Spatial repartition of local plastic processes in different creep regimes in a granular material

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by  A Pons

epl draft
Spatial repartition of local plastic processes in different creep
regimes in a granular material
A. Pons1, T. Darnige1, J. Crassous2, E. Cl´ement1 and A. Amon2
1 Physique et M´ecanique des Milieux H´et´erog`enes, PMMH ESPCI, CNRS (UMR7636) - Univ. P.M. Curie - Univ.
6Paris-Diderot - PSL Research University - 75005 Paris, France
12 Institut de Physique de Rennes (UMR URI-CNRS 6251), Univ. Rennes 1, Campus de Beaulieu, Rennes, France
0
2

n
a PACS 83.80.Fg–Granular solids
J PACS 81.40.Lm–Deformation, plasticity, and creep

1 PACS 83.60.-a–Material behavior
2
Abstract –Granular packings under constant shear stress display below the Coulomb limit, a

] logarithmic creep dynamics. However the addition of small stress modulations induces a linear
t
f creep regime characterized by an effective viscous response. Using Diffusing Wave Spectroscopy,
o
weinvestigatetherelationbetweencreepandlocalplasticeventsspatialdistribution(“hot-spots”)
s
. contributing to the plastic yield. The study is done in the two regimes, i.e. with and without
t
a mechanical activation. The hot-spot dynamics is related to the material effective fluidity. We
m showthatfarfromthethreshold,alocalvisco-elasticrheologycoupledtoanageingofthefluidity
parameter,isabletorendertheessentialspatio-temporalfeaturesoftheobservedcreepdynamics.
-
d
n
o
c
[

Introduction. – Granular packings are often seen as ics changes from a logarithmic to a linear behaviour [13].
1
vrigid below a limit corresponding to a critical ratio be- The physical interpretation stems from the combination
1tweenshearstressandnormalstress(Coulombthreshold) of memory effects and non-linearities, leading to a ”secu-
7[1]. However, the existence of a clear-cut transition be- lar” accumulation of tiny effects, meaning that the creep
6
tween a solid-like and liquid-like behaviour is currently dynamics is revealedat a time scale much larger than the
5
stronglychallenged[2–5]. In the presence of a shear band modulation. Wecallthisbehaviour“rectifiedcreep” inthe
0
.(i.e. afluidzonedwellingsomewhereinthepacking)differ- following as the interpretation of this regime is different
1
ent authors brought evidences for mechanically activated from an Eyring-like activated process. As the ingredients
0
creeping processes taking place in remote regions, below at the origin of this rectified creep are generic for a large
6
1the Coulomb threshold [2,3]. This behaviour led to non- class of soft glassy materials, this effect should be seen in
:local rheological relations proposed to extend the stan- other yield stress fluids displaying creep [14–17].
v
idard local constitutive relations for granular flows [6–9]. In this paper, using a spatially resolved multiple scat-
XFor granular packing sheared in all of its parts below the tering technique [18], we monitor the spatial distribution
rCoulomblimit, dynamicalprocessesleading to a logarith- of hot-spots during the two creep regimes. In parallel, a
a
mic creep occurs [4,5,10]. Interestingly, this creeping dy- spatially resolvedvisco-elasticmodel is solvedassuming a
namics can be mapped onto a simple visco-elastic model direct relation between the hot-spots production rate and
initially designed to render the phenomenology of yield the local fluidity value. The numerical solution is com-
stress fluids displaying ageingin the solidphase [11]. The pared to the experiments. Interpretation of the results in
modeliscentredonadynamicalequationforafluiditypa- the framework of a local rheologicalmodel is provided.
rameter representing an effective visco-elastic relaxation.
