Effect of size polydispersity versus particle shape in dense granular media

We present a detailed analysis of the morphology of granular systems composed of frictionless pentagonal particles by varying systematically both the size span and particle shape irregularity, which represent two polydispersity parameters of the system. The microstructure is characterized in terms of various statistical descriptors such as global and local packing fractions, radial distribution functions, coordination number, and fraction of floating particles. We find that the packing fraction increases with the two parameters of polydispersity, but the effect of shape polydispersity for all the investigated structural properties is significant only at low size polydispersity where the positional and/or orientational ordering of the particles prevail. We focus in more detail on the class of side/side contacts, which is the interesting feature of our system as compared to a packing of disks. We show that the proportion of such contacts has weak dependence on the polydispersity parameters. The side- side contacts do not percolate but they define clusters of increasing size as a function of size polydispersity and decreasing size as a function of shape polydispersity. The clusters have anisotropic shapes but with a decreasing aspect ratio as polydispersity increases. This feature is argued to be a consequence of strong force chains (forces above the mean), which are mainly captured by side-side contacts. Finally, the force transmission is intrinsically multiscale, with a mean force increasing linearly with particle size.

Figure 1

Two types of contact between two polygonal grains: (1) side-vertex contact (left) represented by a single contact point and a unit normal vector perpendicular to the side, and (2) side-side contact (right) represented by two contact points with their parallel unit normal vectors.

Figure 2

Example of a regular pentagon $\delta =0$ transformed into irregular pentagons for two values of the parameter $\delta$ (see text).

Figure 3

Snapshots of dense packings at the end of compression: (a) $(s,\delta )=(0.01,0)$, (b) $(s,\delta )=(0.9,0)$, (c) $(s,\delta )=(0.01,1)$, and (d) $(s,\delta )=(0.9,1)$.

Figure 4

Packing fraction ${\rho }_m$ as a function of size span $s$ for all values of shape polydispersity $\delta$.

Figure 5

Examples of crystalline or quasicrystalline cores in a nearly monodisperse packing of regular pentagons: (A) (red) densest configuration (packing fraction $\simeq$92) involving both positional and orientational order of the pentagons; (B) (orange) “slipped” or slightly randomized examples of the fully ordered configuration shown in the bottom-left inset with a packing fraction $\simeq$0.85; (C) (gray) typical local environment of the pentagons characterized by a local triangular positional symmetry and random orientations of neighboring particles.

Figure 6

Variation of the ratio ${\rho }_{\max }/{\rho }_{\min }$ as a function of the normalized grid spacing for $\delta =0$ and $\delta =1$ (inset).

Figure 7

Radial distributions $g$ as a function of radial distance $r$ normalized by the average particle diameter $\langle d\rangle$ for $\delta =0$ (a) and $\delta =1$ (b) for all values of size span $s$. The inset of (a) shows $g\left(r\right)$ for $s=0.01$ at all values of $\delta$. The inset of (b) shows $g\left(r\right)$ at $s=0.9$ for all values of $\delta$.

Figure 8

Radial volume fraction distribution $\rho$ normalized by the average packing fraction as a function of radial distance $r$ for $\delta =0$ (a) and for $\delta =1$ (b) for all values of size span $s$. The inset of (a) shows $\rho$ at $s=0.01$ for all values of $\delta$. The inset of (b) shows $\rho$ at $s=0.9$ for all values of $\delta$.

Figure 9

Grey level map of particle connectivities and its scale bar for $(s=0.01,\delta =0)$ (a) and $(s=0.9,\delta =1)$ (b). Floating particles are in white and the grey level is proportional to coordination number.

Figure 10

Proportion $P_c$ of particles having $c$ contact neighbors as a function of $s$ for all values of $\delta$.

Figure 11

Coordination number $Z$ as a function of $s$ for all values of $\delta$.

Figure 12

Proportion $k_{ss}$ of side-side contacts as a function of $s$ for all values of $\delta$.

Figure 13

Linear correlation between the coordination number $Z$ and reduced particle size $d_r$.

Figure 14

Proportion $P_f$ of floating particles as a function of $s$ for all values of $\delta$.

Figure 15

Clusters of particles bonded by at least one side-side contact for $(s,\delta )=(0.01,0)$ (a), $(s,\delta )=(0.9,0)$ (b), $(s,\delta )=(0.01,1)$ (c), and $(s,\delta )=(0.9,1)$ (d). Disjoint clusters are represented in different colors (green, orange, light slate gray, and cardinal red). The side-side contacts are marked by a thick black line joining the centers of their partner particles. The floating particles are in white.

Figure 16

Number $N_c$ of clusters as a function of $s$ for all values of $\delta$. The inset shows the mean number ${\langle N_p\rangle }_c$ of particles per cluster as a function of $s$.

Figure 17

(a) Average values of the longest and shortest dimensions of the clusters normalized by the largest particle diameter $d_{\max }$ (a) and their mean aspect ratio $\langle a\rangle$ (b) as a function of $s$ for all values of $\delta$.

Figure 18

(a) Average side-side force $\langle f_{ss}\rangle$ and side-vertex force $\langle f_{sv}\rangle$ as a function of $s$ for all values of $\delta$. (b) The same data normalized by the mean force $\langle f\rangle$ in each sample.

Figure 19

A snapshot of the contact force network for $(s,\delta )=(0.01,0)$ (a) and $(s,\delta )=(0.9,1)$ (b). The normal forces are represented by the thickness of the segments joining particle centers. The clusters of side-side contacts are also shown in different colors.

Figure 20

Mean force $\langle f_d\rangle$ as a function of reduced particle size $d_r$ for all values of $s$ and two values of $\delta$.