epl draft

Crossover from quasi-static to dense flow regime in compressedfrictional granular media.

F. Gimbert1,3, D. Amitrano1 and J. Weiss2

1 Institut des Sciences de la Terre, CNRS-Universite Joseph Fourier, Grenoble, FRANCE, 1381 rue de la Piscine,BP 53, 38041 Grenoble Cedex 92 Laboratoire de Glaciologie et de Geophysique de l’Environnement, CNRS-Universite Joseph Fourier, Grenoble,FRANCE, 54 rue Moliere, BP 96, F-38402 Saint-Martin d’Heres Cedex3 Now at Seismological Laboratory, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125,USA

PACS 62.20.-x – Mechanical properties of solidsPACS 64.60.De – Statistical mechanics of model systemsPACS 91.60.Ba – Elasticity, fracture, and flow

Abstract –We investigate the evolution of multi-scale mechanical properties towards the macro-scopic mechanical instability in frictional granular media under multiaxial compressive loading.Spatial correlations of shear stress redistribution following nucleating contact sliding events andshear strain localization are investigated. We report growing correlation lengths associated toboth shear stress and shear strain fields that diverge simultaneously as approaching the transitionto a dense flow regime. This shows that the transition from quasi static to dense flow regimecan be interpreted as a critical phase transition. Our results suggest that no shear band with acharacteristic thickness has formed at the onset of instability.

Introduction. – The mechanical behavior of granu-lar materials is of wide concern, from natural hazard ingeological context to engineering applications. However,the evolution of properties towards the flowing instabilityis still partially understood.

In case of packing of non-frictional, hard (non-deformable), spherical particles loaded under shear, forcechains, i.e. heterogeneous distributions of contact forceson a scale much larger than the typical particle size, con-trol the mechanical response of the granular assembly [1].For these systems, the concept of jamming [2] providesa powerful framework to analyze the onset of granularflows. These assemblies of non frictional particles exhibitjammed states resisting small stresses without irreversibledeformation, whereas unjammed systems flow under anyapplied shear [3]. The jamming transition for such spheresat zero stress occurs at a critical value of the packing frac-tion φ [2].

We investigate here a different situation, consideringelastic (i.e. non hard) frictional disks loaded under multi-axial compression. This situation is relevant when study-ing geophysical instabilities (e.g. granular gouges withinfault zones, landslides,...) and differs from classical config-

urations used to study the jamming transition [2] mainlyin two ways. First, considering assemblies of frictionalgrains, the parameter that controls whether the grain as-sembly behaves as a jammed or an unjammed state is thefraction of non-rattler grains, i.e. the fraction of grainsthat carry forces, rather than density [4]. Secondly, in-stead of shearing the sample at constant volume, the com-pressive loading conditions here considered imply a con-fining pressure that prevents the non-rattler fraction toevolve freely. As a consequence and contrary to the studyof [4], a percolating strong force newtork remains in theflowing phase, called the dense flow regime [5].

In this letter, we investigate the transition from a quasi-static regime, i.e. a regime where the sample resists tothe applied stress by deforming infinitly slowly, towards adense flow regime, where inertia comes into play.

Loading mode. – We consider 2D compression testsunder multiaxial loading: the axial stress σ1 is increasedwhereas the radial stress σ3, i.e. the confining pressure,is kept constant (see Figure 1). Like this, the sample issheared by increasing the deviatoric stress τ = σ1 − σ3.

p-1

arX

iv:1

208.

1930

v3 [

cond

-mat

.sta

t-m

ech]

15

May

201

3

F. Gimbert et al.

hx

h Y

σ3 σ3

σ1

σ1

Fig. 1: Illustration of the multi axial loading configuration ona sample made of 225 grains (filled circles). The sample hasbeen replicated in all directions (unfilled circles).

