Annular shear of cohesionless granular materials: from inertial to quasistatic regime

0.94 MB·English

by  Georg Koval

Annular shear of
ohesionless granular materials:
from inertial to quasistati
regime
∗
Georg Koval, Jean-Noël Roux, Alain Corfdir and François Chevoir
Université Paris-Est, Institut Navier, Champs sur Marne, Fran
e
(Dated: January 14, 2009)
Using dis
rete simulations, we investigate the behavior of a model granular material within an
annularshear
ell. Spe
i(cid:28)
ally,two-dimensionalassemblies ofdisksarepla
edbetweentwo
ir
ular
9 walls, the inner one rotating with pres
ribed angular velo
ity, while the outer one may expand or
0 shrink and maintains a
onstant radial pressure. Fo
using on steady state (cid:29)ows, we delineate in
0 parameter spa
e the range of appli
ability of the re
ently introdu
ed
onstitutive laws for sheared
2 granularmaterials(basedontheinertial number). Wedis
ussthetwooriginsofthestrongerstrain
rates observed near the inner boundary, the vi
inity of the wall and the heteregeneous stress (cid:28)eld
n
inaCouette
ell. Abovea
ertainvelo
ity,aninertial region developsneartheinnerwall, towhi
h
a
theknown
onstitutivelawsapply,withsuitable
orre
tionsduetowallslip,forsmallenoughstress
J
gradients. Awayfromtheinnerwall,slow,apparentlyunbounded
reeptakespla
einthenominally
4
solid material, although its density and shear to normal stress ratio are on the jammed side of the
1

riti
al values. In addition to rheologi
al
hara
terizations, our simulations provide mi
ros
opi

information on the
onta
t network and velo
ity (cid:29)u
tuations that is potentially useful to assess
]
i theoreti
al approa
hes.
c
s
PACSnumbers: 45.70.Mg,81.05.Rm,83.10-y,83.80.Fg
-
l
r
t
m
I. INTRODUCTION stress ratio, quasistati
strains are possible that are de-
. s
ribed by elastoplasti
models. For large enough shear
t
a strains [11, 12℄, a solidlike material approa
hes the so-
Signi(cid:28)
ant progress in the modeling of dense granular
m
alled
riti
al state, whi
h
oin
ides with the state of
(cid:29)owintheinertialregimehasbeenbroughtaboutbythe I 0
- steady shear (cid:29)ow in the limit of → .
re
ently introdu
ed vis
oplasti
laws [1, 2, 3℄, as identi-
d
n (cid:28)ed in experiments and dis
rete numeri
al simulations On
e
onstitutive laws are obtained on dealing with
o in two-dimensional (2D) [4, 5, 6℄ and three-dimensional homogeneous systems, they should be lo
ally appli
a-
c (3D) [7, 8, 9℄ situations. ble to all possible (cid:29)ow geometries. Of
ourse, they are
[
One typi
ally
onsiders homogeneous assemblies of quite unlikely to provide a proper des
ription of some
1 grains of size d and mass density ρp, under shear stress strongly heterogeneous situations o

urring when strain
σ P
v and average pressure . Denoting the shear rate as is lo
alized near boundaries, in thin layers, on a s
ale
γ˙
0 ,
onstitutive laws are
onveniently expressed as rela- of a few grains. Yet, for smoothly varying stress (cid:28)elds,
06 µti∗ons=bσet/wPeen dimensionlessνquantities: e(cid:27)e
tive fri
tion they mightprovesu
essful,aswasshowne.g., with(cid:29)ows
( ), solid fra
tion , and most noti
eably iner- downin
linedplanes. Those werestudied in the absen
e
2 I =γ˙d ρ /P
p oflateralwallsbothexperimentallyandthroughdis
rete
. tial number , thus res
alingvarious exper-
1 simulations (see [2℄ for a review). Then the stress distri-
imental data into ap
onsistent pi
ture. As the ratio of
0
bution be
omes heterogeneous but the e(cid:27)e
tive fri
tion
9 the inertial to shear times, the latter parameter quanti-
remains
onstant, so that the situation is
omparable to
0 (cid:28)es the inIertia1l0e−(cid:27)2e
ts. For a fri
tional material, a small
homogeneous (cid:29)ows. More remarkably, a three dimen-
v: value of (≤ )
orresponds to the Iquasis1t0a−ti1

rit- sionalversionofthe
onstitutivelaw[7, 13℄ wasfoundto
i
al state regime, while a large value of (≥ )
or-
i I
X responds to the
ollisional regime [10℄. As in
reases, model similar (cid:29)ows between lateral walls, whi
h indu
e
ν
truly three-dimensional stress distributions and velo
ity
r solidfra
tion de
reasesνappro=ximνatelylinearlystarting
a max c pro(cid:28)les [14℄.
from a maximum value (dynamµi
∗ dilatan
y
law), while the e(cid:27)e
tive fri
tion
oe(cid:30)
ient in
reases Othersimplegeometriesaretheverti
al
huteandthe
aµp∗prox=imtaanteφly linearly starting from a minimum value annularshear[2℄. Thepresentpaperinvestigatesthema-
min (dynami
fri
tion law). This yields a vis- terial behavior in the annular (Couette) shear geometry,

oplasti

onstitutivelaw,withaCoulombfri
tionalterm forwhi
hthesampleis
on(cid:28)nedbetweentworough
ylin-
and a Bagnold vis
ous term. ders and sheared by the rotation of the inner one. The
I 0
In theµq∗uasistµat∗i
regime ( → ) this approa
h in- annular shear
ell is a
lassi
al experimental devi
e to
di
ates → min
onstant, independently on the measuretherheologi
alpropertiesof
omplex(cid:29)uids,and
strain, in steady shear (cid:29)ow. Below this minimum hasbeen usedforgranularmaterials,bothin twodimen-
sions[15, 16, 17, 18, 19℄ and in three dimensions [20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34℄. In this
∗ geometry, the stress distribution is well known, as will
Ele
troni
address:
hevoirl
p
.fr be detailedin the following: thenormal stressisapprox-
2
imately
onstantwhiletheshearstressstronµg∗lyde
reases
away from the inner wall. The de
reaseof away from 1
theinnerwallthenexplainsthelo
alizationoftheshear.
0.36788
Wemayevenexpe
tatransµit∗ionµbe∗tweenaninertial(cid:29)ow
near the inner wall, where ≥ min, and a quasistati
di
0.13534
regime further, whi
h analysis would help understand-
ing shearlo
alizationnear awall (in(cid:29)uen
e of shearrate 0.0497d9p
and
on(cid:28)ningpressureonthewidthanddilationofshear P
0.01832
bands), of interest in industrial
ondu
ts [35℄, geote
hni-

