Critical scaling near the yielding transition in granular media

Abram H. Clark,1 Mark D. Shattuck,2 Nicholas T. Ouellette,3 and Corey S. O’Hern1, 4, 5

1Department of Mechanical Engineering and Materials Science,Yale University, New Haven, Connecticut 06520, USA2Benjamin Levich Institute and Physics Department,

The City College of the City University of New York, New York, New York 10031, USA3Department of Civil and Environmental Engineering,Stanford University, Stanford, California 94305, USA

4Department of Physics, Yale University, New Haven, Connecticut 06520, USA5Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA

(Dated: June 28, 2017)

We show that the yielding transition in granular media displays second-order critical-point scal-ing behavior. We carry out discrete element simulations in the overdamped limit for frictionless,purely repulsive spherical grains undergoing simple shear at fixed applied shear stress Σ in two andthree spatial dimensions. Upon application of the shear stress, the systems, which were originallyisotropically prepared, search for a mechanically stable (MS) packing that can support the appliedΣ. We measure the strain γms before each system finds such an MS packing and stops. We showthat the density of MS packings (∝ γ−1

ms ) obeys critical scaling with a length scale ξ that divergesas |Σ−Σc|−ν , where ν ≈ 1.5. Above the yield stress (Σ > Σc), no MS packings that can support Σexist in the infinite-system limit. The MS packings obtained by shear possess an anisotropic fabricand loading of grain-grain contacts. In finite systems, when Σ > Σc, MS packings are characterizedby a normal stress difference that also obeys critical scaling. These results shed light on granu-lar rheology, shear jamming, and avalanches, which display nontrivial system-size dependence nearyielding.

Granular materials are composed of macroscopic grainsthat interact via dissipative contact forces. They areathermal, and thus the constituent grains do not move inthe absence of external forces. Granular media possess anonzero yield stress, below which they are solid-like andabove which they flow. Other particulate, yield-stressmaterials include foams [1], emulsions [2], and colloidalglasses [3] and gels [4]. Although these systems possess arange of other interactions, steric repulsion among parti-cles often plays a dominant role in determining the sys-tem’s mechanical response.

Yield and flow of granular media are controlled bythe ratio Σ = τ/p of the applied shear stress τ to thepressure p, where p is small compared to the stiffnessof grains [5, 6]. In general, grains always move in re-sponse to applied forces [7–9]. When the applied stressis above the yield stress (Σ > Σc), grains flow indefi-nitely, while for stresses below the yield stress (Σ < Σc)grains move temporarily until finding a nearby mechan-ically stable (MS) packing. For frictionless grains, MSpackings generated via shear are not characterized bychanges in the packing fraction φ [9], as in the typicaljamming scenario [10–13]. Even with friction, after aninitial transient regime, φ is similar between sheared [5]and compressed [14, 15] MS packings, suggesting thatcontact structure, not φ, plays a dominant role in deter-mining Σc. Thus, Σc represents the shear stress abovewhich MS packings no longer exist in the infinite-systemlimit. However, accurately determining this value is chal-lenging, since the dynamics at Σ ∼ Σc are often tempo-rally heterogeneous and spatially cooperative [16–21].

In this Letter, we show that the density of MS pack-ings for sheared frictionless granular systems vanishes forΣ > Σc ≈ 0.1085 according to second-order critical scal-ing with a diverging length scale ξ ∝ |Σ − Σc|−ν , whereν ≈ 1.5. For Σ > Σc, MS packings exist for finite systemsof characteristic size L, but these states vanish as L/ξ in-creases. For Σ < Σc, large systems (L > ξ) are equivalentto uncorrelated compositions of small systems. MS pack-ings at finite Σ are characterized by anisotropy in boththe stress and contact fabric tensors, and for Σ > Σc weobserve a normal stress difference that also obeys criticalscaling. Our results may help to explain recent stud-ies [21–25] showing that accurately modeling granularflows requires a cooperative length scale that grows asa power law in Σ−Σc. Moreover, our results are relevantto a broad range of amorphous solids, where second-ordercritical scaling has been observed near yielding [3, 26–28].

