Betweenness centrality illuminates intermittent frictional dynamics
Omid Dorostkar, Karen E. Daniels, Dominik Strebel, Jan Carmeliet
BBetweenness centrality illuminates intermittent frictional dynamics
Omid Dorostkar,
1, 2, ∗ Karen E. Daniels, Dominik Strebel, ∗ and Jan Carmeliet Department of Mechanical and Process Engineering, ETH Z¨urich, Switzerland Department of Engineering Science, University of Oxford, Parks Road, Oxford Department of Physics, North Carolina State University, Raleigh, North Carolina, USA (Dated: February 4, 2021)Dense granular systems subjected to an imposed shear stress undergo stick-slip dynamics withsystematic patterns of dilation-compaction. During each stick phase, as the frictional strength buildsup, the granular system dilates to accommodate shear strain, developing stronger force networks.During each slip event, when the stored energy is released, particles experience large rearrangementsand the granular network can significantly change. Here, we use numerical simulations of 3D,sheared frictional packings to show that the mean betweenness centrality – a property of network ofinterparticle connections – follows consistent patterns during the stick-slip dynamics, showing sharpspikes at each slip event. We identify the source of this behavior as arising from the connectivityand contact arrangements of granular network during dilation-compaction cycles, and find that alower potential for connection between particles leads to an increase of mean betweenness centralityin the system. Furthermore, we show that at high confinements, few particles lose contact duringslip events, leading to a smaller change in granular connectivity and betweenness centrality.
INTRODUCTION
The macroscopic response of a granular systemunder any external loading originates from grain-scaleinteractions [1–4]. When an external load is applied togranular materials, particles form contacts and createforce chains to sustain and transfer the applied load.There has been extensive work addressing the formationof granular networks and seeking appropriate local, grain-scale metrics which aid the estimation and predictionof the system’s global behaviour [ e.g. e.g.
8, 10–13]. For example, failures in alattice-like system under compressive and tensile loadingare shown to take place mainly in locations with largergeodesic edge betweenness centrality than the mean onein the structure [14]. Particle betweenness centrality isalso suggested as a predictor for forces in 2D granularpackings, where the total pressure on each particle inthe system correlates to its betweenness centrality valueextracted from the geometric contact network [15].Frictional instabilities in sheared amorphous systemsthat appear in form of stick-slip dynamics aremanifestations of a sudden transition from a solid- to a fluid-like state, usually accompanied with abruptrelease of energy [16, 17]. In the solid-like stateor stick phase, there are few particle rearrangementswithin the system, and the granular medium is almostjammed under the imposed shear stress. In the fluid-likestate, particles can rearrange and the system is at leastpartially unjammed [16, 18, 19]. Under shear loading,this transition is accompanied with storage and releaseof energy; during stick phase the system’s frictionalstrength increases and energy is accumulated in thesystem. At slip, the stored energy is released throughparticle rearrangement and fragmentation, leading toenergy dissipation [17, 20, 21]. During these cyclicprocess, with respect to change of volumetric strain,the granular system dilates during the stick phase toaccommodate the imposed shear strain, and compactsat slip due to large particle rearrangements [22, 23].Therefore, even though the system is mechanicallyconsidered jammed during the stick phase, there arestill small particle rearrangements owing to an overalldilation in the system, which affects the evolution of boththe particles’ contact network, and the total number ofcontacts [24].As for stick-slip dynamics, we hypothesize that thesystematic dilation-compaction of a dense granularsystem is a key factor controlling the granular networkarchitecture and connectivity between particles. Drawingon tools from network science, we examine the contactstructure of a sheared granular system undergoing stick-slip dynamics. We track the connectivity of particles andextract grain-scale information to examine correlationsbetween the evolution of the granular network and thefrictional instabilities during stick-slip dynamics. Inparticular, we look at particle betweenness centrality;this measure is calculated solely on contact status ofparticles, but depends sensitively on the full network a r X i v : . [ c ond - m a t . s o f t ] F e b of connections. The questions we address in this workare 1) whether there are systematic patterns in granularnetwork architecture during stick-slip cycles, 2) if thereis a relation between the characteristics of frictionalinstabilities and the properties of granular network and3) how we can characterize and measure those patternsand relationships. We focus on the temporal andspatial behaviour of granular network during stick-slipdynamics and measure ensemble particle betweennesscentrality. We discuss the relationship between particlebetweenness centrality and the compaction-dilationcycles, and observe connection between its evolutionand the particles’ freedom to rearrange into a newconfiguration. We discuss that in a 3D sheared system,the patterns of force chains are different from the patternsof chains with high particle betweenness centrality. Wedemonstrate that the time evolution of mean betweennesscentrality can be used as an indicator for slip events,and discuss how denser systems’ decreased freedom torearrange affects this manifestation. METHOD
We simulate our granular system using standard softsphere Discrete Element Method (DEM), implementedin the open source software LIGGGHTS [25, 26]. Thespherical particles have diameters of 90 −
150 mm, drawnfrom a uniform size distribution; they have a Poissonratio of ν = 0 .
