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Nonlinear effects of particle shape angularity in shearedgranular media

Emilien Azéma, Nicolas Estrada, Farhang Radjai

To cite this version:Emilien Azéma, Nicolas Estrada, Farhang Radjai. Nonlinear effects of particle shape angularity insheared granular media. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, AmericanPhysical Society, 2012, 86, pp.1-15. �10.1103/PhysRevE.86.041301�. �hal-00737969�

Nonlinear effects of particle shape angularity in sheared granular media

Emilien Azema,1, ∗ Nicolas Estrada,2, † and Farhang Radjai1, ‡

1LMGC, Universite Montpellier 2-CNRS, Place Eugene Bataillon, 34095 Montpellier cedex 05, France2Departamento de Ingenierıa Civil y Ambiental, Universidad de Los Andes, Bogota, Colombia

(Dated: September 19, 2012)

We analyze the effects of particle shape angularity on the macroscopic shear behavior and textureof granular packings simulated by means of the contact dynamics method. The particles are regularpolygons with an increasing number of sides ranging from 3 (triangles) to 60. The packings areanalyzed in the steady shear state in terms of their shear strength, packing fraction, connectivity,and fabric and force anisotropies, as functions of the angularity. An interesting finding is thatthe shear strength increases with angularity up to a maximum value and saturates as the particlesbecome more angular (below six sides). In contrast, the packing fraction declines towards a constantvalue, so that the packings of more angular particles are looser but have higher shear strength. Weshow that the increase of the shear strength at low angularity is due to an increase of both contact andforce anisotropies, and the saturation of the shear strength for higher angularities is a consequenceof a rapid fall-off of the contact and normal force anisotropies compensated by an increase of thetangential force anisotropy. This transition reflects clearly the rather special geometrical propertiesof these highly angular shapes, implying that the stability of the packing relies strongly on theside-side contacts and the mobilization of friction forces.

I. INTRODUCTION

Granular materials composed of particles of complexshape are common in nature and also in various fieldsof science and engineering. Some examples are angular-shaped particles of soils and rocks, elongated or platyparticles of pharmaceutical products and non-convex par-ticles of metallurgical and sintered powders. These shapecharacteristics strongly affect the rheology and texture ofgranular materials. This has been recently evidenced bya number of numerical and experimental studies carriedout using angular particles [1–12] and by a number of in-vestigations that have focused on other important char-acteristics such as elongation [13–20] or non-convexity[21, 22]. The existing research results suggest that theeffect of shape parameters is often nonlinear and coun-terintuitive as in the case of the unmonotonic relationbetween the elongation of the particles and the packingfraction [13, 14, 18].

Hence, in order to obtain a clear picture of the com-plex behavior exhibited by real granular materials, it iscrucial to understand and quantify the effects of particleshape. However, this is not an easy task, which is whysystematic studies on the subject are scarce. One of theunderlying issues is that it is difficult to control parti-cle shape in experiments. Moreover, introducing parti-cle shape in numerical simulations with discrete elementmethods gives rise to various technical difficulties, bothgeometrical and computational. One example of thesedifficulties involves contact detection and force calcula-tion between particles of arbitrary shape [7, 10, 23–26].

∗Electronic address: emilien.azema@univ-montp2.fr†Electronic address: n.estrada22@uniandes.edu.co

‡Electronic address: farhang.radjai@univ-montp2.fr

The aim of this work is to explore the influence of thedegree of angularity of the particles on the mechanical be-havior of sheared granular packings. We employ the con-tact dynamics method to simulate large two-dimensionalpackings of polydisperse regular polygonal particles. Weconstruct different packings, each of them made up ofparticles with a given number of sides in the range vary-ing from 3 (triangles) to 60. We also simulate a packingof disks, which may be considered as polygons of an in-finite number of sides. Hence, the angularity, defined asthe exterior angle of polygons, varies from 0 for disks to2π/3 for triangles.

The packings are analyzed in the steady state in termsof their shear strength, packing fraction, connectivity,and fabric and force anisotropies, as functions of the an-gularity. A pending issue that we would like to address inthis paper is whether the packing of disks has a singularbehavior compared to the packings of polygons. This isthe case if a discontinuous change (within our statisticalprecision) is observed for a packing property, e.g. theshear strength or packing fraction, between the packingof disks and the packing of polygons of 60 sides, whichare least angular polygons in our simulations. In a simi-lar vein, it is not obvious whether packings composed ofparticles of the lowest numbers of sides, i.e. triangles andsquares, are special as compared to those of larger num-bers of sides whose behavior is expected to be describedby the angularity parameter as deviation from circularshape.

In the following, we introduce in Section II the numeri-cal approach, system characteristics, and loading param-eters. In Section III, we focus on the evolution of shearstrength and packing fraction with angularity. The mi-crostructure is analyzed in Section IV in terms of con-nectivity, and contact and force anisotropies. The finalsection presents the concluding remarks and a summaryof the most salient results.

II. MODEL DESCRIPTION

A. Numerical method

The simulations were carried out by means of thecontact dynamics (CD) method, which is suitable forlarge assemblies of undeformable particles. This methodemerged from a mathematical formulation of nonsmoothdynamics and the subsequent algorithmic developmentsby J. J. Moreau and M. Jean [27–38]. The fundamentaldifference between this method and the common DEM ormolecular dynamics (MD) approach lies in the treatmentof small length and time scales involved in the dynam-ics of granular media. In MD-type DEM, pioneered byP. Cundall, the particles are treated as rigid bodies butthe contacts between particles are assumed to obey aviscoelastic or plastic behavior in which the local strainvariables are defined from the relative particle positionsor displacements ([39–52]). The time-stepping schemesused for the numerical integration of the equations ofmotion imply thus a fine resolution of the small time andlength scales involved in contact interactions.

In the CD method, these small scales are neglectedand their effects absorbed into contact laws together witha nonsmooth formulation of particle dynamics describedat the scale of particle displacements rather than smallelastic response times and displacements. The equationsof motion are integrated by an implicit time-steppingscheme by taking into account the kinematic constraintsresulting from frictional contact interactions. The im-plicit integration makes the method unconditionally sta-ble. Moreover, since in this method the elastic contactdeflections are not resolved, the time step can be largerthan that in the molecular dynamics (MD) method wherethe time step should be small enough to allow for smoothvariations of the overlap at the contact points to ensurenumerical stability. In CD, an iterative algorithm is usedto determine the contact forces and particle velocitiessimultaneously at all potential contacts. A detailed pre-sentation of the CD method is given in Appendix A forpoint contact interactions.

The particle shape enters a CD resolution algorithmthrough the explicit determination of the set of effectivecontacts at the beginning of a time step. For polygonalparticles, two different types of contact can be distin-guished: 1) side-vertex and 2) side-side; see Fig. 1. Aside-vertex contact is a point contact like that betweentwo disks. In this case, the side coincides with the tan-gent common line and the local frame is defined withrespect to this line. In a detection algorithm, such asthe shadow overlap method used in our simulations, aside-vertex contact corresponds to a single corner of apolygon crossing a side of a partner polygon [23, 25, 56].Obviously, ideal contacts with no interpenetration of theparticles (δn = 0) would require infinite precision. Inall numerical methods, the detection of contact betweentwo bodies consists actually in observing an overlap ofthe portions of space they occupy, so that δn ≤ 0. These

i j

!cκ

i

!cκ

j

!nκ

!tκ

κ

(a)

ij

!cκ

i

!cκ

j

!nκ

!tκ

κ

κ′

!cκ′

j

!cκ′

i

(b)

FIG. 1: Side-vertex contact (a) and side-side contact (b) be-tween two polygonal particles.

overlaps are a simple matter of geometrical precision inthe framework of the CD method rather than a strainvariable as in MD. The evolution of a granular system bya CD process is as much sensitive to such imperfectionsas that of a real system to the surface irregularities ofreal particles.

