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PHYSICAL REVIEW E 00, 001300 (2011)

Quasistatic to inertial transition in granular materials and the role of ﬂuctuations

Johan Gaume, Guillaume Chambon,*and Mohamed Naaim

Cemagref, UR ETGR, 2 rue de la Papeterie, FR-38402 St. Martin d’H`

eres Cedex, France

(Received 16 July 2010; revised manuscript received 30 September 2011; published xxxxx)

On the basis of discrete element numerical simulations of a Couette cell, we revisit the rheology of granular

materials in the quasistatic and inertial regimes, and discuss the origin of the transition between these two regimes.

We show that quasistatic zones are the seat of a creep process whose rate is directly related to the existence

and magnitude of velocity ﬂuctuations. The mechanical behavior in the quasistatic regime is characterized by a

three-variable constitutive law relating the friction coefﬁcient (normalized stress), the inertial number (normalized

shear rate), and the normalized velocity ﬂuctuations. Importantly, this constitutive law appears to remain also

valid in the inertial regime, where it can account for the one-to-one relationship observed between the friction

coefﬁcient and the inertial number. The abrupt transition between the quasistatic and inertial regimes is then

related to the mode of production of the ﬂuctuations within the material, from nonlocal and artiﬁcially sustained

by the boundary conditions in the quasistatic regime, to purely local and self-sustained in the inertial regime.

This quasistatic-to-inertial transition occurs at a critical inertial number or, equivalently, at a critical level of

ﬂuctuations.

DOI: 10.1103/PhysRevE.00.001300 PACS number(s): 83.80.Fg, 47.57.Gc

I. INTRODUCTION

One of the most fascinating properties of granular materials

is their ability to either sustain stresses as solids, or to ﬂow as

ﬂuids, depending on the applied solicitation. It has been shown

in several studies [1–3] that, at any given point within the

material, this solid-to-ﬂuid transition is primarily controlled

by the local value of the inertial number I=˙γd (ρ/P)0.5.

This number represents the ratio between a microscopic

inertial timescale d(ρ/P)0.5(Pbeing the pressure, dthe grain

diameter, and ρthe grain density) and the macroscopic time

scale ˙γ−1associated with the shear rate ˙γ. Fluidlike behavior

is obtained for large enough values of Iand corresponds to the

so-called inertial regime. In this regime, the constitutive law of

the material is characterized by a one-to-one relationship, of

the viscoplastic type, between the friction coefﬁcient μ=τ/P

and the inertial number I[4,5]. For low values of I,onthe

contrary, the μ(I) relationship loses its validity and solidlike

behavior is recovered. In this so-called quasistatic regime, and

in agreement with the plastic constitutive laws classically used

in soil mechanics for I→0[6], the mechanical behavior

is generally described as becoming rate independent and

characterized by a constant friction coefﬁcient μs(critical state

theory).

The transition between the inertial and quasistatic regimes

typically occurs for values of Iin the range 10−3–10−2.In

detail, however, the reported Ivalue at the transition appears to

vary between existing studies, and possibly depends on system

size [1,2]. Furthermore, from these studies, it is still not clear

whether this transition occurs sharply at a given value of I,

or progressively as I→0. More generally, the real nature

of the quasistatic-to-inertial transition in granular materials,

and the physical processes involved, still remain largely

unknown. Recently, the mechanical behavior in the quasistatic

regime has been shown to be signiﬁcantly more complex than

described by classical soil mechanics. In particular, continuous

*guillaume.chambon@cemagref.fr

creep in quasistatic zones, incompatible with a supposedly

rate-independent mechanical behavior, has been reported

in several conﬁgurations (such as free-surface ﬂows and

Couette cells) [3,7,8]. Several studies have also evidenced the

existence, in the quasistatic regime, of strong and intermittent

ﬂuctuations characterized by collective particle motions with

large correlation lengths (nonlocal processes) [9–13]. Yet, and

although it is reasonable to think these ﬂuctuations may play

an important role in the macroscopic rheology of the material

[12,14,15], the link between ﬂuctuations and creep has never

been formally proved. Similarly, although the existence of

creep in the quasistatic regime could lead one to think that

the transition toward the “true” solid behavior is in fact

progressive, this issue, as well as the potential connection

between the quasistatic creep and the viscoplastic rheology in

the inertial regime, remains to be properly addressed.

