HAL Id: hal-01731454http://hal.upmc.fr/hal-01731454

Submitted on 14 Mar 2018

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Bending transition in the penetration of a flexibleintruder in a two-dimensional dense granular mediumNicolas Algarra, Panagiotis Karagiannopoulos, Arnaud Lazarus, Damien

Vandembroucq, Evelyne Kolb

To cite this version:Nicolas Algarra, Panagiotis Karagiannopoulos, Arnaud Lazarus, Damien Vandembroucq, EvelyneKolb. Bending transition in the penetration of a flexible intruder in a two-dimensional dense gran-ular medium. Physical Review E , American Physical Society (APS), 2018, 97 (2), pp.022901.<10.1103/PhysRevE.97.022901>. <hal-01731454>

Bending transition in the penetration of a flexible intruderin a 2D dense granular medium

Nicolas Algarra,1 Panagiotis G. Karagiannopoulos,1 Arnaud Lazarus,2 Damien Vandembroucq,1 and Evelyne Kolb1

1Laboratoire PMMH, UMR 7636 CNRS/ESPCI Paris/PSL Research University/Sorbonne Universites,UPMC Univ Paris 06,/Univ. Paris 7 Diderot, 10 rue Vauquelin, 75231 Paris cedex 05, France

2Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7190,Institut Jean Le Rond d’Alembert, F-75005, Paris, France

(Dated: December 20, 2017)

We study the quasi-static penetration of a flexible beam in a two-dimensional dense granularmedium lying on a horizontal plate. Rather than a buckling-like behavior we observe a transitionbetween a regime of crack-like penetration in which the fiber only shows small fluctuations around astable straight geometry and a bending regime in which the fiber fully bends and advances throughseries of loading/unloading steps. We show that the shape reconfiguration of the fiber is controlledby a single non dimensional parameter: L/Lc, the ratio of the length of the flexible beam L to Lc, abending elasto-granular length scale that depends on the rigidity of the fiber and on the departurefrom the jamming packing fraction of the granular medium. We show moreover that the dynamicsof the bending transition in the course of the penetration experiment is gradual and is accompaniedby a symmetry breaking of the granular packing fraction in the vicinity of the fiber. Together withthe progressive bending of the fiber, a cavity grows downstream of the fiber and the accumulationof grains upstream of the fiber leads to the development of a jammed cluster of grains. We discussour experimental results in the framework of a simple model of bending-induced compaction andwe show that the rate of the bending transition only depends on the control parameter L/Lc.

I. INTRODUCTION

Slender structures are extremely flexible and get easilyunstable. The ratio of the bending stiffness of a beamto its axial stiffness in tension or compression scales with(t/L)2, the square of the ratio of the thickness to thelength of the beam [1]. Similarly, over a threshold com-pressive strain of order (t/L)2, the beam undergoes abuckling instability [2]. As a consequence of their highgeometrical aspect ratio, slender structures can be sensi-tive to low forces of various origins and exhibit complexmechanical behaviors due to the associated couplings.This is in particular the case of the elasto-capillary prob-lems (interactions between slender structures and capil-lary forces) which have raised a growing interest in therecent years [3].

In the same spirit, the effect of flexibility on fluid-structure interactions has recently motivated a growingnumber of studies at low [4, 5] or high Reynolds num-ber [6]. Model experiments of reconfigurations have beenperformed with flexible fibers (1D), plates (2D) or assem-bly of plates organized in a circular pattern (3D) thatwere placed initially perpendicular to the incoming flowof air [7], water [8] or 2D soap film [9]. In all these cases,the flexible body experiences shape reconfiguration withself-streamlining and reduction of the surface area ex-posed to the flow, thus resulting in a drastic reductionof the drag force exerted on the object [10]. An immedi-ate field of application of these problems can be found inbio-physical domains, when a passive or an active elas-tic part of a body interacts with the flow [6, 11–13]. Inparticular, in animal locomotion, the flexibility of wingsor fins intervene in the flying of birds and insects or theswimming of fishes or eels. Even in micro-organisms the

propulsion through flagella takes advantage of the flexi-bility. Numerous examples can also be found in the plantdomain [14] where the flexibility of cereal stems, tree’strunk, branches or leaves can be beneficial under windflow for reducing lodging. Even marine algae and plantsin aquatic canopies adapt their shape under water cur-rent [15, 16].

Here we consider an original case of fluid structureinteraction between a slender structure and a granularmedium. More specifically, we study the quasi-static pen-etration of a flexible beam in a two-dimensional densebidisperse mixture of discs lying on a plate. Applica-tions of this problem range from nuclear engineering [17]to biology: growth of roots [18, 19] in structured soilsor locomotion of worms or sandfish lizards in granularmedia [20].

The complex behavior of granular flows has recentlybeen studied through its interaction with rigid intrud-ers [21–33]. The high sensitivity of flexible intruders tolow forces opens here a promising new way of probingthe mechanical behavior of granular media [34–36]. Thecomplexity of such a question of elastogranular mechan-ics (as recently coined in Ref. [35]) will naturally arisefrom the coupling of the low rigidity of the elastic beamwith the discrete nature of the granular flow, its non-localrheology [37] and the emergence of turbulent-like fluctu-ations [38–40] in the vicinity of the jamming transition.

In contrast with two recent works focusing on buck-ling [34, 35], the present study will highlight a bendingtransition of the flexible intruder and an associated tran-sition in the structure of the granular medium.

In the following we first give in section II a brief de-scription of the experimental set-up and tools of analysis;we then discuss in section III the phenomenology of the

2

interaction between an elastic beam and a granular flow,we report our experimental observation of bistability be-tween two different regimes (straight or bent fiber), weidentify the control parameters (rigidity of the fiber andpacking fraction of the granular medium) and propose asimple model ; in section IV we give a specific focus onthe development of the bending transition and show thatit is associated with a clear symmetry breaking of boththe geometry of the fiber and the density of the granularmaterial in the vicinity of the fiber; we then propose insection V a model of bending induced compaction thatallows us to reasonably reproduce the evolution of thefiber deflection in the course of the bending transition; asummary of our main results is finally given in section VI.