Position of the problem. – Laboratory experimentshave been conducted with this multiaxial configuration oneither continuous rocks [6, 7] and discrete materials suchas sand [8] or synthetic analogous materials [9]. In contin-uous materials, the macroscopic instability as been firsttheoretically tackled by the use of the bifurcation the-ory, which considers a transition from an homogeneousto an heterogeneous deformation field materialized by thecreation of a perennial macroscopic shear band present-ing a characteristic value of thickness and spanning thewhole sample [10,11]. Within granular materials, comput-ing the deformation field over a large macroscopic strainwindow, those perennial shear bands appear and seem toshow characteristic sizes either in experiments [8] and sim-ulations [12]. However, this vision is counterbalanced bythe heteregenous and long range correlated kinematics ofquasi-static granular flow [13]: at which temporal and spa-tial scales and at which stage of the loading does the gran-ular assembly deform homogeneously? What are the rele-vant key features of the stress and strain fields associatedto the onset of macroscopic instability? Here we investi-gate this problem through numerical simulations.

Simulation approach. – Our simulations use theMolecular Dynamics discrete element method [14]. Two-dimensional granular assemblies of a number Ng of fric-tional circular grains are considered. To characterize sam-ple size effects [15], we performed 320 simulations withNg = 2500, 80 simulations with Ng = 10000 and 20 simu-lations with Ng = 45000. The results presented here con-cern 10000 grains samples. The grains areas are uniformlydistributed, setting the largest grain diameter Dmax suchthat Dmax = 3Dmin.

The dynamic equations are solved for each grain, whichinteract via linear elastic laws and Coulomb friction whenthey are in contact [16]. The normal contact force fn isrelated to the normal apparent interpenetration δ of thecontacts as fn = kn × δ, where kn is the normal contactstiffness coefficient. The tangential component ft of thecontact force is proportional to the tangential elastic rel-ative displacement, with a tangential stiffness coefficientkt. We set kt = kn. Neither cohesion between grains, norrolling resistance is considered. The Coulomb condition

|ft| ≤ µmicrofn, where µmicro is the grain friction coeffi-cient, requires an incremental evaluation of ft every timestep, which leads to some amount of slip each time one ofthe equalities ft = ±µmicrofn is reached. A normal vis-cous component opposing the relative normal motion ofany pair of grains in contact is also added to the elasticforce fn to obtain a damping of the dynamics.

An isotropic compression of dilute frictionless grainssets builds dense and highly coordinated initial packingsof density φi ≈ 0.85 and backbone coordination number,i.e. coordination number computed over grains that carryforces [17], z∗i = 2Nc/(Ng(1−x0)) = 4, where Nc is thetotal number of contacts and x0 the fraction of rattlersgrains. Then, multiaxial compression tests are performedsetting the particle friction to µmicro = 1.

The external mechanical loading is prescribed on thegrain assembly using periodic boundary conditions. A pe-riodic simulation cell of period h (see section 6.3.3 of [14])is considered. As the simulation cell is rectangular, the

linear operator h can be written as h =

(hx 00 hy

), where

hx and hy correspond to the size of the cell period in theradial and axial direction (see Figure 1). Then, stressesare prescribed solving the dynamic equations of motionof h, i.e. ensuring that the internal stress computed overthe whole grain assembly following equation 1 (see below)counterbalances the prescribed external stresses σ1 andσ3.

The axial stress σ1 is increased at constant rate by im-posing a stress increment δσtr1 at each discretisation time

interval tr =√

mminkn

/25, where mmin is the mass of the

lightest grain. This stress control loading mode avoidsstress relaxations and associated feedbacks that would beobtained under strain controlled loading, i.e. adjustingδσtr1 in order to axialy deform at constant rate ε1, whereε1 is the axial deformation. In this stress controlled case,stress and strain localization structures develop freely.The confining pressure σ3 is kept constant and sized bysetting the contact stiffness κ = kn/σ3 equal to 1000 [14].This value for κ allows to treat elasticity in grain contactsduring reasonable computational times while consideringa relatively low level of deformability of grains that isrelevent for application of geomechanics and geophysics.As examples, compression experiments performed on as-semblies of glass beads of approximate Young’s modulusE = 70 GPa submitted to 100 kPa of confining pressurelead to a κ-value of 700 while, from the knowledge of wavespeed velocities within a granitic Earth’s crust [18], valuesof κ ≈ 1.103 are expected between hundreds of meters toseveral kilometers depth.