The objective of this paper is precisely to explore the

links between creep, ﬂuctuations, and viscoplastic rheology

in order to propose a more consistent description of the

quasistatic-to-inertial transition in granular materials. We will

prove that the creep is effectively related to the existence

of ﬂuctuations within the sample, and that a three-variable

constitutive law between shear rate, friction coefﬁcient, and

ﬂuctuation level can be formulated in the quasistatic regime.

In addition, we will show that this constitutive law remains

valid in the inertial regime, where it can account with good

accuracy for the viscoplastic behavior observed, thus opening

the way toward a uniﬁed treatment of both quasistatic and

inertial regimes.

Our work is based on numerical simulations using the

discrete element method (DEM) [16]. This method allows

us to perform veritable numerical experiments on granular

materials. The conﬁguration simulated is a Couette cell (or

annular shear cell), which presents the speciﬁcity of placing

locally the tested sample in a state of simple shear, but with

a shear stress σrθ that decreases from the inner wall to the

outer sample boundary according to σrθ(r)=σrθ (Ri)Ri2/r2

(where Riis the inner cylinder radius and rthe radial position

of the point considered) [3,10,15,17–19]. This setting is thus

001300-1

1539-3755/2011/00(0)/001300(8) ©2011 American Physical Society

JOHAN GAUME, GUILLAUME CHAMBON, AND MOHAMED NAAIM PHYSICAL REVIEW E 00, 001300 (2011)

particularly well suited to studying the quasistatic-to-inertial

transition, since coexistence between quasistatic and inertial

zones can be observed within the same sample. It is important

to mention that our simulations are very similar to those

reported in [3]. This latter study actually presents a complete

overview of the mechanical response of a granular sample

placed in a Couette cell, which we shall obviously not

reproduce here. In the present paper, we take one step forward

and, building from the results of [3], our analysis of the

simulation data is speciﬁcally focused on insights concerning

sample rheology and the quasistatic-to-inertial transition.

This paper is organized as follows. Section II presents

the simulated system and deﬁnes the mechanical quantities

considered. In Sec. III, after having evidenced the existence of

the quasistatic-to-inertial transition, we revisit the rheological

properties of the inertial regime and propose a new approach to

the rheology of the quasistatic regime. In Sec. IV, an empirical

three-variable constitutive law valid in both the quasistatic

and the inertial regimes is derived. Lastly, conclusions re-

garding the physical nature of these two regimes and of the

transition between them are discussed in Sec. V.

II. SIMULATED SYSTEM

The discrete element simulations were performed using the

commercial software PFC2D (by Itasca) which implements

the original soft-contact algorithm described in [20]. The sim-

ulated Couette cell is two dimensional (Fig. 1), with inner and

outer cylinder radii Ri=0.4 m and Re=0.6 m. The granular

samples are composed of about 7000 circular particles of

average diameter d=4.7 mm [thus (Re−Ri)/d ≈43], with

a grain size distribution polydispersity of ±30% (diameters

ranging from 3 to 6 mm) in order to prevent crystallization.

Shear is applied by rotation of the inner cylinder at an

imposed rotation velocity which was varied between 0.05

and 20 rad s−1. The outer cylinder is ﬁxed, but consists of a

ﬂexible membrane through which a constant radial pressure

P=10 kPa is imposed onto the sample. This setting is

preferable to a rigid wall, in order to accommodate the density

variations undergone by the granular material during shear.

FIG. 1. (Color online) Simulated shear cell. Zones of different

colors within the sample illustrate the shear deformation.

TABLE I. Mechanical parameters used in the simulations. kn: nor-

mal contact stiffness; kt: tangential contact stiffness; μg: intergranular

friction; e: normal restitution coefﬁcient; ρ: particle density.

kn/P kt/knμgeρ

1040.5 0.5 0.1, 0.9 300 kg m−3

Both boundaries are constituted by grains of diameter dto

represent wall roughness.