We use discrete element method simulations in two(2D) and three (3D) spatial dimensions to apply stress-controlled simple shear to systems composed of friction-less bidisperse disks in 2D (2:1 by number with size ratio1:1.4) and spheres in 3D (2:1 by number with size ra-tio 1:1.2) in the overdamped limit. We show the resultsfor 3D in the main text, and the corresponding resultsfor 2D are provided in the Supplemental Material. InFig. 1 (a), we show a schematic of our simulations. Thelateral directions x and z (z is not present in 2D) areperiodic with nondimensional length L = l/D, where l isthe length of the box edge and D is the small grain diam-eter. We study N = L3 grains (from L = 3, N = 27 toL = 12, N = 1728) that interact via purely repulsive, lin-

2

(a)

(b) (c)

(d)

FIG. 1. (a) A schematic of the simulation procedure with thex-, y-, and z-directions indicated. We first create MS packingsunder only a fixed normal force per area −py. We then applya shear force per area τ x and search for an MS packing at agiven Σ = τ/p and nondimensional length L = l/D, wherel is the box edge length and D is the small grain diameter.(b-c) Probability density functions P (γms) for the strain γms

between the initial and final MS packings for (b) Σ < Σc and(c) Σ > Σc and several system sizes. (d) The average strain〈γms〉 between initial and final MS packings plotted versus Lover a range of Σ, where Σc ≈ 0.1085. The solid black lineshows 〈γms〉 ∝ L0.6.

ear spring forces with a characteristic stiffness K. We setK > 103p/D and apply viscous drag forces proportionalto the velocity with damping coefficient B > 5

√mpD,

where m is the grain mass, to the top plate and to eachgrain. With these parameter values, the simulations arein the limits of large grain stiffness and low inertial num-ber [5, 6, 22]. The top plate consists of a grid of rigidlyconnected small particles with gaps that are small enoughto prevent bulk particles from passing through the topplate, but the plate surface is sufficiently rough to pre-vent slip. A no-slip condition is also maintained at thelower boundary.

We begin the simulations with grains placed sparselythroughout the domain and then apply a force per area−py to the top plate and allow the system to compactuntil it reaches an MS packing. The top plate is allowedto shift laterally in the x- and z-directions, such thatthere are no net lateral forces on the plate in the initialMS packing at Σ = 0. We then add a shear force perarea τ x to the top plate and allow the system to evolve.The top plate can move in all coordinate directions untilan MS packing is found such that −py is balanced by anet upward force from the interior grains and τ x is bal-anced by a net lateral force from the interior grains. Wedefine the shear strain γms as the total distance the topplate travels in the x-direction divided by the average ofthe initial and final y-positions of the top plate. We notethat the packing fraction does not change significantly inresponse to the applied stress Σ as shown in the Supple-mental Material.

The statistics of γms allow us to infer the density of MSpackings at each Σ and L. In Fig. 1 (b) and (c), we showthe probability density function P (γms) for two illustra-tive values of Σ over a range of system sizes L, obtainedusing 200 initial MS packings for each L. For small Labove and below Σc, the distributions are roughly expo-nential, P (γms) ≈ 〈γms〉−1 exp(−γms/〈γms〉). This formis indicative of an underlying physical process resemblingan absorption process [29], where the mean travel dis-tance 〈γms(Σ, L)〉 is inversely proportional to the densityof absorbing MS packings as the system is sheared. Fig-ure 1(d) shows 〈γms〉 plotted versus L over a range of Σ,which shows that 〈γms〉 increases monotonically with Σ.When Σ ∼ Σc ≈ 0.1085, then 〈γms〉 increases roughly asa power law in L with exponent 0.6 (solid black line).When Σ < Σc for large L, 〈γms〉 becomes independent ofL and P (γms) develops a peak at γms > 0. For Σ > Σc,P (γms) remains exponential-like for large L, and 〈γms〉increases strongly with increasing L.

In Figure 2(a), we show that the curves in Fig. 1(d)can be collapsed by plotting them in terms of the scaledvariables L−1/|Σ − Σc|ν and 〈γms〉−1/|Σ − Σc|β . Thus,the density of MS states 〈γms(Σ, L)〉−1 obeys the critical

3

(a)

(b)

FIG. 2. (a) The same data shown in Fig. 1 (d) (with the samesymbol and color conventions) collapses for 0.07 < Σ ≤ 0.2when plotted using the scaled variables 〈γms(Σ, L)〉−1/|Σ −Σc|β and L−1/|Σ−Σc|ν , where β and ν are scaling exponentsgiven in Table I. (b) 〈γms(Σ, L)〉−1 versus Σ for different L.The dashed line gives the large-system limit implied by Eq. (1)and the corresponding plot in (a).

