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

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 $\ge 6$, 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.

Figure 1

Illustration of a quadrilateral quadron (shaded), whose diagonals are ${\vec{R}}^q$ and ${\vec{r}}^q$. The intercontact vectors ${\vec{r}}^q$ circulate anticlockwise around grains.

Figure 2

Coordination number vs packing fraction for 15 systems, generated from three different initial states, LIS, IIS, and DIS. Note the convergence for $\mu \to 0$.

Figure 3

Example of an assembly with $\mu =10$, generated from a LIS. Some of the cells contain rattlers, defined as disks that have at most one contact.

Figure 4

Example of an assembly with $\mu =0.01$, generated from LIS.

Figure 5

Example of an assembly with $\mu =10$, generated from DIS.

Figure 6

PDF of the quadron volumes, normalized by the mean grain volume, $v=V_q/{\overline{V}}_g$, for all the 10 systems generated from LIS and DIS.

Figure 7

Mean quadron volume $\overline{v}$ vs mean coordination number $\overline{z}$ for all initial states.

Figure 8

Mean coordination number $\overline{z}$ vs the rattlers-free packing fraction ${\varphi }^{\prime }$.

Figure 9

The rattlers fraction ${\eta }_r$ vs the mean coordination number $\overline{z}$.

Figure 10

The PDF of the normalized quadron volume $u=V_q/{\overline{V}}_q$ for all the 10 systems generated from LIS and DIS.

Figure 11

The cell-order probability $Q\left(e\right)$ for the 10 systems generated from LIS and DIS.

Figure 12

The mean cell order $\overline{e}$ vs the mean coordination number $\overline{z}$.

Figure 13

The normalized cell probability $\overline{e}eQ\left(e\right)$. It is fitted well by a Gaussian form, except for small, but consistent, deviations in the large-$e$ tail (highlighted by a dash-lined ellipse).

Figure 14

The conditional PDFs of the quadron volumes for $e\le 5$ collapse nicely for all the $\mu$ and initial conditions.

Figure 15

The conditional PDFs of the quadron volumes for $e=6,8$ collapse, but not as sharply as those for $e\le 5$ (Fig. 14). This is a result of the under-representation of mechanically unstable elongated large cells, which suppresses the occurrence of small volume quadrons.

Figure 16

The variation of the mean conditional quadron volume, $\overline{v}$, with cell order $e$. The quadron volume of the regular polygon cell (dashed line) forms an upper bound.

Figure 17

The ratio of the mean quadron volume to the quadron volume of a regular polygon, $\gamma$ ($<1$), plotted as a function of cell order $e$. Note that the FCM describes well the geometric effect leading to the decrease in $\gamma$ for $e<5$, but it cannot show the increase at larger values since it does not include the effects of mechanical stability.

Figure 18

The construction of a sample cell configuration generated by the free cell model.

Figure 19

Examples of free cell configurations for $e=6$.

Figure 20

A comparison of the PDF of quadron volumes, generated both with the free cells model and with the DEM numerical experiments.

Figure 21

Simple prediction of the mean quadron volume in terms of the mean coordination number.