25, Young’s modulus of Y = 65 GPa,restitution coefficient of r = 0 .
87 and inter-particlefriction coefficient µ c = 0 .
5. The particle density is2900 kg / m , leading to a DEM time step of 15 × − s.We apply a constant confining stress via two corrugatedplates in the z -direction, with a periodical boundaryacross x -direction and frictionless walls in y -direction (seeFig.1a). The system is sheared using a displacement-control protocol at a constant shear rate of 0 . / s,hence the granular flow is maintained in the quasi-static regime by keeping the inertial number below 10 − [27]. The reference granular model has a sample size of11 × . × . and consists of N = 7996 particles;the sample size is changed later to study its effecton betweenness centrality chains, however keeping thenumber of particles constant.In DEM, the equations of motion are solved for eachparticle: (cid:88) F p = m ˙ u b , (1) (cid:88) T p = I ˙ ω b , (2)where m , I , u p and ω p are the mass, the momentof inertia and the translational and angular velocities of particle, and F p and T p are the forces and torquesacting on particle, respectively. We use the soft sphereDEM approach, in which the particle-particle contact ismodeled with an overlap between them and the contactlaw is described by a combination of different rheologicalelements [24]. Using the nonlinear Hertzian particle-particle contact law, the normal and tangential contactforces are described as [28, 29]: F pn = − k pn δ(cid:15) pn + c pn δu pn , (3) F pt = min ( | k pt (cid:90) tt c, δu pt dt + c pt δu pt | , µ c F pn ) , (4)where k pn and k pt are the normal and tangential springstiffness, c pn and c pt are the normal and tangentialdamping coefficients, δ(cid:15) pn is the overlap, δu pn and δu pt are the relative normal and tangential velocities,and µ c represents the inter-particle friction coefficient,respectively. The normal and tangential spring anddamping coefficients are calculated from [28, 29]: k pn = 43 Y ∗ (cid:112) R ∗ δ(cid:15) pn , (5) k pt = 8 G ∗ (cid:112) R ∗ δ(cid:15) pn , (6) c pn = − (cid:112) (5 / ln ( r ) (cid:112) ln ( r ) + π (cid:113) Y ∗ m ∗ (cid:112) R ∗ δ(cid:15) pn , (7) c pt = − (cid:112) (5 / ln ( r ) (cid:112) ln ( r ) + π (cid:113) G ∗ m ∗ (cid:112) R ∗ δ(cid:15) pn , (8)where r is the restitution coefficient, and Y ∗ , R ∗ , G ∗ and m ∗ are the equivalent Young’s modulus as 1 /Y ∗ =((1 − ν )) /Y + ((1 − ν )) /Y , the equivalent radius as1 /R ∗ = 1 /R + 1 /R , the equivalent shear modulus as1 /G ∗ = (2(2 − ν )(1 + ν )) /Y + (2(2 − ν )(1 + ν )) /Y and the equivalent mass as 1 /m ∗ = 1 /m +1 /m [28, 29].The subscripts 1 and 2 refer to the two specific particlesin contact and ν is the Poisson’s ratio of the particle.We use betweenness centrality b (Fig.1b) as a non-local measure of granular network connectivity. Theparticle betweenness centrality measures the number oftimes that the shortest paths between a pair of otherparticles travel through that particle. Mathematically,the betweenness centrality b of particle n is defined asthe fraction of shortest paths S ij that connect particles i (cid:54) = j (cid:54) = n , going through particle n : FIG. 1. (a) 3-dimensional granular model with frictionless walls at the front and back of the sample, periodic boundaries in x -direction, and corrugated plates on top and bottom. (b) Schematic illustration of a connected network of particles withdifferent betweenness centrality values. The red particle has the highest betweenness centrality value, as it is included in manyshortest paths between other pairs of particles. b ( n ) = (cid:88) i (cid:54) = j (cid:54) = n S ij ( n ) S ij . (9)Since the betweenness centrality of a particle scaleswith the number of pairs of particles, due to thesummation indices, we rescale it by dividing b ( n ) bya factor of ( N − N − /
2, where N is the totalnumber of particles. To calculate particle betweennesscentrality, we use open-source functions provided bythe Brain Connectivity Toolbox and the Boost GraphLibrary [30, 31]. RESULTS
Fig.2a shows the time series of the macroscopicfriction coefficient µ , calculated as the ratio of shearstress to confining stress, and mean particle betweennesscentrality, b , averaged over the whole sample. Thefriction signal exhibits irregular stick-slip dynamics,where the slip events have a variety of magnitudes andrecurrence times. Prior to slip, the sample is in itscritical state; this preslip period is characterized bydeformation being accommodated by microslips mainlydue to small particle rearrangements [32]. The stick-slipcycles show different critical states in which some cyclesexperience many microslips. During the stick phase, the b signal shows slight variations, gradually increasing duringthe approach to the eventual slip event. At slip, we FIG. 2. (a) Time series of macroscopic