scaling relation

〈γms(Σ, L)〉−1 = |Σ− Σc|βf±(

L−1

|Σ− Σc|ν

), (1)

confirming the existence of a correlation length ξ ∝ |Σ−Σc|−ν that grows near the yielding transition. f± arethe critical scaling functions for Σ > Σc and Σ < Σc,Σc ≈ 0.1085 is the critical shear stress, and β ≈ 0.9 andν ≈ 1.5 are critical exponents associated with 〈γms〉−1

and the inverse system size L−1, respectively. Near thecritical point, both branches converge to a power lawrelation with slope β/ν ≈ 0.6, corresponding to the solidblack line in Fig. 1(d). The exponents β and ν are thesame in both 2D and 3D. (See Supplemental Material andTable I.) We note that ν ≈ 1.5 in our stress-controlledsimulations differs from scaling exponents obtained fromprevious rate-controlled [28] and volume-controlled [12]studies of sheared granular media. Note that the scalingform breaks down sufficiently far below the critical point(i.e. for Σ < 0.07).

As L−1/|Σ − Σc|ν becomes small (i.e., L > ξ), f−reaches a plateau. This plateau corresponds to the char-acteristic behavior when Σ < Σc and L is large shownin Fig. 1, namely the flattening of 〈γms(L)〉 and the de-velopment of a peak in P (γms) at γms > 0. We inter-pret this behavior as spatial decorrelation, where largesystems can be approximated as compositions of uncor-related subsystems. When Σ < Σc and L > ξ, P (γms)is consistent with the composition of ∼ (L/ξ)3 exponen-tially distributed random variables, yielding a distribu-tion that is peaked at γms > 0 as in Fig. 1(b), with amean that is independent of L/ξ. Additionally, f+ is fi-nite for L < ξ but tends to zero for small L−1/|Σ−Σc|ν ,meaning that MS packings exist for Σ > Σc and L < ξ,but these packings vanish as L grows above ξ. Figure 2(b) shows the density of MS states 〈γms(Σ)〉−1 over arange of system sizes L, as well as the large-system limitimplied by Eq. (1) and Fig. 2(a).

We quantify the structure of MS packings using thestress and contact fabric tensors,

σαλ =1

V

∑i6=j

rijα Fijλ (2)

Rαλ =1

N

∑i 6=j

rijα rijλ

||rij ||2. (3)

Here, α and λ are Cartesian coordinates, V = l3

is the system volume, rijα is the α-component of theseparation vector between grains i and j, and F ijλ isthe λ-component of the intergrain contact force. Thesum over i and j includes all pairs of contacting parti-cles (excluding grain-wall contacts). These tensors pos-sess eigenvalue-eigenvector pairs {Rk,Rk} and {σk,σk},where the index k = 1, 2, and 3 in 3D (or k = 1, 2 in 2D).The calculation of the stress and fabric tensors for a sin-gle grain is illustrated in Fig. 3 (a) and (b). Anisotropyin these tensors, i.e. when the eigenvalues are unequal,arises from preferential orientation and loading of grain-grain contacts [30, 31].

From continuum mechanics, force balance requires s =σ · n, where s = τ x−py is the applied force per area andn = y is the unit vector normal to the top surface. Thisimplies σxy = σyx = −τ , σyy = p, and σyz = σzy = 0,while σxx, σxz = σzx, and σzz are unconstrained. Fig. 3shows the ensemble averages of the stress tensor ele-ments obtained from simulations, and the distributionsare provided in the Supplemental Material. As expected,we find that σxy/σyy = −Σ, as shown in Fig. 3 (c),and σyz/σyy = 0 (Supplemental Material). In addition,σxz/σyy ≈ 0 (Supplemental Material), which is expectedfrom symmetry, but not determined by force balance.Fig. 3 (d) shows that Rxy/Ryy = −aΣ with a ≈ 0.4. InSupplemental Material, we show that Ryz/Ryy = 0 andRxz/Ryy = 0 within numerical error.

The unconstrained normal stresses σxx and σzz andthe corresponding fabric tensor elements Rxx and Rzz

4

(a) (b)

(c) (d)

FIG. 3. (a,b) Close-up of MS packings illustrating features ofthe (a) stress and (b) fabric tensors in Eq. (3). Eigenvalue-eigenvector pairs from the sum over the contacts (blue andgreen arrows) for the center grain i are denoted with black ar-rows, where the magnitudes of the arrows are proportional tothe eigenvalues and the directions are along the eigenvectors.The angle θ′ denotes rotation of the larger eigenvector awayfrom the compressive direction. (c,d) σxy/σyy and Rxy/Ryyplotted versus Σ, where different colors represent different sys-tem sizes L, showing σxy/σyy = −Σ and Rxy/Ryy = −aΣ,with a ≈ 0.4, for all L.