This phenomenological parameter was directly related to Methods. – The experimental set-up is shown in
the occurrence of mesoscopic plastic events called ”hot fig. 1a. It consists in a cylindrical shear cell (Radius
spots”[5]. Inthevicinityofthedynamicalthreshold,these R = 5cm, height H = 10cm) filled with glass beads of
events combine to provide largescale plastic yields [5,12]. density ρ = 2500 kg/m3 and mean diameter d = 200 µm
Recently,wehaveshownthatprovidingatinystressmod- (rms polydispersity ∆d = 30 µm). A well defined pack-
ulation around a nominal shear stress, the creep dynam- ing fraction Φ = 0.605 0.005 is obtaining using an air
±
p-1
F. Author et al.
(a) (b)
Torque
meter
R
m
h R0
H
H0
Pressured air
Fig.1: (a)Experimentalset-up: shearisimposedbyanhangingmassmconnectedtotheverticalaxis;verticaldisplacementof
thewaterreservoirallowsmodulationoftheappliedstress;acameracollectsbackscatteredlightfromaHe-Nelaserilluminating
thetop of theshear cell. (b) Mean strain as afunction of timefor twoexperiments performed at σ0 =1100Pa, and δ=7.5Pa
, and for two oscillations frequencies: 0.099 Hz (blue line) and 0.168 Hz (green line). The oscillations start at t=1500s (grey
area). γ˙∞ is the slope of the linear part. (c) Imposed stress during an experiment: stress ramp to reach mean stress, σ0,
constant stress during tw and stress modulation characterized by a frequency fσ and an amplitude δ. (d) Steady state strain
rate, γ˙∞, as a function of mean stress, σ0. Color of thesymbol standsfor thevalue of modulation stress rate Rσ =4δf.
fluidized bed as described by Nguyen et al. [4]. topreventlargescaledisturbancesinthepacking,pressur-
Shear is obtained by applying a torque on a stain- izedair is gently flown, just below the fluidization thresh-
less steel four-blade vane (radius R = 1.27 cm, height old providing a packing at the surface bearing almost no
0
H =2.54cm)viaamassmsuspendedfromapulley(see confining pressure. In addition, after the introduction of
0
fig.1a). Usingthis“Atwood-machine”technique,mechan- the vane, we kept the air flowing during 10 min in or-
ical noise inherently coming from any motorized process der to relax the remaining stress perturbations induced
can be suppressed. The mass m hangs partially inside by the vane insertion. The air flow is switched off before
a reservoir filled with water. Thus, by modulating the the start of the experiment. For experiments performed
Archimedesforce,throughtheupanddownmotionofthe with the vane near the surface (h=0 cm), the grains are
reservoirsitting on a verticaltranslation stage, controlled confined with a glass plate in order to obtain a confining
modulation of the torque applied to the granular packing pressure at the top of the vane close to the one existing
can be obtained. A torque probe connected to the vane for experiments done at the insertion height h = 5 cm.
axis measures the applied torque T. The vane rotation The circular glass plate has a central hole to let the vane
angleαis monitoredviaa transversearmwhosedisplace- shaftgothroughandtheverticalconfinementpressurewas
mentis measuredby aninduction probe. Torqueand dis- adjusted by placing loads on the glass plate.
placement signals as well as the vertical translation stage
command are connected to a Labview controller board.
Inadditiontothemechanicalmeasurements,weobtain,
We defined here the mean stress and the mean strain as
σ = T and γ = αR0 respectively. by diffusive wave spectroscopy (DWS) [5,18,19], a spa-
0 2πR20H0 R−R0 tiallyresolvedmapofthe topsurfacedeformations. DWS
For a given experiment, the protocol (fig. 1c) is the fol- is an interference technique using scattering of coherent
lowing(i)stressrampatconstantstressrate(σ˙ =5Pa/s) light by strongly diffusive materials. In our case, a He-Ne
uptothedesiredmeanstressvalueσ0; (ii)constantshear laser (λ = 633 nm) illuminates the top of the shear cell.