friction µ and mean betweenness centrality b (secondary axis). (b) Time series ofsample thickness h (sample size in y -direction in Fig.1) and mean coordination number z . (c-d) Shaded zone from panels (a-b)highlighting the details of several stick-slip cycles. The simulations of this figure are performed under confinement of 500 kPaand shear speed of 0.6 mm/s, and the granular sample has a size of 11 × . × . . consistently observe spikes in b , and sharp drops afterthe slip is complete.The sample thickness h (sample size in y -direction)and average coordination number z also show consistent patterns following stick-slip dynamics (Fig.1). Thegranular sample dilates during the stick phase andcompacts at slip. Due to the dilation during the stickphase, there is a slight decrease in z . At slip, z shows a FIG. 3. Time series of macroscopic friction µ and mean betweenness centrality b (secondary axis) for confining stresses of 1,10 and 100 MPa. The simulations of this figure are performed with shear speed of 0.6 mm/s, and the granular sample has asize of 9 . × . × . . sharp drop; owing to the compaction of granular layer, z recovers and particles again gain more contacts. In thezoomed-in region (see Fig.2c-d), we clearly observe thistrend of b showing gradual increase during stick phaseand a sharp spike at slip. The evolution of b during thecritical state is sensitive to the occurrence of microslips:approaching a major slip event, the increase in b isaffected by occurrence of microslips, where each microslipslightly reduces b . For this reason, wherever there aremicroslips, the gradual increasing trend in b during thestick phase stops and the signal instead undergoes manysmall drops.As shown in Fig.2d, as the granular layer dilates,the mean coordination number z shows a clear decreaseduring the stick phase, followed by a drop and then recovery at slip. This occurs because, as the shearingadvances, the granular system dilates: due to the dilationand increase of porosity (decrease of packing fraction),some contacts are inevitably lost [24]. At slip, thegranular layer undergoes substantial (as compared tothe stick phase) particle rearrangements and ultimatelycompacts. The drop in z shows the comparatively fluid-like (or unjammed) behaviour of granular system, andthe recovery of z is directly caused by compaction anddecrease of porosity during slip [20].It therefore seems that the evolution of b is directlyrelated to the behavior of particles that rearrange andchange contact status: during the stick phase, as thesystem dilates and the pore volume expands, someparticles lose contacts and the granular contact network FIG. 4. Time series of mean coordination number z , and sample porosity n (secondary axis) for confining stresses of 1, 10 and100 MPa. The simulations of this figure are performed with shear speed of 0.6 mm/s, and the granular sample has a size of9 . × . × mm . is partially lost. Where there is contact loss, thespatial distribution of the granular contact network alsobecomes less uniform and the connectivity load i.e. thenumber of shortest paths between particles in the systemfor the surviving contacts, increases on the remainingcontacts, which causes on average an increase of b . It isimportant to note that this change in centrality of theremaining connected particles does not necessarily meana change in the status of their contacts or their position,but only that the shortest path connecting a pair ofother particles can change from a lost path to anotherone that includes these remaining contacts (remainingconnected particles). Betweenness centrality is bydefinition a measure of the centrality of particles withinthe ensemble of shortest paths. Therefore, if particles’connectivity and contact network are formed such that some particles get extreme centrality, the betweennesscentrality increases on average.As was observed for thebehaviour of b during the stick phase, and due to largedrops in z at slip, many connectivity paths are lostand hence the connectivity load increases significantlyon a few remaining chains leading to a sharp increasein b . As the system compacts, the connectivity pathsare recovered, the connectivity load is more evenlydistributed inside the sample, and b drops. An analogyfor this behavior is traffic flow between two cities.Suppose there are two main highways connecting twomajor cities. If highway A gets closed for maintenance,its traffic load will relocate to highway B, which meanscars that were already in highway B are now more inthe center of the flow (more cars, more shortest pathsbetween a pair of them that include the cars in the FIG. 5. Spatial distribution of (a) particle betweenness centrality b , (b) particle potential energy P E and (c) particlecoordination number z . The simulations of this figure are performed