II. EXPERIMENTAL SET-UP

A. The granular medium

The 2D granular material is a bidisperse mixture ofbrass cylinders, in equal mass proportion, forming adense and disordered assembly of rigid disks [27, 28]. Thesmall cylinders in number N1 have an outer diameterd1 = 4 mm, an inner diameter of 2.2 mm and a mass of0.209 g whereas the large cylinders in number N2 = 4

7N1

have an outer diameter d2 = 5 mm, an inner diameterof 2.5 mm and a mass of 0.363 g. Choosing grains asempty cylinders with different internal diameters helpsin distinguishing between large and small grains for pat-tern recognition during image analysis. The choice of thetwo close external diameters and the proportion betweensmall and large grains were chosen to reduce segregationbetween the two populations of grains. This collectionof grains lies on a horizontal glass plate delimited byfour brass walls forming a rectangular frame of widthW = 269.5 mm (54 d2 along the X−axis) and adjustablelength H along the Y−axis (Fig. 1). The total numberof cylinders is kept constant around 6800, but as we var-ied H in the range 457.5 mm (91 d2) to 470.5 mm (94d2), we could adjust the total available cell surface andtherefore the packing fraction φ. In the following experi-ments, the packing fraction φ varies between 80.85% and83.1%, just below the assumed jamming packing fraction,φJ=83.56% ± 0.08%. This value of φJ was determinedfrom a preceding similar experiment where a rigid ob-stacle (cylindrical intruder) was in place of the flexibleone [27]. Before each experiment, grains are carefullymixed such that the topological organisation of grains ischanged without inducing segregation.

The measurement of the static grain-grain friction co-efficient gives µgg=0.32±0.07 [41], while the one for thefrictional contact of grains with the glass bottom leads toµ = µgb=0.49±0.09. The static coefficient of friction µ isobtained from the tangent of the angle at which the incip-ient sliding of a grain on a progressively tilted glass sur-face occurs. In the following, we will use the notation d =N1d1+N2d2N1+N2

= 4.36 mm and m = N1m1+N2m2

N1+N2= 0.265 g for

the weighted average of grain external diameter and massrespectively. We also define an effective density of the

grains as ρ = msh where s = π

4N1d

21+N2d

22

N1+N2is the weighted

average surface occupied by one grain on the bottom glassplate. This effective density ρ = 5835 ± 5 kg.m−3 takesinto account the fact that grains are hollow cylinders.

B. The flexible beam

The flexible beam is cut into a mylar sheet of thicknesst=350 µm, Young modulus E=3.8 GPa and Poisson ratioν=0.4 [42]. The flexible part of the intruder is a longbeam, that we will call further fiber, of length L with arectangular cross-section defined by the sheet thickness tand a height h = 3 mm corresponding to the height of thecylinders of the granular material. One of the intruderextremity is clamped into a rigid metallic part supportedby a transverse arm along the X plane (Fig. 1).The otherfiber extremity is free.

The anchoring point of the intruder is fixed in the lab-oratory frame and is located at an equal distance of thetwo lateral walls delimiting the space for the granularlayer. The basis of the fiber is fixed slightly above thebottom glass plate, so that there is no friction with thebottom. At the beginning of the experiment, the fiberis straight along the Y−axis with no contact with neigh-boring grains. The anchoring point of the fiber is initiallylocated at a distance of 80 mm=16 d2 (or 120 mm=24 d2depending on the batch of experiments) from the backwall, to avoid boundary effects. In this work, we useddifferent fiber lengths between L=1 cm and L=5 cm.

C. Principle of the experiment

A typical experiment consists in translating the gran-ular material supported by the glass plate along theY−axis against the free extremity of the fiber. Theplate velocity V0 is held constant. In the plate framethis is equivalent of having a fiber penetrating the gran-ular material at a constant velocity V0. The experimentis stopped after a plate displacement of typically 260 mm(52 d2).

The velocity of the plate was always kept constant atV0 =5/6 mm/s. This value corresponds to a quasi-staticregime in which the forces are dominated by frictionalcontact forces. Actually we observe that the drag forceexperienced by a rigid cylindrical intruder does not de-pend on velocity in these ranges of values. A CCD cam-era of 1600*1200 pixels placed above the set-up recordsimages at a frequency of 2 Hz, corresponding to a platedisplacement of U0 =5/12 mm=d2/12 between 2 succes-sive images. Ten similar experiments are performed foreach set of parameters L and φ, and grains are carefullyremixed between two consecutive experiments. At theend of an experiment, the bending of the fiber can be solarge that the fiber undergoes plastic deformation at the

3

FIG. 1. Experimental set-up (right panel): the granular layer is contained in an horizontal rectangular plate of width W andlength H that moves at a constant speed along the direction indicated by the blue arrow (Y -axis). Bending of the fiber (leftpanel) produced by the granular flow as observed on a zoom of a picture taken with a CCD camera placed above the setup.The fiber clamping point is fixed in the laboratory frame.

anchoring point. In this case, we replace the fiber by anew one.

D. Segmentation and image processing

For all images of each batch of experiments, a tech-nique of correlation on gray levels has been used for de-termining the grain centers with a sub-pixel accuracy.Then the grains displacements from one image to thenext one have been computed. Furthermore, the full de-flected shape of the fiber is obtained for most images.The delicate challenge for image analysis was to locatethe very thin and elongated fiber amongst the circulargrains surrounding it. For this purpose, the gray-levelimage has been convoluted with different filters to en-hance the contrast of the fiber and disconnect it fromthe neighboring grains. This first step gave a skeleton ofthe fiber, which was further fully reconstructed by start-ing from the fiber anchoring point. From the fiber shape,different informations can be obtained like the free endlateral deflection δ or the local slope θ of the fiber relativeto the Y−axis. This local slope can be computed as afunction of the curvilinear abscissa ξ (normalized by thelength of the fiber), with ξ=0 at the clamped extremityand ξ=1 at the free extremity.

III. PHENOMENOLOGY: FLUID-STRUCTUREINTERACTION IN A GRANULAR FLOW

When immersed in a fluid, a flexible structure interactswith the flow. In particular, depending on the rigidity of

the structure and the flow conditions, it can switch fromthe initial undeformed configuration to a deformed con-figuration that reduces the drag force. Alternatively, fora given bending rigidity, there exists a critical length ofthe structure placed across the flow above which recon-figuration occurs.

A very similar phenomenology can be observed in thepresent experiment which can be regarded as an inter-action between a flexible beam and a granular flow. InFig. 2a we show a series of configurations of a short (rigid)fiber (L = 2 cm) facing the granular flow. Small fluctu-ations are observed around the stable straight configura-tion, as if the fiber was jiggling around its initial position.The slight deflections are here caused by collisions withincoming grains. In Fig. 2b we show the contrasting caseof a slightly longer fiber (L = 3 cm) for the same granularpacking fraction as before. In this case, the fiber expe-riences reconfiguration due to the flow and enters into aregime of bending.

