Finite-Size Analysis of the Collapse of Dry Granular Columns
Teng Man, Herbert E. Huppert, Ling Li, Sergio Andres Galindo-Torres
FFinite Size Analysis of the Collapse of Axisymmetric Dry Granular Columns
Teng Man
Institute of Advanced Technology, Westlake Institute for Advanced Study,18 Shilongshan Street, Hangzhou 310024, Zhejiang Province, China ∗ Herbert E. Huppert
Institute of Theoretical Geophysics, King’s College, University of Cambridge,King’s Parade, Cambridge CB2 1ST, United Kingdom
Ling Li and Sergio Andres Galindo-Torres † School of Engineering, Westlake University, 18 Shilongshan Street,Hangzhou, Zhejiang 310024, China (Dated: December 8, 2020)The collapse of granular columns is potentially connected to the dynamics of complex flows, suchas debris flows, landslides, and particulate flows in chemical engineering and food processing, yetthe link between the microscopic structures of granular assemblies and their macroscopic behaviorsis still not fully understood. In this paper, we focus on the size effect of granular column collapsesusing the polyhedral discrete element method (DEM) to show that the ratio between column radiusand the grain size has a strong influence on the collapse behavior. The finite-size scaling analysis,which is inspired by a phase transition of granular column collapses around a inflection point, wasperformed to obtain a general scaling equation with critical exponents for the run-out distanceand the energy consumption of the granular column collapses. We further formalize a correlationlength scale which exponentially scales with the effective aspect ratio. Such a scaling solution showssimilarities with that of the percolation problem of two-dimensional random networks and can beextended to other similar natural and engineering systems.
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Introduction .– Granular materials are omnipresent inboth natural and engineering systems, and the physicsand mechanics of them are crucial for understandingsome aspects of geophysical flows, natural hazards (suchas debris flows and landslides), food processing, chemicalengineering, and pharmaceutical engineering. Granularmaterials can behave like a solid, a liquid, or a gas indifferent circumstances[1], which increases the difficultyin capturing the macroscopic behavior of them. Besides,collective structures may form inside a granular system,and the existence and the size of such collective structures(i.e. bridging[2], granular agglomerates[3], and contactnetworks[4]) will introduce a strong size effect into themacroscopic behavior of granular materials, which fur-ther increases the complexity of the problem. In recentdecades, breakthroughs have been made to understandthe basic governing principles, especially the constitutiverelationships, of granular materials[1, 5–9], where the be-havior of granular materials or granular-fluid mixtureswere considered to be described by dimensionless num-bers expressed as the ratio between dominating stresses.Due to the similarity and potential links between thecollapse of granular columns and gravity-driven geophys-ical flows, such as debris flows, landslides, and rockavalanches, previous research investigated the collapseof granular columns to analyze the post-failure behav-ior of granular systems [10–14]. Lube et al.[15, 16] and Lajeunesse et al.[17] independently determined relation-ships for both the normalized run-out distance R =( R ∞ − R i ) /R i (where R ∞ is the final radius of the granu-lar pile, and R i the initial radius of the granular column),and the halt time of a collapsed granular column scalewith the initial aspect ratio, α = H i /R i (where H i is theinitial height) of the column, a parameter drawn out of di-mensional analysis. Warnett et al.[18, 19] and Cabrera etal.[20] studied the collapse of granular columns with ex-periments and simulations, respectively, and argued thatthe relative size of a granular column, R i /d , has a stronginfluence on the run-out distance of a collapsed granularcolumn, and to avoid significant size effects, R i /d mustbe larger than 75 for short columns and larger than 50for tall columns.Based on these studies, Man et al.