The interparticle contact laws used in the simulations are

classical [16]. The normal force is the sum of a linear-

elastic and of a viscous contribution (spring-dashpot model),

and the tangential force is linear-elastic with a Coulombian

friction threshold. The corresponding mechanical parameters

are summarized in Table I. Let us mention that the value of the

normal stiffness knwas chosen in order to keep low normal

interpenetrations δat contacts, δ/d 10−3, i.e., to work in the

quasirigid grain limit [2,6]. Concerning the normal restitution

coefﬁcient e, we checked that the results presented below,

and more generally all the macroscopic mechanical quantities

obtained from the simulations, are actually independent of this

parameter (in the range 0.1–0.9), in agreement with previous

studies [2].

One of the main interests of DEM simulations is that

mechanical quantities such as stresses, shear rates, etc., can be

computed at each material point within the sample. Hence, the

rheological behavior of the material can be explored locally,

regardless of the spatial heterogeneities possibly displayed

by these mechanical quantities. In our case, the shear rate

˙γis obtained from the orthoradial velocity proﬁle v(r)

according to ˙γ=r[d(v/r)/d r]. The stress tensor is derived

using the classical Love homogenization formula [21]. In

the following, only mechanical responses obtained in steady

state will be considered, disregarding the transients that occur

10-4 10-3 10-2

δv

10-3

10-2

10-1

δvsp

0.05

0.10

0.25

1.00

5.00

8.00

10.0

20.0

Ω (rad s-1)

FIG. 2. (Color online) Correlation between the spatiotemporal

velocity ﬂuctuations δvsp and the temporal velocity ﬂuctuations δv

in the simulated samples. The different symbols refer to the imposed

values of inner cylinder rotation velocity .

001300-2

QUASISTATIC TO INERTIAL TRANSITION IN ... PHYSICAL REVIEW E 00, 001300 (2011)

0510 15 20 25 30 35 40

r*

0.1

0.2

0.3

0.4

0.5

0.6

μ

0.05

0.10

0.25

1.00

5.00

8.00

10.0

20.0

Ω (rad s-1)

(a)

0510 15 20 25 30 35 40

r*

10-6

10-5

10-4

10-3

10-2

10-1

I

It

(b)

0510 15 20 25 30 35 40

r*

10-4

10-3

10-2

Δ

(c)

FIG. 3. (Color online) Radial proﬁles of (a) friction coefﬁcient μ,

(b) inertial number I, and (c) ﬂuctuation number for the different

imposed values of inner cylinder rotation velocity . The proﬁles are

plotted as a function of the reduced radius r∗=(r−Ri)/d.

at the initiation of shear. Acknowledging this steady-state

condition and the cylindrical symmetry of the system, all the

mechanical quantities computed are subjected to a double,

spatiotemporal averaging procedure ·θ,t over annuli having a

thickness of 1.6dand over time windows that are sufﬁciently

long to integrate both individual and correlated particle

motions.

In addition to average quantities, we will also consider the

orthoradial velocity ﬂuctuations δv =(vθ2t−vθ,t2)1/2.

Note that deﬁned as such, the quantity δv only accounts for

the temporal ﬂuctuations of the spatially averaged velocity

vθ. We chose this ﬂuctuation measure by analogy with

the common practice in ﬂuid turbulence. In addition, as

shown in Fig. 2, we observed that the spatiotemporal velocity

ﬂuctuations deﬁned as δvsp =(v2θ,t −vθ,t2)1/2appear

strongly correlated to the temporal ﬂuctuations δv in our

system. Hence, and although this result would deserve further

discussion beyond the scope of this paper, we argue that the

quantity δv can actually be considered as a good proxy for all

types of ﬂuctuations within our samples.

Compared to the simulations described in [3], the only

notable speciﬁcity of our study lies in the consideration of

complete annular samples, instead of orthoradial periodic

boundary conditions. As a validation of our work, we checked

that the global response of the samples observed in our

simulations, such as the evolutions with rand of the

velocity, density, and stresses, fully agrees with the results

presented in [3]. As already mentioned, the reader is thus

referred to this previous study to get an overall view of the

mechanical behavior of a granular sample in a Couette cell.

In what follows, we only focus on the variables relevant to

describing the macroscopic rheology of the granular material,

namely, the inertial number I(dimensionless shear rate),

the friction coefﬁcient μ(dimensionless shear stress), and a

dimensionless measure of the velocity ﬂuctuations deﬁned

as =δv (ρ/P)0.5.