σ0 applied during tw = 1500 s; (iii) modulation of the A camera imaging the surface at a frame rate of 0.1 Hz
stress around σ0 for at least 2 hours. The modulation collects back-scattered light (fig. 1a). The correlation of
consist in triangular oscillations with an amplitude δ and scattered intensities between two successive images, g , is
I
a frequency fσ. We introduce the modulation stress rate, computed by zones of 16 16 pixels, composing corre-
Rσ =4δfσ to characterize the modulation. lation maps of 370 µm res×olution. The link between the
Let us note that we used the same set-up and protocol value of g and the correspondingdeformations that have
I
than [13] and results presented here include those already occurred inside the sample were extensively described in
presented. The two following points differs from the for- [18,19]. Maximalcorrelation(g >0.99,whiteonFig.2a)
I
mer study. corresponds to an homogeneous deformation below 10−7
Here,twovanepenetrationdepthswereused: h=5cm and vanishing correlation (black on Fig. 2a) corresponds
and 0 cm. During the vane insertion procedure, in order to deformations larger than 10−5.
p-2
Title
125 10
Experimental results. – Figure 1.b shows typi- (a) (b) 4)
cal deformations for two experiments performed at the 2.54cm 100 8-10
stuamdee(mδe=an7s.5trPeass) (foσr0 t=wo11o0sc0iPllaat)ioanndfreoqscuiellnactiieosn. aTmhpelsie- + (t)Nc 5705 64γrain, (
two experiments were performed at a penetration depth cm st
h = 5cm. During the phase at constant stress, we ob- 25 stress mohd(cid:0)u(cid:3)lacti(cid:2)on 2 Mean
serve a slow increase of the deformation, γ(t), similar for 0 0
0 2 4 6 8 10
bothexperiments. Thisinitialdynamicscanbefittedbya Time (103s)
logarithmiccurveasin[4]. Then,whensubmittedtooscil- 250 6 8
(c) 4) 200 (d) 4)
laacttieornizs,edthbeysyastceomnsttraanntsimtseatonastlrianienarractree,epγ˙∞re,gwimheicchhainr-- 200 4γ- (10 150 6γ- (10
cflrueiadsiezsatwiointhbtyhemoescchiallnaitciaolnpfreerqtuurebnactyi.onThhiasssbuebe-ntharelrsehaodldy (t)Nc110500 2strain, (t)Nc100 4 strain,
reportedand interpretedin [13] as a secular drift process. 50 an 50 2 an
Figure 1.d showsthe value of the steady state strainrate, 0 no stress mohd(cid:0)u(cid:1)lacti(cid:2)on 0Me 0 stress mohd(cid:0)u(cid:1)lacti(cid:2)on 0 Me
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
γ˙∞, for all the experiments performed at h=5cm and at
Time (104s) Time (104s)
variousσ andR =4δf . Thoughthedataaredispersed,
0 σ σ 120 4 120 4
aγ˙s∞locnlegaralys iσn0criesasbeeslowwith σ105a0n0dPaR.σ Fasorprheidgihcetredstbryes[s1e3s], 90 (e) 3 γ(t) 90 (f) 3 γ(t)
γ˙∞ keeps increasing with∼σ0 but the influence of Rσ re- (t) 60 2 ain, (t) 60 2 ain,
mains unclear. For these stress values around the stress Nc str Nc str
dynamical threshold (see [4]), the mean shear rate γ˙∞ in- 30 1an 30 1an
e e
deedbecomesverylarge,howevertheexperimentsdisplay M stress modulation M
0 0 0 0
agreatamountofsensitivityto preparation,whichmakes 0 5 10 15 20 0 5 10 15 20
measurements in this limit quite difficult. Time (-) Time (-)
Several experiments performed at either h=0 or 5 cm
were coupled to the DWS technique. Fig. 2a represents Fig. 2: (a) Typical correlation map of the top surface. Two
a typical map of the top surface deformation: in average, typical “hot spots” are visible. The shaded part is not used
in the analysis. (b-d) Comparison of the mean strain, γ(t)
the correlation, g , is larger than 0.99 (white) implying
I
(blue) and the cumulative number of spots, Nc(t) (red) as a
a low and homogeneous deformation except over small
function of time for three experiments performed at the same
areas (black spots), called “hot spots” in [5], which are
mean stress, σ0 = 1000Pa without (b) or with (c-d) stress
characterizedbyrelativelylargeplasticdeformations. Fig-
modulation (δ = 10Pa, fσ = 0.1Hz). Experiments (c) and
ures 2b-d show the mean strain, γ(t) and the cumulative
(d) are performed at h = 0 cm, experiment (b) at h = 5 cm.