Results. –

Macroscopic behaviour. Figure 2 shows the macro-scopic response of a granular sample loaded using δσtr1 =1.10−6σ3. To characterize the dynamical behaviour of thegranular packing, we compute the inertial number I, which

p-2

012345678

x 10−3

ε1

τ/σ3

(a)

0 0.5 1 1.5 2

10−5

10−4

10−3

Ine

rtia

l N

um

be

r

0

0.5

1

1.5

2

τ/σ

3

ε1

Dense flowQuasi−static

(b)

0 2 4 6 8x 10

−3

−2

−1

0

1

2

3x 10

−3

∆ S

/S0

Fig. 2: Evolution of macroscopic parameters during compres-sional testing for a sample of 10000 grains. Color dots corre-spond to color lines on Figures 4(Top) and 5(Top). Dashedlines materialize the limit between quasi-static and dense flowdeformation regimes.

corresponds to the ratio between inertial forces and im-posed forces and is defined as I = ε1

√m/σ3 [14], where

m is the average grain mass. Initially, I is of the order of10−6−10−5 (see Figure 2(a)). Then, when increasing τ to-wards 2σ3, while undergoing brutal fluctuations associatedto large plastic events, I remains lower than 10−4, which isoften considered as the upper bound for quasi-static condi-tions [14,19]. Hence, in the region delimited by τ = 0 andτ ≈ 2σ3, the sample undergoes quasi-static deformation.At values of τ larger than τc ≈ 2σ3, a brutal increase of Iof several orders of magnitude is observed, reaching valuesof the order of 10−3 − 10−2. This indicates the transitiontowards a dense flow regime, where inertia comes into play.This transition is also marked when looking at ε1 versusτ/σ3, where we can see that a drastic change of slope ofthe curve operates around τc (Figure 2(b)). For values ofτ larger than τc, the prescribed axial stress increment δσtr1induces a large amount of axial deformation.

Figure 2(b) also shows the surface variation ∆S/S0 =(hxhy)ε1−(hxhy)0

(hxhy)0of the granular assembly as a function of

ε1, where (hxhy)ε1 corresponds to the sample surface com-puted at a given value of axial deformation ε1 and (hxhy)0corresponds to the initial sample surface. We observe aninitial contracting phase materialized by the decrease of∆S/S0 until a peak of contraction is reached, after whichthe sample dilates continuously. This contracting phaseresults from elastic contacts and would no longer be ob-served in the limit of infinitely rigid grains, i.e. infinitlylarge values of κ [20]. However, in this case, the dense flowtransition is observed at negative values of ∆S/S0, i.e. ata value of packing fraction larger than the initial one.

Thus, the transition to dense flow regime is observedwhen τ reaches a critical value τc ≈ 2σ3, i.e. at a macro-scopic friction µmacro ∼ 0.5. At this transition, stress andstrain concentrations resulting from cooperative effects areexpected, triggering preferential weak zones where flow isfavoured. In the case of our samples made of circulargrains with no rolling resistance at grains contacts, thisleads to a softening of the whole granular assembly andthus to a macroscopic friction µmacro much smaller thanµmicro. According to this, studying the spatial structure

Strain mesh

Stress mesh

−20 −10 0 10 20−20

−10

0

10

20

X coordinate (in Dmax

)

Y c

oo

rdin

ate

(in

Dm

ax)

W=Wmacro

Fig. 3: Left: Delaunay (top) and modified Voronoi (bottom)tesselations for a polydisperse granular material. Right: Coarsegraining analysis on a 2500 grains sample.

of both stress and strain fields is a key point to under-stand the mechanisms that generate the macroscopic in-stability. We thus focus, in this study, on the response ofthe granular assembly to a small stress increment in termsof associated stress concentration and strain localizationstructures that form during mechanical loading.