A side-side contact between two rigid polygons isequivalent to two geometrical constraints and can thusbe represented by two distinct point contacts located onthe common side, which defines the common tangent linebetween the two polygons. For this reason, we refer toside-side contacts as double contacts in contrast to side-vertex contacts to which we refer as simple contacts. Inpractice, a double contact is detected when a double in-trusion occurs between two polygons (two vertices of apolygone crossing the same side of another polygon or atleast one vertice of each of the two polygons crossing aside of the other polygon). The common tangent line isdefined as an intermediate line crossing the overlap zonebetween the two sides involved in the double contact andthe projections of the intruding vertices onto this line areused to define two points representing the double contact.The algorithm is insensitive to the technical details ofthis choice as long as the intrusions are small comparedto particle sizes, i.e. if the neighbor list is frequentlyupdated and the time step is sufficiently small. For ex-ample, in our simulations the intrusion never exceeds 1%of particle diameter.

The two points of a double contact determined bythe detection procedure obey Signorini’s conditions andCoulomb’s friction law; see Appendix A. However, theforces and displacements at the two points are coupledas a result of the rigidity of the particles, which imposesthe equality of the sliding velocities. Let κ and κ′ be two

points belonging to a double contact between two poly-gons, as shown in Fig. 1. The contact frame (~n,~t) iscommon to the two point contacts, but the contact vec-tors ~cκ

i and ~cκ′

j are different. If both contact points arepersistent and nonsliding, the contact normal forces fκ

n

and fκ′

n , and tangential forces fκt and fκ′

t may take in-dependent values compatible with Signorini’s conditions(fκ

n ≥ 0, fκ′

n ≥ 0) and with Coulomb’s law of friction

(|fκt | ≤ µfκ

n , |fκ′

t | ≤ µfκ′

n ). But if one of the two contactsis sliding, then the other contact must be sliding, too,with the equality of the sliding velocities uκ

t = uκ′

t . This

condition implies that fκt and fκ′

t are of the same sign sothat the sliding status is verified not only at each of thetwo contact points (fκ

t = ±µfκn and fκ′

t = ±µfκ′

n ) but

also for the double contact, i.e. fκt +fκ′

t = ±µ(fκ′

n +fκ′

n ).Since the equations of dynamics are based on the rigid-

body degrees of freedom, the equality of sliding velocitiesat the two points representing a double contact is in prin-ciple correctly calculated if the two points are handled asindependent contacts in the iteration process. However,the number of iterations for convergence declines if theequality of the sliding velocities is enforced directly in theiteration process. To do so, Coulomb’s friction law for adouble contact is implemented as follows:

uκt > 0 ⇒

{

fκt = −µfκ

n

fκ′

t = −µfκ′

n and uκ′

t > 0

uκt = 0 or uκ′

t = 0 ⇒

{

−µfκn ≤ fκ

t ≤ µfκn

−µfκ′

n ≤ fκ′

t ≤ µfκ′

n

uκt < 0 ⇒

{

fκt = µfκ

n

fκ′

t = µfκ′

n and uκ′

t < 0(1)

In practice, the inequalities (1) are implemented in thecorrection step when solving the local Signorini-Coulombproblem for a double contact between two particles; seeAppendix A.

The two points attributed to a double contact and thecalculated forces are only intermediate objects. The onlyphysically meaningful forces acting at a double contactare the resultant forces fn = fκ

n + fκ′

n and ft = fκt + fκ′

t .It is easily shown that fn ≥ 0 and |ft| ≤ µfn if the twocontact points obey Signorini’s conidtions and Coulomb’sfriction law. Since only the force resultants and relativedisplacements are material at a double contact, the choiceof the two representative points of a double contact is amatter of technical convenience with no real impact onthe result.

Ideally, vertex-vertex contacts should never occur, butdue to finite precision we do observe ambiguous situa-tions that may be considered as vertex-vertex contacts,as shown in Fig. 2, and that require special treatment.The difficulty lies in the choice of a common tangent lineand two representative points such that the subsequentparticle motions under the effect of contact laws at thosepoints does not lead to further mutual intrusion of theparticles. The intrusion may increase due to both nor-mal and tangential relative displacements with respectto the four sides involved in the vertex-vertex contact.

i j

!nκ

!tκ

κ

κ′

!nκ′

!tκ′

FIG. 2: A vertex-vertex intersection (exagerated) resolvedinto two side-vertex contacts.

This means that a vertex-vertex contact may be resolvedeither into two side-vertex contacts or into two side-sidecontacts and treated as described previously. An exam-ple is shown in Fig. 2 where two side-vertex contacts aredefined to represent the intersecting vertices (exageratedon the figure). This is a simple and robust method al-though alternative methods for the choice of the commonline and local frame may be proposed.

B. Description of the packings and the simpleshear test

We prepared 13 different packings, each comprising10000 regular polygons with the same number of sidesns ∈ [3, 4, .., 10, 11, 17, 30, 40, 60]. Additionally, we buildone more packing composed of the same number of disks.The angularity α = 2π/ns varies from 0 for disks to 2π/3for triangles. In order to avoid long-range ordering, weintroduce size polydispersity by varying the circumra-dius of the polygons from 0.6〈d〉 to 2.4〈d〉, where 〈d〉 isthe mean circumradius, with a uniform distribution byvolume fractions.

The particles are initially placed in a semi-periodicbox 100〈d〉 wide, using a geometrical procedure [57, 58].Next, all packings are sheared by imposing a constantconfining stress σwall and a constant horizontal velocityvwall on the upper wall, as schematized in Fig. 3. Toavoid strain localization at the boundaries and to guar-antee that the shear strain is uniformly distributed in thebulk, the particles in contact with the walls are “glued”to them, and the gravity is set to zero. The friction co-efficient µs between particles is set to 0.4.

Since we are interested in the quasi-static (rate-independent) behavior, the particle inertia should be neg-ligible compared to the confining pressure. From theshear rate γ = vwall/ywall and σwall an “inertia parame-ter” I is defined by [59]

I = γ〈d〉

√

ρ

σwall, (2)

x

y

σwall

vwall

FIG. 3: Schematic representation of the simulated shear test;the dashed lines represent periodic boundaries. vwall is thehorizontal velocity of the wall and σwall is the confining pres-sure. The arrows inside the box represent the velocity field.

where ρ is the mass density. Experiments and simula-tions show that this condition is fulfilled when I < 10−3.In all our tests we have γ = 10−6/∆t, and σwall =10−4ρ(〈d〉∆t)2. Hence, I ∼ 10−4, which means that oursheared samples can reasonably be considered to be in aquasistatic state

The samples are sheared up to a large cumulative shearstrain γ = xwall/ywall = 4, where xwall is the horizontaldisplacement of the upper wall and ywall is its verticalposition. Figure 4 shows the stress ratio τwall/σwall andthe normalized volume of the packing V/〈d〉2, as func-tions of the shear strain γ, for four different values of α,where τwall is the tangential stress at the moving wall.We see that the packings are in the steady state up tosmall fluctuations around a mean both for τwall/σwall

and V/〈d〉2. In the following sections, all average quanti-ties represent the last 50% of cumulative shear strain sothat they truly characterize the behavior of the systemin the steady state [60]. Video samples of the simula-tions analyzed in this paper can be found at www.cgp-gateway.org/ref016.