The radial proﬁles within the sheared samples of the

quantities I,μ, and are shown in Fig. 3. The observed

decrease of the friction coefﬁcient μwith ris fully explainable

by the geometrical heterogeneity of the shear stress inside

the Couette cell: μ∝(r/Ri)−2(the pressure being constant

in the sample; see [3]). In parallel, both the inertial number

Iand the velocity ﬂuctuations also decrease with r.

These two quantities display roughly exponential trends, with

characteristic lengths that remain quasiconstant for all tested

values of the rotation velocity . (In detail, however, the

localization width, i.e., the characteristic length associated

with the exponential decrease of I, may show a slight increase

with ;see[3].)

III. GRANULAR RHEOLOGY

A. A marked quasistatic-to-inertial transition

The mechanical behavior of the tested material is high-

lighted when representing directly the friction coefﬁcient μor

the ﬂuctuation number as a function of the inertial number

Ifor all locations within the sample and all rotation velocities

(Fig. 4). As already noted in [3], the plot of μversus I

[Fig. 4(a)] clearly evinces the existence of a marked rheological

transition at a given value of I,It≈5×10−3in our case. For

I>I

t, all the obtained data points collapse on a master curve

and, therefore, deﬁne a single μ(I) relationship regardless of

the values of rotation velocity and radius r.ForI<I

t,on

the contrary, there appears to be no one-to-one relationship

001300-3

JOHAN GAUME, GUILLAUME CHAMBON, AND MOHAMED NAAIM PHYSICAL REVIEW E 00, 001300 (2011)

10-6 10-5 10-4 10-3 10-2 10-1

I

0.2

0.3

0.4

0.5

μ

0.05

0.10

0.25

1.00

5.00

8.00

10.0

20.0

It

Ω (rad s-1)(a)

10-6 10-5 10-4 10-3 10-2 10-1

I

10-4

10-3

10-2

Δ

0.05

0.10

0.25

1.00

5.00

8.00

10.0

20.0

It

Ω (rad s-1)(b)

FIG. 4. (Color online) (a) Friction coefﬁcient μas a function

of inertial number I. The different symbols refer to the imposed

values of inner cylinder rotation velocity . The blue dotted curve

represents Eq. (1), while the black dashed one corresponds to Eq. (6).

(b) Fluctuation number as a function of inertial number I.The

dashed line represents Eq. (2).

between μand I. Interestingly, this transition at I=Itis also

clearly visible on the evolution of the ﬂuctuation number

with I[Fig. 4(b)]. Similarly, a one-to-one relationship (I)

is obtained for I>I

t, while no such relationship exists for

I<I

t.

Following [3], we identify this transition observed at I=It

with the quasistatic-to-inertial transition. The data points

corresponding to the inertial (I>I

t) and to the quasistatic

(I<I

t) regime will now be examined independently, in order

to exhibit the rheological properties of each of these regimes.

Let us recall that due to the decrease of the inertial number

Iwith r, zones lying in the inertial and in the quasistatic

regime may simultaneously coexist within our samples. In

what follows, these cases of coexistence between inertial and

quasistatic zones will prove to be particularly informative in

terms of rheological behavior. In detail, Fig. 3(b) shows that

such a coexistence is actually observed only above a particular

value of the rotation velocity, t≈1rads

−1.For>

t,

I>I

tat the sample’s inner boundary and an inertial zone thus

develops around the inner cylinder, surrounded by a quasistatic

zone outside. The thickness of the inertial zone progressively

decreases with , and vanishes at t.For<

t, the whole

sample lies in the quasistatic regime. Note that a quasistatic

zone, either alone or in coexistence with an inertial zone, was

always present in all our simulations.

B. Inertial regime

The rheological behavior observed for I>I

tis fully con-

sistent with the results obtained in previous studies dedicated

to the inertial regime of granular materials [1,2]. In particular,

the μ(I) relationship in Fig. 4(a) can be well ﬁtted by the

empirical expression proposed in [4]:

μ=μs+μl−μs

I0/I +1,(1)

with parameters on the same order as those obtained from

experimental data (μs=0.26, μl=0.62, I0=0.07). Note

that a simpler, alternative expression to describe this μ(I)

relationship will be proposed hereinafter. Independent of the

particular ﬁtting law used, the existence of such a one-to-one

relationship between the friction coefﬁcient μand the inertial

number Iindicates that the inertial regime is characterized

by a rate-dependent rheological behavior similar to that of

a complex ﬂuid. As a macroscopic signature of this rate-

dependent behavior, Fig. 3(a) shows that as soon an inertial

zone exists around the inner cylinder, the friction coefﬁcient

proﬁle in the sample (and thus the global torque on the inner

cylinder) increases with the rotation velocity .

