Evolution of the contact network during tilting cycles of a granular pile under gravity

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Evolution of the contact network during tilting cycles of a granular pileunder gravity
S. Deboeuf
∗
, J.-P. Vilotte, A. Mangeney
Institut de Physique du Globe de Paris, F-75252 Paris Cedex 5, France
L. Staron
Department of Applied Mathematics and Theoretical Physics, Cambridge CB3 OWA, UK
O. Dauchot
Commissariat a l’Energie Atomique, F-91191 Saclay, France
We perform discrete numerical simulations of a granular pile undergoing quasi-static tilting cycles in the gravityﬁeld. The volumic deformation and the granular fabric exhibit hysteretic evolutions with the slope of thefree surface. When exploring the range of metastable slopes -between the angle of repose and the angle of avalanche-, the contact network is strongly affected, as evidenced by the qualitative change of the hysteresis. Aspeciﬁc contribution to the observed hysteresis of the weak contacts is underlined: they carry the memory of the proximity of the slope destabilization.1 INTRODUCTIONWhen inclined in the gravity ﬁeld, a granular slopestarts ﬂowing when the surface slope reaches theavalanche angle
θ
a
, and comes to rest afterwards atthe angle of repose
θ
r
< θ
a
. In between these twoangles, the slope is metastable: avalanches can betriggered by ﬁnite amplitude perturbations (Daerr &Douady 1999). Recent numerical studies investigatedthe mechanisms of destabilization of granular pilesduring continuous load in the gravity ﬁeld (Staron,Vilotte, & Radjai 2002). Far before the avalanche,the authors observe the occurrence of local instabil-ities leading to the compaction of the pile. At theapproach of the stability limit, they evidence dila-tancy simultaneously with the percolation of clustersof critical contacts, that are essentially weak contacts.These results lead them to identify the major role of spatial correlations in the range of metastable slopes
[
θ
r
;
θ
a
]
. Similar correlations were also experimentallyobserved in (Deboeuf, Bertin, Lajeunesse, & Dauchot2003; Kabla, Debregeas, di Meglio, & Senden 2004).The metastability of the slope can be seen as a sig-nature of a major modiﬁcation of the internal state of the granular packing. In order to investigate the natureof these modiﬁcations, and to study the inﬂuence of this metastability on the following evolution, we sim-ulate by contact dynamics the quasi-static cyclic evo-lution of a granular pile tilted in the gravity ﬁeld to-wards the slopes
[
θ
r
;
θ
a
]
. Details of the numerical pro-cedure are given in section 2. An analysis of the con-tact network in section 3 shows that the volumic strainand the granular fabric evolutions are hysteretic. Theshape of the hysteresis is strongly affected when thepile slope
θ
explores the metastable range
[
θ
r
;
θ
a
]
. Insection 4 we observe the contributions of the strongand weak contacts and we evidence the speciﬁc roleof memory played by the latter.2 NUMERICAL SET-UP
Figure 1. Measure of the surface slope of the pile during therotation cycles of amplitude
θ
rev
= 10
◦
and
θ
rev
= 18
◦
.
The discrete numerical simulations are performedusing the contact dynamics method (Moreau 1994).The grains are perfectly rigid, and interact via fric-tional contacts governed by a Coulomb friction law(
µ
= 0
.
5
). The granular pile is made of a two dimen-sional rectangular box ﬁlled with
4000
circular grains
of mean diameter
D
, with a thickness of
≈
35
D
, anda width of
≈
120
D
. The solicitation consists in quasi-static rotations in the gravity ﬁeld at a rate of
0
.
001
◦
pertime step.Initially,theslopeincreasesfrom
θ
= 0
◦
to a maximal inclination angle
θ
rev
. At that stage, twosuccessive cycles
[
−
θ
rev
;
θ
rev
]
are performed. All thesimulations are averaged over
50
granular piles dif-fering in the initial grains arrangments, and showingcharacteristic angles
θ
r
15
◦
and
θ
a
21
◦
.To study whether exploring the metastable regimeinﬂuences the evolution of the granular pile, two val-ues of the maximal inclination angle
θ
rev
are investi-gated. Figure 1 shows the time evolution of the pileslope averaged over
50
simulations. The smallest am-plitude is chosen in order to keep inclination belowthe metastable range:
θ
rev
= 10
◦
< θ
r
. The largest am-plitude is chosen such that the metastable regime isexplored during each cycle, still below the minimalavalanche angle to prevent the pile from avalanching:
θ
rev
= 18
◦
∈
[
θ
r
;
θ
a
]
.3 EVOLUTION OF THE CONTACT NETWORK3.1
Volumic strains of the granular pile
Figure 2. Estimation of the volumic strain as a function of thepile slope (same symbols as ﬁgure 1).