Multi-scale analysis. We first characterize the spatialextent of regions of stress concentration by means of acoarse graining analysis [21, 22]: an averaged shear stressrate < τ > [23] is computed at different stages of mechani-cal testing (cf color dots on Figure 2) over a time window Tand over a broad range of spatial scales L, from the micro-scale corresponding to the scale of the mesh element, tothe macroscale corresponding to sample size. Subsystemsof the granular assembly are selected by means of squareboxes of size W (see Figure 3) and the average scale L is

computed as L = 1Nbox

∑Nboxk=1 Lk, where Nbox is the num-

ber of boxes of size W and Lk is the scale associated tothe box number k, computed as the square root of thesum of mesh element surfaces. The standard deviationof Lk-values is maximum for the smallest subsystem ofaverage size L = 1.1Dmax, and corresponds to 0.2Dmax

when considering the Voronoi triangulation used to com-pute stresses and to 0.13Dmax when considering the De-launay triangulation used to compute deformations (seeFigure 3). We checked that no overlap of scales Lk occursthrough successive values of W .

For a given assembly of grains lying within the box num-ber k, the stress tensor is computed as in [24] writing

σkij =1

sk

∑gk

∑ck

(rcki − rgki )fgkckj (1)

where sk = L2k is the surface associated to the grain as-

sembly, fgkckj is the jth component of the contact force

exerted on grain gk at contact ck, rcki is the ith componentof the position vector of ck, and rgki is the ith componentof the position vector of the center of mass of the grain gk.Considering two successive configurations, the stress ratetensor is obtained by differentiating the respective stresstensor components in time. The time resolution used isT =

√Ng×100× tr, which corresponds to the travel time

p-3

F. Gimbert et al.

10−210

−110

010

110

0

νγ = 1.3

L ( ∆νγ L δ + C)/L

sδ

<δ γ

>[(∆

ν γLδ +C

)/L

sδ ]−ργ

∆0.650.420.280.180.120.070.030.005 10

−110

010

110

2

100

L/Dmax

<τ>.

ρτ = 0.38

100

101

102

103

100

ντ = 1.3

<τ>

[(∆

ντL

δ+

C)/

Lsδ]−

ρτ

L (∆ν

τ Lδ + C)/L

s

δ.

∆ = 0.42

∆ = 0.005

100

101

102

100

L/Dmax

<τ>.

tT2T4T8T16T

100

101

102

100

L/Dmax

<τ>.

=t 4 T

= 16 Tt

Fig. 4: Multi-scale analysis performed on the shear stress ratefield τ . A selection of corresponding fields is shown on the rightside: an arbitrary color scale (not shown) has been chosen foreach snapshot. Top: < τ > versus L for decreasing values of ∆.Configuration locations on the stress-strain curve are shown onFigure 2. The timescale T is used to compute values of < τ >.The inset displays data collapse with respect to ∆ (equation 2).We find ντ = 1.3, C = 0.5 and δ = 1. Bottom: < τ > versusL at ∆ = 0.005 when increasing the timescale t from t = T tot = 16T . For graphical convenience, all computed values havebeen normalized by the ones computed at the micro-scale.

of elastic waves through the granular assembly [15].