[21] investigatedthe collapse of axisymmetric granular columns and theirresulting deposition morphology with a wide range ofinter-granular and particle/boundary frictional coeffi-cients, and concluded that the normalized run-out dis-tance, R , scales with an effective aspect ratio, α eff = (cid:112) / ( µ w + βµ p )( H i /R i ), where µ w is the frictional co-efficient between particles and the boundary, µ p is theinter-particle frictional coefficient, and β = 2 . a r X i v : . [ c ond - m a t . s o f t ] D ec like regime. Here, α eff can be seen as a ratio between theinertial effect and the frictional effect during the collapseof granular columns.In this paper, we conducted a systematic study of ax-isymmetric granular column collapses with the polyhe-dral discrete element method (DEM) to study the in-fluence of relative column size on the run-out distanceto obtain a scaling solution to describe granular columncollapses with different sizes. The paper is organized asfollows. We introduce the simulation set-up and the con-tact law in DEM and introduce the method we use tocalculate the run-out distance in simulations. We thenpresent and analyze the simulation results, the finite-sizescaling, and the correlation length scale associated withthe collapse of granular columns. Finally, we present ourconcluding remarks in the summary. FIG. 1. Simulation set-up. (a) shows the initial state ofthe granular column; (b) shows the final deposition of thecollapsed column in a 3-D view; and (c) represents Voronoi-based particles in the DEM simulation; (d) shows the methodfor measuring the run-out distance of a collapsed granu-lar column. The x − axis is the radial position r , and the y − axis is the percentage of number of particles located within( r − ∆ r/ , r + ∆ r/
2) divided by the radial position, P dm . Wemeasure the final radius of a collapsed granular column when P dm ( r ) ≤ P maxdm . Simulation set-up .– We performed DEM simulations ofthe granular column collapses with spheropolyhedra par-ticles. The model has been validated with experimentsin previous research[21]. In the simulation, we createdparticles in a certain cylindrical domain[22], with initial height, H i , and initial radius, R i [FIG. 1(a)]. To intro-duce randomness, 20% of the particles were randomlyremoved from the simulation to create a granular pack-ing with an initial solid fraction of φ s =0.8. The averageparticle size d is 0.2 cm. The coefficient of restitution ofparticle collisions is 0.1. The material properties are setto be the same as that of quartz sand (density of particles2.65 g/cm ). This paper entirely focuses on simulationswith granular columns of circular cross-sections. Moregeneral cross-sections are currently under investigationand will be presented in a subsequent publication. Theparticle/boundary frictional coefficient µ w was set to 0.4,while the inter-particle frictional coefficient µ p was 0.2,0.4, and 0.6.We treat the relative column size ( R i /d = 2, 2.5, 3.75,5, 7.5, 10, 12.5, 15, 17.5, 20, 30) as a key parameter.Within one set of simulations with the same R i /d , we var-ied the initial height H i and the inter-particle frictionalcoefficient µ p to obtain the general collapse behavior witha wide range of initial conditions. After releasing parti-cles to the horizontal plane, the granular material willform a pile of loosely packed grains [FIG. 1(b)], with fi-nal packing radius, R ∞ . Thus, the normalized run-outdistance, R = ( R ∞ − R i ) /R i , could be obtained. WithVoronoi-based spheropolyhedra particles (FIG. 1(c)), weimplement the Hookean contact model with an energydissipation term to calculate the interactions among con-tacting particles[22]. In this paper, we take the normalstiffness K n = 1 × N/cm, and tangential stiffness K t = 5 × N/cm. The motion of particles is then cal-culated by step-wise resolution of Newton’s second law.
Determination of the run-out distance .– In simulations,the measurement of the final packing radius is more com-plicated than that in experiments. In cases with smallparticle/boundary and interparticle frictional coefficientsbut large initial aspect ratios, the spread of particles isfar-reaching and leads to sparse (single layer) coverageof the area, especially at the front edge. In these cases,it is difficult to determine the edge/boundary and hencethe final run-out distance. Thus, we measured the fi-nal radius with a histogram of particle distribution foreach simulation. The following figure (FIG. 1(d)) givesan example of how we measured the run-out distance.In FIG. 1(d), the x-axis is the radial position r , and y -axis is the percentage of number of particles locatedwithin ( r − ∆ r/ , r +∆ r/
2) divided by the radial position, P dm ( r ) = (1 /r )[ N ( r − ∆ r/ , r +∆ r/ / (Σ r N )], where ∆ r is the bin width of the histogram, and N ( r − ∆ r/ , r +∆ r/
2) is the number of particles located between r − ∆ r/ r +∆ r/
2, and Σ r N is the total number of particles inone simulation. FIG. 1(d) shows the normalized particlenumber distribution (i.e. deposition morphology) of asimulation with µ w = 0 . µ p = 0 . R i = 4 cm, and H i = 32 cm. It shows that most particles locate within r ≤