number of “hotspots”,Nc(t), overthe surface of interest, Thegreystripshowswhenthemodulation ison. (e)-(f)Mean
S, as a function of time for three experiments performed strain,γ(t)(blue)andcumulativefluidity,Nc(t)(red)obtained
at the same mean stress, σ = 1000Pa. For the experi- bythenumericalresolutionofeqs.(4-5)takingintoaccountthe
0
ment shown in Fig. 2c no modulation is performed unlike Couettegeometryofthesystem,without(e)andwith(f)stress
the ones shown in Fig. 2b and d. Experiments b and d modulationaroundthesamemeanstress. Parameters: a=1,
differ in the depth of the vane, respectively h=5 cm and σ0 =0.8σD, f0 =1, modulation: δ=0.005σD and fσ =0.2.
h=0 cm. One can note that the “hot-spots” are present
atthe top surface evenwhen the vane is buriedinside the
or rectified-regime). For the latter, the “hot-spots” pro-
packing (fig. 2b) which indicates that the localized relax-
duction rate inducing similar plastic deformations when
ation process is spreading over the whole material.
compared to the relaxation-regime values, is significantly
Fig. 3a shows N (t) normalized by S/σ as a function
c 0 smaller(Fig.3a). Thechangeoftheslopeisvisibleinboth
the global plastic deformation for experiments performed
experiments at h=0 and 5 cm. Finally, two experiments
atthesamestressforeitherh=0cm(green)orh=5cm
with stress modulation for each depth are superimposed,
(orange). Without modulation (fine lines in Fig. 3.a), the
showing the reproducibility of the measurements.
proportionality between N (t) and γ(t) observed already
c
in [5] is recovered, with a smaller slope in the case of Rheologicalmodel.– Theobservedlinearcreepwas
h=5cm(fineorangeline)comparedtothecaseh=0cm interpreted in [13] as a secular drift, i.e. the accumula-
(finegreenline). Whenthestressmodulationisturnedon, tion of tiny effects over a very long time. The ingredi-
wealsoobtain,afteratransientregime,aproportionalre- ents necessary for this sub-threshold fluidization by me-
lationshipbetweenN (t)andγ(t)butwithasmallerslope. chanical fluctuations to take place are the combination of
c
This proportionnality shows that the macroscopic plas- shear rejuvenation and memory effect. Such ingredients
tic deformation is the result of the accumulation of local are taken into account in the simplest mathematical way
plastic events. The coefficient of proportionality depends in a model put forwards by Derec et al. [11], to render
thus on the value of h but also on the regimes (relaxation the macroscopic phenomenology of aging complex fluids.
p-3
F. Author et al.
10 x104 50 The rheological equations (1) and (2) present rescaling
(a) h=0cm (b)
parameters which are for stress σ = G a/r, for de-
D
8 40
formation γ = σ /G, for fluidity R /σ and for time
S 0 D σ Dp
/0 6 h=5cm σ0 30 σD/Rσ. Then the dimensionless equations are as follows
σ
c4 Nc 20 σ˙ =γ˙ fσ (4)
N no stress no stress −
2 modulation 10 modulation f˙= a(f2 γ˙2) (5)
wmiothd ustlaretisosn wmiothd ustlaretisosn − −
00 2 4 6 8 00 1 2 3 4 Notethatinthisdimensionlessrepresentationtheonlyre-
Mean strain, γ (10-4) rain γ maining material parameter is a and the only control pa-
rametersdescribingthe stress arethe dimensionless mean
Fig. 3: Normalized cumulative numberof hot-spots as a func- stress and amplitude: σ0/σD and δ/σD. This system of
tion of the mean strain for 5 different experiments performed equationscanthenbesolvednumericallytoreproducethe
atσ0=1000Pa. Greenlines: h=0cm,orangelines: h=5cm. experimental protocol. In presence of stress modulation,
Fine line: no modulation, thick lines: with modulation. Dash the resulting plastic deformation exhibits a transient log-
lines are guide to the eye. (b) Corresponding graph obtained arithmic response followed by a linear regime (Fig. 2f).