al situations [36℄ and te
tonophysi
s [37℄.
0.00674
Followingpreviousdis
retesimulations[38,39,40,41℄,
we investigate the rheology and the mi
rostru
ture of 0.00248
granular materials in this geometry. We
onsider two-
r
dimensional,slightlypolydisperseassembliesof
ohesion- 0.0 0.2 0.4 0.6 0.8 1.0
less fri
tional disks. This allows to vary the shear state
and provides a

ess to mi
ros
opi
information at the RRiint RRoext
s
ale of the grains and of the
onta
t network, hardly
FIG. 1: Annular shear geometry (bla
k grains
onstitute the
measurable experimentally. We pres
ribe the shear rate
rough walls).
and the pressure, allowing global dilation of the shear

ell. To save
omputation time, we implement periodi
boundary
onditions. Allalongthispaper, weshall
om-
pare our results with the homogeneous shear
ase [4℄. Sin
e we have not observed any in(cid:29)uen
e of the outer
Se
. II is devoted to the des
ription of the simulated wall on the behavior of the sheared material
lose to the
R 2R R = 2R
system, its preparation and the de(cid:28)nition of dimension- inner wall for o ≥ i, we have set o i, in the
less
ontrol parameters. Se
. III des
ribes the in(cid:29)uen
e results presented in this paper. The geometry is then
R /d
oftheshearvelo
ityandofthesystems
aleontheshear de(cid:28)ned by the sole value of i .
lo
alizationneartheinnerwallthroughtheradialpro(cid:28)les
Animportantfeatureofourdis
retesimulationsisthe
of various quantities. Se
. IV shows the validity of the

ontrol of the normal stressexerted by the outer wall on
σ (R ) = P
previous
onstitutivelawforinertialregime,andanalyzes rr o
the grains, as done in [41℄. We pres
ribe ,
its limit in quasistati
regime. Se
. V then explains how
Rt˙hro=ug(hPtheσra(dRial))m/gotion of thge outer wall, given by:
the
onstitutive law is able to predi
t various quantities o rr o p p
− , where is a vis
ous damping
measured in Se
. III.
parameter. In steady state, the mR˙otio=n0of the outer wall
Preliminary and
omplementary results are presented os
illatesaround a mean value (h oi )
orresponding
in [42℄.
to a pres
ribed value of the normal stress at this point
σ (R ) = P
rr o
(h i ). Su
h
ontrol of the radial stress is
applied in
ylinder shear apparatus aimed at studying
II. SIMULATED SYSTEMS
the behavior of soils near an interfa
e [43℄. It di(cid:27)ers
from most experiments and dis
rete simulations, where
A. Annular shear the volume is (cid:28)xed in twodimensions [16, 18, 31, 38, 40,
44℄, or dilatan
y is possible through the free surfa
e in
The simulated systems are two dimensional (2D) as three dimensions [20, 22, 24, 25, 26, 28, 30, 32, 33, 34℄.
indi
ated in Fig. 1. The granular material is a dense as- We use the standard spring-dashpot
onta
t law de-
n d
sembly of dissipative disks of average diameter and s
ribed in [4℄, whi
h introdu
es the
oe(cid:30)
ients of resti-
m 20% e µ
average mass . A small polydispersity of ± pre- tution and fri
tion , and the elasti
sti(cid:27)ness pa-
k k
n t
vents
rystallization. rameters and . Dis
rete simulations are
ar-
The granular material is subje
ted to annular shear ried out with standard mole
ular dynami
s method, as
between two
ir
ular rough walls. The outer wall (ra- in [4, 9, 45, 46, 47, 48℄. The equations of motion are
R
dius o) does not rotate, while the inner wall (radius dis
retized using Gear's order three predi
tor-
orre
tor
R Ω
i) moves at the pres
ribed rotation rate . The wall algorithm [49℄.
roughness, whi
h redu
es sliding, is made of
ontiguous Tode
reasethe
omputationtime,wehaveintrodu
ed
θ
glued grains with the same
hara
teristi
sas the (cid:29)owing periodi
boundary
ondition along , exploiting the an-
grains (polydispersity and me
hani
al properties). We gular invarian
e [50℄. This redu
es the representation of
r θ r =R 0 θ Θ
i

all and theradialandorthoradialdire
tions. the annular shear
ell to an angular se
tor ≤ ≤
r=R θ < π 0 θ 2π
o
and respe
tively
orrespond to the
enters of the ( ) instead of the whole system ≤ ≤ .
Θ = 2π/N N
grains whi
h
omposethe inner and outer walls. As the , where is an integer. The des
ription of
grains of the inner wall form one rigid body, the motion this method, together with the analysis used to
hoose
Θ
of ea
h of them
ombines a translation of its
enter with the values of a