27 cm. Thus, we take the final deposition radius as R ∞ = 27 cm, and the normalized run-out distance canbe calculated as R = ( R ∞ − R i ) /R i = 5 . FIG. 2. (a) shows the relationship between the normalizedrun-out distance, R = ( R ∞ − R i ) /R i , and the effective as-pect ratio, α eff = α (cid:112) / ( µ w + 2 µ p ), with relative column size R i /d = 2.5 ( ) and 17.5 ( ), respectively, to show that thecritical inflection point ( α c , R c ) changes as we change therelative size of the column. (b) plots R against the effectiveaspect ratio, α eff for all the column sizes. Results and discussions .– After simulating granularcolumn collapses with various column sizes and frictionalcoefficients, we can obtain the relationship between R and α . Similar to what we have seen in our previ-ous research[21], for columns with the same relative size R i /d , less friction leads to larger run-out distances. Thisis because higher friction results in higher energy dissipa-tion, which further shortens the run-out distance of thecollapse of granular columns. As we change the x − axisto the effective aspect ratio, α eff , normalized run-out dis-tance with the same R i /d but different frictional coeffi-cient collapse onto one curve in R − α eff space [FIG. 2(a)and (b)].Additionally, we can see that, in both FIG. 2(a) and(b), significant variations exist between simulations withdifferent column sizes. For granular columns with thesame initial aspect ratio, granular columns with largerrelative column size R i /d have longer run-out distances.This shows that the collapse of the granular column hasa significant size effect, which was also observed in the re-search of Warnett et al.[18] and Cabrera et al.[20]. How-ever, no quantitative studies has been made to univer-sally include different frictional coefficients and bound-ary conditions to describe the run-out behavior and fewresearchers pointed out the physical nature of the initialaspect ratio α other than obtaining it from dimensionalanalysis[21].Further, changing the relative column size also funda-mentally influences the shape of R − α eff . For simula-tions with the same R i /d , a threshold α c of α eff existsto divide the R − α eff relationship into two groups (FIG.2(a)). When α eff < α c , R approximately scales with α eff .When α eff > α c , R approximately scales with ( α eff ) . with a rather sharp division between the two. Here, thetransitional aspect ratio α c and the corresponding tran-sitional normalized run-out distance R c can be seen asthe critical aspect ratio and the critical run-out distance, respectively (FIG. 2(a)). Both α c and R c vary as wechange the size of the granular column. For instance,in FIG. 2(a), the transition happens at α c ≈ , R c ≈ R i /d = 2 .
5, and happens at α c ≈ , R c ≈ . R i /d = 17 . FIG. 3. (a) shows the relationship between transitional aspectratio, α c , and the relative column size, R i /d . (b) plots thetransitional normalized run-out distance, R c against R i /d We show that, in FIG. 2(a), a critical transition point( α c and the corresponding R c ) exists and also varies withdifferent relative size R i /d . Thus, R − α eff curve is dic-tated by the position of α c and R c in a way that R = f ( α eff − α c , R c , R i /d ) . (1)Both α c and R c decrease with an increase of R i /d . InFIG. 3, based on the relationship between α eff and R inFIG. 2(b), we plot the relationship between α c and R i /d and the relationship between R c and R i /d , respectively.Thus, we could write α c and R c as functions of the rela-tive column size, where, when R i /d approaches infinity,both α c and R c converge to α c ∞ and R c ∞ , as shown by α c = α c ∞ + a (cid:18) R i d (cid:19) b , (2a) R c = R c ∞ + a (cid:18) R i d (cid:19) b , (2b)where α c ∞ = 0 . R c ∞ = 0 .
732 are the fitted criticalaspect ratio and the corresponding critical run-out dis-tance, respectively, when the relative column size R i /d goes to infinity, and both parameters are fitted values.Also, a = 16, b = − . a = 5 .
4, and b = − . α c and R c arethe dashed curve in FIG. 3(a) and the dash-dot curve inFIG. 3(b).The power-law decay of both α c and R c with respectto R i /d inspires us to perform finite size scaling of therun-out distance of the collapse of granular columns. Asshown in FIG. 4(a) and the insert of FIG. 4(a), the nor-malized run-out distance, R indeed exhibits excellent fi-nite size scaling, suggesting that the transitional aspectratio, α c , is critical. All the normalized run-out distance FIG. 4. (a) shows the relationship between R ( R i /d ) . and A = ( α eff − α c ∞ )( R i /d ) − /ν , where ν = 0 .