In parallel, and also in good agreement with previous

studies [1,2], the relationship between the ﬂuctuation and

inertial numbers and Iobserved for I>I

tis well ﬁtted

by a power law with an exponent of 0.5 [Fig. 4(b)]:

=CII1/2,(2)

where CI≈0.12. The existence of this one-to-one relationship

(I) can be interpreted as the ﬂuctuations being created locally

by the granular agitation resulting from the shear rate [2,15].

Actually, as will be discussed later, we propose that it is

precisely this property of locality for the ﬂuctuations that

constitutes the “intrinsic” deﬁnition of the inertial regime.

C. Quasistatic regime

In contrast to the inertial regime, the quasistatic regime

is characterized by the absence of one-to-one relationships

between μand Iand between and I. We also observe in

Fig. 3(a) that the friction coefﬁcient radial proﬁles are inde-

pendent of the rotation velocity for <

t, i.e., when the

whole sample lies in the quasistatic regime. Nevertheless, these

properties do not imply that the mechanical behavior in this

regime is rate independent, as would be predicted by classical

soil mechanics constitutive laws. First, Fig. 3(b) clearly shows

that signiﬁcant deformation rates exist within the quasistatic

zones, both for >

tand for <

t. When coexistence

between inertial and quasistatic zones occurs (for >

t), the

radial proﬁles of Iare actually completely continuous across

the two zones. Hence, as already noted in [3], the quasistatic

zones in the Couette cell appear to undergo a continuous creep

001300-4

QUASISTATIC TO INERTIAL TRANSITION IN ... PHYSICAL REVIEW E 00, 001300 (2011)

which is incompatible with a rate-independent mechanical

behavior. (With a rate-independent behavior, we would rather

expect the material to remain immobile in the quasistatic zones,

except in particular localization layers concentrating all the

deformation [19].)

Second, even if there is no unique relationship between μ

and Ifor I<I

t, these two variables do nevertheless show clear

correlations [Fig. 4(a)]. In particular, all data points obtained

in quasistatic zones that coexist with an inertial zone (case

>

t) appear to collapse on a master curve which smoothly

connects with the μ(I) relationship obtained for I>I

t.For

<

t, the data points follow distinct paths in the μ-Ispace

according to the value of , but all these paths remain globally

parallel to the master curve just described for >

t. Similar

correlations are observed between and Iin Fig. 4(b).In

fact, the evolution of the ﬂuctuation number with Iin

the quasistatic regime strongly resembles that of the friction

coefﬁcient μ. Data points corresponding to >

tcollapse

on a master curve, while data points obtained for <

t

follow distinct but approximately parallel paths.

These correlations and the similarity between the evolutions

of μand with Isuggest the existence, in the quasistatic

regime, of a unique relationship between these three quantities.

To check this hypothesis, the three variables are represented in

a three-dimensional (3D) plot in Fig. 5. Although not clearly

evident on a planar representation, we observe that all the data

points obtained in the quasistatic regime effectively appear

to deﬁne a single surface in this plot. More quantitatively,

a principal component analysis of the data set made of the

triplets [log10(I),log10 (),μ] has been performed. It shows

that more than 99.9% of the data dispersion is explained

by the two largest eigenvalues of the correlation matrix (Fig. 5).

This constitutes a formal proof that the three variables are

effectively linked by a unique relationship.

123

0.01

0.1

1

10

100

%

PCA

analysis

10 −3

10−2

0.16 0.24 0.28

0.2 μ

Δ

I

10 −2

10 −3

10 −4

10 −5

FIG. 5. (Color online) 3D plot representing the triplets constituted

by the values of inertial number I(log10 scale), ﬂuctuation number

(log10 scale), and friction coefﬁcient μobtained in the quasistatic

regime (I<I

t). The histogram represents the three eigenvalues of the

data-set correlation matrix inferred through principal value analysis

(PCA). The surface plotted as an eyeguide corresponds to the plane

deﬁned by the two largest eigenvalues of the PCA.