III. SHEAR STRENGTH AND PACKINGFRACTION

The shear strength of a granular material is charac-terized by the coefficient of internal friction µ∗, whichrequires the stress tensor σ at any stage of deformationcalculated from the simulation data, giving access to thecontact network and forces. We start with the internalmoment tensor Mp of each particle p, defined by

Mpij =

∑

c∈p

f ci rc

j , (3)

where f ci is the i component of the force exerted on par-

ticle p at contact c, rcj is the j component of the position

vector of the same contact, and the summation runs overall contacts c of particle p. The average stress tensor σ

in a volume V of the granular assembly is defined by [61]

σ =1

V

∑

p∈V

Mp =1

V

∑

c∈V

f ci ℓc

j , (4)

0 1 2 3 4

γ0

0.1

0.2

0.3

0.4

0.5

0.6

τ w/σ

w

ns = 3

ns = 5

ns = 10

Disks

(a)

0 1 2 3 4

γ4000

6000

8000

10000

12000

14000

V/<

d>

2

ns = 3

ns = 5

ns = 10

Disks

(b)

FIG. 4: (Color online) Stress ratio τwall/σwall (a) and nor-malized volume of the packing V/〈d〉2 (b) as functions of theshear strain γ for four different values of α.

where ℓc is the intercenter vector joining the centers ofthe two touching particles at the contact c. Remark thatthe first summation runs over all particles whereas thesecond summation involves all contacts in the volume V ,with each contact appearing only once. The mean stressin 2D is given by p = (σ1+σ2)/2 and the deviatoric stressis q = (σ1 − σ2)/2, where σ1 and σ2 are the principalstresses. The coefficient of internal friction in the steadystate is defined by

µ∗ =q

p, (5)

Figure 5 shows the evolution of µ∗ as a function of theangularity α of the particles. The shear strength firstincreases with α from µ∗

0 ≃ 0.29 and then saturates forparticles having 6 or less number of sides (α ≥ 1.25) forwhich µ∗

≥1 ≃ 0.47. The data are well fit to an exponentialfunction:

µ∗ = µ∗0 + (µ∗

≥1 − µ∗0)(1 − e−α/αc), (6)

with αc ≃ 0.6. The fast increase of µ∗ with α and itssaturation is rather unexpected as it indicates that smalldeviations of the shape from disk have stronger effect onµ∗ than the larger variations of angularity for low numberof sides. This point will be discussed in more detail whenwe analyze below the microstructure and force transmis-sion.

Figure 6 shows the packing fraction ν∗ as a functionof α. We see that the packing fraction declines from

0 0.5 1 1.5 2 2.5α

0.2

0.3

0.4

0.5

0.6

µ∗

FIG. 5: (Color online) Coefficient of internal friction µ∗ asa function of the angularity α evaluated directly from thenumerical data (full squares) and predicted by Eq. (8) (emptysquares). The error bars represent the standard deviation inthe steady state.

0.0 0.5 1.0 1.5 2.0 2.5α

0.79

0.80

0.81

0.82

0.83

ν

FIG. 6: Steady-state value of the packing fraction ν∗ as afunction of the angularity α. The error bars represent thestandard deviation in the steady state.

ν∗0 ≃ 0.828 (for the disk packing) and saturates to ν∗

≥1 ≃0.798. It is remarkable that the packing fraction followsan opposite trend to that of the shear strength. Theseresults represent a new example in which a decrease inpacking fraction is accompanied by an increase in shearstrength, as it was previously observed for packings ofelongated and non-convex particles [18, 22].

In the following, we analyze the microstructural prop-erties of our packings of polygonal particles with the aimof identifying the origins of their shear strength.

IV. MICROMECHANICAL ANALYSIS

A. Connectivity

Figure 7 shows a snapshot of the contact network inthe steady state for three samples of polygonal particleswith ns = 10, 5 and 3, as well as for the disk packing.We see that the contact network topology varies stronglywith angularity. For example, the floating particles areorganized in groups in the disk packings whereas theyare mostly isolated in the case of triangular particles.

On the other hand, the contact network becomes moreconnected as the angularity increases. At lowest order,the connectivity of the particles is characterized by theproportion κ of non-floating particles and the coordina-tion number z (average number of force-bearing contactsper particle). Figure 8 shows κ and z as functions of α.We see that κ and z decline (from 0.85 to 0.68 and from3.25 to 3.15, respectively) as α increases, in accordancewith the decrease of packing fraction shown in Fig. 6.But the trend is reversed beyond α ≃ 1 for both z andκ. In particular, we observe that z increases up to 3.5which is higher than that in the disk packing. This in-crease suggests that the sharp corners of very angularparticles allow for deep contacts between neighbors thatare unreachable for less angular particles. These resultsshow that, for large angularities, the packings are looserbut better connected.

The connectivity of the particles may be characterizedin more detail by specifying the proportion Pc of particleshaving exactly c contacts. Remark that only the force-bearing contacts are concerned. We have P0 = P1 = 0.Figure 9 shows Pc for c = 2, . . . , 8 as a function of αin the steady state. For all values of α, in exception totriangles and squares (i.e. for α = 2π/3 and α = π/2in the figure), P3 prevails and it remains nearly constantbelow α ≃ 1.25. Beyond α = 1.25, it declines rapidlycontrary to all proportions Pc which increase with α.We also observe that P4 decreases slightly with α forα < 1.25 whereas in the same range P2 increases almostin the same proportion. Hence, as the angularity be-comes higher, an increasing number of particles are equi-librated by two opposite forces mostly acting at the side-side contacts. Finally, it is interesting to notice that theproportions Pc of particles with more than four contactsremain nearly constant below α = 1.25 but they increaseonly slightly in number for squares and triangles. In thisway, even a slight increase in angularity (with disk asreference shape) has a strong effect on the connectivityand mechanical behavior as we already remarked withrespect to the evolution of µ∗ and ν∗ in Figs. 5 and 6.

B. Anisotropy of the contact and force networks

The shear strength of granular materials is generallyattributed to the buildup of an anisotropic structure in-duced by shearing. This anisotropy is basically relatedto the distribution of contact normals n. Therefore, wemay obtain a full description of the state of anisotropyby a partition of various mechanical quantities accordingto the directions of contact normals. This amounts to re-placing the space direction used in continuum mechanicsfor the representation of the stress and strain fields bythe contact orientation.