under confinement of 10 MPa and shear speed of 0.6mm/s, and the granular sample has a size of 11 × . × . . middle of the highway), and their betweenness centralityincreases. A closure of highway A is similar to the lossof contacts at slip in our model.We additionally observe that b can change byalterations of the confining stress, which controls theconnectivity of the granular system. As shown inFig.3, by increasing the confining stress, the evolutionof b neither shows a clear gradual increase duringthe stick phase nor a sharp increase at slip for veryhigh confinements. Particularly at slip events, theamplitudes of the spikes in b are very small, and almostindistinguishable from the background values. Anotherimportant observation in Fig.3 is the change in the sizeof slip friction drops, defined as the change in µ , where athigh confinements, lower slip friction drops are observed.Slip friction drop demonstrates the ability of a granularsystem to release the accumulated shear stress relative toits confinement. A smaller drop in friction for sampleswith higher confinement (lower porosity) is due tolower potential and freedom for particle rearrangements:during a frictional instability, the system can releasesmaller amount of stress relative to its confinement, as compared to a weakly confined system.In Fig.4 we demonstrate the evolution of meancoordination number, z , and porosity, n , defined as theratio of pore volume to total volume of the sample.At low confinement, during the stick phase n increasesas the sample dilates and z decreases as the systemloses contacts. The sharp drops are clear for z at slipevents. This consistent behaviour of z and n are lessobservable at higher confinements. On the other hand,with increasing confinement, the mean value of n and z decreases and increases, respectively. The observationsin Fig.4 demonstrate that particles in samples withhigher confinement have lower freedom and potentialfor rearrangement during stick-slip dynamics, supportingour hypothesis as the underlying mechanism for thebehaviour of b in Fig.3.For comparison with more traditional measuresof granular micromechanics, we consider the spatialdistributions of potential energy, betweenness centrality,and coordination number (Fig.5). As an approximation,we calculate contact potential energy as P E = F pn /k pn + F pt /k pt , and distribute this quantity evenly between the FIG. 6. Spatial distribution of particle betweenness centrality b for samples with different measurements in x - and y -directionbut fixed size in z -direction, as presented in the legend table. The simulations of this figure are performed under confinementof 10 MPa and shear speed of 0.6 mm/s. The snapshots are taken from the front of the samples and are representative of thewhole sample depth ( z -direction). two particles in contact. The total potential energyof a particle is the sum of energies gained from allits contacts. An interesting observation in Fig.5 isthat, unlike the rather uniform distribution observedfor coordination number, both particle betweennesscentrality and potential energy form chain-like patterns. However, contrary to the energy chains that formdiagonally to resist the applied load (and whichcorrespond to conventional notions of force chains [33,34]), the betweenness centrality chains show horizontalpatterns, aligned with the flow direction. A quantitativestudy of their properties is beyond the scope of this work, FIG. 7. (a) Mean coordination number z and mean betweenness centrality b (secondary axis) for 4 samples described in Fig.6.The original simulations are performed for sample I under confinement of 10 MPa and shear speed of 0.6 mm/s, and theother samples are loaded relative to sample I with either constant pressure (confining stress, shown with continuous line) orconstant force (shown with dashed line). (b) Complimentary Cumulative Distribution Function (cCDF) of particle betweennesscentrality for 4 points during the stick phase for 4 samples described in Fig.6 but we will next undertake a qualitative investigation ofthe betweenness centrality chains structures, which wefind to arise from the geometry of the sample.To investigate the origin of horizontal alignment ofbetweenness centrality chains in Fig.5a, we seek tounderstand whether they are formed along the directionof applied shear stress i.e. x -direction (see Fig.1), orare related to smaller dimension of the sample along y -direction (Fig.1). To shed more light, we performsimulations on granular assemblies with variable aspectratio: different sizes in y -direction but the same numberof particles. Therefore, the length of the sample in x -direction is smaller for the samples with larger size in y -direction (Fig.6). Note that the boundary conditions in x -direction still remain periodic for all