Results are shown on Figure 4 (Top), from the earlystages of biaxial testing up to τc ≈ 2σ3. The curves areselected with respect to the control parameter ∆ definedas ∆ = τc−τ

τc, i.e. ∆ decreases as approaching τc. At

the early stages of macroscopic deformation, a decrease of< τ > with L is observed at small scales while for L-valueslarger than a crossover scale l∗τ a plateau is observed. Thismeans that shear stress rate fields are heterogeneous forl << l∗τ , and homogeneous for l >> l∗τ . Examples ofassociated fields are provided on the right hand side ofFigure 4: at ∆ = 0.42, the computed snapshot shows aroughly homogeneous stress field at large spatial scales, i.e.scales larger than l∗τ ≈ 3Dmax in that case (see crossoverscale observed on the corresponding curve), while the re-gion of stress localization observed on the top-left part ofthe snapshot exhibits a spatial extension of the order ofl∗τ ≈ 3Dmax. We interpret l∗τ as the associated correla-tion length [22]. As macroscopic deformation proceeds,l∗τ grows until reaching the entire size of the system at τcwhere a power law scaling < τ >∼ L−ρτ is observed, withρτ = 0.38. The cut-off remaining on the scaling at ∆→ 0is a finite size effect [15]. At that stage, large and stronglylocalized structures characterize the shear stress field (seesnapshot computed at ∆ = 0.005). These results suggesta progressive structuring of the stress field as approach-

ing the transition to the dense flow regime, associated tothe divergence of the correlation length l∗τ . It can be veri-fied from a collapse analysis (inset of Figure 4(Top)) thatl∗ = l∗τ diverges as

l∗ ∼ Lδs∆νLδs + C

(2)

where Ls is the square root of the sample area andν = ντ = 1.3 ± 0.1 is the exponent of divergence. Pa-rameters δ and C characterize the finite size effect [15]. Asimilar analysis performed on other moments < τ q > ofthe shear rate [15] confirms the divergence of l∗τ ∼ ∆−ντ

at ∆ → 0, and reveals the multi-fractality of the shearrate field at the critical point. These particular featuresof the shear stress field are observed only at the specifictimescale t = T corresponding to the travel time of elasticwaves. This multi-scale behaviour is no longer observedat larger timescales (Figure 4(Bottom)), as a clear depar-ture from power law is observed for t > T , associatedto the progressive homogenization of the correspondingfields, i.e. decrease of l∗τ , as t increases (see also associ-ated snapshots). Hence, the multi-scale properties of theshear stress rate field are only observed at the time scalecorresponding to the time of propagation of the elastic in-formation throughout the sample. Beyond this time, a lossof scaling properties is observed, explained by the super-position of several uncorrelated events in time, consistantwith a spatially correlated stress structure associated withlittle memory, limited to the travel time of an elastic wave.

To study whether similar observations can be reportedon the shear strain field, we consider a delaunay triangula-tion performed on the grain centers (Figure 3), after hav-ing removed the rattlers grains from the grain set. Then,we compute the partial derivatives at the mesh scale asεij = 1/2(∂ui/∂xj +∂uj/∂xi), where (u1, u2) and (x1, x2)are respectively the incremental displacements and spatialcoordinates of grain centers. The coarse graining analysisis performed similarly than previously for stresses, hereby averaging partial derivatives at corresponding spatialscales. An average shear strain rate [23] is thus obtainedas a function of L. While not shown here, if one usesthe constant timescale T to compute incremental displace-ments, the correlation length associated with the shearstrain rate field does not diverge as approaching the tran-sition to the dense flow regime. Thus, the structure ofthe total strain field does not form simulteanously in timewith the stress field. Intuitively, this would be the caseif one would consider only the elastic component of thestrain. Here, for Ng = 10000, ∼ 104 stress incrementsare prescribed during the propagation time of an elasticwave throughout the sample. Hence, a multitude of con-tacts, in our case about 5% of the whole contact network,are then sliding, although elastic interactions did not havetime to travel across the entire sample.Despite this, a pro-gressive structuring of the shear strain field is observedwhen considering constant macroscopic deformation win-dows δε1 = δεp = 1.10−5 to compute the scaling of < δγ >

p-4

10−210

−110

010

110

0

νγ = 1.3

L ( ∆νγ L δ + C)/L

sδ

<δ γ

>[(∆

ν γLδ +C

)/L

sδ ]−ργ

∆0.650.420.280.180.120.070.030.005 10

−110

010

110

2

100

L/Dmax

<δ γ

>ρ

γ = 0.24

10−2

10−1

100

101

100

νγ = 1.3

L (∆ν

γ Lδ + C)/L

s

δ<δ γ

>[(

∆ν

γLδ+

C)/

Lsδ]−

ργ

∆ = 0.42

∆ = 0.005

100

102

100

L (in Dmax

)