75 in our analysis.The insert of FIG. (a) plot the same relationship in a log-log coordinate system. (b) shows the relationship between E ( R i /d ) . and A , where E = (cid:104) E k (cid:105) / (cid:104) E D (cid:105) is the ratio be-tween the time-averaged kinetic energy and the time-averageddissipated energy during the collapse of granular columns.The insert of FIG. (b) plot the same relationship in a log-logcoordinate system. The markers in these four figures are thesame as that in FIG. 2(b). data collapse nicely onto a master curve in the form[23] R = (cid:18) R i d (cid:19) − β /ν F r (cid:34) ( α eff − α c ∞ ) (cid:18) R i d (cid:19) /ν (cid:35) , (3)with the limiting scaling of the normalized run-out dis-tance being R ∼ α eff − α c ∞ , where ν = 1 .
33 and β = 0 .
25 are obtained to best collapse all the data,and β /ν ≈ . A = ( α eff − α c ∞ )( R i /d ) − /ν as the x − axis. Since the system is axisymmetric, the scalingsolution for the collapse of granular materials shows sim-ilar phenomena in the scaling solutions for connectiv-ity of two-dimensional continuous random networks[23],where the scaling parameters ν = 1 .
33 and β = 0 . α eff − α c ∞ , ξ/H i ∼ ( α eff − α c ∞ ) − ζ , (4)where ζ is a fitting parameter for the length scale ξ ,and ξ/H i denotes the degree of occupation of shearedgranular media along the height of initial granular col-umn. Later we will show that this length scale is notlimited to the normalized run-out distance. To checkwhether the finite size scaling and the length scale de-scribed by Eq. 3 and Eq. 4 are universal, we also per-form the finite size analysis of quantities related to energyconsumptions during the collapse of granular columns. For each simulation, we obtained the kinetic energyat time t , E k ( t ) and calculated the time averaged ki-netic energy using (cid:104) E k (cid:105) = (1 /τ ) (cid:82) τ E k ( t ) dt , where τ isthe terminating time of the granular column collapse.We then calculated the time averaged dissipated energy (cid:104) E D (cid:105) = (1 /τ ) (cid:82) τ [ E p ( t ) + E k ( t ) − E p (0)] dt , where E p ( t )is the potential energy of the system at time t and E p (0)is the initial potential energy. Then, the ratio betweentime-averaged kinetic energy and the time-averaged dis-sipated energy, E = (cid:104) E k (cid:105) / (cid:104) E D (cid:105) , can be calculated. InFIG. 4(b) and the insert of FIG. 4(b), we plot the rela-tionship between E ( R i /d ) β /ν and A , where β = 0 . β , and write this relationshipin the following form E = (cid:18) R i d (cid:19) − β /ν F E (cid:34) ( α eff − α c ∞ ) (cid:18) R i d (cid:19) /ν (cid:35) , (5)which demonstrates that the ratio between time-averagedkinetic energy and the time-averaged potential energyshows perfect finite size scaling with the same lengthscale as proposed by Eq. 4. The relationship between E ( R i /d ) β /ν and A follows a power-law scaling as shownin the insert of FIG. 4(b). FIG. 5. The relationship between ξ/H t and α eff − α c ∞ when ξ and H t are measured at the beginning of the granular collapse,where ξ is the correlation length scale, and H t is the heightof the system at the time we take the measurement. Weonly measured ξ for four different system sizes and plot theresults using circle markers. Meanwhile, the fitted power-lawrelationships are denoted using dashed lines. We further investigate the length scale ξ associatedwith the collapse of granular columns. We choose foursets of simulations with different relative system sizes.We consider grain pairs along the vertical direction atthe very beginning of the collapse when time t = 0.05 s,and define the corresponding correlation G ( z ), which de-scribes the correlation between particle pairs with verti-cal distance z , based on the equation proposed by Barkerand Mehta[2, 24], by G ( z ) = (cid:104) ∆ z i ∆ z j δ ( | z ij | − z ) Θ ( | t ij − / | ) (cid:105)(cid:104)| ∆ z i |(cid:105) , (6)where ∆ z i and ∆ z j are the vertical displacements ofparticle i and j after one time step (e.g. ∆ z i = v i ∆ t where v i is the velocity of particle i and ∆ t is the timestep), z ij is the