During each loading and unloading phase, localinstabilities occur intermittently within the granularpile. The effect of these rearrangments of the grainsis apparent when plotting the volumic strain
V
=(
V
−
V
0
)
/V
0
, where
V
is the volume of the pile and
V
0
its initial volume, as a function of the slope of thepile (Fig. 2).Irrespective of the value of
θ
rev
, cycles produce anoverall densiﬁcation of the pile. Note that the consol-idation is of the same order (
10
−
3
) as via avalanch-ing from (Barker & Mehta 2000). For small cycles(
θ
rev
= 10
◦
), the granular pile is always contractant:
V
decreases. On the contrary, for large cycles (
θ
rev
=18
◦
), the granular pile exhibits compaction and di-latancy stages. Indeed for slopes in the metastablerange
[
θ
r
;
θ
a
]
, the rearrangments cause the granularpile to dilate:
V
increases, as already observed byStaron
et al.
. Interestingly, the dilatancy stage occur-ring when
θ
∈
[
θ
r
;
θ
a
]
, changes dramatically the over-all behaviour of the pile over the complete solicita-tion: the compaction is twice more efﬁcient for largecycles than for small ones.3.2
Coordinancy of the grains
Figure3.Coordinancyofthegrainsasafunctionofthepileslope(same symbols as ﬁgure 1).
For both values of the amplitude
θ
rev
, the coordina-tion number averaged over all the grains
Z
,
i.e.
themean number of contacts per grains, varies less than
1%
(Fig. 3). Nevertheless, it exhibits a weak hystere-sis according to the sense of rotation:
Z
tends to de-crease during load, but to increase during unload.Small and large cycles are identiﬁed by two differ-ent states: the value of the coordination number de-pends on
θ
rev
. During the small cycles,
Z
remains ap-proximately at its initial value, in contrast with thelarge cycles, for which a variation of the coordinationnumber occurs when exploring the metastable regimeduring the ﬁrst load phase. Exploring the metastablerange of slopes
[
θ
r
;
θ
a
]
enhances the susceptibility forgrains to rearrange and deeply modiﬁes the state of the granular packing according to its initial one.3.3
Granular fabric of the pile
To analyse further the micro-mechanical state of thegranular pile, we investigate the statistics of the ori-entation of contacts. To do so, we compute the fabrictensor
t
using the following deﬁnition:
t
ij
= 1
n
cn
c
α
=1
n
αi
n
α j
,
(1)where
i
and
j
denote the space dimension,
n
is theunity vector normal to the contact surface, and
n
c
isthe total number of contacts
α
. The anisotropy of thegranular pile is analysed with respect to the slope in
terms of the anisotropic intensity
Λ
(
= 2
×
the devia-toric component of
t
) and the direction of anisotropy
Φ
with respect to the normal direction to the free sur-face (Fig. 4 and 5).
Figure 4. Anisotropic intensity of the contact network as a func-tion of the pile slope (same symbols as ﬁgure 1).Figure 5. Anisotropic direction of the contact network as a func-tion of the pile slope (same symbols as ﬁgure 1).
During the small cycles, a rather isotropic state isobserved, and only small variations with
θ
are ob-served (
Λ
0
.
05
): the granular fabric is only weaklyaffected by the solicitation and remains close to theinitial one. On the contrary, for large cycles, the gran-ular fabric evolves signiﬁcantly up to
Λ
0
.
12
. Notethat during large cycles, the asymmetry of the initialstate is removed on the granular fabric, by contrastwith small ones.Irrespective of the value of
θ
rev
, the direction of anisotropy tends to approach that of the free surface.For large inclinations,
Φ
tends towards
±
45
◦
. This be-haviour corresponds with the mechanisms of creationof contacts in the direction of compressive stress, andthe loss of contacts in the direction of extension. In-deed in the particular case of a hydrostatic inﬁniteslope inclined in the gravity ﬁeld, continuous me-dia mechanics predicts that the principal direction of stress is
±
45
◦
with respect to the free surface.Besides these general observations, both
10
◦
and
18
◦
-cycles exhibit hysteresis in the evolution of thegranular fabric. Furthermore when the granular pileexplores the metastable regime, the shape and the am-plitude of the hysteresis are dramatically changed,pointing out the peculiar effect of this regime onthe granular packing evolution. In particular, the fab-ric evolves less rapidly during unloading than duringloading.Altogether, this makes the state of the pile at anangle
θ
depend strongly on the history: the previousproximity of the destabilization is apparent on the ob-served characteristics. Knowing that dense granularmedia exhibit two contact subnetworks, strong andweak, depending on the intensity of the force trans-mitted at the contacts (Radjai, Wolf, Jean, & Moreau1998), the role of the respective contributions of thetwo subnetworks is now investigated.4 BI-MODAL RESPONSE OF THE FABRICA vertical gradient of the contact forces is related tothe gravity ﬁeld, such that the force intensity averagedover
h
-altitude contacts obeys:
f
(
h
)
∝
h
. Thereforea
h
-altitude contact is deﬁned as strong (resp. weak) if it transmits a normal force larger (resp. smaller) thanthe normal force averaged over all
h
-altitude con-tacts. The statistical analysis of the fabric is now re-stricted to strong and to weak contacts, from whichthe anisotropic direction of the strong (
Φ
s
) and theweak subnetwork (
Φ
w
) are calculated.