<δ γ

>

δ ε1

δ εp

2 δ εp

4 δ εp

8 δ εp

16 δ εp

10−1

100

101

102

100

L/Dmax

<δ

γ>

10−2

10−1

100

100.1

100.3

100.5

αγ = 0.4

L (ξα

γ Lδ + C)/L

s

δ<δ

γ>[(

ξαγL

δ +C

)/L

sδ]−

ργ

=δε1 2 δεp

=δε1 εp16 δ

Fig. 5: Multi-scale analysis performed on the incremental shearstrain field δγ. A selection of corresponding fields is shown onthe right side: an arbitrary color scale (not shown) has beenchosen for each snapshot. Top: < δγ > versus L for decreasingvalues of ∆ towards the critical point. The deformation scaleδε1 = δεp = 1.10−5 is used to compute values of < δγ >. Theinset displays data collapse with respect to ∆ (equation 2). Wefind νγ = 1.3, C = 0.5 and δ = 1. Bottom: < δγ > versus L at∆ = 0.005 when increasing δε1 from δε1 = δεp to δε1 = 32δεp.

The inset displays data collapse with respect to ξ =δε1−δεpδεp

(equation 3). We find αγ = 0.4. For graphical convenience, allcomputed values have been normalized by the ones computedat the micro-scale.

(Figure 5). A divergence of the correlation length l∗γ sim-ilar to the one observed on the shear stress rate field isobtained as approaching the transition to the dense flowregime, as we find νγ = ντ = 1.3 from a collapse analysis(equation 2). When considering larger macroscopic defor-mation windows δε1 > δεp, the multi-scale properties ofthe deformation field are no longer observed. This obser-vation is in agreement with the shrinkage of the distribu-tions of the fluctuation velocities at increasing timescalesobserved by [13]. In this study, δε1 = δεp is the char-acteristic deformation value we need to consider in or-der to observe on the deformation field the critical be-haviour already reported previously on the stress field atthe timescale T of an elastic wave propagation. As thematerial softens when τ increases, the corresponding timeof integration at constant deformation window δε1 = δεpdecreases as the critical point is approached. This timecan either be smaller or larger than the elastic wave trav-eling time T , depending on the imposed loading rate δσtr1 .However, whatever the loading rate considered, δε1 = δεpremains equal to 1.10−5, pointing out that a given amountof plastic activity has to operate in order to observe multi-scale properties within the incremental shear strain field.As the correlation lengths l∗τ and l∗γ diverge the same way,

0.8

0.85

0.9

0.95

1

<Q

(δ ε

1)>

10−8

10−7

10−6

10−5

10−4

10−3

10−4

10−2

100

δ ε1

<χ

4(δ

ε1)>

β = 1.8

Fig. 6: Susceptibility analysis performed in the quasi-static re-gion on the sliding contacts belonging to the major (red curves)and minor (black curves) network. The vertical dashed line in-dicates the deformation value δεp = 1.10−5.

the progressive structuring of the strain field is probablydirectly related to the progressive structuring of the stressfield.

All the presented scalings are undifferently obtainedwhatever the loading rate δσtr1 considered as soon as itis smaller than δσtr1 = 1.10−6. Considering larger valuesfor δσtr1 , i.e. for example δσtr1 = 5.10−6, no specific pointmaterialized by the divergence of the correlation lengthcan be reported [15]. Indeed, the mechanical behaviour ofthe granular assembly is in that case related to a denseflow regime from the initial stages of deformation, prov-ing that the scalings observed here are associated to thetransition from a quasi-static to a dense flow regime ofdeformation.