The most basic descriptor of anisotropy is the prob-ability distribution P (n) of the contact normals, whichis generically nonuniform. In two dimensions, the unitvector n is described by a single angle θ, and the prob-

(a)

(b)

(c)

(d)

FIG. 7: Snapshots of the contact network for three samples ofpolygonal particles with ns = 10(b), 5(c) and 3(d), and forthe disk packing (a). The floating particles (i.e., particles withone or no contact) are drawn in light grey and the contactsare represented by line segments joining the centers of massof the particles with the contact points.

ability density P (θ) of contact orientations θ providesthe required statistical information about the contactnetwork. A local frame (n, t) can be attached to eachcontact, where t is an orthonormal unit vector; see Fig.10. The local geometry associated with the two contactneighbors is characterized by the branch vector ℓ join-ing the particle centers. It can be projected in the localcontact frame: ℓ = ℓnn + ℓtt. Note that, in contrast tocircular particles, for which ℓt = 0, in a packing of polyg-

0 0.5 1 1.5 2 2.5α

0.65

0.70

0.75

0.80

κ

(a)

0 0.5 1 1.5 2 2.5α

3.0

3.2

3.4

3.6

z

(b)

FIG. 8: (a) Proportion κ of non-floating particles as a functionof the angularity α. (b) Coordination number z as a functionof the angularity α. The error bars represent the standarddeviation in the steady state.

0 0.5 1 1.5 2 2.5α

0

0.1

0.2

0.3

0.4

0.5

Pc(2)

Pc(3)

Pc(4)

Pc(5)

Pc(6)

Pc(7)

Pc(8)

FIG. 9: (Color online) Connectivity of particles defined asthe proportion Pc(c) of particles with exactly c contacts as afunction of α in the steady state.

onal particles this component is nonzero. In the sameway, the contact force f can be expressed in terms of itsnormal and tangential components: f = fnn + ftt.

Along with P (θ), the anisotropy of the packing canbe further characterized by the angular averages of thecomponents of the branch vectors and contact forces asa function of the orientation θ: 〈ℓn〉(θ), 〈ℓt〉(θ), 〈fn〉(θ)and 〈ft〉(θ). These functions describe the general state ofanisotropy, and both experiments and simulations showthat, in a sheared granular material, they tend to take asimple unimodal shape, which can be well approximatedby the lowest-order Fourier expansion [3, 10, 18, 20, 22,

−→

ℓ

−→

f

−→n

−→

t

c

FIG. 10: (Color online) Local contact frame.

62–71]:

P (θ) = 12π{1 + ac cos 2(θ − θc)}

〈ℓn〉(θ) = 〈ℓn〉{1 + aln cos 2(θ − θln)}〈ℓt〉(θ) = 〈ℓn〉alt sin 2(θ − θlt)〈fn〉(θ) = 〈fn〉{1 + afn cos 2(θ − θfn)}〈ft〉(θ) = 〈fn〉aft sin 2(θ − θft),

(7)

where ac is the contact orientation anisotropy, aln is thenormal branch anisotropy, alt is the tangential branchanisotropy, afn is the normal force anisotropy, and aft isthe tangential force anisotropy. The angles θc, θln, θlt,θfn, and θft are the corresponding privileged directions.These directions can all be different, but they coincidewith the principal stress direction θσ in a sheared gran-ular material, as illustrated in Fig. 11.

The anisotropies ac, aln, alt, afn and aft are interest-ing not only as descriptors of the granular microstructureand force transmission, but more fundamentally becausethey add together to build the shear strength of the mate-rial. Indeed, from the expression (4) of the stress tensor,the following relationship can be easily established be-tween the anisotropy parameters and the stress ratio q/p[3, 18]:

q

p≃

1

2(ac + aln + alt + afn + aft), (8)

where the cross products between the anisotropy param-eters have been neglected. The stress ratio q/p given bythis expression from the anisotropy parameters measuredfrom the numerical data is shown in Fig. 5 as a functionof α together with those given by direct measurement.We see that Eq. (8) provides a nice approximation of theshear strength for all values of α [73].

The evolution of the five anisotropies with α isshown in Fig. 11. The normal and tangential branchanisotropies, aln and alt, are negligible in comparison tothe other anisotropy parameters. This is due to the ab-sence of shape eccentricity of the particles [18, 72] andto the low span in the particle size distribution [69]. Theother anisotropies, ac, afn, and aft, grow as α increases

(a)

(b)

FIG. 11: (Color online) Evolution of the anisotropy param-eters ac, aln and alt (a), and afn and aft (b) as functionsof the particle shape angularity α. Error bars represent thestandard deviation in the steady state. The polar diagrams ofthe corresponding angular distributions are shown for α ≃ 1(i.e., ns = 6) together with their Fourier expansion (i.e., Eqs.7).

from zero (for the disk packing) up to α ≃ 1.25 (forthe hexagon packing). This increase of all anisotropiesunderlies the observed increase in the internal angle offriction in this range. On the other hand, the increase ofthe anisotropies reflects the increasing number of side-to-side contacts, which capture the strong force chains andform column-like structures, which can be stable withoutsidewise support; see below.

For polygons with fewer than six sides (α ≥ 1.25), arapid decrease of ac and afn occurs whereas aft grows atthe same time. As it is observed in Fig. 11, this increaseof aft is large enough to compensate additively (See Eq.(8)) the decrease of ac and afn, so that the shear strengthremains nearly constant in this range of α, as observedin Fig. 5.

The decrease of ac for α ≥ 1.25, is related to the in-crease of the coordination number z as shown in Fig.8(b). Indeed, higher values of z imply higher dispersionof contact orientations. On the other hand, the increaseof aft may be attributed to the fact that the rotationalmobility of the particles are strongly reduced as a resultof enhanced angular exclusions due to shape angularity sothat the particles tend to slide rather than rolling with astrong increase of friction mobilization [70]. At the sametime, afn declines naturally as the friction forces takemore actively part in force transmission. This is, indeed,what we observe in Fig. 12 showing the mean normalforce 〈fn〉 and mean tangential force 〈|ft|〉, as well asthe proportion kslide of sliding contacts (i.e., contacts in

(a)

(b)0 0.5 1 1.5 2 2.5

α0

0.1

0.2

Ksl

ide

FIG. 12: (Color online) (a) Mean normal force 〈fn〉 and meantangential force 〈|ft|〉 normalized by p〈d〉, as functions of theangularity α; (b) Proportion kslide of sliding contacts as afunction of α. The error bars represent the standard deviationin the steady state.

which |ft| = µ|fn|), as functions of α. Both 〈fn〉 and〈|ft|〉 initially increase with α, but 〈fn〉 declines beyondα & 1 whereas 〈|ft|〉 keeps increasing. The proportionof sliding contacts rises as the particles become increas-ingly angular and takes values as high as 0.2, i.e. nearly4 times above those measured in the packing composedof disks (α = 0).

C. Role of side-to-vertex and side-to-side contacts

As it was mentioned in the previous subsection, thedistinctive features of a material composed of polygo-nal particles are explained by the possibility of formingside-side contacts. It is thus interesting to investigatethe relative roles of the two types of contacts, i.e. side-vertex (sv) and side-side (ss) contacts, with respect tothe shear strength and anisotropy. Fig. 13 shows theproportions ksv and kss of sv and ss contacts, respec-tively, as a function of α. Irrespective of angularity, thesv contacts prevail in the contact network. However, ksv

decreases from 1 for the disks (α = 0) down to ≃ 0.75for α & 1.25 and remain practically constant for moreangular particles.