vertical distance between two parti-cles, z is the vertical displacement between particles,and t ij = (cid:112) ( x i − x j ) + ( y i − y j ) is the horizontal dis-tance between two particles. Here, δ ( ) is a rectangu-lar function, where the function is equal to one when z − . d p < | z ij | ≤ z + 0 . d p and equal to zero else-where, and Θ( ) is the Heaviside step function. Thisdefinition ensures that the averages run over all displace-ments of sphere pairs in the vertical direction. Thecorrelation function G ( z ) shows how particles influenceeach other in z − direction. The displacement correla-tion function can be fitted with an exponential equation G ( z ) = A · exp( − z/ξ ), where ξ can be seen as a corre-lation length scale associated with the displacement cor-relation among particles during the collapse of granularcolumns, and A is a fitting parameter. The occupationof correlated grains across the height of a granular col-umn, represented by ξ/H t where H t here is the height ofthe column at the time when we measure the correlationlength scale (since ξ is measured at the very beginningof the collapse, H t ≈ H i ), implies the collective motionof particles during a collapse of granular columns. Thus,we plot the relationship between ξ/H t and α eff − α c ∞ inFig. 5, and determine that the ratio between correlationlength scale and the system height shows power-law de-cay as we increase α eff − α c ∞ , which is consistent withthe finite size scaling of both run-out behavior and ra-tio of kinetic and dissipated energies of granular columncollapses, and ζ = 0 .
89 in Fig. 4 best fits the power-law decay. When α eff approaches α c ∞ , ξ/H t starts todeviate from the power-law relationship. This is due tothe fact that, when the initial aspect ratio is too small,only several layers of particles present along the heightof a granular column. To further understand this be-havior, we have to continue increase the relative size ofthe column with millions of DEM particles, which is notcomputationally feasible under current condition. Summary .– Previous research concluded that we couldcombine the influence of initial and boundary conditionsand the initial column aspect ratio to determine the nor-malized run-out distance with a physics-based effectiveaspect ratio, α eff , which introduced a universal relation-ship to link the behavior of granular column collapses inthree different collapsing regimes (quasi-static, inertial,and liquid-like)[21]. In this paper, we further investi-gated the size effect associated with the collapse of ax-isymmetric dry granular columns based on the previousresearch. Our research is performed with DEM simula-tions of Voronoi-based spheropolyhedron particles. Wefound that the transition point in R − α eff space, whichdistinguishes the inertial collapse regime from the quasi-static collapse regime, varies as we change the relativesystem size R i /d . Both R c and α c experience power- law decay with respect to R i /d , which implies possiblefinite-size scaling for the normalized run-out distance R .Similar to the finite-size analysis of the jamming tran-sition of granular materials[25], where the stress scaleswith the solid fraction φ − φ c ∞ , we took the previouslydetermined effective aspect ratio α eff as the key param-eter, and discovered that both the run-out distance ofgranular column collapse and the energy consumption ofit follows strong finite-size scaling. Interestingly, the scal-ing parameters we discovered for the finite-size scalingof the run-out distance of granular column collapses aresimilar to those presented in percolation problems of two-dimensional random networks[23], which may imply thatthe behavior of granular column collapses is strongly in-fluenced by the contact network presented inside the col-umn during the collapse. Additionally, to better under-stand the scaling of granular column collapses, we furtheranalyze the correlation length scale at the very beginningof the collapse. Simulation results show that, as we in-crease α eff − α c ∞ , the length scale ξ/H t shows a power-law decay with scaling parameters ζ ≈ .