When the 3 cm long fiber has been highly tilted, fluc-tuations of the fiber positions are again observed but ofa very different nature: in Fig. 3a we show the grad-ual bending of the deformed fiber due to an accumula-tion of incoming grains and in Fig. 3b its sudden par-tial recovery. In Fig. 4, we represent the correspond-ing grain displacements in the plate frame (by subtract-ing the plate displacement U0 between 2 consecutive im-ages) as if the fiber was penetrating the granular mate-rial. Fig. 4a shows the displacement field between thetwo images just before the unloading event. It is charac-terized by long range displacements in front of the fiberand small recirculations on both sides. Fig. 4b shows thedisplacement field corresponding to the unloading phase,with the fiber returning from position B to position C. In

4

(a)

(b)

FIG. 2. (a) Fluctuations of an elastic beam of length L = 2cm in a granular medium of packing fraction φ = 80.94%(jiggling regime). The clamping point of the fiber is locatedin (0,0). The granular flow direction is along decreasing Y -axis. (b) Gradual bending of an elastic beam of longer lengthL = 3 cm (bending regime). The granular packing fraction isthe same as in (a).

(a) (b)

FIG. 3. Individual avalanche, loading and unloading of afiber of length L = 3 cm in a granular medium of packingfraction φ = 80.94% for a highly bent configuration (regimeIII).(a) Successive fiber shapes during a loading phase (platedisplacement of 2U0 = d2/6 between each fiber shape fromA to B). (b) Unloading phase from B to C with a suddenelastic return of the fiber during one increment U0 of platedisplacement.

(a)

(b)

FIG. 4. Map of the displacement field of grains (amplified bya factor of 20) in the plate frame between 2 consecutive imagesfor the loading and for the unloading phases of Fig. 3. Theplate displacement U0 between two consecutive images hasbeen subtracted from each grain displacement. (a) Betweenthe two images just before the unloading event (b) during theavalanche BC.

contrast with the loading phase, the elastic return of thefiber is associated to huge recirculations on both sides.Therefore the sudden change of the fiber conformationcan be associated to a large reconfiguration of the grains,in other words a burst of the velocity field of the granularflow. Such an alternation of smooth and regular loadingstages interspersed with sudden unloading events is typ-ical of granular avalanches and can also be associated tothe turbulent-like velocity fluctuations reported in densegranular flows [38–40].

In Fig. 5 we summarize our observations of the straight(jiggling regime) or deformed conformation (bendingregime) of the fiber when varying its length L as wellas the packing fraction φ of the granular medium.

In the present experiment, the flexible structure is ini-tially facing the flow in a geometry of penetration. There-fore one would naturally introduce the critical force forfiber’s buckling. In the case of a non embedded fiber, thisforce is given by the Euler force FC with clamped-free ex-tremities. It drastically depends on the fiber length L as:

5

FC(L) =π2

4× EI

L2(1)

where I is the quadratic moment of the fiber corre-sponding to a non embedded fiber buckling in the (X,Y )plane. Taking into account the rectangular cross section

of the fiber yields I = h t3

12 . Thus varying L is a way tovary the maximum acceptable loading force (or equiva-lently the maximum penetrative force) in a case of a pureaxial loading. For example, the typical bending rigidityfor a beam of length L= 3 cm and thickness t= 350 µm isEI = 4.07 10−5 N.m2 and the corresponding Euler forcewould be FC(L) ≈ 0.11 N. For comparison, the bend-ing force produced by the fiber tip displacement over adistance δ is:

Fb(L,α) =αEIδ

L3(2)

with α=3 for a force only acting at the tip perpen-dicularly to the fiber and α=8 for a perpendicular forcehomogeneously distributed over the fiber length. Thusthe bending force will be Fb(L) ≈ 0.02 N for a fiberlength L = 3 cm and a fiber tip displacement producedby one grain size d (the typical size of obstacle encoun-tered by the fiber), that is δ=d and α=3. This value ofFb(L) is much smaller than the corresponding bucklingforce for the same geometrical parameters of the fiber. Inany cases, the thickness t of the fiber is small comparedwith the diameter d of a grain with t/d ≈ 0.08, suchthat the probability of a force purely acting along theaxis of the beam will be negligible. Therefore the mainmechanism producing reconfiguration of the fiber will beattributed to bending and not buckling, contrarily to therecent experiment of [34].

In studies of interactions between fluids and flexiblestructures it is customary to design a non dimensionalnumber that compares drag and elastic restoring forcesin order to determine the stability of the flexible object [5,7]. In the present context, a first naive scaling analysismay consist in building such a Cauchy-like number fromthe comparison between the friction force associated tothe contact between the bottom plate and an individualgrain. Thus the friction force is fn = µρgπd2h/4 whereρ is the effective density of an average grain of diameterd and µ is the static coefficient of friction between thebasis of a grain and the glass bottom plate as defined insection II. The bending force associated to a deflectionδ of the tip of the fiber equivalent to one grain size d isgiven by equation (2) with fb=Fb(L,α=3), such that thecorresponding Cauchy-like number is written as:

cGY =fnfb

= µπρgd

E

(L

t

)3

(3)

This analysis corresponds to a very dilute gas-like regimewhere the flexible beam only experiences collisions with

FIG. 5. Phase diagram in the plane L − φ. Experiments vsprediction. The symbols represent the experiments performedby varying the fiber length L or granular packing fraction φwith green diamonds for the bending regime and red circlesfor the jiggling regime. The blue curve is the predicted elasto-granular length from eq.(9) that separates the jiggling fromthe bending regimes with the associated error bars.

isolated grains, such that we took α=3. Thus the transi-tion between straight and deformed configurations of thebeam would occur at cY ≈ 1 for a critical length:

LGc ≈ t(

E

µπρgd

)1/3

(4)

With the experimental parameters used in the presentset-up, we get LGc ≈ 7.5± 0.5 cm, i.e. below this lengthscale the flexible intruder should not experience bendingreconfiguration.

However, as shown in Figs. 2-3, fully bent configu-rations were actually observed for much shorter fibers(L = 3 cm) than LGc . This diluted regime argument thusunderestimates the drag force exerted by the granularflow. In order to account for the effect of a denser liquid-like flow, one can consider the friction force associated tothe grains present in the area that a slightly bent beamshould sweep over to recover its initial straight shape. Fora tip deflection of δ, the swept area Aδ is of the order ofAδ ≈ δL/2 and the corresponding number of grains thatare pushed back is φAδ/s where s is the weighted averagesurface occupied by one grain on the bottom plate. Theassociated friction force will be Fn = µφρghδL/2. Com-paring this friction force Fn to the elastic restoring forcegiven by equation (2) with α=8, that is Fb = 8EIδ/L3

for an homogeneous and perpendicular loading on thewhole fiber leads to:

cLY =FnFb

= µ3φρgt

4E

(L

t

)4

, (5)

and

LLc ≈ t(

4E

3µφρgt

)1/4

(6)

6

which gives now a slightly lower value of the criticallength, LLc ≈ 5.5± 0.25 cm for φ = 0.815, but whichremains much larger than the experimental observationsdespite the change of scaling exponent. This means inparticular that the friction forces exerted by the grainsare underestimated in the present argument.

