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

Abstract

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.

Full text

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
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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 .
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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 =(vθ2t−vθ,t2)1/2.
Note that defined as such, the quantity δv only accounts for
the temporal fluctuations of the spatially averaged velocity
vθ. 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 =(v2θ,t −vθ,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
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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
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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
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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
β1log10 −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
001300-6

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).
[1] G. D. R. MiDi, Eur.Phys.J.E14, 341 (2004).
[2] F. da Cruz, S. Emam, M. Prochnow, J. N. Roux, and F. Chevoir,
Phys.Rev.E72, 021309 (2005).
[3] G. Koval, J. N. Roux, A. Corfdir, and F. Chevoir, Phys.Rev.E
79, 021306 (2009).
[4] P. Jop, Y. Forterre, and O. Pouliquen, Nature 441, 727 (2006).
[5] Y. Forterre and O. Pouliquen, Annu. Rev. Fluid Mech. 40,1
(2008).
[6] J. N. Roux and G. Combe, C.R. Phys. 3, 131 (2002).
[7] T. S. Komatsu, S. Inagaki, N. Nakagawa, and S. Nasuno, Phys.
Rev. Lett. 86, 1757 (2001).
[8] P. Jop, Y. Forterre, and O. Pouliquen, Phys. Fluids 19, 088102
(2007).
[9] F. Radjai and S. Roux, Phys. Rev. Lett. 89, 064302 (2002).
[10] G. Chambon, J. Schmittbuhl, A. Corfdir, J.-P. Vilotte, and
S. Roux, Phys.Rev.E68, 011304 (2003).
[11] O. Pouliquen, Phys. Rev. Lett. 93, 248001 (2004).
[12] P. Mills, P. G. Rognon, and F. Chevoir, Eur. Phys. Lett. 81, 64005
(2008).
[13] O. Pouliquen and Y. Forterre, Phil. Trans. Math. Phys. Eng. Sci.
367, 5091 (2009).
[14] L. Bocquet, W. Losert, D. Schalk, T. C. Lubensky, and J. P.
Gollub, Phys. Rev. E 65, 011307 (2001).
[15] G. I. Tardos, S. McNamara, and I. Talu, Powder Technol. 131,
23 (2003).
[16] F. Radjai and F. Dubois, Discrete-Element Modeling of Granular
Materials (ISTE, London, and Wiley, New York, 2011).
[17]C.T.Veje,D.W.Howell,andR.P.Behringer,Phys.Rev.E59,
739 (1999).
[18] D. M. Mueth, G. F. Debregeas, G. S. Karczmar, P. J. Eng, S. R.
Nagel, and H. M. Jaeger, Nature 406, 385 (2000).
[19] A. Zervos, I. Vardoulakis, M. Jean, and P. Lerat, Mech. Cohes.
Frict. Mat. 5, 305 (2000).
[20] P. A. Cundall and O. D. L. Strack, G´
eotechnique 29,47
(1979).
[21] B. Cambou and M. Jean, Microm´
ecanique des Mat´
eriaux
Granulaires,Trait´
e MIM, S´
erie G´
eomat´
eriaux (Hermes Science,
London, 2001).
[22] L. Bocquet, J. Errami, and T. C. Lubensky, Phys.Rev.Lett.89,
184301 (2002).
[23] K. A. Reddy, Y. Forterre, and O. Pouliquen, Phys. Rev. Lett.
106, 108301 (2011).
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