Abstract
Granular materials transmit stress via a network of force chains. Despite the importance of these chains in characterizing the stress state and dynamics of the system, there is no common framework for quantifying their properties. Recently, attention has turned to the tools of network science as a promising route to such a description. In this paper, we apply a common network-science technique, community detection, to the force network of numerically-generated packings of spheres over a range of interparticle friction coefficients and confining pressures. In order to extract chain-like features, we use a modularity maximization with a recently-developed geographical null model (Bassett et al. in Soft Matter 11:2731–2744, 2015), and optimize the technique to detect sparse structures by minimizing the normalized convex hull ratio of the detected communities. We characterize the force chain communities by their size (number of particles), network force (interparticle forces), and normalized convex hull ratio (sparseness). We find that the first two are highly correlated and are therefore largely redundant. For both pressure P and interparticle friction \(\mu \), we observe two distinct transitions in behavior. One, for \(\mu \lesssim 0.1\) the packings exhibit more distinguishability to pressure than at higher \(\mu \). Two, we identify a transition pressure \(P^*\) at which the frictional dependence switches behaviors. Below \(P^*\) there are more large/strong communities at low \(\mu \), while above \(P^*\) there are more large/strong communities at high \(\mu \). We explain these phenomena by comparison to the spatial distribution of communities along the vertical axis of the system. These results provide new tools for considering the mesoscale structure of a granular system and pave the way for reduced descriptions based on the force chain structure.
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Bassett, D.S., Owens, E.T., Porter, M.A., Manning, M.L., Daniels, K.E.: Extraction of force-chain network architecture in granular materials using community detection. Soft Matter 11, 2731–2744 (2015)
Dantu, P.: Contribution l’étude méchanique et géométrique des milieux pulvérulents. In: Proceedings of the Fourth International Conference on Soil Mechanics and Foundation Engineering, London, pp 144–148 (1957)
Liu, C.H., Nagel, S.R., Schecter, D.A., Coppersmith, S.N., Majumdar, S., Narayan, O., Witten, T.A.: Force fluctuations in bead packs. Science 269, 513–5 (1995)
Howell, D., Behringer, R.P., Veje, C.: Stress fluctuations in a 2D granular couette experiment: a continuous transition. Phys. Rev. Lett. 82, 5241–5244 (1999)
Newman, M.E.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Arévalo, R., Zuriguel, I., Maza, D.: Topology of the force network in the jamming transition of an isotropically compressed granular packing. Phys. Rev. E 81, 041302 (2010)
Walker, D.M., Tordesillas, A.: Topological evolution in dense granular materials: a complex networks perspective. Int. J. Solids Struct. 47, 624–639 (2010)
Walker, D., Tordesillas, A.: Taxonomy of granular rheology from grain property networks. Phys. Rev. E 85, 011304 (2012)
Bassett, D.S., Owens, E.T., Daniels, K.E., Porter, M.A.: Influence of network topology on sound propagation in granular materials. Phys. Rev. E 86, 041306 (2012)
Peters, J., Muthuswamy, M., Wibowo, J., Tordesillas, A.: Characterization of force chains in granular material. Phys. Rev. E 72, 041307 (2005)
Zhang, L., Wu, J.-Q., Zhang, J.: Force-chain identification in quasi-2D granular systems. In: AIP Conference Proceedings, Powders and Grains, pp. 397–400 (2013)
Kondic, L., Goullet, A., O’Hern, C.S., Kramar, M., Mischaikow, K., Behringer, R.P.: Topology of force networks in compressed granular media. Europhys. Lett. 97, 54001 (2012)
Kaczynski, T., Mischaikow, K.M., Mrozek, M.: Computational Homology. Springer, New York (2004)
Radjai, F., Wolf, D., Jean, M., Moreau, J.-J.: Bimodal character of stress transmission in granular packings. Phys. Rev. Lett. 80, 61–64 (1998)
Porter, M.A., Onnela, J.-P., Mucha, P.J.: Communities in networks. Not. Am. Math. Soc. 56, 1082 (2009)
Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 103 (2010)
Navakas, R., Džiugys, A., Peters, B.: Application of graph community detection algorithms for identification of force clusters in squeezed granular packs. Modern Build. Mater. Struct. Tech. 1–4 (2010)
Navakas, R., Džiugys, A., Peters, B.: A community-detection based approach to identification of inhomogeneities in granular matter. Phys. A Stat. Mech. Appl. 407, 312–331 (2014)
Owens, E.T., Daniels, K.E.: Sound propagation and force chains in granular materials. Europhys. Lett. 94, 54005 (2011)
Herrera, M., McCarthy, S., Slotterback, S., Cephas, E., Losert, W., Girvan, M.: Path to fracture in granular flows: dynamics of contact networks. Phys. Rev. E 83, 061303 (2011)
Mukhopadhyay, S., Peixinho, J.: Packings of deformable spheres. Phys. Rev. E 84, 011302 (2011)
Saadatfar, M., Sheppard, A.P., Senden, T.J., Kabla, A.J.: Mapping forces in a 3D elastic assembly of grains. J. Mech. Phys. Solids 60, 55–66 (2012)
Brodu, N., Dijksman, J.A., Behringer, R.P.: Spanning the scales of granular materials through microscopic force imaging. Nat. Commun. 6, 6361 (2015)
LAMMPS. http://lammps.sandia.gov
Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)
Chen, Y., Best, A., Butt, H.-J., Boehler, R., Haschke, T., Wiechert, W.: Pressure distribution in a mechanical microcontact. Appl. Phys. Lett. 88, 20–23 (2006)
Silbert, L., Ertas, D., Grest, G., Halsey, T., Levine, D.: Geometry of frictionless and frictional sphere packings. Phys. Rev. E 65, 031304 (2002)
Blumenfeld, R., Edwards, S.F., Ball, R.C.: Granular matter and the marginal rigidity state. J. Phys. Condens. Matter 17, 11 (2005)
Owens, E.T., Daniels, K.E.: Acoustic measurement of a granular density of modes. Soft Matter 9, 1214–1219 (2013)
Jutla, I.S., Jeub, L.G.S., Mucha, P.J.: A generalized Louvain method for community detection implemented in MATLAB. http://netwiki.amath.unc.edu/GenLouvain
Newman, M.: Fast algorithm for detecting community structure in networks. Phys. Rev. E 69, 066133 (2004)
Silbert, L.E.: Jamming of frictional spheres and random loose packing. Soft Matter 6, 2918 (2010)
Makse, H.A., Johnson, D.L., Schwartz, L.M.: Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160–4163 (2000)
Zhang, H.P., Makse, H.A.: Jamming transition in emulsions and granular materials. Phys. Rev. E 72, 11301 (2005)
Newman, M., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004)
Janssen, H.A.: Versuche über Getreidedruck in Silozellen. Zeitschr. d. Vereines deutscher Ingenieure 39, 1045–1049 (1895)
Acknowledgments
We are grateful for support from the National Science Foundation (DMR-1206808) and the James S. McDonnell Foundation. The simulations were performed at the NC State High Performance Computing Center. We are grateful to Leo Silbert, Danielle Bassett, and Mason Porter for valuable conversations.
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This research has been supported by the James S. McDonnell Foundation and the National Science Foundation (DMR-1206808).
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Huang, Y., Daniels, K.E. Friction and pressure-dependence of force chain communities in granular materials. Granular Matter 18, 85 (2016). https://doi.org/10.1007/s10035-016-0681-6
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DOI: https://doi.org/10.1007/s10035-016-0681-6