Fig. 14 shows a snapshot of the normal force networkdisk packing as well as three snapshots of the packingswith ns = 10, 5, and 3. The force lines connect particle

0.0 0.5 1.0 1.5 2.0 2.5α

0.0

0.2

0.4

0.6

0.8

1.0

ksv

kss

FIG. 13: (Color online) Proportions of side-side (ss) and side-vertex (sv) contacts as functions of the angularity α. The er-ror bars represent the standard deviation in the steady state.

centers to the contacts and their thickness is proportionalto the normal force. For ns = 10 and ns = 5, the ss con-tacts appear often in distinctive force chains. But forns = 3 (triangles), despite approximately the same pro-portion kss, the ss contact forces are much more diffuseand intricately mixed with sv contacts. This visual im-pression is consistent with the decrease of an observed inFig. 11.

The stress tensor can be partitioned as a sum of twotensors representing the respective contributions of svand ss contacts by considering the expression (4) of thestress tensor and restricting the summation to each con-tact type:

σ = σsv + σss, (9)

where

(σsv)ij =1

V

∑

c∈A(sv)

ℓifj ,

(σss)ij =1

V

∑

c∈A(ss)

ℓifj , (10)

where A(sv) and A(ss) are the sets of sv and ss contacts,respectively. Fig. 15 displays the evolution of q/p, qsv/p,and qss/p as a function of α. It is seen that qsv/p is nearlyconstant and ≃ 0.24, except for the packing of triangularparticles in which qsv/p ≃ 0.35. In contrast, qss/p firstincreases with α from 0 to ≃ 0.3 for pentagons and thendeclines to 0.2 for squares and 0.1 for triangles. Thisshows that the variation of the shear strength is mostlygoverned by the contribution of side-side contacts, evenif their proportion is low. This profound effect of facetedgrain shapes on stress transmission has been previouslyshown, both experimentally and numerically [17–20].

Along the same lines, we may also evaluate the partialcontact and force anisotropies acγ , alnγ , altγ , afnγ , andaftγ , where γ stands either for ss or for sv. Since theprivileged directions of the partial angular functions de-scribing the γ contacts and forces are practically the same

(a)

(b)

(c)

(d)

FIG. 14: (Color online) Snapshots of the packing of disks (a)and the packings of polygons with ns = 10(b), 5(c), and 3(d)in the steady state. The sv contacts are in red (dark grey)and ss contacts are in green (light grey). The line thicknessis proportional to the normal force.

as the overall privileged direction for all contacts andforces, the total contact and force anisotropies are givenby the sum of the corresponding partial anisotropies. Thepartial contact and forces anisotropies are shown in Figs.16 and 17 as a function of α together with the totalanisotropies. Note that Eq. (8) is also verified whenrestricted to γ contacts. We see that acsv ≃ afnsv ≃ 0.2

0.0 0.5 1.0 1.5 2.0 2.5α

0

0.1

0.2

0.3

0.4

0.5

0.6q/pq

sv/p

qss

/p

FIG. 15: (Color online) Total shear strength (q/p) and partialshear strengths for side-vertex (sv) and side-side (ss) contactsas functions of the angularity α, together with the predictedvalues by Eq.8 (empty symbols). The error bars represent thestandard deviation in the steady state.

0.0 0.5 1.0 1.5 2.0 2.5α

0

0.1

0.2

0.3

0.4

0.5

a c

ac

acsv

acss

FIG. 16: (Color online) Partial contact orientationanisotropies acss and acsv of ss and sv contacts as functionsof the angularity α. The error bars represent the standarddeviation in the steady state.

and aftsv ≃ 0.05 for all α. In other words, the variationof the total anisotropy is mainly governed by that of theanisotropies developed by side-side contacts. The stressplateau discussed previously for the whole contact net-work for higher angularity is due to the fall-off of acss

and afnss for squares and triangles compensated by theincrease of the partial tangential force anisotropy of side-side contacts aftss. This shows the crucial role of side-side contacts in stress transmission and mobilization ofinternal friction for most angular particles.

V. CONCLUDING REMARKS

In this paper, we investigated the effect of particleshape angularity for the quasistatic behavior of shearedgranular materials by means of contact dynamics simu-lations. The particles are regular polygons characterizedby their angularity. The macroscopic and microstruc-tural properties of several packings of 104 particles insimple shear conditions were analyzed as a function of

0 0.5 1 1.5 2 2.5α

0

0.1

0.2

0.3

0.4

0.5a

fna

fnvsa

fnss

(a)

0 0.5 1 1.5 2 2.5α

0.0

0.1

0.2

0.3

aft

aftvs

aftss

(b)

FIG. 17: (Color online) Partial normal force anisotropiesafnss and afnsv (a) and partial tangential force anisotropiesaftss and aftsv, (b) of ss and sv contacts as functions of theangularity α. The error bars represent the standard deviationin the steady state.

angularity in the steady state.

We expected the steady-state internal friction coeffi-cient to decrease rapidly for decreasing angularity andtend to a nearly constant value close to that of a diskpacking. Instead, our numerical simulations reveal anearly constant value of the internal friction coefficientfor most angular polygons (triangles, squares and pen-tagons) and decreasing rapidly as angularity is reduced.A similar behavior was also observed for the packing frac-tion and several descriptors of the microstructure suchas the coordination number and anisotropy parameters.This counterintuitive observation shows that a slight in-crease in angularity (with disk as reference shape) has astrong influence on the mechanical behavior. In this re-spect, the effect of a low angularity seems to be as strongas that of surface roughness and friction coefficient be-tween particles.

For polygons with the highest angularity, i.e. for poly-gons of 3, 4 and 5 sides, a different mechanism is ob-served. In particular, the coordination number declinesas angularity increases except for highly angular particleswhere it rises. In the latter case, the contact orientationanisotropy and normal force anisotropy decline as angu-larity increases whereas the tangential force anisotropyincreases. The compensation between these effects leadsto a nearly constant shear strength. The friction mobi-

lization appears as a key parameter for the shear strengthof angular particles. Il grows smoothly with angularityand, mainly at side-side contacts, it is responsible for theincreasing shear strength of the material.

In this work, the friction coefficient between particleswas kept at a constant value for all angularities. It wouldbe highly instructive to assess the proper role of frictionby varying this parameter systematically for each angu-larity. A similar investigation can also be performed withirregular polygons in 2D and polyhedra in 3D, making itpossible to explore the implications of these results in thecontext of practical applications

We specially thank Alfredo Taboada for fruitful dis-cussions and Frederic Dubois for assistance with theLMGC90 platform used for the simulations. We acknowl-edge financial support by the Ecos-Nord program (GrantNo. C12PU01).

APPENDIX A: CONTACT DYNAMICS METHOD

In this appendix, we briefly describe the CD methodin 2D by adapting a detailed description given in [53].The implementation of the CD method with polygonalparticles is described in Section II.