Figure 6. Fabric anisotropic direction of strong contacts as afunction of the pile slope (same symbols as ﬁgure 1).
The evolution of
Φ
s
as a function of the pile slopeexhibits also hysteresis (Fig. 6). However, by contrastwith all previous observations, the shape of the hys-teresis is the same for both small and large cycles.Exploring the metastable regime does not affect thestrong contact subnetwork.
Figure 7. Fabric anisotropic direction of weak contacts as a func-tion of the pile slope (same symbols as ﬁgure 1).
On the contrary, the evolution of
Φ
w
as a func-tion of the pile slope (Fig. 7) is very different ac-cording to the cycle amplitude
θ
rev
. For small cy-cles (
θ
rev
= 10
◦
),
Φ
w
exhibits almost no hysteresis,and the hysteretic behaviour of the pile (see Fig. 5) istherefore governed by the strong contacts only. Dur-ing small cycles,
Φ
w
remains approximately alwaysnormal to
Φ
s
, as observed by Staron
et al.
in the caseof continuous load. On the contrary, for large cycles,a signiﬁcant hysteretic behaviour is observed in theevolution of
Φ
w
. It consists in an early rotation of
Φ
w
,which leads to a remarkable orientation of the fabricof the two contact subnetworks:
Φ
s
and
Φ
w
are nowaligned during a large part of the cycle, when explor-ing the metastable regime.Two conclusions can be drawn from these observa-tions:
•
For small cycles, the hysteretic behaviour of thepile can be seen as a result of the strong contactsubnetwork contribution;
•
For large cycles, the weak contact subnetwork becomes strongly affected, and the hysteretic be-haviourofthewholepileisstronglymodiﬁedde-spite the identical response of the strong subnet-work.5 CONCLUSIONSWhen performing quasi-static tilting cycles in thegravity ﬁeld, the overall behaviour of a granular pileis towards densiﬁcation at constant coordination num-ber. However differences of consolidation are ob-served between small and large cycles, due to thedilatancy stage occurring in the metastable range of slopes, that tends to enhance rearrangments of grains.Accordingly the evolution of volumic strains on geo-material samples could be measured for the detectionof previous proximities of the destabilization. Furtheranalyses conﬁrm the peculiar effect of the metastableregime on the grain packing organization: the hys-teretic evolution of the fabric is deeply modiﬁed whenexploring the metastable regime. Finally, this hys-teretic behaviour is the result of a complex interplaybetween the strong and weak contact subnetworks.Such a speciﬁc interaction has to be considered wheninvestigating the mechanical properties. For instance,it suggests to distinguish strong and weak contactswhen appreciating stability limit analyses.REFERENCES
Barker, G. & Mehta, A. 2000. Avalanches at rough surfaces.
Phys. Rev. E 61
(6).Daerr, A. & Douady, S. 1999. Two types of avalanche behaviourin granular media.
Nature
399.Deboeuf, S., Bertin, E. M., Lajeunesse, E., & Dauchot, O. 2003.Jamming transition of a granular pile below the angle of re-pose.
Eur. Phys. J. B
36.Kabla, A., Debregeas, G., di Meglio, J.-M., & Senden, T. J.2004. X-ray observation of micro-failures in granular pilesapproaching an avalanche. in prep.Moreau, J.-J. 1994. Some numerical methods in multibody dy-namics: Application to granular materials.
European Jour-nal of Mechanics A/Solids
: 93–114.Radjai, F., Wolf, D. E., Jean, M., & Moreau, J.-J. 1998. Bimodalcharacter of stress transmission in granular packings.
Phys. Rev. Lett. 80
(1).Staron, L., Vilotte, J.-P., & Radjai, F. 2002. Preavalanche insta-bilities in a granular pile.
Phys. Rev. Lett. 89
(20).

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