ording to the size of the systems, is
ΩR~e Ω
i θ
velo
ity and a rotation rate . presented in App. A. The list of simulated geometries is
3
0.0025
given in Tab. I. We noti
e that the studied systems are havebeen studied systemati
ally forall systems: ,
0.025 0.25 0.5 1.0 1.5 2.5 0.00025
mu
h larger than in previous dis
rete simulations. , , , , and . The value was
also
onsidered in a few
ases.
k P
n
Moreoverthe stresss
ales and may be
ompared
κ = k /P
TABLE I: List of simulated geometries. n
through the dimensionless number . Let us
h
n Ri/d Θ/2π
all the normal de(cid:29)e
tion of the
onta
t (or apparent
R25 1500 25 1/4 interpenetration of undeformed disks). hB/deing inversely
R50 3100 50 1/8 proportional to the relativPe dκe(cid:29)e
tion of the
on-
R100 8000 100 1/12 ta
tsfora
on(cid:28)ningstress , is
alled
onta
t sti(cid:27)ness
R200 15700 200 1/24 number [4℄. A large value
orresponds to rigid grains,
while a small value
orresponds to soft grains. It was
shown [41℄0t4hat it has no in(cid:29)uen
e on the results on
e it
ex
eeds , whi
h is the value
hosen in all our dis
rete
simulations (rigid grain limit).
In the following (both text and (cid:28)gures), the length,
B. Dimensional analysis d m
mass, time and stress are made dimensionless by , ,
m/P P
and , respe
tively.
In our dis
rete simulations, the system is
ompletely pTable II gives the list of material parameters.
des
ribed by alist of independent parametersasso
iated
to the grains and to the shear state. As a way to redu
e
TABLE II:List of material parameters.
the numberof parameters,itis
onvenientto usedimen-
sional analysis, whi
h guarantees that all the results
an µ e kt/kn κ
be expressed as relations between dimensionless quanti- poly±di2s0p%ersity 0.4 0.1 0.5 104
ties.
d m
The grains are des
ribed by their size and mass ,
e µ
their
oe(cid:30)
ientsofrestitution andfri
tion , andtheir
k k
n t
elasti
sti(cid:27)ness parameters and . It was shown in
kt/kn C. Steady shear states
previous dis
rete simulations [4, 46, 51℄, that and
e
havenearlyno in(cid:29)uen
e on dense granular(cid:29)ows. Con-
k /k 0.5 e 0.1
sequently, t n was (cid:28)xed to and was (cid:28)xed to . Foragivensample,the(cid:28)rststep
onsistsindepositing
µ µ=0
Thein(cid:29)uen
eof ,espe
iallynear ,hasbeenshown the grains without
onta
t and at rest between the two
in [4℄. In this paper, werestri
touranalysisto the value distantwalls. Applyinganormalstressattheouterwall,
µ=0.4
, ex
ept for the dis
ussion of tµhe=
o0nstitutive law we
ompresstheassemµb=ly0ofgrains,
onsidering(cid:28)rstthat
where the
ase of fri
tionless grains ( ) will be also theyarefri
tionless( ),soastogetaverydenseini-
µ
analyzed. Results for other values of may be found tial0s.8ta5te. Ex
eptnearthewalls,itssolidfra
tionis
lose
in [42℄. to , near the random
lose pa
king of slightly poly-
The shear state is des
ribed by the pres
ribed normal dispersed disks [52℄. When the granular material sup-
P
stress on the outer wall , the rotation rate of the inner ports
ompletely the applied normal stress, the grains
Ω R R
wall , the radius i and o of the two walls, and the are at rest and the dense system is ready to be sheared.
g
vis
ous damping parameter p. We have not observed Westartto shearthematerial(now
onsideringthatthe
g
any in(cid:29)uen
e of p, on
e the shear zone is lo
alized near grains are fri
tional) imposing the rotation of the inner
theinnerwallandseparatedbyarelativelythi
klayerof disk. Afteratransient,thesystemrea
hesasteadystate,
materialfrom the outer wall. The dimensionless number
hara
terized by
onstant time-averaged pro(cid:28)les of solid
g /√mk 0.1
p n remains of order in all our simulations, so fra
tion, velo
ity and stress. In pra
ti
e, the stabiliza-
R
that the time s
ale of the (cid:29)u
tuations of o is imposed tion of the pro(cid:28)les depends on the
onsVid∆erted variab∆let.
θ
by the material rather than the wall, and that the wall If we take the inner wall displa
ement (where
sti
ks to the material. Consequently, the shear state is is the simulation time) as a shear length parameter, the
R /d R =2R
des
ribedbythegeometri
parameters i ( o i), stressesVus∆uatlly5present a short transient on a distan
e
θ
and by the dimensionlesstangentialvelo
ity ofthe inner around ≤ . HoweVve∆r,tthe5s0tabilization of the solid
θ
wall (also
alled shear velo
ity): fra
tionratherrequires ≈ ,mostlybe
auseofthe
very dense initial state. Consequently we
onsider that
V ∆t 100
θ
the
onditiontorea
hasteadystateis ≥ . This
ΩR m
i
Vθ = d P, (1) pVrθo
edure provides an initial state with a shear velo
ity
r . As a way to guarantee an initial state
onsistent for
the
omparisons between dis
rete simulations with dif-
V
θ
whi
h is similar to the notion of inertial number, but ferent , the pro
edure is (cid:28)rst applied with the highest
V V V
θ θ θ
at the s
ale of the whole system. A small value of value of , and then is progressivelyde
reased.

orrespondstothequasistati
regime,whilealargevalue In steady state, we
onsider that the statisti
al distri-
V
θ

orrespondsto the
ollisional regime. Seven values of bution of the quantities of interest (stru
ture, velo
ities,
4
t θ
for
es...) areindependentof and , sothatweaverage Fig. 2a shows the
oarse-grained pro(cid:28)les of the nor-
θ 200 σ R
rr 50
bothinspa
e(along )andintime(
onsidering time mal stress
omponent in geometry for di(cid:27)erent
V ∆t 200 V
θ θ
steps distributed over the distan
e ≥ ). Then wall velo
ities , while Fig. 2b shows theσrrartio between
we
al
ulate the pro(cid:28)les of solid fra
tion, velo
ity and the orthonormal and the normal stresses σθθ. The nor-
σ
rr
stress
omponents a

ording to the averaging pro
edure mal stress is nearly
onstant and equal to the
on-
P ∂σ /∂r
rr
des
ribed in App. B. (cid:28)ning pressure . The term in the momentum
σ
rr
Beyond the number of a
quisition points, the
onsis- equation (2) smoothes the pro(cid:28)le, whi
h might ex-
σ
rr
ten
y of the averagedvalues depends on the shearstrain plain the absen
e of (cid:29)u
tuations of . A
rude es-
a