1. Contact laws

Let us consider two particles i and j with a contact ata point κ within a granular material. We assume thata unique common line (plane in 3D) tangent to the twoparticles at κ can be geometrically defined so that thecontact can be endowed with a local reference frame de-fined by a unit vector ~n normal to the common line and aunit vector ~t along the tangent line with an appropriatechoice of the orientations of the axes.

Geometrically, a contact potentially exists if the gapδn between two particles is so small that within a smalltime interval δt (time step in numerical simulations) acollision may occur between the two particles. If thecontact is effective, i.e. for δn = 0, a repulsive (positive)normal force fn may appear at κ with a value dependingon the particle velocities and contact forces acting on thetwo partners by their neighboring particles; see Fig. 18.But if δn is positive (a gap), the potential contact is noteffective and fn at the potential contact κ is identicallyzero. These disjunctive conditions can be described bythe following inequalities:

{

δn > 0 ⇒ fn = 0δn = 0 ⇒ fn ≥ 0.

(A1)

The important point about this relation between δn andfn, called Signorini’s conditions, is that it can not bereduced to a (mono-valued) function.

Signorini’s conditions imply that the normal force van-ishes when the contact is not effective. But the normal

i

j

!cκ

i

!cκ

j

!nκ!t

κ

κ

FIG. 18: Geometry of a contact κ between two particles i andj with contact vectors ~cκ

i and ~cκj , and contact frame (~nκ,~tκ).

force may vanish also at an effective contact. In partic-ular, this is the case for un = δn > 0, i.e. for incipientcontact opening. Otherwise, the effective contact is per-

sistent and we have un = δn = 0. Hence, Signorini’sconditions can be split as follows:

δn > 0 ⇒ fn = 0

δn = 0 ∧

{

un > 0 ⇒ fn = 0un = 0 ⇒ fn ≥ 0

(A2)

We see that for an effective contact, i.e. for δn = 0,Signorini’s conditions hold for the variables un and fn.

Like Signorini’s conditions, the Coulomb law of dryfriction at an effective contact point can be expressed bya set of alternative inequalities for the friction force ft

and the sliding velocity ut:

ut > 0 ⇒ ft = −µfn

ut = 0 ⇒ −µfn ≤ ft ≤ µfn

ut < 0 ⇒ ft = µfn

(A3)

where µ is the coefficient of friction and it is assumed thatthe unit tangent vector t points in the direction of thesliding velocity so that ~ut · ~t = ut. Like Signorini’s con-ditions, this is a degenerate law that can not be reducedto a (mono-valued) function between ut and ft.

Signorini’s conditions (Eq. A2) and Coulomb’s frictionlaw (Eq. A3) are represented as two graphs in Fig. 19 foran effective contact between two particles. We refer tothese graphs as contact laws in the sense that they char-acterize the relation between relative displacements andforces irrespective of the rheology (visco-elastic or plas-tic nature) of the particles. These contact laws shouldbe contrasted with force laws (employed in MD simu-lations), which describe a functional depedence betweendeformations (attributed to the contact point) and forcesthat is extracted from the material behavior of the par-ticles. The force laws often employed in MD may also beconsidered as a “regularization” of the contact laws, inwhich the vertical branch in Signorini’s and Coulomb’sgraphs is replaced by a steep linear or nonlinear function.

un

fn

0

ut

ft

0

µfn

−µfn

(a) (b)

FIG. 19: Graphs of (a) Signorini’s conditions and (b) Coul-mob’s friction law.

2. Augmented contact laws

The use of contact laws in the CD method is consistentwith the idea of a discrete model defined only at the scaleof particle motions and involving no small sub-particlelength or force scales inherent to the force laws. But sucha “coarse-grained” model of particle motion implies non-

smooth dynamics, i.e. possible discontinuities in particlevelocities and forces arising from collisions and variationsof the contact status (effective or not, persistent or not,sliding or not). Such events occur frequently in granularflows and hence the approximation of the contact forcefn during δt is a measure problem in the mathematicalsense [32, 54]. A static or regular force fs is the densityof the measure fs dt with respect to time differential dt.In contrast, an impulse p generated by a collision has nodensity with respect to dt. In other words, the forces atthe origin of the impulse are not resolved at the scale δt.In practice, however, we can not differentiate these con-tributions in a “coarse-grained” dynamics, and the twocontributions should be summed up to a single measureand the contact force is defined as the average of thismeasure over δt.

In a similar vein, the left-limit velocities u−n and u−

t

at time t are not always related by a smooth variation(acceleration multiplied by the time step δt) with theright-limit velocities u+

n and u+t at t + δt. Hence, we

assume that the contact laws (Eq. A2) and (Eq. A3) aresatisfied for a weighted mean of the relative left-limit andright-limit velocities:

un =u+

n + en u−n

1 + en, (A4)

ut =u+

t + et u−t

1 + et. (A5)

The physical meaning of the coefficients en and et is bestillustrated by considering a binary collision between twoparticles. A binary collision corresponds to an effectivecontact occuring in the interval [t, t + δt] and a persis-tent contact in the sense of the mean velocity un. Inother words, we have un = 0 and thus −u+

n /u−n = en.

Hence, en can be identified with the normal restitutioncoefficient. In the same way, for ut = 0, correspondingto a nonsliding condition (adherence of the two particles

during their contact), implies −u+t /u−

t = et, which is thetangential restitution coefficient. We see that, when Sig-norini’s and Coulomb’s graphs are used with the meanvelocities given by equation (A12), a contact is persistentin terms of u+

n (i.e. u+n = 0) only if en = 0.

When a collision is not binay, the generated impulsespropagate through the contact network so that a con-tact may experience several successive impulses duringδt. Such events can be resolved for a sufficiently smalltime increment δt or they may be tracked according toan event-driven scheme. The event-tracking strategy is,however, numerically inefficient, of limited applicabilityand in contradiction with the scope of the CD methodbased on coarse-grained dynamics. The use of mean ve-locities (Eq. A12) with the contact laws, should thus beconsidered as a generalization of restitution coefficientsto multiple collisions and contact networks for which theright-limit veocities u+

n and u+t are not simply given by

the left-limit velocities multiplied by the coefficients ofrestitution as in binary collisions but by combining thecontact laws with the equations of dynamics.

3. Nonsmooth motion

The rigid-body motion of the particles is governed byNewton’s equations under the action of imposed external

bulk or boundary forces ~Fext, and the contact reaction

forces ~fκ exerted by neighboring particles at the contactpoints κ. An absolute reference frame with unit vectors(x, y) is assumed, and we set z = x × y. Each parti-cle is characterized by its mass m, moment of inertia I,

mass center coordinates ~r, mass center velocity ~U , angu-lar coordinates θ, and angular velocity ωz. For a smoothmotion (twice differentiable), the equations of motion ofa particle are

m ~U = ~F + ~Fext

I ω = M + Mext

(A6)

where ~F =∑

κ~fκ and M = z ·

∑

κ ~cκ× ~fκ where ~cκ is thecontact vector joining the center of mass to the contactκ and Mext represents the moment of external forces.