by numerical resolution of eq. (4-5) taking into account the
As we established before [13], this is qualitatively what
geometry of the system. Same parameters as Fig 2.e-f.
is observed experimentally. The model allows thus to re-
cover the general experimental behaviour as far as global
This model was used successfully to interpret experimen- deformation is concerned.
talstrainrelaxationcurvesandrectifiedcreepin previous Several works have shown experimentally a direct re-
works [4,13,20]. A central parameter in this model is a lation between the rate of plastic events and the fluidity
fluidityvariablef representingthe rateofvisco-elasticre- variable f [5,21]. However, it is not clear a priori that
laxation for the shear stress σ : the mechanical driving would generate the same modes
of plastic relaxation in the packing in the presence of the
σ˙ =Gγ˙ fσ (1) mechanicalperturbations. And indeed, in fig. 3a, one can
−
identify a systematic change of slope for the relation be-
G is the shear elastic modulus. The rheological complex-
tweenthecumulatednumberof”hot-spots”andthemean
ity stems from the dynamicalequation undertakenby the
strain in the presence of stress modulation. This exper-
fluidity variable. Many forms were suggested by Derec et
imental fact leads to a central question on the relation
al. [11], but in the context of granular creeping flow the
between the rate of “hot spots” production and the fluid-
experiment results point on a simple expression :
ity parameter. This is the central object of the incoming
discuccion.
f˙= af2+rγ˙2 (2)
−
Spatial response. – Provided the assumption of a
where dimensionlessparametersa andr representrespec-
constant linear relationship between the rate of hot-spots
tively ageing and shear-induced rejuvenation processes.
production and the fluidity, the local and global distribu-
Thismodelnaturallyinducesalogarithmicrelaxationun-
tionofhot-spoteventscanbe computed. First,the radial
der constant shear and also a dynamical threshold σ =
D heterogeneity of the stress field is taken into account in
G a/r. In the context of granular matter this thresh-
the model, i.e. σ(r) =σ(R )R2/r2. However, here we do
oldvalue shouldbe proportionalto the confining pressure 0 0
p notaccountforthe verticalheterogeneitiesasforexample
to get a Coulomb dynamical friction coefficient. Experi-
would be the case for experiments with the vane buried.