Close to the jamming packing fraction, interactions ofthe flexible beam with the flow are actually expected tostrongly alter the structure of the granular medium [43].Transient jammed clusters can thus form in the vicinity ofa wall [44] or a rigid intruder [27, 28], and more generallyby the application of shear stress [45].

In the present context, the elastic return of the de-flected fiber to its straight conformation not only inducesthe displacement of the grains present in the swept overarea but also the building of a jammed cluster. The areaof the latter can be estimated by a simple conservationargument analogous to the three-phase model developedfor the accumulation of grains in front of a rigid obstacle[27, 28]. We assume that the grains which are initiallycontained in the area Aδ at the initial packing fractionφ and which are pushed back by the slightly bent fiberwill all contribute to the formation of the jammed clus-ter. Therefore, for defining a criterion for bending orjiggling, we hypothesize that there is no recirculation ofgrains induced by the elastic return of the fiber and thatthe connected cluster of grains is at the homogeneousjamming packing fraction φJ and is contacting the fiberupstream along its whole length. Thus the area AJ ofthe cluster that reaches the jamming packing fraction φJafter it has absorbed the excess of grains swept over bythe elastic fiber is:

AJ (φJ − φ) = Aδ φ =δL

2φ . (7)

In this close-to-jamming regime, we can thus build aCauchy number with a drag force that results from thefriction force of the jammed cluster FJ = µρghφJAJ .Here, the vicinity of the jamming density limit thus sim-ply induces an amplification of the friction force by afactor φJ/(φJ − φ) so that we get for the Cauchy num-ber:

cJY =φJ

φJ − φcLY = µ

φφJφJ − φ

3ρgt

4E

(L

t

)4

, (8)

and for the critical length:

LJc ≈ t(φJ − φφφJ

4E

3µρgt

)1/4

. (9)

This critical length can be viewed as an elasto-granularlength as also defined in [35] in analogy with other workson elasto-capillary phenomena [3].

The stability of the flexible beam thus appears tobe controlled by two parameters: the rigidity (herethe length) and the packing fraction of the granularmedium. Using the value φJ = 0.8356 determined inRef. [27], this leads, for an initial density φ = 0.815 to

LJc≈ 2.2 ± 0.1 cm, a value that is more consistent withour experimental observations. In the phase diagram ofFig. 5 we see that the expression of the elasto-granularlength Lc = LJc (φ) from eq. (9) gives indeed a reason-able account of the transition between the straight ordeformed conformation of the fiber.

IV. DEVELOPMENT OF THE TRANSITION

In contrast to buckling instability, the switching of theflexible beam from its initial straight conformation to abent one is gradual and requires reorganization of thegranular medium. We focus here on the development ofthis transition.

A typical sequence (zoomed on the fiber) is shown inFig. 6 for an intermediate fiber length L = 3 cm and agranular packing fraction φ = 80.94%. The initial loca-tion of the fiber is along the vertical of the figure (Y -axis). In a first stage, the fiber experiences only smalllateral deflections and fluctuates around its straight ini-tial position. For example, in Fig. 6a, the fiber is slightlydeflected on the left side and the lateral displacementof the free extremity is almost not visible. The granularmaterial is still homogeneous, while grains are contactingthe fiber on both sides. After a given traveling distance,the fiber bends irreversibly on one side with no possibilityof returning back to its straight shape. While progress-ing inside the granular material, the fiber continues tobend (Fig. 6b) with a clear lateral deflection on the rightside, indicated by the horizontal (red) arrow along theX-axis. Associated to this deflection, we clearly observethe appearance of a cavity downstream of the fiber. Asthe whole granular material still flows along the Y−axis,the fiber continues to bend and the free extremity mightreach its maximum lateral deflection (Fig. 6c) when mostof the fiber axis is placed perpendicularly to the initialflow. At this stage, the area of the cavity is larger. Inthe following we will focus on this bifurcation in the pen-etration behavior by introducing some quantitative de-scriptors.

A. Deflection of the fiber tip

One way to quantitatively characterize the penetrationof the fiber relative to the granular medium is to com-pute the lateral deflection δ (along the X axis) of the freeextremity of the fiber as a function of the distance ` trav-eled by the plate (or equivalently the displacement of thefiber anchoring point relative to the granular medium).The evolution of δ as a function of ` associated to theexperiment shown in Figs. 6abc is shown in Fig. 6d.

In order to give a more systematic account of this phe-nomenon, we report in Fig. 7 the evolution of the de-flection δ as a function of the penetration distance ` ex-pressed in diameters of a large grain (d2) for ten indepen-dent experiments performed for a fiber length L = 3 cm

7

(a)

(b)

(c)

(d)

FIG. 6. Irreversible deflection of the bending fiber of lengthL = 3 cm in a granular medium of packing fraction φ =80.94% for different penetration distances `. (a) ` = 15.1 d2(b) ` = 31.7 d2 (c) ` = 38.5 d2 (d) Lateral deflection δ of thefree fiber extremity as a function of the penetration distance `.The horizontal dotted lines represent the maximum excursionof δ during regime I. The three successive regimes I, II andIII are separated by the vertical blue dotted lines.

(a)

(b)

FIG. 7. (a) Lateral deflection δ of the free extremity of thefiber as a function of the plate displacement (or penetrationdistance) ` for 10 independent experiments performed withthe same fiber length L = 3 cm and granular packing fractionφ = 80.94%. Both quantities are given in diameters of alarge grain d2. All the 10 curves reach the same maximalvalue δMax, with `α the penetration distance obtained forδMax/2.(b) Same curves as (a) expressed as a function of thereset penetration distance ` − `α. The orange dash curve isthe average over the different experimental curves.

and a granular packing fraction φ = 80.94%. At the endof an experiment, the fiber is observed to bend eithertoward the right (along X > 0) or toward the left side(along X < 0). In order to compare experiments, theδ values corresponding to experiments where the fibereventually bends toward the left side have been invertedin Fig. 7, such that the final value of δ is always positive.