umulated during the a
quisition of data. We
on
en- timate of the
entrifugal e(cid:27)e
ts may be given, if the
trateourinterestontheregionwherethesystemmaybe last term of equation (2) is negle
ted, and, anti
ipat-

onsideredinasteadystate,whi
ho

ursatlargeenough ing on Se
. IIIB and Se
. IIIC, a
onstant solid fra
tion
ν 0.8
shear strain. Based on the observation of the transients, ≃ is assumed and an exponential velo
ity pro(cid:28)le
γ˙∆t 10 v (r)=V exp( (r R )/ℓ) ℓ 2 6
θ θ i
we
onsider that this is true when ≥ . Be
ause − − , with between and :
ofthestrainlo
alization,theregionofinterestislo
ated
R R + R
i i steady
wnehaerreththeeinvnaelurewoaflRl astnedadylimisitgeivdentoin Ta→b. III. |σrr(Ri)−1|≤ 2πνℓRiVθ2. (4)
R =50 σ (R ) 1 0.05 V =
i rr i θ
TABLE III: Limit of the steady state region. Minimum and 1Conseqℓu=ent5ly, forV = 2.5,| − |≤ for
θ
maximum values
orrespond to global quasistati
regime and and . For , the
entrifugal e(cid:27)e
ts might
V =2.5
θ
to respe
tively. be
ome signi(cid:28)
ant, howeverit has not been observed.
σ σ
rr θθ
The radial and orthoradial stresses are nearly
Rsteady r Ri . 10
R25 7−17 equal for − . This has already been observed
R50 9−25 in oth5e%r
on(cid:28)gurations (plane shear [4, 5, 9℄ within less
R100 13−35 than , in
lined plane [5, 32, 46℄), and was previously
R200 18−52 reportedinannularshear[40,44℄. Thisverysmallnormal
stressdi(cid:27)eren
e isnotexplained yet. The (cid:29)u
tuationsof
σ r R & 15
θθ i
for − probably re(cid:29)e
t the frozen disorder
beyond the steady zone, where the material is mu
h less
deformed than
loser to the inner wall, so that the time
averagingisunsu(cid:30)
ient. Consequentlythese(cid:29)u
tuations
III. LOCALIZED SHEAR STATES Vθ
in
rease as de
reases.
σ (r)
rθ
The shear stress pro(cid:28)Vles shown in Fig. 3a1(/fro2r
Inthisse
tion,weshowtheshearlo
alizationnearthe θ
di(cid:27)erent shear velo
ity ) are
onsistent with the
inner wall through the radial pro(cid:28)les of di(cid:27)erent quanti-
de
rease of (3). The os
illations about the mean value
ties. In App. C we fo
us on internal variables asso
iated
Z aredue to the materialstru
turationnear the inner wall
to the
onta
t network [53, 54℄ (
oordination number
M (see Se
. IIIC) and to the frozen disorder in the very
and mobilization of fri
tion ) and to the (cid:29)u
tuations
slowlysheared regions, whi
h is beyond the steady zone.
of the motion of the grains, translational or rotational.
V Fig.3bshowsthedependen
eoftheshearstressatthe
θ S V
We systemati
ally dis
uss the in(cid:29)uen
e of . θ
inner wall on shear velo
ity . Below a
ertain value
V .0.025 S
θ
( ), tends toa(cid:28)nitelimit. Consequently,the
σ (r) V
rθ θ
shear stress pro(cid:28)les be
ome independent of .
A. Stress (cid:28)eld
This behavior
hara
terizes the global (that is to say, in
∂ = 0 ∂ = 0 the whole system) quasistati
regime, where the stresses
dwii
tItnhso[us5tt5ea℄radadyivaa(lr∂i(cid:29)taotwion(vor)f1=na/nro02rn)mu,laa
lrosntstrhienesuasurσm(cid:29)rromwrees
lah(ta∂enθdi
tsoptrhe)e-, (H
aaonnwdteaovntehdre,rSfosrtianVt
eθrev&aasre0isa.0bw2lei5ts,h)idnVoeθr.ntoiaPtlrdeeev(cid:27)pieoe
untsds wobneo
rtokhmseerveesplioog
rntiite(cid:28)yd-.
σvelo
ity pro(cid:28)le, and a de
rease of the shear stress a similar dependen
e of the shearstress on the shearve-
rθ
asso
iated to the
onservation of the torque: lo
ityin other
on(cid:28)gurations(see [2℄ forareview). More
spe
i(cid:28)
ally, the experimental measurementof the torque
4νv2 ∂σ σ σ
θ = rr + rr− θθ, asafun
tionoftherotationrateintheannularshearge-
π r ∂r r (2) ometryindi
ates atransitionfrom arate independent to
a rate dependent regime [20, 23, 27, 33,S56=℄. SOur+reαsVulβts
σrθ = S(Ri)2,
an beSaqpsproximated by a fun
tion like qsα θβ,
r (3) where is the global quasistati
limit value, and
β 1/2
aretwo
onstants. Wenoti
ethat is
loseto rather
ν(r) v (r) 2
θ
where and are the solid fra
tion and orthora- than asmightbenaïvelyexpe
ted fromBagnold'srhe-
S
dial velo
ity pro(cid:28)les, is the shear stress at the inner ology. Wenoti
ethatintheexperimentof[20℄,thetran-
S =σ (R ) σ V 0.3
rθ i ii θ
wall ( ) and are positive for
ompression. sition o