FIG. 6. (Color online) Instantaneous snapshots of particle ve-

locities in the simulated Couette cell. White (light) particles are

characterized by a velocity that exceeds a ﬁxed threshold (0.4 m s−1

in this example), whereas red (dark) particles have a velocity

below this threshold. (a) =5, (b) =8, (c) =10, and (d)

=20 rad s−1.

Hence, in spite of the absence of one-to-one relationships

between μand Iand and Iin the quasistatic regime,

these quantities are nevertheless strongly correlated through

a three-variable relationship of the form I=f(μ,). Since

it only involves local variables, this relationship, for which

we shall propose an empirical expression in what follows,

can be viewed as a local constitutive law characteristic of the

quasistatic regime. Note, however, that the existence of this

law does not imply that the rheology of the material is local.

Indeed, and unlike in the inertial regime, the ﬂuctuations in

the quasistatic regime appear to be essentially produced by

nonlocal processes. Qualitative observation of the particle ve-

locities shows that ﬂuctuations tend to organize into short-lived

“bursts” that emerge at the boundary of the quasistatic zone

(either at the inner cylinder or at the interface with the inertial

zone) and then “propagate” into the material (Fig. 6). We also

remark that when the ﬂuctuation level at the boundary of a

quasistatic zone is ﬁxed, i.e., when the quasistatic zone coexists

with an inertial zone, then the complete radial proﬁle of the

ﬂuctuations inside this quasistatic zone is also ﬁxed (cf. the col-

lapse of the data points corresponding to >

tin Fig. 4(b)].

These observations indicate that ﬂuctuations in the quasistatic

regime are in fact essentially sustained by the boundary

conditions applied at the periphery of the quasistatic zones.

IV. EMPIRICAL CONSTITUTIVE RELATIONS

A. Derivation of a three-variable relationship

From the principal component analysis presented above,

it can be deduced that a linear function in terms of the

001300-5

JOHAN GAUME, GUILLAUME CHAMBON, AND MOHAMED NAAIM PHYSICAL REVIEW E 00, 001300 (2011)

variables log10(I), log10(), and μwould constitute a good

approximation for the three-variable constitutive relationship

characterizing the quasistatic regime (at least for the values of

I,μ, and covered in our simulations). However, an even

better empirical expression for this relationship can be derived

by analyzing the dependence between Iand at constant

values of μ. Such an analysis is possible for <

t, i.e.,

when the whole sample lies in the quasistatic regime, owing

to the fact that the friction coefﬁcient μis independent of

in this case. Therefore, studying the dependence between I

and for data determined at constant values of radius ris

equivalent to considering constant values of μ.

Figure 7(a) shows that for each value of friction coefﬁcient

μ, the evolution of the ﬂuctuation number as a function of

the inertial number Iapproximately follows a power-law of

the form

=CQS Iβ(μ),(3)

with a prefactor CQS ≈0.59 independent of μ, and an

exponent βincreasing with μ. Furthermore, the dependence

of this exponent βwith μappears to be essentially linear

[Fig. 7(b)]:

β(μ)=β0+β1μ, (4)

with β0≈0.29 and β1≈1.80. Quantitatively, in terms of

least-squares error, these empirical expressions (3) and (4)

provide a ﬁt to the surface deﬁned by the quasistatic data

points in Fig. 5which is 20% better than the ﬁt obtained

with the linear relationship derived from principal component

analysis.

10-6 10-5 10-4 10-3 10-2

I

10-3

10-2

Δ

0.284

0.266

0.249

0.239

0.221

0.206

0.2 0.24 0.28

μ

0.65

0.7

0.75

0.8

β

μ

b

a

FIG. 7. (Color online) (a) Fluctuation number as a function

of inertial number Iat ﬁxed values of friction coefﬁcient μ(see

legend). The represented data come from simulations in which the

whole sample lies in the quasistatic regime (<

t). Dotted lines

correspond to the best power law ﬁts Eq. (3) obtained for each value

of μ. The dashed line represents the extrapolation of the relationship

between and Iestablished in inertial regime Eq. (2). (b) Evolution

of power-law exponent βwith friction coefﬁcient μ. The dashed line

represents the best linear ﬁt Eq. (4).

