We use a DEM method to simulate the shear flow of dense assemblies of 3D monosized frictional spherical grains under a fixed normal stress  P in steady-state, either dry or  in the presence of a small amount of an interstitial liquid, which gives rise to capillary menisci and attractive forces.  Dry grain assemblies are used as a reference system for which the rheological properties - in particular the approach to the quasistatic critical state – are rather well known and can be measured with good accuracy. The system behavior is characterized by the dependence of internal friction coefficient, solid fraction, normal stress differences and internal state parameters on two dimensionless control parameters: the inertial number, I and the reduced pressure, P*, comparing confining forces to contact tensile strength. Capillary forces have a significant effect on the macroscopic behavior of the system, up to P* values of several unities, especially for longer force ranges associated with larger menisci. We relate this effect to fabric anisotropy parameters of contact and distant interactions. The sensitivity of the results to the chosen model for liquid bridge formation and volume distribution for bonding and debonding pairs has also been studied

CHEVOIR, François

KHAMSEH, Saeed

ROUX, Jean-Noel

I-Revues

21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
Numerical simulation of the ow of wet granular
materials
S. Khamseh, J.-N. Roux and F. Chevoir
. Universite Paris-Est, Laboratoire Navier, 2 allee Kepler, Cite Descartes,
77420 Champs-sur-Marne, France
Resume :
Nous etudions par simulation numerique discrete (DEM) le comportement d'assemblages de billes
spheriques en ecoulement de cisaillement a contrainte normale P contro^lee, en presence de forces
capillaires creees par une faible quantite de uide intertitiel, sous forme de menisques joignant les
grains voisins (etat pendulaire). Nous portons une attention particuliere a l'approche de la limite
quasi-statique et caracterisons le comportement du materiau en exprimant le coecient de frottement
interne et la densite en fonction de deux parameters de contro^le sans dimension, le nombre d'inertie
I et la pression reduite P* (qui compare les forces de connement et la resistance a la traction d'un
menisque). Les forces capillaires ont un eet notable sur la rheologie jusqu'a des valeurs de P* de
plusieurs unites, particulierement dans le cas de force attractives a plus longue portee, pour les volumes
de menisques plus importants. Cet eet est relie a l'anisotropie de la texture du reseau des contacts et
des interactions a courte distance.
Abstract :
We use a DEM method to simulate dense assemblies of frictional spherical grains in 3D steady shear
ow under controlled normal stress P, either dry or in the presence of a small amount of an inter-
stitial liquid, which gives rise to capillary menisci and attractive forces. We pay special attention to
the quasi-static limit of slow ow. The system behavior is characterized by the dependence of internal
friction coecient and solid fraction on two dimensionless control parameters : the inertial number, I
and the reduced pressure, P*, comparing conning forces to contact tensile strength. Capillary forces
have a signicant eect on the macroscopic behavior of the system, up to P* values of several uni-
ties, especially for longer force ranges associated with larger menisci. We relate this eect to fabric
anisotropy parameters of contact and distant interactions.
Mots clefs : Granular material ; Wet grains ; Capillary force
1 Introduction
Signicant recent progress in the understanding of the rheology of granular materials in dense ows
has recently been gained from the consideration steady shear ows [1, 2, 3], under constant normal
stress P . Constitutive relations are then conveniently written for internal friction  and solid fraction
, as functions of an adequate reduced form of shear rate _ , the inertial number I = _
q
m
aP (m
and a denoting particle mass and diameter). If cohesion is introduced, another dimensionless control
parameter should be introduces [4, 5] : the reduced pressure P  is dened as the ratio P  = Pa
2
F0
of
applied pressure P on grains with diameter a to tensile strength of cohesive contacts, force F0.
We report here on simulations of spherical grain assemblies in which cohesion stem from capillary
forces acting in the narrow gaps between neighboring grains, where menisci of a wetting uid are
formed.
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21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
2 Model and Simulation
We simulate homogeneous Couette ows, as a uniform shear rate _ = dvx=dy is imposed to mono-sized
spherical grain assemblies. We use periodic, Lees-Edwards boundary conditions (no wall), and the cell
height H (in the y direction) is allowed to uctuate to keep normal stress yy = P constant [2, 3, 5].
Particles interact via Hertz-Mindlin elasticity and Coulomb friction (with coecient  = 0:3) in their
