Topological and geometric measurements of force-chain structure

Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts into a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Recent work applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force-chain structure, with the goal of identifying geometric and topological differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here we discuss a trio of related but fundamentally distinct measurements of the mesoscale structure of force chains in two-dimensional (2D) packings, including a statistic derived using tools from algebraic topology, which together provide a tool set for the analysis of force chain architecture. We demonstrate the utility of this tool set by detecting variations in force-chain architecture with pressure. Collectively, these techniques can be generalized to 3D packings, and to the assessment of continuous deformations of packings under stress or strain.

Figure 1

Community detection techniques extract force chains from granular media. (a) An image of the single-layer photoelastic disk experiment. Particles are confined in two dimensions and are vertically compressed via the application of brass weights to the top of the configuration. (b) The internal stress pattern in the photoelastic disks exhibits force-chain structure. Brightness indicates the strength of forces between particles. (c) The corresponding weighted graph overlaid on the photoelastic image, where the thickness of the line segments (edges) are proportional to the normal force between the two particles they connect (nodes). (d) Community structure extracted from the force-weighted network as a function of the resolution parameter $\gamma$ in the modularity quality function. Colors indicate distinct communities, and warmth corresponds to network force, the amount by which the total interparticle forces exceed $\gamma$ times the mean interparticle force.

Figure 2

Three metrics of force-chain structure. (a) The gap factor of a force chain measures the discrepancy between the topology of the contact network of a force chain and its physical layout. Force chains for which the two are poorly correlated display branching or curving behavior. (b) The hull ratio of a force chain is the ratio of the area of the constituent particles to those of its convex hull. Curved or branched chains (top) have low hull ratio, while straight or clustered chains (bottom) have a high hull ratio. (c) The topological compactness factor (TCF) measures mesoscale connectivity properties of the contact graph that contain physically relevant information. (Left) Binary contact graph for a force chain. (Middle) A compact region (gray triangle) is a cluster of particles in the chain which are connected to all of their neighbors in a triangular grid, while branch points (red) are particles with more than two neighbors, none of which share an edge. (Right) After collapsing the leaves (green) in the contact graph to a single vertex, we can enumerate fundamental cycles (purple, yellow, blue, orange) in the graph to measure the prevalence of compact regions versus branch points of the chain.

Figure 3

Comparisons of hull ratio and TCF to gap factor. Scatter plots of (a) hull ratio and (b) TCF against the gap factor. Points correspond to the mean of each measure across optimizations of the modularity quality function for each packing, each pressure, and choice of the resolution parameter between 0.1 and 2.1. The hull ratio is strongly correlated with the gap factor (Spearman's $\rho =-0.72$), while the TCF is more weakly correlated with the gap factor ($\rho =-0.37$). The striped regions in (a) correspond to packings with small numbers of communities, and are explained in detail in the text.

Figure 4

Measurements of force-chain structure across resolutions. (a) The systemic gap factor as a function of resolution parameter $\gamma$ across varying pressure on laboratory and (b) simulated packings. Curves are means across trials, and error bars indicate one standard deviation. The maxima of the gap factor correspond to the optimal resolution parameters for identifying community structure reminiscent of force chains; but in both the laboratory and simulated packings, the gap factor is relatively homogeneous across pressure. (c) The weighted hull ratio as a function of resolution parameter $\gamma$ and pressure for the laboratory and (d) simulated packings. The optimal $\gamma$ is that which minimizes $R_s$, and agrees relatively well with the optimal $\gamma$ found from the gap factor in both experiments and simulations. However, the laboratory weighted hull ratio can distinguish pressure across packings for large values of $\gamma >1.3$. The simulated hull ratio also partially distinguishes pressures for high values of $\gamma$, but is less well differentiated. (e) The systemic topological compactness factor $T_s$ as a function of the resolution parameter $\gamma$ for the laboratory and (f) simulated packings. $T_s$ provides a strong discrimination of packing pressures at the optimal resolution obtained from the hull ratio. The high values of the TCF for $\gamma \approx 0.1$ is the result of the small number of compact communities which rapidly fragment as $\gamma$ increases.

Figure 5

Community structure for the experimental packings at the optimal and pressure-stratifying resolution parameters. (a) Near the optimal resolution parameter, found by maximizing the gap factor or minimizing the hull ratio, the communities have force-chain structure. (b) For larger values of $\gamma$, communities are small and sparsely distributed. However, the hull ratio indicates that their structure is reflective of packing pressure. (c) The topological compactness factor $T_s$ best distinguishes packings by pressure at $\gamma \approx 1$, where the communities are long and branch irregularly and are also reminiscent of force chains.