urs for ≃ (after appropriate res
aling),
5
(a) (a)
1.3 0.5
1.2
0.4
V = 2.5
1.1 V = 1.0
0.3
rr 1.0 r V = 0.25
0.2
0.9
0.8 0.1 V = 0.025
V = 0.0025
0.7 0.0
0 10 20 30 40 50 0 10 20 30 40 50
r-R r-R
i i
(b) (b)
1.3 0.50
1.2 0.45
rr 1.1 0.40
/ 1.0 S 0.35
0.9 0.30
0.8 0.25
0.70 10 20 30 40 50 0.20 -3 -2 -1 0 1
10 10 10 10 10
r-R
i V
FIG.2: (Coloronline)(a) Normalstress σrr(r)anσdθθ(b/)σrrra(trio) FIG. 3: (Color online) (a) Shear stress pro(cid:28)les σrθ(r) (solid
pVVberθθtow==(cid:28)elee10sn..50f,t2oh5(re,Ndn()i(cid:4)o(cid:27)Vr)eθmrVe=anθlt2=a.s5nh.0de.2Gat5rhe,eovm(eo◮lreot)t
hriVyotinθeRos=r.5m00.(a.H5l,)s(tVr(cid:7)eθ)ss=Vesθ0=.0012.05,,((◭•)) wtltihinoaeenllsfu)voenfal
onVt
diθiotn((cid:28)ies:tesmaSV
iθ
=-.olor(0dgb.ai2)nr6giSt+hhtemo0ai.rE1
qs3st.Vr
θe(a03sl.se5)7)a(..dtGTathsheheoeemidnseonltlierindryewslRi)an5lfe0lo.rSredpairs(cid:27)efseuernenn
ts-t
whi
h is not far from what is observed in Fig. 3b. How-
S(V )
θ
ever, we alsopoint out that the
urve, here shown
R
50
for geometry , in fa
t depends on the geometry.
v (r) V
θ θ
The normalizationof by shearvelo
ity allows
to
learly visualize the in(cid:29)uen
e of this latter parameter
B. Velo
ity (cid:28)eld oVn the0v.0e2lo5
ity pro(cid:28)les. In the global quasistati
regimVe
θ θ
( ≤ ),thereisnoin(cid:29)uen
e,whileforin
reasing
0.025
above , an in
rease of the lo
alization width is ob-
The shear lo
alization near the inner wall is revealed
vθ(r) served,
onsistently with experimental observations [25℄.
bythestrongde
reaseofvelo
itypro(cid:28)les shownon
Fig. 4. The de
ay apvpe/aVrs=toebxep[ni
ae(lry aRpp)roxbi(mratRed)b2]y γ˙(r) = r ∂ (vθ(r))
aGaussianfun
tion θ θ − − i − − i , The shear rate is equal to − ∂r r . We
ω(r)
as shown on Fig. 4. We noti
e however that there is a denote the pro(cid:28)leof the averageangular velo
ityof
V 2.5
θ
sliding velo
ity for the higher value of ( ), whi
h is thegrains. Aspreviouslyreportedindis
retesimulations
apparent in Fig. 5a. Previous studies in 2D systems [21, ofgranular(cid:29)ows[4℄,theaverageangularvelo
ityisequal
ω(r)= γ˙(r)/2
25, 39, 40, 44, 57℄ found an exponential shape, while a tohalfthelo
alshearrate(orvorti
ity) − .
gaussian de
ay was observed in three dimensional (3D) Os
illations of the averageangular velo
ity are observed
systems for non spheri
al or polydispersed grains [22℄. inthe3or4(cid:28)rstgrainlayersneartheinnerwall(Fig.5),
The agreementbetween the measurementofthe velo
ity as previously noti
ed by [40℄. They may be due to the
pro(cid:28)les in 3D experiment (using 3D MRI velo
imetry in frustration of the rotation of the (cid:29)owing grains in
on-
the bulk or CIV at the free surfa
e) [2℄ and 2D dis
rete ta
twiththegluedgrainsofthewalls(whi
hrotatewith
Ω
simulations is satisfa
tory [32℄. angular velo
ity ).
6

0
10 C. Solid fra
tion
ν(r) V =
θ
Fig. 6 shows the solid fra
tion pro(cid:28)les for
-1 0.025 2.5 V
10 V and . In the global quasistati
regime ( θ ≤
V 0.025 Vθ
/ ), the pro(cid:28)le be
omes independent of , while a
v de
rease of the solid fra
tion is observed for in
reasing
V
-2 θ
10 . The material is signi(cid:28)
antly dilated ne5ar the inner
wall[17,39,40℄,andisstru
turedinabout layers
lose
V
θ
to the inner wall, with a higher amplitude for low .
-3 This was previously observed in various shear geome-
10
0 5 10 15 20 25 tries[22,39, 58, 59, 60℄. Thisstru
turationofthegranu-
larmaterial
ertainlya(cid:27)e
tstheslidingoflayersofgrains,
r-R
i withsigni(cid:28)
ant
onsequen
esontheme
hani
albehavior
V near the wall. As previously reported [40, 44℄, indepen-
θ V ν
FIG. 4: (Color online) In(cid:29)uen
e of the shear velo
ity on θ
t0h.0e0v2e5lo
i(cid:4)ty pVroθ(cid:28)l=es0v.θ2(5r) (◮semVi-θlo=gar0it.5hmi
(cid:7)s
Valθe)=. (1H.)0Vθ◭= tdoewntalrydoafvtahleueinν(cid:29)muaexn
(e
loosfe to, 0so.8li2d, tfrhae
tsioolnid frain

