Spatial force correlations in granular shear flow. I. Numerical evidenceGregg Lois,1Anaël Lemaître,2and Jean M. Carlson11Department of Physics, University of California, Santa Barbara, California 93106, USA2Institut Navier–LMSGC, 2 allée Képler, 77420 Champs-sur-Marne, France共Received 16 December 2006; revised manuscript received 18 April 2007; published 3 August 2007兲We investigate the emergence of correlations in granular shear flow. By increasing the density of a simulatedgranular flow, we observe a transition from a dilute regime, where interactions are dominated by binarycollisions, to a dense regime characterized by large force networks and collective motions. With increasingdensity, interacting grains tend to form networks of simultaneous contacts due to the dissipative nature ofcollisions. We quantify the size of these networks by measuring two-point force correlations and find dramaticchanges in the statistics of contact forces as the size of the networks increases.DOI: 10.1103/PhysRevE.76.021302 PACS number共s兲: 81.05.RmI. INTRODUCTIONGranular materials exhibit a wide range of fascinating be-haviors 关1兴, but predictive theories linking the microscopicgrain interactions to macroscopic properties remain a topicof much debate. Of particular interest are constitutive rela-tions for granular flow, which are important for engineeringand geophysical applications 关2,3兴 and challenge the tenetsof conventional statistical physics 关4兴. Because granular ma-terials are athermal, their dynamics always occur far fromequilibrium, and a proper formulation of constitutive rela-tions relies on the construction of nonequilibrium statisticaltheories that must be sensitive to the interactions betweengrains.Interactions in realistic granular materials arise due tograin elasticity and friction, but are complicated by variousother mechanisms, including humidity 关5–9兴, grain shapes关10–12兴, and fracture processes occurring within the material关13兴. A great deal of theoretical and computational progresshas been made using the simple approximation that grainsare spherical and perfectly dry 关14兴. In this case a purelyrepulsive force arises when two grains come into contact dueto the deformation of grains and friction between grains.Understanding the nature of grain forces and dynamics,even in this relatively simple case, has proven difficult. Atvery low densities it can safely be assumed that only binaryinteractions occur, and constitutive relations can be deter-mined by statistically tracking the repulsive force created ineach interaction. This is the basis of kinetic theory, which hasbeen successfully applied to granular flows 关15,16兴. How-ever, for very large densities, it is observed that multigraincontacts always occur 关17–19兴 and contact forces are trans-mitted through “force chain networks” formed by the topol-ogy of the contact network 关14,21–23兴. For these high den-sities the forces between contacting grains still arise fromgrain deformation and friction, but the extent of the interac-tions is not localized and depends on properties of the forcechain networks.The presence of force chain networks calls into questiontheories that assume localized interactions and has inspirednew models based on properties of the force chains 关24–30兴.However, although force networks can be visualized, it hasproven difficult to measure quantitative correlations betweencontact forces 关31–34兴. This led to the speculation that forcechain networks are simply a perceived correlation, until re-cently, when spatial correlations were measured between theaveraged contact forces in a quasistatic experimental shearflow at high density 关23兴.This discovery raises important questions about theproper assumptions to make when constructing theories ofgranular materials. For very dilute systems only binary col-lisions occur, and force chain networks do not play a role; forvery dense systems force networks are the dominant micro-scopic interaction. In order to understand the origin of mac-roscopic properties in granular flows, it is necessary to pin-point exactly how and when correlations appear.In this paper we measure spatial correlations of the totalforce on grains undergoing shear deformation. This measure-ment defines a characteristic length scale␰quantifying thesize of force networks arising from clusters of simulta-neously contacting grains. We find that␰grows with packingfraction, diverges at a finite packing fraction, and has mea-surable effects on the contact forces between grains. Thecorrelation measurement also provides a natural boundarybetween dilute flows where only binary collisions occur anddense flows where force networks emerge.We begin in Sec. II by briefly outlining the phenomenol-ogy of granular shear flow to define the exact regime we willbe studying. In Sec. III we discuss the numerical algorithmused to simulate shear flow. In Sec. IV we introduce thespatial force correlation measurement, which provides anatural definition for a correlation length, and in Sec. V weshow how the size of the correlation length affects contactforces between grains.II. GRANULAR SHEAR FLOW: BASIC CONSIDERATIONSIf no external force is applied to a dry granular material inthe absence of gravity, it quickly loses all its kinetic energyin dissipative collisions, and each grain comes to rest. If thisoccurs for a dilute system there are no residual contacts be-tween any grains and the total energy is zero. However, forgranular materials with larger densities, there are contactsbetween grains in the relaxed state and a nonzero residualenergy remains due to grain deformation and friction.If a shear stress is then applied to the system, motionoccurs only if the stress is large enough to overcome thePHYSICAL REVIEW E 76, 021302 共2007兲1539-3755/2007/76共2兲/021302共12兲 ©2007 The American Physical Society021302-1energy stored in the contacts. The minimum stress needed toinitiate motion is called the yield stress; it is zero below acritical packing fraction␯cand is an increasing function ofpacking fraction above␯c关35–37兴.For granular shear flows with␯⬎␯c, previous researchhas demonstrated that the stiffness of the grains plays animportant role at all values of the shear rate 关38兴. This isbecause grains are not able to rearrange to a configurationwhere no contacts exist and the system moves between dif-ferent configurations