Coherent Force Chains in Disordered Granular Materials Emerge from a Percolation of Quasilinear Clusters

Dense granular materials and other particle aggregates transmit stress in a manner that belies their microstructural disorder. A subset of the particle contact network is strikingly coherent, wherein contacts are aligned nearly linearly and transmit large forces. Important material properties are associated with these force chains, but their origin has remained a puzzle. We classify subnetworks by their linear connectivity, and show the emergence of a percolation transition at a critical linearity at which the network is sparse, coherent, and contains the force chains. The subnetwork at critical linearity closely reflects the macroscopic stress and explains distinctive features of granular mechanics.

Figure 1

Definition of cluster linearity. The linearity of the blue triplet is $r_t={\mathit{n}}_1·{\mathit{n}}_2$. The linearity $r$ of the cluster is the minimum of $r_t$ over all triplets in a connected cluster [Eq. (1)]. The end particles A and B are either at the boundaries, or in triplets of $r_t\le 0$.

Figure 2

Linearity percolation. (a) Percolation probability $P(r)$ for 2D isotropic compression for different system sizes $L$ (in units of $d_p$), the inset showing the collapse for different system sizes when the abscissa is plotted as $(r-r_c)N^{1/d\nu }$. The dashed line marks the value of $r_c$, which here is 0.609; it is in general a function of the area fraction. (b) Finite size scaling of the mean cluster size $S(r)$. (c) Percolation probability $P(r)$ for anisotropic forcing, in two and three dimensions. The geometries and boundary conditions are described in Sec. III of the Supplemental Material [24].

Figure 3

Statistics of contact forces in subnetworks of different linearity. (a) Spatial correlation function of the contact force $C_F(\ell )$ in the vicinity of critical linearity; the red line is the power law fit $a{\ell }^b$ at critical linearity ($a=0.11$, $b=-0.65$). (b) Variation of average contact force with linearity $r$, for random seeds (black squares) and seeds with a minimum contact force of the global mean $⟨F_n⟩$ (green circles). The radius of gyration $R_g$ of nonpercolating clusters is also shown. The dashed line marks critical linearity $r_c=0.609$. The results in $\mathbf{a}$ and $\mathbf{b}$ are for 2D isotropic compression with system size $L=100d_p$. (c) Probability distribution of the incremental normal contact force due to a point force $F_L$ acting on the surface of a 2D rectangular bed under gravity (see Sec. III. D in the Supplemental Material [24]), in clusters of different linearity; here $r_c$ is 0.66.

Figure 4

Clusters of critical linearity reflect the macroscopic stress. (a) Orientation distributions of clusters of different linearity, $P_{\mathrm{cl}}(\theta ,r)$, and the pair contact vector for different forcing: (a) Isotropic compression ($r_c=0.6089$), (b) uniaxial compression ($r_c=0.63$), and (c) plane shear ($r_c=0.62$). In (b) and (c) the red dashed lines indicate the principal directions of compression. (d) Schematic of a 3D granular column; the column is static when the velocity $V_w$ is zero, and it is sheared in the horizontal direction when $V_w$ is nonzero. (e),(f) Variation of the mean critical cluster angle [see Fig. S2.3(b) [24] ] and axial shear stress with depth $z$ in a 3D granular column of lateral dimension $2W=20d_p$ when the column is, respectively, static, and sheared in the $y$ direction. Here, $\overline{z}\equiv z/(2W)$ and ${\overline{\sigma }}_{yz}\equiv {\sigma }_{yz}/(\rho gW)$ are the scaled depth and vertical shear stress, where $\rho$ is the bulk density of the granular material in the column, and $g$ the gravitational acceleration.