0.6 0.7 0.8 0.9 11

(log10 Δ - log10 CQS)/log10 I

0.15

0.2

0.25

0.3

0.35

0.4

μ

0.05

0.10

0.25

1.00

5.00

8.00

10.0

20.0

Ω (rad s-1)

FIG. 8. (Color online) Friction coefﬁcient μas a function of the

composite variable (log10 −log10 CQS)/log10 I. All data points

determined from the simulations are represented, the different

symbols referring to the values of imposed rotation velocity .The

dashed line represents the prediction of Eq. (5).

Equations (3) and (4) can be rewritten in a more compact

form as

μ=1

β1log10 −log10 CQS

log10 I−β0.(5)

As a further validation of this expression, Fig. 8shows

that all data points determined in the quasistatic regime,

including those corresponding to >

twhich were not

taken into account in the establishment of Eqs. (3) and (4),

effectively collapse on a single master curve when plotted

in terms of μversus (log10 −log10 CQS)/log10 I. Hence,

we argue that Eq. (5), or equivalently Eqs. (3) and (4), can

be regarded as good approximations of the three-variable

constitutive relationship characterizing the quasistatic regime.

We emphasize, however, that these equations are purely

empirical at this stage, and that we cannot rule out the possible

existence of alternative expressions that would produce an

even better ﬁt to the data.

B. Extrapolation to the inertial regime

An important point not mentioned above is that Fig. 8

also includes the data determined in the inertial regime.

Interestingly, these data appear to collapse on the same master

curve as those corresponding to the quasistatic regime. This

unexpected result seems to indicate that Eq. (5), established in

the quasistatic regime, would also remain valid in the inertial

regime. To check this prediction, Eq. (5) can be combined

with Eq. (2) relating and Iin the inertial regime, to yield the

following relationship between the variables μand Ialone:

μ=μ0−μ1

log10 I,(6)

with μ0=(1/2−β0)/β1≈0.12 and μ1=(log10 CQS −

log10 CI)/β1≈0.38. As shown in Fig. 9, this expression

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QUASISTATIC TO INERTIAL TRANSITION IN ... PHYSICAL REVIEW E 00, 001300 (2011)

10-5 10-4 10-3 10-2 10-1

I

0.2

0.3

0.4

0.5

μ

extrapolated data

raw data

FIG. 9. (Color online) Relationship between friction coefﬁcient μ

and inertial number Ifor the data points obtained in the inertial regime

(I>I

t) and for the data points extrapolated from the quasistatic

regime (see text). The dashed curve represents Eq. (6).

effectively provides an excellent ﬁt to the μ(I) relationship

observed in the inertial regime, with fewer parameters than

Eq. (1) used previously [see also Fig. 4(a)]. Furthermore,

we emphasize that in this case, the two parameters μ0and

μ1were not adjusted on the inertial data, but directly derive

from Eqs. (2) and (5), the latter having been established using

quasistatic data only.

Hence, the three-variable constitutive law between I,μ,

and characteristic of the quasistatic regime turns out to

be also valid in the inertial regime. This three-variable law

actually includes the μ(I) relationship characterizing the

inertial regime, which emerges as soon as ﬂuctuations begin

to be governed by the local law (2). In consequence, a virtual

continuation in the quasistatic regime of the inertial μ(I)

relationship can be drawn assuming that Eq. (2) remains valid

for I<I

t. This continuation is shown in Fig. 9using directly

Eq. (6), and through extrapolated data points which have been

computed from quasistatic results as the intersects, for each

value of friction coefﬁcient μ, between relationships (3) and

(2)(seeFig.7). Note that the value of the exponent βused to

compute these intersects was the best-ﬁtting value obtained

for each friction coefﬁcient, and not the linear approximation

given by Eq. (4). It is interesting to note that these extrapolated

data points ﬁgure in the exact continuity of the data points

obtained in the inertial regime, which constitutes further

evidence that the quasistatic and inertial regimes are

effectively governed by the same underlying constitutive law.