From Fig. 6, Fig. 7 and from the corresponding ex-perimental images, we recover the three stages identifiedabove:

- In a first regime (I), the lateral deflection δ is ob-served to fluctuate around zero. The fiber continuouslypenetrates the granular medium and keeps in averageits initially straight shape. When the fiber is contacting

8

grains at its tip, it is slightly deflected on one side or theother but the maximal amplitude of δ does not exceedthe radius of a grain. Thus the extremity of the fiberexperiences an erratic and fluctuating motion, depend-ing on the local arrangement of the grains encounteredby the fiber. There is only few interactions between thefiber and individual grains, and the wake left behind thefiber is reduced to a small elongated cavity almost notvisible.

- After a penetration distance `1 (indicated by the leftvertical blue dotted line in Fig. 6d) that strongly dependson the realization, the fiber irreversibly bends towardsone side (left or right). We checked the absence of biasin the direction of deflection. This regime II is character-ized by a progressive and continuous bending of the fiber,shown by the regular increase of δ with the penetrationdistance `. This increase appears to be quite similar forthe different experiments of Fig. 7.

- A third regime appears when the extremity of thefiber roughly lies perpendicular to the direction of theplate translation X, i.e. θ(ξ = 1) ≈ π/2. Then the cor-responding lateral deflection reaches its maximal valueδMax. For the examples of Fig. 7, the fiber then fluc-tuates around this new bent configuration. In the caseof a pinned rigid fiber, the maximum possible value ofthe deflection would be δMax = L. In our case, the an-choring point is clamped and the fiber is flexible. Thenthe observed δMax is slightly smaller than L, but has aconstant value independent of φ. For example, the aver-age experimental values are δMax = (0.952± 0.005)L forL = 3 cm.

As it appears clearly in Fig. 7a, we observe two fluctu-ating regimes separated by a transition that in contrastappears to be very deterministic. This deterministic as-pect appears even more clearly in Fig. 7b after a resettingof the abscissa. Starting from the observation that themaximal deflection δMax only depends on the length ofthe fiber, we could indeed locate the traveling distance`α corresponding to the deflection value δMax

2 , the halfof the average maximal deflection for each experimentα. Then we could plot in Fig. 7b the deflection δ as afunction of the reset penetration distance distance `−`α.The curves collapse rather well such that the average overthe ten shifted experiments can be computed (Fig. 8a).Interestingly the first part of the bending regime can befitted by an exponential growth (orange curve superim-posed on the averaged experimental deflection in Fig. 8a).The same plot is presented in semi-log scales in Fig. 8b.This fit provides a characteristic length λ for the kineticsof bending according to the formula δ = δ0exp(`/λ). Forthe example of Fig. 8, λ=(4.89 ± 0.04)d2. The length λhas been obtained for all batches of experiments wherebending occurred. Its value normalized by the Lc valuefrom eq. (9) associated to the packing fraction φ is plot-ted in Fig. 9 red circles as a function of the rescaled fiberlength, that is ε = L−Lc

Lc. The ε value can be viewed as

an order parameter: For ε < 0, no irreversible bendingoccurs, the fiber stays in the jiggling regime, while for

(a)

(b)

FIG. 8. Average of the lateral deflection δ of the free extrem-ity of the fiber over the ten experiments of Fig. 7 as a functionof the rescaled plate displacement `− `α. Both quantities arenormalized by the diameter of a large grain d2 (bottom andright axes) or by the fiber length (left and top axes). Theorange curve is an exponential fit of the bending. (a) normalscales (b) semi-log scales.

ε > 0, bending occurs.

B. Asymmetry of the packing fraction

As clearly visible in Fig. 6, when the fiber enters thedeflected regime and the lateral deflection is larger thana grain size, we observe the formation of a cavity emptyof grains downstream of the fiber. A similar trend wasobserved in the framework of the penetration of a rigidobstacle in a granular layer [27].

Conversely, we expect an accumulation of grains, morespecifically the formation of a jammed cluster, upstreamof the fiber. Such an evolution is indeed observable inFig. 10 where we present three successive snapshots of anexperiment of penetration of a fiber of length L = 3 cmin a granular medium of packing fraction φ = 80.94%.

In the present experiment, we recall that the anchoringpoint of the flexible intruder is fixed in the laboratory

9

FIG. 9. Characteristic length scale λ for the rate of bendingas a function of ε = L−Lc

Lc. Red circles are λ values extracted

from the exponential fit of deflection δ vs penetration dis-tance `. Blue triangles are derived from the A values of thecompaction-induced bending model coupled with the elasticasimulations.

frame while the granular medium lies on a glass platethat is displaced at a constant velocity with U0, the typi-cal displacement of the plate between 2 successive images.The grains that accumulate upstream of the fiber are thusexpected to be slowed down with respect to the meanflow. In Fig. 10, we show the distribution of the slowest(red-labeled) grains, here selected on a simple thresholddisplacement criterion: We labeled the grains i whoseamplitudes of displacement between 2 successive imagesin the laboratory frame are ui < uT = 0.048U0. As ex-pected, we observe a strong clustering of these slow grainsupstream of the flexible beam. Although only qualitative,this observation supports the idea of the development ofa jammed cluster of grains ahead of the fiber.

The penetration of the flexible beam in the granularmedium thus leads to a clear symmetry breaking in thelocal packing fraction. In order to quantify this effect,we computed the coarse-grained packing fraction in thevicinity of the fiber.

Here we adapted the computation of the coarse-grainedpacking fraction [46] to define over a coarse-grainedlength scale σ a left and a right packing fraction of grainson either side of the fiber. As sketched in Fig. 11a, wedefine two domains D+ and D− from either side of thefiber such that the distance of any point of a domain tothe fiber remains below a cut-off value rcut. The twopacking fractions φ+ and φ− are then computed as:

φ± =1

α±

∑gi∈D±

Ai exp

(− d

2i

σ2

), (10)

where Ai is the area occupied by the grain gi and di is itsdistance to the fiber. Here the parameter σ is the coarse-graining length and we use rcut = 4σ. The normalization

FIG. 10. Growth of a “cluster” of slow grains (red-labeled) forthree successive images starting from a penetration distance`=24.1d2 for a granular packing at φ = 80.94% and L = 3 cm.

factor α± is obtained by the expression:

α± =

∫~r∈D±

exp

(−d(~r)2

σ2

)dxdy , (11)

where d(~r) is the distance to the fiber of a point ~r.In Fig. 11b, we show the evolution of the lateral pack-

ing fractions φ+ and φ− obtained for a coarse-graininglength σ = d2 after resetting the abscissa as in Fig. 7 byusing the distance `α and then averaging over 10 experi-ments. The bending transition appears very clearly to beaccompanied by i) a strong decrease of the downstreampacking fraction φ− down to values approaching zero andii) a slight increase of the upstream φ+ up to a plateauvalue about φ+max ≈ 0.7.