depend on Σ and L. We show σxx/σyy − 1 ≡ 2εx inFig. 4 (a) and Rxx/Ryy − 1 ≡ 2ρx in Fig. 4 (b). ForΣ < Σc, εx and ρx begin at some finite value and tendto zero in the large-system limit. For Σ > Σc, εx andρx increase with Σ. Both quantities obey critical scalingrelations. For the stress anisotropy, we find

εx(Σ, L) = |Σ− Σc|∆h±(

L−1

|Σ− Σc|ν

), (4)

as verified in Fig. 4(c). The critical exponent for εx is∆ ≈ 1.5, and Σc ≈ 0.1085 and ν ≈ 1.5 are the same as inEq. 1 for 〈γms(Σ, L)〉−1. In the Supplemental Material,we show that εz, ρx, and ρz also obey similar scalingrelations with identical exponents in both 2D and 3D.

Thus, the ensemble-averaged stress tensor of MS pack-ings in 3D at a given Σ is

σ = P

1 + 2εx −Σ 0−Σ 1 00 0 1 + 2εz

. (5)

We first consider only the x-y plane, which is decoupledfrom z in Eq. (5), and its eigenvalue-eigenvector pairs

(a) (b)

(c)

FIG. 4. (a,b) The normal anisotropies in (a) the stress tensor,σxx/σyy − 1 ≡ 2εx and (b) the fabric tensor, Rxx/Ryy − 1 ≡2ρx are plotted versus Σ for different values of L. (c) Thesame data from panel (a), plotted as the scaled normal stressanisotropy 2εx/|Σ − Σc|∆ versus scaled inverse system sizeL−1/|Σ−Σc|ν . The dashed, black lines in panels (a) and (b)show the large-system limit that is implied by the collapseddata in (c) and additional data in Supplemental Material.

TABLE I. The critical stress Σc and critical exponents ν, β,and ∆ shown in Figs. 2 (a) and 4 (c) and Eqs. (1) and (4) in 2Dand 3D. Errors are estimated by varying the exponents andexamining how well the data collapse in Figs. 2(a) and 4(c).

d Σc ν β/ν ∆/ν

2 0.1085± 0.005 1.5± 0.2 0.6± 0.05 1± 0.13 0.1125± 0.005 1.5± 0.2 0.6± 0.05 1± 0.1

{σ1,σ1} and {σ2,σ2}. The internal stress anisotropy isΣi = τi/pi, where pi = (σ1 + σ2)/2 and τi = (σ1 − σ2)are the internal pressure and shear stress, respectively.From the form of the stress tensor in Eq. (5), Σi =√

Σ2 + ε2x/(1 + εx), and σ1 is oriented at an angle thatdeviates from the compression direction by an angle θ′,as shown in Fig. 3 (a). An expansion of Σi for small εxgives θ′ = εx/(2Σ) + O[(εx/Σ)3]. Thus, when εx = 0,Σi = Σ and σ1 and σ2 are aligned with the compres-sion and dilation directions, respectively. However, Σi isminimized by a positive, nonzero value of εx = Σ2 withΣmini = Σ/

√1 + Σ2.

5

Thus, in finite-sized systems, MS packings at Σ > Σcare characterized by a symmetry breaking, where theeigenvectors of the internal stress and fabric tensors arerotated by an angle θ′ from the principal axes of the ap-plied shear deformation. MS packings with θ′ > 0 may bepreferable in finite systems since those with large Σi arelikely rarer in configuration space, and θ′ > 0 reduces Σi.The excess lateral forces can also be interpreted as weakforces that buttress force chain networks. However, inthe large-system limit, symmetry dictates that the fab-ric and stress tensors should be aligned with the axesof the applied shear deformation, and thus MS packingswith nonzero θ′ do not occur. We argue that the criticalscaling behavior near the yield stress arises from a com-bination of (1) the decreasing density of MS packings atlarge Σi and (2) a propensity for the principal axes of theinternal stress and fabric tensors of MS packings to berotated relative to the axes of the applied deformation infinite-sized systems.

This research was sponsored by the Army ResearchLaboratory and was accomplished under Grant NumbersW911NF-14-1-0005 and W911NF-17-1-0164 (A.H.C.,N.T.O., and C.S.O.). The views and conclusions con-tained in this document are those of the authors andshould not be interpreted as representing the official poli-cies, either expressed or implied, of the Army ResearchLaboratory or the U.S. Government. The U.S. Govern-ment is authorized to reproduce and distribute reprintsfor Government purposes notwithstanding any copyrightnotation herein. M.D.S. acknowledges support from theNational Science Foundation Grant No. CMMI-1463455.

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