Taxonomy of granular rheology from grain property networks

We construct complex networks from symbolic time series of particle properties within a dense quasistatically deforming granular assembly subjected to biaxial compression. The structure of the resulting networks embodies the evolving structural rearrangements in the granular material, in both contact forces and contact topologies. These rearrangements are usefully summarized through standard network statistics as well as building block motifs and community structures. Dense granular media respond to applied compression and shear by a process of self-organization to form two cooperatively evolving structures comprising the major load-bearing columnlike force chains, and the lateral trusslike three-cycle triangle topologies. We construct networks summarizing their individual evolution based on relationships between symbolic time series indicating a particle's chronological force chain and three-cycle membership histories. We test which particle membership histories are similar with respect to each other through the information theory-based measure of Hamming distance. The complex networks summarize the essential structural rearrangements, while the community structures within the networks partition the material into distinct zones of deformation, including interlacing subregions of failure inside the shear band. The taxonomy of granular rheology at the mesoscopic scale distills the inelastic structural rearrangements throughout loading history down to its core elements, and should lay bare an objective and physics-based formalism for thermodynamic internal variables and associated evolution laws.

Figure 1

(a) The shear stress-strain curve (solid blue) of the sample throughout loading and the number of three-particle buckled segments of force chains which have buckled by at least $1^{\circ }$ over a strain interval of $\Delta {\varepsilon }_{yy}=0.0015$ (dashed green). Observe precipitous drops in the shear stress coinciding with sudden bursts in the number of buckled force chain segments. (Inset) The displacement field of change in particle positions across $0.0381\le {\varepsilon }_{yy}\le 0.0458$ clearly identifying the shear band. (b) Evolution with axial strain of the particle population, expressed as a percentage of the total number of particles in the sample, that are part of a force chain (solid blue) and part of a three-cycle support (dashed green). Force chain particles using the detection algorithm in [14, 17] are approximately 36% of the particles in the sample, consistent with predictions in Radjai et al. [29, 30]. Observe sudden drops in the percentage of particles in three-cycle topologies as dilatation develops in the sample over $0.0346\le {\varepsilon }_{yy}\le 0.0361$ and then remains above 50% in the near-steady “critical state” regime. These trends are consistent with previous studies of three-cycle topologies from network metrics of clustering coefficient and subgraph centrality [12, 13].

Figure 2

The granular assembly where each particle is colored by their particle load vector magnitude. Low to high corresponds to color from cool (dark blue) to hot (light yellow), respectively: (a) onset of loading (${\varepsilon }_{yy}=0.00$); (b) after failure (${\varepsilon }_{yy}=0.04$). The extracted force chain network from the assembly (c) onset of loading (${\varepsilon }_{yy}=0.00$) (d) after failure (${\varepsilon }_{yy}=0.04$). At the onset of loading, the isotropic initial conditions result in a force chain network with no obvious directionality; however, as loading progresses, force chains increasingly align in the direction of maximum principal (most compressive) stress direction to resist the load. Observe that after failure some of the force chains appear buckled in the shear band region.

Figure 3

The sensor for a typical particle in the assembly recording its (a) force chain membership status, (b) three-cycle participation status, (c) conjoined force chain and three-cycle status. (d) A force chain at ${\varepsilon }_{yy}=0.057$. The particle load vector magnitude is represented by the particle color, from low (dark blue) to high (light yellow). The bottom force chain particle (green) is conjoined, as indicated, with two three-cycle topologies. Notice the particle's side neighbors in blue, two to its lower left and one to its upper right, all in three-cycle topologies, prop up the chain as well as frustrate relative rotation at contacts.

Figure 4

The three-cycle arrangements within the granular assembly where each particle is colored by their particle load vector magnitude (dark blue is low value and hotter colors to light yellow are higher valued). (a) Onset of loading (${\varepsilon }_{yy}=0.00$); (b) after failure (${\varepsilon }_{yy}=0.04$). Note that at the onset of loading the three-cycle population is high and they are densely located throughout the sample. After failure, the three-cycle population has significantly decreased and they are sparsely represented in the area of strain localization (shear band). The conjoined three-cycle and force chain particles; (c) onset of loading (${\varepsilon }_{yy}=0.00$); (d) after failure (${\varepsilon }_{yy}=0.04$).

Figure 5

Community structure of the reconstructed networks where particles are colored by community membership. (a) Hamming distance network of particles force chain status results in three communities; (b) three-cycle participation network using Hamming distance has 17 communities.

Figure 6

Community structure of the reconstructed networks where particles are colored by community membership. (a) Conjoined three-cycle and force chain status Hamming distance network resolves into five communities. The conjoined status time series network is detecting the interlacing of the two most dominant communities within the shear band in space (b) and time (c). The particles displayed in (b) for axial strain ${\varepsilon }_{yy}=0.1146$, situated approximately midlength of the shear band (rotated and zoomed in on for clarity), show a typical spatial distribution for the two most dominant communities in the band that in (c) show a typical pair of particle time series, one from each of these two communities (solid line from the dark blue community and dashed line from the light blue community), displaying out-of-phase cycles between the two states of jamming (conjoined status 1) and unjamming.

Figure 7

The ratio of particles in the conjoined status network communities of Figs. 6a and 6b (dark blue community 1 represented by solid blue line; light blue community 2 represented by dashed green line) which are (a) force chain particles, (b) members of three-cycle formations, (c) conjoined force chain and three-cycle particles. In all three figures the right ordinate scale refers to the dashed green curve and the left ordinate scale to all other curves. Observe that the ratio of particles in the conjoined force chain and three-cycle status for these two communities appears to oscillate out of phase with each other suggesting that the networks have detected the jamming-unjamming interplay of shear band dynamics. The ratios of the remaining communities show insignificant contributions to the force chain (a) and conjoined (c) series.

Figure 8

(a) The constitutive model in [19] is based on internal variables and evolution laws describing the nonaffine deformation of this idealized model of a structural rearrangement of force chain buckling, under lateral confinement from neighboring contacts initially in three-cycle formations. (b) This buckling process involves an evolution in both contact forces and contact topologies, here reducing the number of three-cycle formations that the central particle is part of from six to one. (c)–(d) Example of a force chain buckling event in the simulation, showing loss of energy (colors go from hotter light yellow to colder dark blue-green) and misalignment as well as changes in the lateral support of contacting neighbors.