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Force network ensemble for the triangular lattice: A tale of tiles

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Force network ensemble for the triangular lattice: A tale of tiles

Brian P. Tighe,1Adrianne R. T. van Eerd,2 and Thijs J. H. Vlugt3

1

Universiteit Leiden, 2300 RA Leiden, The Netherlands 2

Utrecht University, 3508 TA Utrecht, The Netherlands 3

Delft University of Technology, 2628 CA Delft, The Netherlands 共Received 27 July 2009; published online 27 October 2009兲 关doi:10.1063/1.3207833兴

In granular materials, contact forces between neighboring grains are organized in disordered force networks. At present, it is unclear whether or not theories based on statis-tical mechanics can correctly predict the statistics of these force networks.

The force network ensemble1is a convenient model sys-tem for studying force networks. This ensemble comprises all sets of noncohesive contact forces on a fixed underlying contact network and a fixed load, i.e., global stress tensor, for which all grains are in static force and torque balance. Each force network is assigned an equal a priori probability. The corresponding phase space can be sampled using the Monte Carlo method. Figure1共a兲shows a typical force network for the frictionless triangular lattice. Each force network has a complementary representation known as a reciprocal tiling

关Fig.1共b兲兴 in which each grain maps to a particular tile and each contact force to a tile face. The face is oriented at a␲/2 rotation to the contact force f with a length proportional to f. Due to local force balance, the tile faces form closed loops and, because forces come in action-reaction pairs, all tiles fit together without any gaps. The local pressure of a single grain is equal to its tile perimeter. One can show that fixing the global stress tensor 具␴ˆ典 leads to a conservation of the total tile areaA.

In Ref. 2, we derived an analytical expression for the distribution of the local pressure on individual grains, by maximizing the entropy while conserving具␴ˆ典 and A. Figure

1共c兲 shows that this theoretical prediction is in excellent agreement with the numerical result from the force network ensemble. Due to these constraints, large local stresses obey Gaussian statistics, in sharp contrast to the common belief that exponential force statistics are characteristic for granular materials. This observation is robust to changes in contact network 共including disordered networks兲 and finite friction coefficient.

1J. H. Snoeijer, T. J. H. Vlugt, M. van Hecke, and W. van Saarloos,Phys.

Rev. Lett. 92, 054302共2004兲.

2B. P. Tighe, A. R. T. van Eerd, and T. J. H. Vlugt,Phys. Rev. Lett. 100,

238001共2008兲. 0 6 12 18 24 30 -18 -15 -12 -9 -6 -3 0 numerics theory

p/〈 f 〉

log

10

ρ

(a) (b) (c)

FIG. 1. 共Color兲 共a兲 Typical force network for a frictionless triangular lattice. Line thickness is proportional to contact force. All grains are in static force balance. Periodic boundaries are used.共b兲 Reciprocal representation of 共a兲. Tile edges correspond to contact forces 共rotated by␲/2兲 with lengths proportional to their magnitude. Due to Newton’s laws, the tile faces form closed loops and all tiles fit together without any gaps. The local pressure of a single grain is equal to its tile perimeter. Fixing the global stress tensor leads to a conservation of the total tile area. The animation shows the sampling of force networks in the force network ensemble.共c兲 Theoretical 共dashed line兲 and numerical 共solid line兲 local pressure probability distribution␳共p兲 for the frictionless triangular lattice共enhanced online兲.关URL: http://dx.doi.org/10.1063/1.3207833.1兴

CHAOS 19, 041107共2009兲

1054-1500/2009/19共4兲/041107/1/$25.00 19, 041107-1 © 2009 American Institute of Physics

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