10

(a) (b)

(c) (d)

FIG. 11. Development of a packing fraction contrast betweenthe two sides of the bending fiber. (a) Schematic diagram ofthe zones upstream and downstream of the fiber for calcu-lation of φ+ and φ−. (b) Lateral packing fractions φ+ andφ− (sliding window averaged over 10 points and over 10 ex-periments) as a function of the rescaled penetration distance`−`α normalized by d2 for a fiber length L = 3 cm in a granu-lar medium with a macroscopic packing fraction φ = 80.94%(indicated by the blue horizontal line in (c) and (d)). Ef-fect of the choice of the coarse-grained length scale σ on thecalculation of (c) φ+ and (d) φ−.

Note that this value is far lower than the jammingpacking fraction of the present granular medium φJ ≈0.8356 estimated in Ref. [27]. However, the value of themaximum packing fraction of a granular cluster is knownto be significantly altered (decreased) in presence of awall [47]. A clear illustration of this effect is given inFigs. 11c and 11d where we show the respective evolu-tion of the upstream and downstream packing fractionsφ+ and φ− with increasing values of the coarse-graininglength σ. We see that the larger the coarse-graininglength, the larger the maximum upstream packing frac-tion φ+, which for σ = 4d2 reaches a plateau value closeto the expected jamming packing fraction.

By comparing Fig. 11b with Fig. 8, we observe thatthe symmetry breaking in the packing appears when thefiber deflection typically exceeds the half diameter of agrain. These results thus give a quantitative support toour phenomenological observations: the bending transi-tion of the flexible beam is associated to a structuraltransition of the flowing granular medium with the for-mation of jammed cluster upstream of the fiber and thedevelopment of a cavity downstream of the fiber.

FIG. 12. Schematic diagram of the formation of a clusterat the jamming packing fraction φJ ahead of the fiber and acavity empty of grains behind the fiber. Aggregation of grainsin front of the cluster through the advance of a compactionfront and cluster erosion through recirculations of grains (redarrows) on both sides of the flexible intruder. The rest of thegranular layer is at the packing fraction φ.

V. A SIMPLE MODEL OF BENDING INDUCEDCOMPACTION

A. Bending induced compaction

As discussed in the previous section, the bending tran-sition of the flexible beam is associated with an accu-mulation of grains upstream of the fiber, leading to thegrowth of a cluster at the jamming density φJ as well asthe development of a cavity empty of grains downstreamof the fiber (see the schematic diagram of Fig. 12). Thegradual bending of the flexible beam in the course of thepenetration of the granular medium tends to increase thesection of the intruder across the granular flow. We makethe simple assumption that a larger intruder cross-sectioncollects a larger number of grains in the cluster. Thusthe number of grains dNc

+ entering the cluster for aninfinitesimal penetration distance of d` is assumed to beproportional to the deflection δ along X. This accumu-lation of grains in the cluster, which acts as a solid blockconnected to the fiber, increases the force acting againstthe elastic restoring force on the fiber.

In the spirit of the compaction front observed in thework of Waitukaitis and al. [48] and the three phase-model we developed for the penetration of a rigid intruderin a granular medium [27] we propose that the clustergrows faster for a packing fraction approaching the jam-ming transition. The compaction front (see Fig. 12) de-limits the interface between the cluster at φJ and therest of the granular layer at φ in front of the fiber. Foreach penetration step d`, the compaction front advancesover a distance of d`F greater than d`. Indeed there is anamplification factor linked to the approach to the jam-ming transition: φ

φJ−φ , thus resulting in d`F= φφJ−φd` for

11

a one dimensional growth of the cluster as illustrated inFig. 12. Therefore the number of grains arriving in the

cluster will evolve like: dNc+

d` ∝ φφJ−φδ.

However the compaction of a cluster of grains upstreamof the fiber induces long range effects in the granularmedium [37, 49]. In particular, it is accompanied hereby recirculation flows on both sides of the intruder thattend to erode the cluster and convect grains downstreamof the fiber. These recirculations are visible in the mapof displacement fields of Fig. 4 in the plate frame, whenthe displacement of the plate has been subtracted fromeach grain displacement. In our previous work on thepenetration of rigid intruders [28], it was observed thatthe lateral extent of recirculations was proportional tothe diameter of the cylindrical intruder. Adapting thisargument to the flexible intruder of spanwise length δleads to a grain flux recirculating on both sides of thefiber which is proportional to δ. The number of grainseroded from the cluster for each penetration step is thus

expected to scale like dNc−

d` ∝ δ. Hence the evolution ofthe number of grains Nc in the cluster is the balance oftwo terms, an accretion term and an erosion term that

both depend on δ with dNc

d` = dNc+

d` −dNc

−

d` .We assume that the friction force resulting from con-

tacts between grains and between grains and fiber re-mains negligible with respect to the friction force result-ing from the contact grains and the supporting plate. Wealso assume that during bending the accretion term (A+)is larger than the erosion term (A−), such that the clus-ter grows and induces an increment of the friction forceFc with dFc

d` ∝dNc

d` ∝ δ. During this transition regimewe expect a quasi-static loading where the elastic forcedue to the bending Fb compensates the friction force in-duced by the jammed cluster of grains upstream of thefiber. This leads to the following evolution law:

dFbd`

= A+ · δ −A− · δ = A · δ (12)

Note that A is a phenomenological parameter whichhas the dimension of a pressure.

Given the knowledge of the constitutive relation Fb(δ)that relates the bending force of the flexible beam withits lateral deflection, Eq. (12) should thus give us an evo-lution law for the lateral deflection δ with respect to thepenetration of the fiber `.

In the limit of small deflection and for a uniform per-pendicular force density along the fiber, we can use thelinear elasticity result Fb = 8EIδ/L3 and we get:

dδ

d`=A L3

8EIδ =

δ

λ(13)

We thus recover an exponential growth of the deflectionof the fiber with respect to the penetration. This wasobserved experimentally in Fig. 8 where δ = δ0exp(`/λ)for the early stage of the bending. The identification

FIG. 13. Bending-induced compaction model (blue curve)superimposed on the experimental evolution of the averagedlateral deflection (black curve) as a function of the rescaledpenetration distance with the same experimental parametersas in Fig. 7. The best A value obtained from the model isA=2200±750 Pa for this example.

of the exponential fit for the experimental values of δwith the evolution law of eq. 13 gives the length λ, whichcharacterizes the kinetics of bending:

λ =8EI

A L3(14)

However the experimental evolution of the deflectionwith penetration eventually shows a saturation which cannot be reproduced in the framework of the linear approx-imation of the elastic bending of the fiber, only valid forsmall deflection.

