Topology of the force network in the jamming transition of an isotropically compressed granular packing

The jamming transition of an isotropically compressed granular packing is studied by means of molecular dynamics simulations. The system is shown to undergo a critical transition which is analyzed by looking at the topological structure of the force network. At the critical packing fraction there is a sudden growth of the number of polygons in the network. Above the critical packing fraction the number of triangles keeps growing while the number of the rest of polygons is weakly reduced. Then, we prove that in the jammed regime, there is a linear relationship between the number of triangles and the coordination number. Furthermore, the presence of these minimal structures is revealed to be connected with the evolution of some important topological properties, suggesting its importance to understand the physical properties of the packing and the onset of rigidity during the compression.

Figure 1

(Color online) Evolution of the granular packing during the compression process for $\varphi =0.596$ (Fig. 1), $\varphi =0.741$ (Fig. 2), $\varphi =0.829$ (Fig. 3), ${\varphi }_c=0.838$ (Fig. 4), $\varphi =0.843$ (Fig. 5), and $\varphi =0.849$ (Fig. 6). The magnitudes of the normal forces are proportional to the widths of the chains.

Figure 2

(Color online) (a) Coordination number versus the packing fraction $\left(\varphi \right)$ along the whole compaction process. The coordination number is calculated without taking into account rattlers, particles without contacts. (b) Average deformation $\left(\delta \right)$ rescaled with the radius of the greater disks $\left(R_1\right)$ versus the packing fraction. The continuous line is a linear fit.

Figure 3

(Color online) Excess coordination number versus the packing fraction above the critical point. The continuous line is a fit to $Z-Z_c\propto {\left(\varphi -{\varphi }_c\right)}^{\beta }$ with $\beta =0.33$, ${\varphi }_c=0.838$, and $Z_c=3.00$. The inset shows the same results in logarithmic scale.

Figure 4

(Color online) (a) Distribution of connectivities obtained for different values of $\varphi$ as indicated in the legend. (b) $l^{\ast }$, the average shortest path length normalized by $\sqrt{N}/2$ versus the packing fraction $\varphi$. The dotted lines indicate $l^{\ast }=1$ and $\varphi ={\varphi }_c$.

Figure 5

(Color online) Number of the different polygons versus $\varphi$. Inset: the same data in semilogarithmic scale. Different symbols are used for triangles (◻), squares (○), pentagons (△), hexagons (▽) and heptagons (◇). Triangles are the only polygons which grow in number after the transition.

Figure 6

(Color online) Number of triangles (NT) vs Z. Different symbols are used for three typical runs. The continuous line is a fit of the asymptotic linear dependence.

Figure 7

(Color online) A region of a compressed packing $\left(\varphi =0.892\right)$ showing the effect on the network of taking several values of the parameter $f$.

Figure 8

(Color online) Number of polygons in a highly packed configuration $\left(\varphi =0.892\right)$ as a function of the parameter $f$. The inset shows the same data in semilogarithmic scale. The legend indicates the symbols used for triangles, squares, pentagons, hexagons, and heptagons.

Figure 9

Clustering coefficient as a function of the parameter $f$ for the same highly packed configuration than the one presented in Fig. 8.

Figure 10

The excess of contacts EC obtained for a packing of $\varphi =0.892$ as a function of the parameter $f$.

Figure 11

(Color online) (a) Size of the giant component in the network (GC) resulting after applying distinct values of the force threshold $f$. GC is given in number of grains and normalized with the total number of grains in the sample $\left(N\right)$. (b) Average shortest path length normalized by $\sqrt{N}/2$ for values of $f$ going from 0 to 5. In both figures different symbols are used for different packing fractions as indicated in the legends.