An understanding of the characteristic value δεp canbe obtained using a four-point dynamic susceptibilityχ4 [25,26] analysis on the inter-particles contact network.From a contact configuration that we refer as “initial”,selected at a value of axial deformation denoted εinit1 , wecompute the self-overlap order parameter Qεinit1

(δε1) =1Nc

∑Nci=1 wi, where Nc is the number of contacts that are

not sliding in the initial configuration and wi is a step-function cutoff that equals 1 if no sliding event has beenrecorded on contact i over the whole deformation windowεinit1 rightarrow epsiloninit1 + δε1, and 0 otherwise. Thefirst two moments Q(δε1) =< Qεinit1

(δε1) > and χ4(δε1) =

Nc[< Qεinit1

(δε1)2 > − < Qεinit1(δε1) >2

]of Qεinit1

(δε1)(calculated from sample-to-sample fluctuations) are thencomputed in the quasi-static region (τ < τc). Doing this,we evaluate from an initial configuration the number andthe associated spatial heterogeneity of sliding events nu-cleation as axial deformation increases. Figure 6 shows< Q(δε1) > and < χ4(δε1) >, where < . > here meansan average over all the values of ∆ (since no significantvariation of Q(δε1) and χ4(δε1) is observed in the quasi-static region) computed by considering separately the ma-jor and minor force networks. The major force network isdefined by selecting contact forces larger than the aver-age. By construction, < Q(δε1) > is initialy equal to 1.As δε1 increases, < Q(δε1) > decreases but never reaches0, meaning that a considerable amount of contacts never

p-5

F. Gimbert et al.

slide. About 35% (respectively 55%) of the contacts ofthe minor (respectively major) network did not slide atthe end of the test, meaning that the permanent deforma-tion is extremely localized and that rigid bodies remainthroughout the whole test [27]. The value of < χ4(δε1) >indicates, with respect to increasing axial deformation, thevariability in the nucleation of new contact slidings. Atlow values of δε1, it increases as < χ4(δε1) >∼ δεβ1 withβ = 1.8, meaning that spatially correlated sites of contactsliding events are nucleating. As δε1 exceeds the thresh-old value δεp = 1.10−5, < χ4(δε1) > saturates, as all thespatially correlated contacts located close to the coulombcriteria, i.e. susceptible to slide, have been destabilized.At this stage, only 4% (respectively 3%) of contacts haveslided at least one time in the minor (respectively major)network.

To conclude, in dense granular assemblies, incremen-tal stress and strain fields are both characterized by agrowing correlation length that diverges as approachingthe onset of macroscopic instability, which can thereforebe identified as a critical point. A similar behavior hasbeen reported in compressive failure of continuous mate-rials [22, 28]. We interpret these stress and strain specificstructures as resulting from dynamic stress redistributionsinduced by the local dissipation of elastic energy materi-alized by contact slidings. At macroscopic instability, alocal contact sliding event induces correlated elastic stressperturbations up to the scale of the whole granular assem-bly. These features can only be observed when carefullyexamining characteristic timescales for stresses, and char-acteristic macroscopic strain increments for strains. Thesecharacteristic timescales may drastically be affected whenconsidering different inital packing properties of the gran-ular assemblies, e.g. considering low coordinated and/orloose inital samples, which has not yet been investigatedin the present study.