B. Elastica

In order to account for the large deflection regime untilthe saturation behavior, we set up a nonlinear numericalmodel of our experimental system. The fiber is modeledby a unidimensional inextensible centerline parameter-ized in a 2D space by the rotation angle along the ar-clength between the local tangent of the deformed andundeformed curve. Our model is geometrically exact inthe sense that there is no restriction on the amount ofdeflection the inextensible fiber can take (the geometri-cal limit is that self-interactions are not allowed in ourmodel). The equilibrium configuration of the weigthlessfiber under an external density of force along its arclengthis given by a 2D’s Kirchhoff equation [1]. To model thecontacts of the grain on the fiber, we choose a density offorces that remains orthogonal to the fiber upon defor-mation (note that because of this non-conservative po-sitional loading, the equilibrium equation cannot be de-rived from a mechanical energy formulation). Finally, we

12

ensure a free boundary condition at one end and a tor-sional spring with a given stiffness on the other end toaccount for the possible imperfections of the experimen-tal clamping.

Using finite differences [50], the second order partialdifferential equilibrium equation is discretized in a set ofN nonlinear algebraic equations with N unknowns thatare the rotational degrees of freedom of the discretizedstructure. Here, N is the number of discrete elementsmodelling the fiber that has been set to N = 200 to en-sure numerical convergence. For a given discrete densityof forces applied on the nodes of the discrete fibers, theequilibrium configuration is computed thanks to a clas-sic Newton-Raphson algorithm. Note that the intensityof the density of forces has to be applied gradually from0 to the desired intensity to ensure proper convergenceof the Newton-Raphson algorithm. This numerical pro-cess gives us the nonlinear relation Fb(δ) that we wereseeking.

As shown by the blue curve in Fig. 13, this compu-tation now allows us to reasonably reproduce the fullbending transition (regime II) of the fiber induced bythe gradual compaction of grains upstream of the fiber.By tuning the value of A to a given value, it is possi-ble to reproduce the experimental evolution of deflectiontill its maximum value. The model does not allow toreproduce the behavior beyond (regime III), in particu-lar the plateau observed for this fiber length L=3 cm, asthe elastica calculations are based on the same type ofloading whatever the deflection, i.e. an orthogonal repar-tition of forces along the whole fiber length. This strongassumption is certainly no more valid in regime III whenthe fiber adopts a hook shape, that is when the angleθ(ξ = 1) exceeds π/2.

C. Discussion of the model

The values of A that match the curves of deflectionlike in Fig. 13 have been obtained for the different exper-iments performed with various fiber lengths L or pack-ing fractions φ. For getting a physical meaning of A interms of length, we compute the corresponding λ valuesfrom Eq. 14. The λ value characterizes the typical pen-etration distance necessary to bend the fiber in regimeII once the bifurcation occurs. In Fig. 9 these λ values(blue triangles) are plotted as a function of the relativeexcess length above Lc, that is ε = L−Lc

Lc. The values of λ

(red circles) obtained from the direct fit δ = δ0exp(`/λ)of Fig. 8 are also plotted on the same curve for compar-ison. For both derivations, the λ values are of the orderof Lc but both curves also exhibit the same complex andnon-monotonous evolution with ε. For large ε, that is forlong fiber lengths, the λ value seems to be governed bythe excess length L− Lc. However for ε→ 0, that is forfiber length approaching the critical length for bending,λ increases by a factor of 2. For a fiber length just atthe transition L = Lc (ε = 0) one would expect a diver-

gence of λ, as Lc provides the limit for the non-bendingcase. The increase of λ observed for smaller ε might beinterpreted in this framework. In any cases, the Fig. 9provides a intriguing relationship between λ, the lengthscale characterizing the rate of bending, and Lc definingthe critical fiber length for bending.

The elastica-derived model, though based on simpleand minimal arguments, well captures the full range ofthe bending transition (regime II) (Fig. 13), even if itcan not describe the deflection in regime III. More re-fined and complex loadings of the fiber (like non orthog-onal forces or non homogeneous amplitudes or locationsof point forces...) might be investigated to improve thematching of the model with experimental deflections.

Moreover, even in the bending regime II, the observednon-monotonous evolution of λ with ε probably indicatesthat there are two competing effects with increasing fiberlengths. Following the work of [34] on the buckling ofelastic beams in granular media, it is tempting to con-sider that the fiber can be decoupled in two parts: onepart of effective length Leff (probably related to Lc) thatmight supports the force exerted by the cluster, while thecomplementary part of length L − Leff would be sim-ply led by the granular flow like a pinned rod. Directmeasurements of forces and torques exerted on the fiberare needed to clearly identify the way the fiber is loaded.Complementary elastica simulations would provide a sys-tematic way of quantifying the role of the fluctuating anddiscrete nature of force transmissions specific of granu-lar material at the grain-fiber contacts. The approach tojamming introduces a further complex step in this newand intriguing fluid-structure interaction between and aflexible fiber and a dense granular flow.

VI. CONCLUSION

This work presented a new fluid/structure interactionbetween a granular flow close to the jamming transi-tion and a flexible fiber in a geometry of penetration.We identified a bending transition occuring for fiberlonger than a characteristic length, that we called elasto-granular length Lc as in the very recent work of [35]. Inour case, the reconfiguration of the fiber shape was dueto bending and not buckling but one can imagine to pro-mote mechanisms of buckling by progressively increasingthe aspect ratio between the fiber thickness and the graindiameter.

The bending transition was associated with a symme-try breaking in the packing fraction, with the formationof a cluster upstream of the fiber and a cavity emptyof grains downstream of the fiber. We proposed an ex-pression for Lc that combines both rigidities of the flexi-ble fiber and granular material approaching the jammingtransition. The elasto-granular length Lc controlled thetransition between the bending and jiggling regimes forthe fibers embedded in the granular medium. But sur-prisingly Lc was also observed to determine the kinetics

13

of the bending once the transition started.

ACKNOWLEDGEMENTS

We acknowledge the crucial technical help of T.Darnige as well as enlightning discussions with P.

Gondret and A. Seguin. We also acknowledge all thegraduated students who have contributed to this work,Marguerite Leang, Yuka Takehara (from the group of K.Okumura), Francois Postic as well as Katherin Lugin-buhl, Dawn Wendell and Nadia Cheng from the group ofA. Hosoi through the MIT-France Seed Fund project on”Flexible Objects in Granular Media”.

[1] B. Audoly and Y. Pomeau, Elasticity and geometry:From hair curls to the nonlinear response of shells (Ox-ford press, 2010).

[2] Z. Bazant and L. Cedolin, Stability of structures: Elastic,inelastic, fracture and damage theories (World Scientific,2010).