V. DISCUSSION AND CONCLUSIONS

The analysis presented in this paper sheds new light on the

rheological behavior of granular materials in the quasistatic

regime. Our results clearly show that this regime is charac-

terized both by a friction coefﬁcient independent of the shear

rate, and by a local, three-variable constitutive relationship

between the inertial number, the friction coefﬁcient, and the

normalized velocity ﬂuctuations. Importantly, this constitutive

law directly relates the occurrence of creep in quasistatic zones

to the existence of ﬂuctuations. It is only in the absence

of ﬂuctuations that the material is jammed and I=0for

whatever the level of stress. As soon as ﬂuctuations exist, the

material can ﬂow even under very small applied stresses, with

an apparent viscosity that is a direct function of the ﬂuctuation

amplitude [see Eq. (3)]. The key role played by the ﬂuctuations

in the rheology of quasistatic granular materials has already

been hypothesized by several authors [13,15,22,23]. Our

study thus formally demonstrates this assumption by showing,

directly from local mechanical data, that ﬂuctuations must be

accounted for in the constitutive relationship. An interesting

perspective for future work would now be to go beyond

the purely empirical approach presented here, and develop

a theoretical framework capable of yielding a three-variable

constitutive law compatible with our results.

Another important outcome of this study is the fact that

the three-variable constitutive law obtained in the quasistatic

regime appears to be also valid in the inertial regime. When

combined with the expression governing the ﬂuctuations in

the inertial regime Eq. (2), this constitutive law yields a

one-to-one μ(I) relationship which, though of a different form

from the relationships previously proposed in the literature

[2,4], provides an excellent ﬁt to the inertial data. Hence,

the distinction between the quasistatic and inertial regimes

cannot be related to the rate-dependent or rate-independent

character of the mechanical behavior. Both regimes are in

fact characterized by the same underlying constitutive law

involving the variables I,μ, and .

In spite of this similarity in mechanical behavior, the

quasistatic and inertial regimes are nevertheless separated by

an abrupt transition at I=It. Elaborating from our results,

we propose that the “true” origin of this transition is in fact

related to the mode of production of the ﬂuctuations within

the material. As already pointed by other workers [1,2], in

the inertial regime the ﬂuctuations necessary for the ﬂow are

created locally by the ﬂow itself. This is the meaning of Eq. (2),

and explains the possibility of reducing the three-variable

constitutive law to a local relationship between μand Ialone.

On the contrary, in the quasistatic regime, the ﬂuctuations

result from nonlocal processes [9–11] and, as shown by our

results, are essentially sustained by sources localized at the

boundaries of the quasistatic zones (either at the wall or at the

interface with the neighboring inertial zone). In particular,

when a quasistatic zone coexists with an inertial one, the

ﬂuctuations within the former and, as a consequence, its

apparent rheology (namely, the apparent relationship between

μand I), are completely controlled by the latter. In the absence

of boundary sources, on the contrary, the ﬂuctuations in the

bulk would rapidly die off, and Iwould tend to 0. Eventually,

the three-variable relationship presented in this paper will thus

need to be complemented by a nonlocal evolution equation for

the ﬂuctuations in quasistatic regime.

As a conclusion, the quasistatic-to-inertial transition in

granular materials thus appears to correspond to a transition

between a regime where the ﬂuctuations are governed by

nonlocal processes and a regime where ﬂuctuations are

produced locally. This transition occurs at a critical value of

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JOHAN GAUME, GUILLAUME CHAMBON, AND MOHAMED NAAIM PHYSICAL REVIEW E 00, 001300 (2011)

Ior equivalently [according to Eq. (2)], at a critical value

of ,t≈8×10−3in our case, which can be interpreted

as the ﬂuctuation level above which the mechanisms for

long-range propagations of the ﬂuctuations become inefﬁcient.

Following this interpretation, the parameters tand Itshould

thus represent intrinsic characteristics of the material, whose

value should be independent of the considered system. This

prediction tends to be conﬁrmed by the results reported in [2,3],

in which different system sizes and system geometries result

to apparently constant values of Iat the quasistatic-to-inertial

transition. Yet, the uncertainties associated with the determi-

nation of Itand the different methods employed to deﬁne this

parameter (from global or local measurements) render difﬁcult

the comparisons among existing studies; further work would

be needed to conclude on the intrinsic nature of It.

ACKNOWLEDGMENT

Financial support from the French National Agency for

Research is acknowledged (ANR project MONHA).

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