We introduce novel sets of measures with the goal of describing dynamical properties of force networks in dense particulate systems. The presented approach is based on persistent homology and allows for extracting precise, quantitative measures that describe the evolution of geometric features of the interparticle forces, without necessarily considering the details related to individual contacts between particles. The networks considered emerge from discrete element simulations of two dimensional particulate systems consisting of compressible frictional circular disks. We quantify the evolution of the networks for slowly compressed systems undergoing jamming transition. The main findings include uncovering significant but localized changes of force networks for unjammed systems, global (system-wide) changes as the systems evolve through jamming, to be followed by significantly less dramatic evolution for the jammed states. We consider both connected components, related in loose sense to force chains, and loops, and find that both measures provide a significant insight into the evolution of force networks. In addition to normal, we consider also tangential forces between the particles and find that they evolve in the consistent manner. Consideration of both frictional and frictionless systems leads us to the conclusion that friction plays a significant role in determining the dynamical properties of the considered networks. We find that the proposed approach describes the considered networks in a precise yet tractable manner, allowing to identify novel features which could be difficult or impossible to describe using other approaches.

"Evolution of Force Networks in Dense Particulate Media"
M. Kramar, A. Goullet, L. Kondic, K. Mischaikow
available on arXiv.org at arXiv:1404.6279

We present mathematical models based on persistent homology for analyzing force distributions in particulate sys- tems. We define three distinct chain complexes of these distributions: digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.

"Quantifying force networks in particulate systems"
M. Kramar, A. Goullet, L. Kondic, K. Mischaikow
available on arXiv.org at arXiv:1311.0424

We utilize the tools of persistent homology to analyze features of force networks in dense granular matter, modeled as a collection of circular, inelastic frictional particles. The proposed approach describes these networks in a precise and tractable manner, allowing us to identify features that are difficult or impossible to characterize by other means. In contrast to other techniques that consider each force threshold level separately, persistent homology allows us to consider all threshold levels at once to describe the force network in a complete and insightful manner. We consider continuously compressed system of particles characterized by varied polydispersity and friction in two spatial dimensions. We find significant differences between the force networks in these systems, suggesting that their mechanical response may differ considerably as well.
Read more

"Persistence of force networks in compressed granular media"
M. Kramar, A. Goullet, L. Kondic and K. Mischaikow
Physical Review E 87, 042207 (2013)

Using numerical simulations, we investigate the evolution of the structure of force networks in slowly compressed model granular materials in two spatial dimensions. We quantify the global properties of the force networks using the zeroth Betti number B0, which is a topological invariant. We find that B0 can distinguish among force networks in systems with frictionless vs. frictional disks and varying size distributions. In particular, we show that 1) the force networks in systems composed of frictionless, monodisperse disks differ significantly from those in systems with frictional, polydisperse disks and we isolate the effect (friction, polydispersity) leading to the differences; 2) the structural properties of force networks change as the system passes through the jamming transition; and 3) the force network continues to evolve as the system is compressed above jamming, e.g., the size of connected clusters with forces larger than a given threshold decreases significantly with increasing packing fraction.

"Topology of force networks in compressed granular media"
Kondic, L.; Goullet, A.; O'Hern, C. S.; Kramar, M.; Mischaikow, K.; Behringer, R. P., Europhysics Letters, Volume 97, Issue 5, pp. 54001, 6 pp. (2012).