trioenasiens
, ( ) , ( ) , ( ) , ( )
Vθ =1.5 N Vθ =2.5 the
riti
al state for fri
tional disks with a similar poly-
vθ/Vθ =,ex(p[)−0.34(r−.RTi)h−e0s.o0l0id15l(inre−iRndi)i2
a]tes the fun
tion dispersity [53℄) away from the inner wall, and remains
onethe fun
tionvθ/Vθ =exp[−0.21(r−Ri)−,0a.0n0d2t(hre−dRasih)2e]d.
lose to its larger initial value 0.85 in the region where
Geometry R50. the material has not been sheared enough.
(a)
(a)
0.90
0.8
V = 0.025
0.7 0.85
0.6
0.80
0.5
-
2, 0.4 0.75 V = 2.5
. / 0.3
0.70
0.2
0.1 0.65
0.0
0.60
-0.1 0 10 20 30 40 50
0 1 2 3 4 5 6 7
r-R
r-R i
i
(b)
(b)
0.90
0.006
0.85 V = 0.025
0.005
0.80
0.004
-
2, 0.003 0.75 V = 2.5
/
.
0.70
0.002
structuration
0.65 near inner wall
0.001
0.60
0.000 0 5 10 15
0 1 2 3 4 5 6 7
r-R
r-R i
i
V
θ
FIG. 6: (Color online) In(cid:29)uen
e of shear velo
ity on the
FIG. 5: (Color online) In(cid:29)uen
e of the inner wall on the
ol- ν(r)
ω(r) stru
turation near the inner wall. Solid fra
tion pro(cid:28)les
lapse between the average angular velo
ity (hollow sym-
bols) and the shear rate γ˙(r) (full symbols) (a) Vθ =2.5 and (a)inthewholesystemand(b)intheregi3odn
losetotheinner
(b) Vθ =0.025. Geometry R50. wall. The solid line is0a.n5daverage overR50, while the dotted
line is an average over . Geometry .
7
IV. CONSTITUTIVE RELATIONS 1. Dynami
fri
tion law
I & 0.02 µ∗
In Se
.VIII and App. C, it has been shown that shear In the inertial regime, fIor , in
reases ap-
velo
ity θ, if small enough, no longer in(cid:29)uen
es the proximately linearly with and nearly independently of
radial pro(cid:28)les of various quantities (see Fig. 3, 4, 24 the geometry (Fig. 8a):
and 25). Then, the whole system is in the quasistati
V γ˙
regime. When θ in
reases, the shearrate in
reasesin µ∗(I) µ∗ +bI,
thewholesample. Abovea
ertainlevelofshearrate,in- ≃ min (5)
ertiale(cid:27)e
tshavesigni(cid:28)
ante(cid:27)e
tonthematerialbehav- µ∗ 0.26 b 1
with min ≃ and ≃ . The agreement with the
ior, whi
h
hara
terizes the inertial regime. Considering
dynami
fri
tionlawmeasuredinthehomogeneousplane
theshearstressdistributionintheannulargeometryand
shear geometry is ex
ellent [4, 6, 48℄. In
ontrast, for
the de
ay of the velo
ity away from the inner wall, we I
lower values of , a deviation from this linear relation is
expe
tthattheinertialzonebeginsattheinnerwalland
that its thi
kness in
reases when Vθ in
reases (Fig. 7). observed, depending on the geometryµ∗(Fig. 8b). The ef-
fe
tivefri
tionbe
omessmallerthan min,andthisdevi-
R
i
ationin
reasesas de
reases,thatistosayasthestress
R
i
function gradient in
reases. Re
ipro
ally, as in
reases, that is
of and P to say as the stress distribution be
omes more homoge-
P
neous, the results of the annular geometry tend to the
ones of the plane shear geometry. This reµv∗eals that the
simple relation between e(cid:27)e
tive fri
tion and inertial
I
inertial
number does not depend on the stress distribution in
regime
the inertial regime, and isthen quite general(see [5, 32℄
quasistatic for(cid:29)owsdownanin
linedplane)µ,w∗hileitfailsinthequa-
regime sistati
regime. In plane shear, min may be
onsidered
as the internal fri
tion inµth∗e
riti
al state [9, 53℄. This
is the maximum value of supported by the granular
material, before it starts to (cid:29)ow quasistati
ally. With a
heterogeneous stress distribution, the granular material
FIG. 7: Inertial and quasistati
zones.
is able to (cid:29)ow below this level.
λ
in
We
all the width of the inertial zone. Using
Eqn. (3) and (5), we dedu
e that:
In this se
tion, we analyze the relations between dif-
ferentdimensionlessquantitiesin the inertialregimeand
howtheyarea(cid:27)e
ted bythetransitiontothequasistati
λ (V ,R )= S(V ,R )/µ∗ 1 R .
in θ i θ i min− i (6)
regime.VWe>re0s.0tr2i5
touranalysistolargeenoughshearve- (cid:18)q (cid:19)
θ
lo
ity ( ), so that a wide enough inertial zone
exists
lose to the inner wall. We also
onventionally de(cid:28)ne the width of the shear
λ v (R +λ ) = V /10
loc θ i loc θ
zone through . Fig. 9a and
λ (V ) λ (V ) R
in θ loc θ 50
b shows and in geometry . We no-
λ V
in θ
ti
ethat smoothlyin
reasesfromzerowith , while
A. Inertial number and me
hani
al behavior λloc Vθ
seems to saturate at a low value for low (global
V
θ
quasistati
regime) and at a high value for high (this
Dis
retesimulationsofhomogeneousplaneshear(cid:29)ows
is related to an apparent velo
ity dis
ontinuity near the
[4℄ haverevealedthatthe
onstitutivelawofdense gran-
wall, suggesting in
reasing
ollisional e(cid:27)e
ts in the (cid:28)rst
V 0.3 1
ular (cid:29)ows may be des
µri∗bed through theσdependen
y layers),with asudden in
reasefor θ between and .
of the e(cid:27)e
tive fri
tion (ratio of shear to normal
P ν We noti
e that in the experiment of [25℄, the shear zone
V
stresses) and of the solid fra
tion on the inertial θ
I = γ˙ m/P invades the whole gap for high enough .
number (a 2D equivalent of the de(cid:28)nition In a given geometry, for a small enough shear velo
ity
V
given in Se
. Ip), where all the quantities are measured θ, the inertial zonedisappears, and the whole system is
lo
ally. The annularshear(cid:29)owsbeing heterogeneousν,(rw)e in the quasistati
regµim∗e. Fig. 10 then shows agaIin that
mµ∗e(ars)ur=e tσhe(rre)la/tσion(sr)betweeIn(rt)he=loγ
˙a(rl)quamn/tiσties(,r) , the e(cid:27)e
tive fri
tion is no more a fun
tion of .
rθ rr rr
and (or
γ˙(r)/ σ (r)
rr in dimensionless unit). Epa
h simulation
providpes dynami
dilatan
y and fri
tion laws in a range 2. Dynami
dilatan
y law
of inertial number. In the following, we try to analyze
ν
the granular material as a
ontinuum,
onsequently we We observe a linear de
rease of solid fra
tion as a
I
do not take into a

ount the (cid:28)ve (cid:28)rst layers where wall fun
tion of inertial number , independently of the ge-
ν
stru
turatione(cid:27)e
tsaresigni(cid:28)
ant(seeFig.5andFig.6). ometry in the inertial regime (Fig. 11 and 12), and
8
(a) (a)
0.45 20
0.40 =0.4
0.35 15
0.30
n
0.25 i 10