contacts as in [6], supplemented by the Maugis force [7] for capillary attraction (inter-particle distance
D, meniscus volume V and surface tension  ) assuming complete wetting :
F cap: = a [1  1q
1 + 4VaD2
] (1)
The present study is limited to the pendular state (small saturation) [8]. The liquid is conned within
(constant volume) menisci joining neighboring grains, which form as soon as two particles touch [10]
and disappear once the inter-particle distance exceeds the rupture distance, D0 = V
1=3. Macroscopic
properties were shown not to be sensitive to assumptions about meniscus volumes [9] in such models.
Capillary forces F cap:, for solid contacts (h  0) stay equal to F0 = a .
Elastic properties are such that contact deections are kept very small (rigid particle limit [3]). By
denition I ! 0 corresponds to the quasistatic limit. Our simulations investigate the interval 10 3 <
I < 10 0:5 , while P  values range from innity (no cohesion) down to 0:43 (corresponding, e.g., to
a = 1m;P = 100kPa and   = 73mJ:m 2 , the surface tension of water). V=a3 = 10 3 (thus setting
D0=a to 0:1) used in our simulations, corresponds to saturations slightly below 1%.
3 Macroscopic observations
Macroscopic friction coecient  (Fig. 1a), measured as a time average of the ratio of the shear stress
to the normal stress ( =< j12j22 >), is recorded along with solid fraction  (Fig. 1b) for dierent
values of I and P . As I decreases and the quasistatic limit of I ! 0 is approached,  decreases to
0(P ) and  increases to 0(P ), characterizing quasistatic plastic ow.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.001 0.01 0.1 1I
µ
*
P* = 0.43
P* = 1.0
P* = 10.0
P* = ∞
(a)
0.5
0.52
0.54
0.56
0.58
0.6
0.001 0.01 0.1 1I
Φ
P* = ∞
P* = 10.0
P* = 1.0
P* = 0.43
(b)
Figure 1 { Macroscopic friction coecient  (a) and solid fraction  (b) versus inertial number I
for dierent values of reduced pressure P  when rupture distance of meniscus is D0=a = 0:1.
If (0   ) / I as I tends to zero, this corresponds to a divergence of eective viscosity in constant
density shear ow as (0   ) 1= (in the range 0.4 - 0.5 in our results).
While qualitatively similar results were observed in cohesionless and cohesive systems [5], the eect
of capillary forces on the system behavior is considerably stronger in the present case, as  increases
from about 0.35 to more than 0.6 between P  innite and P  = 1.
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21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
This is due, to some extent, to the longer attraction range (here D0=a = 0:1), compared to previous
2D results [5], as conrmed by Fig. 2 : the increase of internal friction and the decrease of density
caused by cohesion are reduced for smaller D0.
0.75
0.8
0.85
0.9
0.95
1
1.05
0.001 0.01 0.1 1I
µ
*
D0 =0.1
D0 =0.01
(a)
0.505
0.51
0.515
0.52
0.525
0.53
0.001 0.01 0.1 1I
Φ
D0 =0.1
D0 =0.01
(b)
Figure 2 { Macroscopic friction coecient  (a) and solid fraction  (b) versus inertial number I
with P  = 0:43 for two dierent values of rupture distances 0.1 (red circle points) and 0.01 (blue
squares).
To understand the microscopic origin of such rheological features, we now investigate properties of
contact networks and forces.
4 Microscopic observations
4.1 Coordination numbers
The coordination number of all interacting pairs z, can be written as z = zc + zd with zc the average
number of contacts per grain and zd that of distant interactions through the liquid bridges. z and zc
are plotted for dierent values of I and P  in Fig. 3a.
2
2.5
3
3.5
4
4.5
5
5.5
0.001 0.01 0.1 1I
z
c
P
*
= 0.43
P
*
= 1.00
P
*
= 10.0
P
*
= ∞ 
(a)
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
7
0.001 0.01 0.1 1I
z
P* = 0.43
P* = 1.00
P* = 10.0
(b)
Figure 3 { Coordination number for pairs in contact zc (a) and for all interactions z (b) versus inertial
number I for dierent values of P  with rupture distance D0=a = 0:1.
3
21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
z decreases for larger values of I that correspond to lower solid fractions. Fig. 3b plots the contact
coordination number zc. zc increases for lower values of P
 , but already strongly departs from its value
in the dry case at quite large P , especially near the quasistatic limit. Enduring liquid bridges prevent
contacting grains from moving apart, and, for small I, grains are at least contacting two neighbors
in cohesive systems, while a signicant proportion (> 6%) are \rattlers\ (no force-carrying contact)
without cohesion.
Contact coordination number zc is therefore, not systematically increasing with density, which is larger
for smaller cohesive forces. However, the coordination number associated to close neighbors, separated
by a gap smaller than h, as shown in Fig. 4, is directly correlated to  for h=a > 2:5  10 3 . This
function is described by power law z(h)   z(0) / (ha )0:6 in range 0 < h=a < 0:06. On comparing
zd = z   zc (Figs. 3a, 3b) to z(D0) (Fig. 4), one deduces the proportion xM = zdz(D0) zc of pairs at
distance lower than D0 joined by a meniscus : xM varies between 0.61 and 0.68 as I increases from