For a nonsmooth motion with time resolution δt involv-ing impulses and velocity discontinuities, an integratedform of the equations of dynamics should be used. Hence,the equations of dynamics should be written as an equal-ity of measures:

m d~U = d~F ′ + ~Fext dtI dω = dM′ + Mext dt

(A7)

where d~F ′ =∑

κ d~f ′κ and dM′ = z ·∑

κ ~cκ×d~f ′κ. Thesemeasure differential equations can be integrated over δt

with the definitions of ~F and M as approximations of

the integral of d~F ′ and dM′. With these definitions, the

integration of equation (A7) over δt yields

m (~U+ − ~U−) = δt ~F + δt ~Fext

I (ω+ − ω−) = δt M + δt Mext(A8)

where (~U−, ω−) and (~U+, ω+) are the left-limit and right-limit velocities of the particle, respectively.

The equations of dynamics can be written in a com-pact form for a set of Np particles by using matrixrepresentation. The particles are labelled with integersi ∈ [1, Np]. The forces and force moments F i

x, F iy,Mi

acting on the particles i are arranged in a single high-dimensional column vector represented by a boldface let-ter F belonging to R

3Np . In the same way, the externalbulk forces Fext,x, Fext,y,Mext applied on the particlesand the particle velocity components U i

x, U iy, ωi are rep-

resented by column vectors Fext and U , respectively. Theparticle masses and moments of inertia define a diagonal3Np × 3Np matrix denoted by M . With these notations,the equations of dynamics (A8) are cast into a single ma-trix equation:

M(U+ − U−) = δt(F + Fext) (A9)

4. Transfer equations

Since the contact laws are expressed in contact vari-ables (un, ut, fn and ft), we need to express the equa-tions (A9) in the same variables. The contacts are la-belled with integers κ ∈ [1, Nc], where Nc is the totalnumber of contacts. Like particle velocities, the contactvelocities uκ

n and uκt can be collected in a column vector

u ∈ R2Nc . In the same way, the contact forces fκ

n andfκ

t are represented by a vector f ∈ R2Nc . We would like

to transform the equations of dynamics from F and U tof and u. The formal transformation of matrix equations(A9) is straightforward. Since the contact velocities u

are linear in particle velocities U , the transformation ofthe velocities is an affine application:

u = G U (A10)

where G is a 2Nc × 3Np matrix containing basically in-formation about the geometry of the contact network. Asimilar linear application relates f to F :

F = H f (A11)

where H is a 3Np×2Nc matrix. We refer to H as contact

matrix. It contains the same information as G in a dualor symmetric manner. It can easily be shown that H =GT where GT is the transpose of G. This property canbe inferred from the equivalence between the power F ·Udeveloped by “generalized” forces F and the power f ·udeveloped by the bond forces f . In general, the matrixH is singular and, by definition, its null space has adimension at least equal to 2Nc − 3Np.

The matrix Hiκ can be decomposed into two matricesHiκ

n and Hiκt such that

uκn =

∑

i

HT,κin U i

uκt =

∑

i

HT,κit U i (A12)

and

F i =∑

κ

(Hiκn fκ

n + Hiκt fκ

t ) (A13)

Using these relations, the equations (A9) can be trans-formed into two equations for each contact κ:

uκ+n − uκ−

n = δt∑

i,j

HT,κin M−1,ij {

∑

λ

(Hjλn fλ

n + Hjλt fλ

t ) + F jext}

uκ+t − uκ−

t = δt∑

i,j

HT,κit M−1,ij {

∑

λ

(Hjλn fλ

n + Hjλt fλ

t ) + F jext}

(A14)We now can make appear explicitly linear relations be-

tween the contact variables from equations (A14) anddefinitions (A12). We set

Wκλk1k2

=∑

i,j

HT,κik1

M−1,ijHjλk2

, (A15)

where k1 and k2 stand for n or t. With this notation,equations (A14) can be rewritten as follows:

1 + en

δt(uκ

n − uκ−n ) = Wκκ

nnfκn + Wκκ

nt fκt

+∑

λ( 6=κ)

{Wκλnnfλ

n + Wκλnt fλ

t }

+∑

i,j

HT,κin M−1,ijF j

ext (A16)

1 + et

δt(uκ

t − uκ−t ) = Wκκ

tn fκn + Wκκ

tt fκt

+∑

λ( 6=κ)

{Wκλtn fλ

n + Wκλnt fλ

t }

+∑

i,j

HT,κit M−1,ijF j

ext (A17)

The coefficients Wκκk1k2

for each contact κ can be calcu-lated as a function of the contact network geometry andinertia parameters of the two partners 1κ and 2κ of thecontact κ. Let ~cκ

i be the contact vector joining the cen-ter of mass of particle i to the contact κ. The followingexpressions are easily established:

Wκκnn =

1

m1κ

+1

m2κ

+(cκ

1t)2

I1κ

+(cκ

2t)2

I2κ

,

Wκκtt =

1

m1κ

+1

m2κ

+(cκ

1n)2

I1κ

+(cκ

2n)2

I2κ

, (A18)

Wκκnt = Wκκ

tn =cκ1ncκ

1t

I1κ

+cκ2ncκ

2t

I2κ

,

where cκin = ~cκ

i ·~nκ and cκ

it = ~cκi ·~t

κ are the components ofthe contact vectors in the contact frame. The coefficientsWκκ

k1k2are inverse reduced inertia.

An alternative representation of equations (A16) and(A17) is the following:

Wκκnnfκ

n + Wκκnt fκ

t = (1 + en)1

δtuκ

n + aκn, (A19)

Wκκtt fκ

t + Wκκtn fκ

n = (1 + et)1

δtuκ

t + aκt . (A20)

The two offsets aκn and aκ

t can easily be expressed fromthe equations (A16) and (A17). The equations (A19) and(A20) or equations (A16) and (A17) are called transfer

equations [55]. It is easy to show that

aκn = bκ

n − (1 + en)1

δtuκ−

n +

(

~F 2κ

ext

m2κ

−~F 1κ

ext

m1κ

)

· ~nκ.(A21)

aκt = bκ

t − (1 + et)1

δtuκ−

t +

(

~F 2κ

ext

m2κ

−~F 1κ

ext

m1κ

)

· ~tκ.(A22)

The effect of the approach velocity (left-limit velocity)(uκ−

n , uκ−t ) appears in these equations as an impulse de-

pending on the reduced mass and the restitution coeffi-

cient. The effect of contact forces ~fλi acting on the two

touching particles i are represented by bκn and bκ

t givenby

bκn =

1

m2κ

∑

λ( 6=κ)

~fλ2κ

· ~nκ −1

m1κ

∑

λ( 6=κ)

~fλ1κ

· ~nκ,(A23)

bκt =

1

m2κ

∑

λ( 6=κ)

~fλ2κ

· ~tκ −1

m1κ

∑

λ( 6=κ)

~fλ1κ

· ~tκ. (A24)

The transfer equations (A19) and (A20) define a systemof two linear equations between the contact variables ateach contact point. The solution, when the values of an

and at at a contact are assumed, should also verify thecontact laws (A2) and (A3). Graphically, this means thatthe solution is at the intersection between the straightline (A19) and Signorini’s graph on one hand, and be-tween (A20) and Coulomb’s graph, on the other hand.