89. This studyis based on our previous work where we introduced aphysics-based dimensionless number (ratio between iner-tial effects and frictional effects) to describe the behaviorof granular column collapses. In this study, we furtherexpand our analysis to include the size effect, which iscrucial to applications in engineering, such as chemicalengineering, food processing, civil engineering. However,the granular system we are studying is still axisymmetric,and our preliminary results show intriguing phenomenawhen the cross-section of a granular column is no longeraxisymmetric. Further investigations will be presentedin future publications.The authors acknowledge the financial support fromWestlake Institute of Advanced Study and Westlake Uni-versity and thank the Westlake University Supercom-puter Center for computational resources and related as-sistance. H.E. Huppert acknowledges with gratitude thehospitality of his co-authors while he was at WestlakeUniversity ∗ Also at School of Engineering, Westlake University † [email protected][1] G. D. R. MiDi, On dense granular flows, Euro Phys J E , 341 (2004).[2] A. Mehta, Granular Physics (Cambridge UniversityPress, 2007).[3] T.-T. Vo, P. Mutabaruka, S. Nezamabadi, J.-Y. Delenne,and F. Radjai, Evolution of wet agglomerates inside in-ertial shear flow of dry granular materials, Phys. Rev. E , 032906 (2020).[4] L. Zhang, Y. Wang, and J. Zhang, Force-chain distri-butions in granular systems, Phys. Rev. E , 012203(2014).[5] P. Jop, Y. Forterre, and O. Pouliquen, A constitutive law for dense granular flows, Nature , 727 (2006).[6] M. Trulsson, B. Andreotti, and P. Claudin, Transitionfrom the viscous to inertial regime in dense suspensions,Physical Review Letters , 118305 (2012).[7] T. Man, Rheology of Granular-Fluid Systems and Its Ap-plication in the Compaction of Asphalt Mixtures , Ph.D.thesis, University of Minnesota (2019).[8] T.-T. Vo, Rheology and granular texture of viscoinertialsimple shear flows, Journal of Rheology , 1133 (2020).[9] T.-T. Vo, S. Nezamabadi, P. Mutabaruka, J.-Y. Delenne,and F. Radjai, Additive rheology of complex granularflows, Nature Communications , 10.1038/s41467-020-15263-3 (2020).[10] R. Zenit, Computer simulations of the collapse of a gran-ular column, Physics of Fluids , 031703 (2005).[11] E. L. Thompson and H. E. Huppert, Granular columncollapses: further experimental results, Journal of FluidMechanics , 177 (2007).[12] L. Lacaze and R. R. Kerswell, Axisymmetric granular col-lapse: a transient 3d flow test of viscoplasticity, PhysicalReview Letters , 108305 (2009).[13] L. Rondon, O. Pouliquen, and P. Aussillous, Granularcollapse in a fluid: role of the initial volume fraction,Physics of Fluids , 073301 (2011).[14] P.-Y. Lagr´ee, L. Staron, and S. Popinet, The granu-lar column collapse as a continuum: validity of a two-dimensional Navier–Stokes model with a µ (I)-rheology,Journal of Fluid Mechanics , 378 (2011).[15] G. Lube, H. E. Huppert, R. S. J. Sparks, and M. A.Hallworth, Axisymmetric collapses of granular columns,Journal of Fluid Mechanics , 175 (2004).[16] G. Lube, H. E. Huppert, R. S. J. Sparks, and A. Freundt, Collapses of two-dimensional granular columns, PhysicalReview E , 041301 (2005).[17] E. Lajeunesse, J. Monnier, and G. Homsy, Granularslumping on a horizontal surface, Physics of Fluids ,103302 (2005).[18] J. Warnett, P. Denissenko, P. Thomas, E. Kiraci, andM. Williams, Scalings of axisymmetric granular columncollapse, Granular Matter , 115 (2014).[19] Warnett and Jason, Stationary and rotational axisym-metric granular column collapse, University of Warwick(2014).[20] M. Cabrera and N. Estrada, Granular column collapse:Analysis of grain-size effects, Physical Review E ,10.1103/PhysRevE.99.012905 (2019).[21] T. Man, H. E. Huppert, L. Li, and S. A. Galindo-Torres,Universality and deposition morphology of granular col-umn collapses, arXiv preprint arXiv:2002.02146 (2020).[22] S. A. Galindo-Torres and D. Pedroso, Molecular dy-namics simulations of complex-shaped particles usingvoronoi-based spheropolyhedra, Physical Review E ,061303 (2010).[23] S. A. Galindo-Torres, T. Molebatsi, X.-Z. Kong,A. Scheuermann, D. Bringemeier, and L. Li, Scaling so-lutions for connectivity and conductivity of continuousrandom networks, Phys Rev E , 041001 (2015).[24] G. C. Barker and A. Mehta, Vibrated powders: Struc-ture, correlations, and dynamics, Phys. Rev. A , 3435(1992).[25] H. Liu, X. Xie, and N. Xu, Finite size analysis ofzero-temperature jamming transition under applied shearstress by minimizing a thermodynamic-like potential,Phys. Rev. Lett.112