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

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.

Structural evolution of granular systems: theory

“…Such time dependence is expected because the contact event rate is likely to be sensitive to the intergranular force distribution, which is position-and time-dependent. Indeed, it has been argued [10,19] that the structural evolution of a granular system is a self-organisation process coupled to the evolution of the intergranular forces. Therefore, these equations and their extension form an important step towards a complete understanding of the structure-forces co-evolution.…”

“…To confirm that clapping is a genuine phenomenon and study the effect of the driving on it, we analysed a long-time DEM simulation of a bi-axial compression. The simulated system consisted of 22,381 discs, whose diameters were distributed log-normally [9,10]. The disc interaction was harmonic, with tangential and normal spring constants k t and k n , respectively, obeying k t ∕k n = 1∕4 .…”

“…The contact network is a graph containing cells, which are the smallest voids enclosed by grains (aka irreducible loops), and they are characterised by the number of grains enclosing it-its order. We develop here a theory for the evolution of the cell order distribution (COD), which has been argued [8] and shown [9,10] to converge to a universal form. This theory can be extended to model other structural descriptors.…”

Statistical properties of cell stresses in 2D granular solids

“…With this definition, the total volume of the system is a sum of all the cell volumes, . This definition differs from that in Matsushima and Blumenfeld [10,11], where a cell is defined by the shortest (irreducible) closed loop of contact points around a void. Nevertheless, this difference is of no major consequence in disc systems.…”

“…Nicot and Darve [9] proposed a constitutive model, focusing on 6-cell structures. Matsushima and Blumenfeld [10,11] studied the structural properties of cells in details and showed that the cell shape is elongated the most when the cell order is around six, because elongated cells of higher cell order are less stable mechanically. Recently, Wanjura et al [12] developed a model for cell structure evolution, based on the statistics of cell making and breaking during quasi-static dynamics.…”