ments [4] seems to indicate that under large shear, major
Eqs. (4) and (5) are then integrated numerically for
reorganizations may occur in the packing and one should
each r giving local values of the fluidity f(r) and strain
not consider anymore a and r as stress independent pa-
γ(r). Because of the cylindrical geometry, the total rate
rameters. Recently, Pons et al. [13] have shown theoret-
of hot-spots occurrence is then expected to be propor-
ically that the threshold of a visco-elastic fluid describes
R
tional to f(r)rdr. The cumulated number of spot
bythismodelwillbedestroyedbyvanishinglysmallstress R0
N (t) can then be obtained numerically (within a mul-
fluctuations around a bias by a secular effect. Conse- c R
tiplicative constant) by integrating this rate over time:
quently, below the threshold (σ < σ ), an effective vis-
0 D t R
cosity is expected of the form: Nc(t)= 0 R0f(τ,r)rdτdr. Inparallel,γ(r)isintegrated
over r in order to obtain the rotation angle α(t) of the
R R
Gσ σ2 vane. Thus we can obtained numerically the mean strain
η = D 2 1 0 . (3)
ωδ s (cid:18) − σD2 (cid:19) γ(t)=α(t)R0/(R−R0) corresponding to the one experi-
mentallymeasured. Thosetwoquantitiesarecomparedto
In the following, we will consider the present model in the dimensionless time in Fig. 2e-f. In the case of Fig. 2e
itssimpleform(aandrconstant)toseeifatleastqualita- no stress modulation is considered and we recoverthe be-
tivelythesalientexperimentaloutcomescanberecovered. haviourofthe modelwhenno spatialdependence istaken
p-4
Title
No stress Modulation With stress Modulation
into account [4,5]. In the case of Fig. 2f, stress modu-
lation are imposed from t = 0 and in this last case we (a)
t(cid:4)(cid:9) t(cid:4)(cid:9)
observea change of creep regime after a transientregime, t(cid:4)(cid:10) t(cid:4)(cid:10)
U) t(cid:4)(cid:8)(cid:6) U) t(cid:4)(cid:8)(cid:6)
inagreementwiththe experimentalgraphsof Fig.2b and A t(cid:4)(cid:8)(cid:11) A t(cid:4)(cid:8)(cid:11)
d. Finally, Fig. 3b shows the relationship between N (t) y ( t(cid:4)(cid:8)(cid:12) y ( t(cid:4)(cid:8)(cid:12)
c t t
andγ(t)obtainednumericallyineachcase. This relation- uidi uidi
ship is the same as the one observed in experiments: the Fl Fl
integralof the fluidity is proportionalto the plastic defor-
mation and for the rectified creep, a strong reduction of 11 22 33 44 1 2 3 4
the slope is observed when the transient is terminated. (c) (d)
Therefore, the outcome of the simulations is qualita- s t(cid:4)(cid:8)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7) t(cid:4)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7)
t t(cid:4)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7) t(cid:4)(cid:9)(cid:6)(cid:6)(cid:6)(cid:6)(cid:7) t(cid:4)(cid:8)(cid:6)(cid:6)(cid:6)(cid:6)(cid:7)
tively very similar to the experimental results. In par- po)U t(cid:4)(cid:8)(cid:6)(cid:6)(cid:6)(cid:6)(cid:7) t(cid:4)(cid:9)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7) )U t(cid:4)(cid:8)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7)
sA A t(cid:4)(cid:9)(cid:6)(cid:6)(cid:6)(cid:6)(cid:7)
ticular, the change of slope in Fig. 3a is reproduced in ot y ( oty ( t(cid:4)(cid:9)(cid:5)(cid:6)(cid:6)(cid:6)(cid:7)
tfiheeldrhdeuoelotgoictahlemgoedoemlwethreynisthteahkeenterinogtoenaecitcyouonftt.heMsticreross- dial hdensit dial hdensit
a a
scopically, this reduction originates from a difference in r r
spatial distribution of “hot spots” for the two cases as we
1.5 2 2.5 3 3.5 1.5 2 2.5 3 3.5
show numerically and experimentally in Fig. 4. Fig. 4a Radial position, r/R0 ion / 0
(resp. b) shows the radial evolution of the fluidity pa-
(e) (f)
rameter without (resp. with) stress modulation at dif-
ferent time of the numerical integration. For the simple
relaxationregime (logarithmic creep), the fluidity is quite
homogeneous across the cell and thus rather independent
of the local stress. The fluidity decreases uniformly with
time, reflecting the decrease of the strain rate. On the
contrary, for the rectified regime (Fig. 4b), the fluidity is
quite different close or further away from the centre. It Fig. 4: (a-b) Numerical resolution of eq. (4-5) taking into ac-
reaches rapidly a permanent regime with a high density count thegeometry of thesystem: evolution of theradial spa-
tial fluidity over time whitout and with stress modulation, re-
value close to the centre while far from it, the density be-
spectively. (c-d)Experimentalresultsdisplayingtheevolution
haviour is close to the one observed without modulation.