0.20
5
0.15
=0
0.10
0
0.05 -3 -2 -1 0
10 10 10 10
0.00
0.00 0.05 0.10 0.15
V
I (b)
(b) 11
0.45
inertial regime
near inner wall 10
0.40
c 9
o
0.35 l
quasistatic regime
8
0.30 Ri
7
0.25
0.20 6 -3 -2 -1 0
10 10 10 10
0.15 -4 -3 -2 -1 V
10 10 10 10
λin
I FVIG. 9: (a) Width of the inertial zone as a fun
tion of
θ
0.30 (c) res,eanstsdtehdeu
fuedn
ftrioomn:Eλqinn.=(65)0a(npd1F+ig0..35aV.θ0.T57h−e s1o)l.id(bl)inWe ridepth-
λ V
loc θ
of the lo
alization zone as a fun
tion of . Geometry
0.25 R50
.
0.20

0.15 0.45
inertial regime
0.10 0.40 near inner wall
0.35 quasistatic regime
0.05 in the whole sample
0.30
0.00 -4 -3 -2 -1
10 10 10 10 0.25
0.20
I
0.15 quasistatic regime
FIG. 8: (Color online) Dynami
fri
ti≈on1law (a) in linear
s
ale (the solid lineµin=di
0ates aµslo=pe0.4 ) for parti
le
o- 0.10 -5 -4 -3 -2 -1 0
e(cid:30)
ient of fri
tion and . Dynami
fri
tion 10 10 10 10 10 10
µ = 0.4 µ = 0
law in semi-logarithmi
(cid:4)s
aRle25for•(bR)50 N Ra1n0d0 (
H) R200. I
VDθi(cid:27)=er2e.n5t geometries: ( ) , ( ) , ( )(cid:3) , ( ) . µ∗
. Comparison with plane shear [4℄ ( ). FIG. 10: (Color online) E(cid:27)e
tive fri
tion as a fun
tion of
I ⋆ V = 0.00025 H V = 0.0025
θ θ
th•e Viner=tia0l.0n2u5mb(cid:4)er V(=)0.25 ◮ V =,0(.5) (cid:7) V = 1.0,
θ θ θ θ
( ) , ( ) , ( ) , ( ) ,
◭ V = 1.5 N V = 2.5
θ θ
tends to a maximum value νmax, whi
h identi(cid:28)es to the µ(∗)=0.26 , ( ) R50 . The solid line
orresponds to
. Geometry .
solidfra
tionin the
riti
alstate. We
anthenwritethe
dynami
dilatan
y law:
ν(I) ν aI,
max plane shear geometry is ex
ellent [4, 48℄. However, far
≃ − (7)
from the walls, in the region where the material is less
ν 0.82 a 0.37
max
with ≃ and ≃ . The agreement with deformedand soremainsin its initialdense state, higher
ν
thedynami
dilatan
ylawmeasuredinthehomogeneous values of are observed. Fig. 12 also indi
ates that the
9
inner wall indu
es further dilation. Those di(cid:27)eren
es are likely related to some pe
uliar-
ities of assemblies of fri
tionless grains [9℄. The qua-
0.86
sistati
limit, in su
h materials, is only approa
hed for
I
0.84 =0 µm∗u
h smaller values of than in the fri
tional
ase, and
min is itself
onsiderably lower. As a
onsequen
e on
0.82
may expe
t awider inertial zone. Moreover,asthe
riti-
0.80
alsolidfra
tion
oin
ideswiththerandom
losepa
king
value [9℄, no solid-like region of the system
an be pre-
0.78
vented from (cid:29)owing be
ause of its density.
0.76 =0.4
0.74
4. Comparison with previous studies
0.72
0.00 0.05 0.10 0.15
Thevalidityofthe
onstitutivelaw,on
esuitablygen-
I
eralized to three dimensions, was su

essfully tested in
FIG.11: (Coloro≈nl−in0e.)37Dynami
dilatan
ylaw(the s(cid:4)olidRli2n5e (cid:29)ows down a heap between lateral walls [7, 14℄. In
in•diR
a5t0e aNsloRpe100 H R2)00for di(cid:27)erent geometries: ( ) , that
ase the velo
ity (cid:28)eld, as dedu
ed from numeri-
((cid:3)) ,( ) ,( ) . Comparisonwithplaneshear[4℄
al
omputations in whi
h the vis
oplasti
law was im-
( ). plemented, exhibits a more
omplex three-dimensional
stru
ture. Predi
ted velo
ities at the free surfa
e agreed

losely with experimental results.
Thus, the appli
ability of the
onstitutive law as a re-
0.86
lation between lo
al values of non-uniform strain rate
0.84 and stress (cid:28)elds, whi
h we just established in 2D an-
nular shear (cid:29)ow, was previously
he
ked in the 3D
ase
0.82
of a laterally
on(cid:28)ned gravity-driven (cid:29)ow. The validity
0.80 of su
h an approa
hshould be restri
ted to situations in
V=0.025
0.78 whi
h the
haral
teristi
length for stress or strain rate
variations, say , is signi(cid:28)
antly larger than the grain
l=R
0.76 wall effect size. In annularshear, one has i, whereasthe (cid:28)nite
w
0.74 V=2.5 width of thel =
hwannel wRas found in [7, 14℄ to
ontrol
i
the gradients, . As , in units of grain diameters,
w
0.72 -5 -4 -3 -2 -1 varies between 25 and 200 here, while the interval of
10 10 10 10 10
extends between 16.5 and 500 in [7, 14℄, similar levels of
I heterogeneity are explored.
FIG. 12: (Color online) Dynami
dilatan(cid:4)
yRla2w5 in• seRm50i-
logarithmi
s
ale. Di(cid:27)erent geometries: ( ) , ( ) ,
N R100 H R200 (cid:3) B. Internal variables
( ) , ( ) . Comparison with plane shear [4℄ ( ).
νmax =0.82
The dashed line
orresponds to .
We now dis
uss how internal variables, whi
h pro(cid:28)les
I
aredis
ussedinApp.C,s
alewiththeinertialnumber ,
revealing lo
al state laws,
onsistent with the one mea-
sured in homogeneousshear (cid:29)owZs.= Z eIf
3. Fri
tionless grains We observe a relation like max − (with
Z 3 Z I
max
≈ ) between
oordination number and on
As shown in Fig. 8a and Fig. 11, the mi
ros
opi
fri
- Fig. 13, nearly independent of the geometry.
µ
tion
oe(cid:30)
ient has a signi(cid:28)
ant in(cid:29)uen
e on the
on- We do not observe a general relation between the mo-
M I
stitutive law parameters. Those (cid:28)gures also reveal good bilization of fri
tion Rand , but an asympMtoti