[3] B. Roman and J. Bico, J. Phys. Condens. Matter 22,493101 (2010).

[4] M. J. Shelley and J. Zhang, Annual Review of Fluid Me-chanics 43, 449 (2011).

[5] A. Lindner, M. J. Shelley, O. du Roure, and E. Nazock-dast, Ann. Rev. Fluid Mech 50, 1 (2017).

[6] S. Ramananarivo, R. Godoy-Diana, and B. Thiria, Proc.Natl. Acad. Sci. 108, 5964 (2011).

[7] F. Gosselin, E. de Langre, and B. A. Machado-Almeida,J. Fluid Mech. 650, 319 (2010).

[8] L. Schouveiler and A. Boudaoud, J. Fluid Mech. 563, 71(2006).

[9] S. Alben, M. Shelley, and J. Zhang, Nature 420, 479(2002).

[10] S. Vogel, Life in moving fluids: the physical biology offlow (Princeton University Press, 1994).

[11] N. Coq, O. du Roure, J. Marthelot, D. Bartolo, andM. Fermigier, Phys. Fluids 20 (2008), 10.1063/1.2909603.

[12] J. Xiao and X. Chen, J. Roy. Soc. Interf. 10 (2013),10.1098/rsif.2013.0399.

[13] M. K. Jawed, N. K. Khouri, F. Da, E. Grinspun, andP. M. Reis, Phys. Rev. Lett. 115, 168101 (2015).

[14] E. de Langre, A. Gutierrez, and J. Cosse, C.R.Mecanique 340, 35 (2012).

[15] M. Luhar and H. M. Nepf, Limnology and Oceanography56, 2003 (2011).

[16] S. Barsu, D. Doppler, J. J. S. Jerome, N. Riviere,and M. Lance, Physics of Fluids 28, 107101 (2016),http://dx.doi.org/10.1063/1.4962309.

[17] G. C. Buster, M. R. Laufer, and P. F. Peterson, Nucl.Eng. Design 303, 153 (2016).

[18] E. Kolb, C. Hartmann, and P. Genet, Plant and Soil360, 19 (2012).

[19] D. M. Wendell, K. Luginbuhl, J. Guerrero, and A. E.Hosoi, Exp. Mech. 52, 945 (2012).

[20] A. E. Hosoi and D. I. Goldman, in Annual review of fluidmechanics, vol 47, Annual Review of Fluid Mechanics,Vol. 47, edited by S. Davis and P. Moin (2015) pp. 431–453.

[21] R. Albert, M. A. Pfeifer, A. L. Barabasi, and P. Schiffer,Phys. Rev. Lett. 82, 205 (1999).

[22] I. Albert, J. G. Sample, A. J. Morss, S. Rajagopalan, A.-L. Barabasi, and P. Schiffer, Phys. Rev. E 64, 061303(2001).

[23] M. B. Stone, R. Barry, D. P. Bernstein, M. D. Pelc, Y. K.Tsui, and P. Schiffer, Phys. Rev. E 70, 041301 (2004).

[24] R. Candelier and O. Dauchot, Phys. Rev. Lett. 103,128001 (2009).

[25] R. Candelier and O. Dauchot, Phys. Rev. E 81, 011304(2010).

[26] A. Fiege, M. Grob, and A. Zippelius, Granul. Matt. 14,247 (2012).

[27] E. Kolb, P. Cixous, N. Gaudouen, and T. Darnige, Phys.Rev. E 87, 032207 (2013).

[28] E. Kolb, P. Cixous, and J. C. Charmet, Granul. Matt.16, 223 (2014).

[29] A. Seguin, Y. Bertho, F. Martinez, J. Crassous, andP. Gondret, Phys.Rev. E 87, 012201 (2013).

[30] A. Seguin, C. Coulais, F. Martinez, Y. Bertho, andP. Gondret, Phys. Rev. E 93, 012904 (2016).

[31] A. H. Clark, L. Kondic, and R. P. Behringer, Phys.Rev.E 93, 050901(R) (2016).

[32] R. A. L. de la Cruz and G. A. Caballero-Robledo, J. FluidMech. 800, 248 (2016).

[33] A. Merceron, A. Sauret, and P. Jop, Phys. Rev. E 93,062904 (2016).

[34] A. R. Mojdehi, B. Tavakol, W. Royston, D. A. Dillard,and D. P. Holmes, Ext. Mech. Lett. 9, 237 (2016).

[35] D. J. Schunter Jr., M. Brandenbourger, S. Perriseau, andD. P. Holmes, arxiv:1706.07849 (2017).

[36] N. Algarra, M. Leang, A. Lazarus, D. Vandembroucq,and E. Kolb, EPJ Web Conf. 140, 14002 (2017).

[37] M. Bouzid, A. Izzet, M. Trulsson, E. Clement,P. Claudin, and B. Andreotti, Eur. J. Phys. E 38, 125(2015).

[38] F. Radjai and S. Roux, Phys. Rev. Lett. 89, 064302(2002).

[39] G. Combe, V. Richefeu, G. Viggiani, S. A. Hall, A. Ten-gattini, and A. Atman, AIP Conf. Proc. 1542, 453(2013).

[40] K. Saitoh and H. Mizuno, Soft Matter 12, 1360 (2016).[41] P. Cixous, Blocage et ecoulement d’un milieu granulaire

dense et desordonne autour d’un obstacle rigide, Theses,Universite Pierre et Marie Curie - Paris VI (2009).

[42] A. Boudaoud, P. Patricio, Y. Couder, and M. Ben Amar,Nature 407, 718 (2000).

[43] P. Mills, P. G. Rognon, and F. Chevoir, EPL 81 (2008).[44] P. G. Rognon, T. Miller, B. Metzger, and I. Einav, J.

Fluid Mech. 764, 171 (2015).[45] D. Bi, J. Zhang, B. Chakraborty, and R. P. Behringer,

Nature 480, 355 (2011).[46] C. Goldenberg, A. Tanguy, and J.-L. Barrat, Europhys.

Lett. 80, 16003 (2007).[47] K. W. Desmond and E. R. Weeks, Phys. Rev. E 80,

051305 (2009).[48] S. R. Waitukaitis, L. K. Roth, V. Vitelli, and H. M.

Jaeger, EPL (Europhysics Letters) 102, 44001 (2013).

14

[49] K. A. Reddy, Y. Forterre, and O. Pouliquen, Phys. Rev.Lett. 106, 108301 (2011).

[50] A. Lazarus, J. T. Miller, and P. M. Reis, J. Mech. Phys.Solids 61, 1712 (2013).