10 3 to 10 0:5 , similar to some experimental results [10].
4
5
6
7
8
0 0.02 0.04 0.06 0.08 0.1
h/a
z(
h
)
P
*
= 10.0
P
*
= 1.0
P
*
= 0.43
4.8
5
5.2
5.4
0 0.0025 0.005
Figure 4 { Coordination number of close neighbors for I = 10 3, growing with gap thickness (dis-
tance) h. Blown-up detail corresponds to small h range.
4.2 Agglomeration
As observed and reported in [4, 5], adhesive forces entail particle agglomeration, and grains that stick
to one another form clusters that are transported by the ow for some distance before they are broken
or restructured. As a consequence, the distribution of the age c of contacts, as shown in Fig. 5, is
such strain intervals _c) reach values of several units. This suggests that contacting pairs survive
full tumbling motion in the average ow (Fig. 6). The average contact age increases as P  decreases,
indicating stronger cohesion eects produce longer lasting contacts. Little dierence is noted between
I = 0:1 and 0:01, showing little microstructural or geometric changes take place in this interval, which
is not far from the quasistatic limit.
4.3 Fabric
Macroscopic friction is related to fabric anisotropy [11]. The distribution of unit vectors ~n normal to
the surface of interacting grains can be expressed [11] with a few spherical harmonics terms and fabric
parameters F12, (F11   F22), (F33   1=3), F13 and F23, dened, as usual, as :
F = hnni (2)
Figs. 7a and 7b plot the fabric parameters for contact and distant interactions. In both cases F13 and
F23 are negligible. (F
c
11   F c22) and (F c33   1=3) are also small. The most noteworthy contributions to
4
21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 2 4 6 8 10 12 14 16 18
τ
c
γ
.
P
(τ
c
γ. )
I= 0.1: P
*
= 0.43
P
*
= 10
I= 0.01: P
*
= 0.43
P
*
= 10
Figure 5 { Distribution of age of contacts
P ( c _) for two dierent I and P . Age of
contacts  c is normalized by shearing time
1= _.
Figure 6 { Conguration of a system of
particles in steady shear ow with I =
0:01 and P  = 0:43. All particles that have
at least one contact with  c _  5 are rep-
resented by a red color.
shear stress at small I are associated to the additive eect of contacting pairs with ' near 3=4, with
F c12 =  0:03, mostly carrying repulsive forces, on the one hand ; and of distant attractive interactions,
oriented near ' = =4, with F d12 = 0:14, on the other hand. As sketched in Fig. 8, those repulsive and
attractive contributions respectively correspond to approaching and receding pairs according to the
average ow kinematics. This specic fabric anisotropy of distant interactions thus explains, to some
extent, the important inuence of the attractive forces of nite range to the material rheology.
-0.06
-0.04
-0.02
0
0.02
0.001 0.01 0.1 1I
F
c
12
F
c
11
-F
c
22
F
c
33
-1/3
(a)
-0.04
0
0.04
0.08
0.12
0.16
0.001 0.01 0.1 1I
F
d
12
F
d
11
-F
d
22
F
d
33
-1/3
(b)
Figure 7 { Fabric parameters of contact interactions (a) and distant interactions (b) versus I for
P  = 0:43
5 Conclusions
Quantitative measurements of the rheological properties of model wet granular materials at low sat-
uration thus reveal an important, unexpected inuence of distant attractive force on the behavior.
Many other features should be exploited (such as normal stress dierences, related to non-negligible
values of (F d11 F d22) and (F d33  1=3). Further studies, extending to larger saturations should reveal a
5
21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aou^t 2013
rich landscape of interesting rheophysical properties.
Figure 8 { Sketch of approaching (' near 3=4) and receding (' near =4) grain pairs in shear ow.
References
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1-24.
[2] Hatano, T. 2007 Power-law friction in closely packed granular materials. Phys. Rev. E 75,
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[3] Peyneau, P. -E., Roux, J.-N. 2008 Frictionless bead packs have macroscopic friction, but no dila-
tancy. Phys. Rev. E 78, 011307.
[4] Gilabert, F. A., Roux, J.-N., Castellanos, A. 2007 Computer simulation of model cohesive powders :
Inuence of assembling procedure and contact laws on low consolidation states. Phys. Rev. E 75,
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[5] Rognon, P. G., Roux, J.-N., Naam, M., Chevoir, F. 2008 Dense ows of cohesive granular materials.
J. Fluid Mech. 596, 21-47.
[6] Agnolin, I., Roux, J.-N. 2007 Internal states of model isotropic granular packings. I. Assembling
process, geometry, and contact networks Phys. Rev. E 76, 061302.
[7] Pitois, O., Moucheront, P., Chateau, X. 2000 Liquid Bridge between Two Moving Spheres : An
Experimental Study of Viscosity Eects. Journal of colloid and interface science 231, 26-31.
[8] Mitarai, N., Nori, F. 2006 Wet granular materials. Advances in Physics 55, Nos. 1-2, 1-45.
[9] Richefeu, V., Radja, F., El Youssou, M. S. 2006 Stress transmission in wet granular materials.
Eur. Phys. J. E 21, 359-369.
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Numerical simulation of the flow of wet granular materials

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