5. Iterative resolution

In order to solve the system of 2Nc transfer equations(in 2D) with the corresponding contact law relations, weproceed by an iterative method which converges to thesolution simultaneously for all contact forces and veloc-ities. We first consider a single-contact problem whichconsists of the determination of contact variables fκ

n , fκt ,

uκn and uκ

t at a single contact given the values of the off-sets aκ

n and aκt at the same contact. The solution is given

by intersecting the lines representing transfer equationswith Signorini’s and Coulomb’s graphs. The intersec-tion occurs at a unique point due to the positivity of thecoefficients Wκκ

k1k2(positive slope). In other words, the

dynamics removes the degeneracy of the contact laws.Notice, however, that the two intersections can not be

established separately when Wκκnt 6= 0. To find the lo-

cal solution, one may consider the intersection of transfer

equations with the force axis, i.e. by setting un = ut = 0.This yields two values gκ

n and gκt of fκ

n and fκt , respec-

tively:

gκn =

Wκκtt aκ

n −Wκκnt aκ

t

WκκnnW

κκtt − (Wκκ

nt )2, (A25)

gκt =

Wκκnnaκ

n −Wκκtn aκ

t

Wκκtt Wκκ

nn − (Wκκtn )2

. (A26)

It can be shown that the denominator is positive. If gκn <

0, then the solution is fκn = fκ

t = 0. This correspondsto a breaking contact. Otherwise, i.e. if gκ

n ≥ 0, wehave fκ

n = gκn. With this value of fκ

n , we can determinethe solution of the Coulomb problem. If gκ

t > µfκn , the

solution is fκt = µfκ

n and in the opposite case, i.e. ifgκ

t < −µfκn , the solution is fκ

t = −µfκn (sliding contact).

Otherwise, i.e. when −µfκn < gκ

t < µfκn , the solution is

fκt = gκ

t (rolling contact).In a multicontact system, the terms bκ

n and bκt in

the offsets aκn and aκ

t depend on the forces and veloc-ities at contacts λ 6= κ; see equations (A21), (A21),(A23) and (A24). Hence, the solution for each con-tact depends on all other contacts of the system andit must be determined simultaneously for all contacts.An intuitive and robust method to solve the sys-tem is to search the solution as the limit of a se-quence {fκ

n (k), fκt (k), uκ

n(k), uκt (k)} with κ ∈ [1, Nc].

Let us assume that the transient set of contact forces{fκ

n (k), fκt (k)} at the iteration step k is given. From

this set, the offsets {aκn(k), aκ

t (k)} for all contacts can becalculated through the relations (A21) and (A22). Thelocal problem can then be solved for each contact κ withthese values of the offsets, yielding an updated set of con-tact forces {fκ

n (k +1), fκt (k +1)}. This correction step is

equivalent to the solution of the following local problem:

Wκκnnfκ

n (k + 1)− {aκn(k)−Wκκ

nt fκt (k + 1)}

S←→ fκ

n (k + 1),

Wκκtt fκ

t (k + 1)− {aκt (k)−Wκκ

nt fκn (k + 1)}

C←→ fκ

t (k + 1).

Remark that this force update procedure does not requirethe contact velocities uκ

n(k + 1), uκt (k + 1)} to be calculated

as the offsets depend only on the contact forces. The set{fκ

n (k), fκt (k)} evolves with k by successive corrections and

it converges to a solution satisfying the transfer equationsand contact laws at all potential contacts of the system. Theiteration can be stopped when the set {fκ

n (k), fκt (k)} is stable

with regard to the force update procedure within a prescribedprecision criterion εf :

| fκ(k + 1)− fκ(k) |

fκ(k + 1)< εf ∀κ. (A27)

Finally, from the converged contact forces, the particle ve-locities {~U i} can be computed by means of the equations ofdynamics (A8).

The iterative procedure depicted above provides a robustmethod which proves efficient in the context of granular dy-namics. The information is treated locally and no large ma-trices are manipulated during iterations. The number Ni ofnecessary iterations to converge is strongly dependent on theprecision εf but not on δt. The number of iterations is sub-stantially reduced when the iteration is initialized with a glob-ally correct guess of the forces. This is the case when the

forces at each time step are initialized with the forces com-puted in the preceding step.

The uniqueness of the solution in a multicontact systemwith rigid particles is not guaranteed at each step of evolu-tion. We have 3Np equations of dynamics and 2Nc contactrelations. The unknowns of the problem are 3Np particlevelocities and 2Nc contact forces. The indeterminacy arisesfrom the fact that the 2Nc contact relations are inequations.Thus, the extent of indeterminacy of the solution reflects allpossible combinations of contact forces accommodating thosecontact relations. The degree of indeterminacy may be high,but it does not imply significant force variability since the so-lutions are strongly restrained by the contact laws. In prac-tice, the issue is more to find a mechanically admissible solu-tion (verifying the contact laws and equations of dynamics)than indeterminacy. In other words, the variability of the so-lution is often below the precision controlled by εf when theforces are computed at each time step from the forces at thepreceding step.

6. Time-stepping scheme

In CD method, the global problem of the determination offorces and velocities, as described above, is associated with atime-stepping scheme. This scheme is based on the fact thatthe first condition of Signorini’s relations in (A2) is the onlycondition referring to space coordinates. Both the equationsof dynamics and contact laws are formulated at the velocitylevel, and the first condition of Signorini is accounted for byconsidering only the effective contacts where δn = 0. Hence,the contact network is defined explicitly from particle posi-tions and it will no more evolve during the time interval δt.But the treatment of forces and velocities is fully implicit,and the right-limit velocities {~U i+, ωi+} should be used toincrement particle positions.

These remarks devise the following time-stepping scheme.Let t and t + δt be the considered time interval. The config-uration {~ri(t)} and particle velocities {~U i(t), ωi(t)} are givenat time t. The latter play the role of left-limit velocities{~U i−, ωi−}. The contact network {κ,~nκ,~tκ} is set up fromthe configuration at time t or from an intermediate configu-ration {~ri

m} defined by

~rim ≡ ~ri(t) +

δt

2~U i(t). (A28)

When this configuration is used for contact detection, otherspace-dependent quantities such as the inverse mass parame-ters Wκκ

k1k2and external forces ~U i

ext should consistently bedefined for the same configuration and at the same timet + δt/2. Then, the forces and velocities are iteratively de-termined for the contact network and the right-limit particlevelocities {~U i+, ωi+} are calculated. The latter correspond tothe velocities at the end of the time step t + δt:

~U i(t + δt) = ~U i+, (A29)

ωi(t + δt) = ωi+. (A30)

Finally, the positions are updated by integrating the updatedvelocities:

~ri(t + δt) = ~rim +

δt

2~U i(t + δt), (A31)

θi(t + δt) = θim +

δt

2ωi(t + δt). (A32)

This scheme is unconditionally stable due to its inherentimplicit time integration. Hence, no damping parameters atany level are needed. For this reason, the time step δt canbe large. The real limit imposed on the time step is cumula-tive round-off errors in particle positions since the latter areupdated from the integration of the velocities. Although theexcessive overlaps have no dynamic effect in the CD method,

they falsify the geometry and thus the evolution of the sys-tem. A sufficiently high precision or a large enough numberof iterations is required to avoid such errors. The time stepis not a precision parameter but a coarse-graining parameterfor nonsmooth dynamics. It should be reduced if the impulsedynamics at small time scales is of interest.

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