unique, initial-state-independent relation between the mean coordination number and the rattler-free packing fraction. The relation is supported numerically for a range of different systems. (iii) We collapse the quadron volume distributions from all systems onto one curve, and we verify that they all have an exponential tail. (iv) The nature of the quadron volume distribution is investigated by decomposition into conditional distributions of volumes given the cell order, and we find that each of these also collapses onto a single curve. (v) We find that the mean quadron volume decreases with increasing intergranular friction coefficients, an effect that is prominent in high-order cells. We argue that this phenomenon is due to an increased probability of stable irregularly shaped cells, and we test this using a herewith developed free cell analytical model. We conclude that, in principle, the microstructural characteristics are governed mainly by the packing procedure, while the effects of intergranular friction and initial states are details that can be scaled away. However, mechanical stability constraints suppress slightly the occurrence of small quadron volumes in cells of order ≥6, and the magnitude of this effect does depend on friction. We quantify in detail this dependence and the deviation it causes from an exact collapse for these cells. (vi) We argue that our results support strongly the view that ensemble granular statistical mechanics does not satisfy the uniform measure assumption of conventional statistical mechanics. Results (i)–(iv) have been reported in the aforementioned reference, and they are reviewed and elaborated on here.

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Outline

Information

Phys. Rev. E 95, 032905– Published 20 March, 2017
  • Received 3 April 2016
  • Revised 11 February 2017

©2017 American Physical Society

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