The last question that arises is to whether a limit indecreasing correlation length l∗γ on the shear strain fieldis reached at ∆ → 0 for values of δε1 much larger thanδεp, which would characterize the thickness of a perennialmacroscopic shear band potentially formed at the onset ofinstability. To investigate this, we hypothesize that, closeto the critical point (∆ → 0), l∗ varies as

l∗γ ∼Lδs

Lδsξαγ + C

(3)

where ξ =δε1 − δεp

δεpand αγ is the exponent of diver-

gence with respect to ξ. This hypothesis is tested from acollapse analysis (inset of Figure 5). We find αγ = 0.4.This shows that l∗γ keeps decreasing as the considered de-formation window size δε1 is increased, showing that thecorrelation length only depends on the value of δε1 andthat no intrinsic scale of saturation, potentially associatedto a shear band thickness, can be identified at the onsetof macroscopic instability.

∗ ∗ ∗

We thank Gael Combe for having provided the discreteelement model and for fruitfull discussions. We thank JeanBraun for having provided efficient routines to computeVoronoi tesselations. All computations were performed atSCCI-CIMENT Grenoble.

REFERENCES

[1] M.E. Cates, J.P. Wittmer, J.P. Bouchaud, andP. Claudin, Phys. Rev. Lett., 81 (1998) 9.

[2] A. Liu and S. Nagel, Nature, 396 (1998) 21.[3] G. Combe and J-N. Roux, Phys. Rev. Lett., 85 (2000)

3628.[4] D. Bi, J. Zhang, B. Chakraborty, and R.P.

Berhinger, Nature, 480 (2011) 355.[5] O. Pouliquen and F. Chevoir, C.R. Physique, 3 (2002)

163.[6] K. Mogi, J. Geoph. Res., 72 (1967) 5117-5131.[7] B. Haimson and C. Chang, Int. J. Rock Mech. Min.

Sciences, 17 (2000) 285-296.[8] J. Desrues and G. Viggiani, Int. J. Numer. Anal.

Meth. Geomech., 28 (2004) 279.[9] S. A. Hall, D. M. Wood, E. Ibraim, and G. Viggiani,

Granular Matt., 12 (2010) 1-14.[10] J.W. Rudnicki and J.R. Rice, J. Mech. Phys. Solids,

23 (1975) 371.[11] B.C. Haimson and J.W. Rudnicki, J. Struct. Geol., 32

(2010) 1701.[12] A. Tordesillas, Phil. Mag., 87 (2007) 32.[13] F. Radjai and S. Roux, Phys. Rev. Lett., 89 (2002)

064302.[14] F. Radjai and F. Dubois, Discrete Numerical Modeling

of Granular Materials, edited by Wiley-ISTE 2011.[15] See supplementary material for details and extended re-

sults.[16] P.A. Cundall and O.D.L. Strack, Geotechnique, 29

(1979) 47.[17] I. Agnolin and J-N. Roux, Phys. Rev. E., 76 (2007) 6.[18] T.M. Brocher, BSSA, 98 (2008) 2.[19] G.D.R. Midi, Eur. Phys. J. E., 14 (2004) 341.[20] G. Combe and J-N. Roux, Deformation Charactersitics

of Geomaterials, 3me Symposium sur le Comportementdes sols et des roches tendres, Lyon, 22-24 Septembre2003, In di Benedetto et al., (2003) 1071.

[21] D. Marsan, H. Stern, R. Lindsay, and J. Weiss,Phys. Rev. Lett., 93 (2004) 178501.

[22] L. Girard, D. Amitrano, and J. Weiss, J. Stat. Mech.,(2010) P01013.

[23] the invariant x, where x represents τ for the shear stressfield or γ for the shear strain field, is computed as x =xI − xII , where xI and xII are the principal componentsof the stress or strain tensor.

[24] E. DeGiuli and J. McElwaine, Phys. Rev. E, 84 (2011)041310.

[25] A. R. Abate and D. J. Durian, Phys. Rev. E, 76 (2007)021306.

[26] A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J.Durian, Nature Phys., 3 (2007) 260.

p-6

[27] K. Szarf, G. Combe, and P. Villard, Powder Tech.,208 (2011) 239.

[28] L. Girard, J. Weiss, and D. Amitrano, Phys. Rev.Lett., 108 (2012) 225502.

p-7