goInh-
i
agreement with homogeneous shear simulations [4℄. The vergen
e for growing toward a relation ≈
I
solid fra
tion remains a linearly de
reasing fun
tion of (Fig.14). Forthisquantity,thereisnosatisfa
toryagree-
a
(with afast
hangein the quasistati
limit). The slope ment with the homogeneousshear
ase.
ν 0.85 δv
max θ
is not a(cid:27)e
ted, while in
reases to ≃ . The dy- Weanalysethe(cid:29)u
tuationsoforthoradialvelo
ity
γ˙ I
nami
fri
tionlawkeepsthesametenden
ybutisshifted normalized by the natural s
ale as a fun
tion of
toward smµa∗ller val0u.e1s1of fri
tion. The lineaIr ap0p.r0o1xima- (Fig. 15). In the quasistati
regime, the development
tion with min ≃ (Eqn. (5)) fails for ≤ . We of
olle
tiveand intermittentmotions(see [28, 29℄ in an-
noti
e that the range of validity of the dynami
fri
tion nular shear and [61℄ for a re
ent review) explain the
law is mu
h larger than for fri
tional grains, and that it in
rease of these relative (cid:29)u
tuations. For higher values
I δv /γ˙ 1
θ
does not seem to depend on the geometry. ofthe inertialnumber , weobservethat → . On
10

3
3.0 10
2.5
2
2.0 10
Z ’
/
1.5 v
1
1.0 10
0.5
0.0 -4 -3 -2 -1 100 -5 -4 -3 -2 -1
10 10 10 10 10 10 10 10 10
I I
Z
FIG.13: (ColoronlineI)Coordinationnumber asafun
tion FIG.15: (Color onlinIe)Relative (cid:29)u
tuations as a fun
tion of
ZRof5=t0h,e2(N.i9n)5erR−ti1a70l0.6,n5u(IHm0).b6e5Rr)20fo0(r,th(d(cid:3)ei(cid:27))seoprlieldanntliengeseohrmeeaperrter[i4see℄s.n.tVs(θ(cid:4)th=)e2Rfu.52n5.
,ti(o•n) δRthv2eθ0/0i(,nγ˙e(cid:3)dr)tpi=alaln1neu+msh0be.e0ar7rI[−40℄(..t7hV)e.θs(=o(cid:4)li2)d.5Rl.i2n5e, (re•p)rRes5e0n,ts(Nt)heRf1u0n0,
t(iHon)

0
10
∂ v (r) µ∗ SR2
( θ )= min i,
∂r r br − br3 (8)

M S
where the shear stress at the inner wall depends both
V R
θ i
on and on (see Se
. IIIA). As shown from the
V
θ
measurements drawnin Fig. 16b, for a largevalue of ,
S R
i
ishighinsmallgeometriesandstronglyde
reasesas
-1 in
reases. We now integrate this relation over the range
10
of validity of the dynami
fri
tion law, this is to say in
-4 -3 -2 -1 Ri Rin = Ri +λin
10 10 10 10 the inertial zone → , from whi
h we
get:
I
M SR2 µ∗
FIG.14: (Coloronline)MobiIlizationoffri
tion asa fun
- v (r)= i + minrln(r)+cr.
θ
tion of thMe =ine0r.7ti3aIl0.n2u9mber (the solid line rep(cid:4)reseRn2t5s th•e 2br b (9)
fun
tion fordi(cid:27)erent geometries: ( ) ,( )
R50 N R100 H R200 Vθ =2.5 c
, ( ) , ( ) . .
The
onstant isdeterminVe+dbythevalueofthevelo
Vity
at the inner wall,
alled θ , whi
h is smaller than θ,
revealing some sliding at the wall as previously noti
ed
(see Se
. IIIB).
thewholeδ,vwe/γp˙r=op1os+etcoI−ddes
ribethedependen
ybythe
θ
equation δv ∝(Fγ˙i0g.4. 15). Experimental re- SR µ∗
sults [22, 26℄ show that . Dividing this relation V+ = i + minR ln(R )+cR .
byγ˙,wegetanexponentequalto−0.6,
losetoexponent θ 2b b i i i (10)
d= 0.7
− , dedu
ed from the previous (cid:28)t.
On the whole, the velo
ity pro(cid:28)le is equal to:
V. CONSEQUENCES FOR THE SHEAR r S R 2 µ∗ r
v (r)=V+ +r i 1 + min ln .
LOCALIZATIONBAENHDAVTIHOERMACROSCOPIC θ θ Ri "2b (cid:18) r (cid:19) − ! b (cid:18)Ri(cid:19)#
V+ (11)
Using the
onstitutive law established in Se
. IV, we An absolute measurement of θ happens to be di(cid:30)-
now show that it is possible to understand some obser-
ult,
onsidering the wall e(cid:27)e
t that disturbs the mate-
vations des
ribed in Se
. III. rialbehaviorinalayerofafewgrainsneartheinnerwall
P γ˙
Still using dimensionless units, sin
e the pressure is (like in Fig. 5 for the shear rate ). Consequently, we

onstant in the system and the shear stress is given by obtain this quantity from a (cid:28)t, and a
omparison with
Eqn. (3), the dynami
fri
tion law Eqn. (5) provides the the measured velo
itypro(cid:28)lesis shownon Fig. 16b. The
v (r)
θ
following equation for the velo
ity pro(cid:28)le : agreementisex
ellent, suggestingon
emorethevalidity