samples, meaningthere is no influence of sample length on simulatedbehavior. Starting with sample I and by increasingthe sample size in y -direction towards sample IV, thebetweenness centrality chains become more diagonal orvertical. This shows that sample size in y -direction affects and guides the patterns of betweenness centralitychains, since particles have more possibilities for formingcontacts in sample IV. In addition, we also observethat, while sample I shows betweenness centrality chainswith some extreme b values (red chains), b decreasesfor samples with larger sample size in y -direction.The observations in Fig.6 confirms that if particlesare provided with more options for forming contacts,the betweenness centrality chain patterns become moreuniformly distributed, even though some extreme chainsare present in the sample. This point inferred from thespatial patterns of b is consistent with the mechanismexplained for the temporal evolution of b during the stick-slip dynamics. As we progress during the stick phase,particles have less options to form new connectivitypaths, and therefore extreme b values appear, leading toan increase of b . This behaviour reaches its ultimate stateat occurrence of slip, where because of the loss of manycontacts, the connectivity paths are lost, and centralityincreases on the remaining chains, leading to a spike in0 b . Fig.7a quantitatively confirms that b decreases withincreasing sample size in y -direction, whereas z increases.This interesting observation, that with higher numberof contacts the mean betweenness centrality is lower,implies that in a sample with larger size in y -direction,there are more possibilities for each particle to make thecontact with its neighbouring particles, such that theconnectivity of particle is more uniformly distributed and b is smaller. Note that, with changing sample size in y -direction from sample I to IV, we perform the simulationswith two loading mechanisms: one at constant pressure(confinement) and the other at constant force, and theobservations are valid for both loading protocols (Fig.7a).We also show in Fig.7b the complimentary CumulativeDistributions Functions (cCDF) of b for 4 samplesdescribed in Fig.6. The distributions are made for 4random points during a stick phase considered to berepresentative for the whole simulation period. ThecCDFs in Fig.7b show that, the extreme betweennesscentrality chains in Fig. 6 are caused by only a smallportion of particles, as the extreme tails in Fig.7b showthe deviation of distributions at around 10 percent. Wehighlight here that our observations for spatial patternsof betweenness centrality chains, and the relation ofaspect ratio (different relative sample sizes in x - and y -direction) with b and z are consistent with thebehaviour of mean betweenness centrality during stick-slip cycles. When particles have more possibilities formaking contacts, betweenness centrality is smaller; thissituation occurs, for instance, at the beginning of a stickphase or in samples with lager sample size in y -direction.On the other hand, when connectivity is limited, whetherdue to a loss of contacts at slip or to a thinner sample,the centrality experiences higher values. CONCLUSIONS
We model stick-slip dynamics in a sheared granularsystem and study the evolution of network connectivityduring frictional intermittent failures using particlebetweenness centrality. The mean particle betweennesscentrality shows sensitivity to the friction level duringstick-slip cycles, controlled by the coordination numberand the freedom of particles for rearrangement. Inhigh porosity samples, as occurs at lower confiningpressure due to dilation of the granular system, themean betweenness centrality increases gradually duringthe stick phase and spikes at slip. The mean betweennesscentrality drops along with the compaction phasefollowing slip. With increasing confinement and decreaseof porosity, both the contact loss during the stick phaseand the substantial rearrangements during the slip phasebecome smaller; therefore, the connectivity of networkof particles only slightly changes, leading to a less prominent change of mean betweenness centrality. Thelower freedom of the particles to rearrange at very highconfinement also limits the granular sample’s ability torelease stored shear stress (relative to its confinement)during the frictional instabilities, as the friction dropsare smaller at slip. Our results in this work showthat betweenness centrality, a metric that solely dealswith geometric contact network, can be an indicator forapproach of frictional instabilities in sheared granularsystems.
ACKNOWLEDGEMENTS
Authors thank Empa for infrastructural supports.KED is grateful for the support of the James S.McDonnell Foundation.
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