oftheradialhotspotsdensityovertime(experimentsofFig.2c
Theradialdistributionofhot-spotsproductioncanalso
and d respectively). The vertical axis unit is arbitrary but
be obtained experimentally and and are shown in Fig. 4c
identical for both experiments. (e) Schematic of the mechani-
(without modulations) and d (with modulations). The
cal response in absence of modulation: the fluidityis spatially
different curves correspond to the time evolution of the uniform in the cell (pink uniform background); the stress and
density. The density is calculated by averaging the num- thestrainhavethesamespatialdependence. (f)Rectifiedcase:
ber of “hot spots” over a time windows of 5000s and for the fluidityis non uniform and decreases as ∝1/r2.
a 5mm wide annulus. We first observe that this density
decreaseswithtimeforbothexperiments. Then,although
both in numerical solution (Fig. 4a) and in experiment
thedataarequitenoisyduetothesmallstatistics,itseems
(Fig. 4c). Yet, it uniformly slowly decreases with time.
that this radial density stays homogeneous for the simple
We thus obtain γ˙(r,t) = f(t)σ0(r) 1/r2. This case is
relaxation(Fig.4c)whiletheapparitionof“hotspots”de- G ∝
schematically represented in Fig. 4e.
creasesmore rapidly far from the center ofthe cell during
On the other hand, in the rectified case, as recalled in
the rectified creep (Fig. 4d), which qualitatively corrobo-
the description of the rheological model (eq. 3), the local
rates the outcome of the numerical measurements.
∗
stationary value f (r) of the fluidity is for σ σ [13]:
0 D
Discussion. – To interpret this difference of be- ≪
ωδ(R )R2
haviour in the radial spatial repartition of fluidity in the f∗(r)= 0 0, (7)
two regimes, we use the analytical results obtained from σD√2 r2
theperturbativeanalysisdonein[13],buthere,weexplic-
If we suppose that a stationary solution is reached across
itlytakeintoaccountthespatialdependenceofthestress.
the cell, we obtain for the local strain rate γ˙(r) =
In absence of modulation, the fluidity is mostly governed f∗(r)σ0(r) 1/r4. In this case, represented in Fig. 4f,
by its initial value f0 [4]: the Gstrain ∝is highly localized in the vicinity of the blades.
f0 f0 Because of the strong decrease of the fluidity with r in
f(r,t)= (6)
1+af 1 σ02(r) t ≈ 1+af0t the rectified case, the total activity in the cell for a given
0 − σD2 mean strainacrossthe cellis smaller inthe case ofFig.4f
(cid:16) (cid:17)
Consequently, when σ σ , the non-rectified fluidity than of Fig. 4e. This is the origin of the decrease of the
0 D
≪
is independent of the radial position, a result recovered slope in Fig 3 in the rectified regime. Note that further
p-5
F. Author et al.
refinement of the discussion taking into account that the the spatialpropagationofthe plasticactivity throughthe
stationarysolution, in the rectified case, is not reachedat material bulk and thus of non-locality.
large r does not change the picture. The duration of the Finally, we must underline that most of the present re-
transients are nevertheless of major importance in such sults were obtainedrather far fromthe dynamicalthresh-
experiments andit canbe problematic to conclude onthe old, where essentially, we could individualize the hot-spot
nature of the creep. Indeed, the time for the transient to apparition. Closer to the threshold and possibly due to
∗ ∗
reachthestationarysolutionf istypically1/f whichdi- spatialcouplingandavalanchingevents,itwouldbeinter-
verges when the targeted fluidity decreases. Actually, the esting to see if quantitatively the simple local description
∗
distinctionbetweenarectifiedcreepofverysmallf anda still holds.
logarithmic creep could be pointless experimentally. The
timetoreachthestationarysolutionmaybecomeincreas-
∗∗∗
ingly long while environmental background noise, always
presents in practice, may eventually hinder the latter. ThisworkwasfundedbyaCNESfundamentalresearch
grant, a CNES post-doctoral grant and ANR Jamvibe.
Conclusion.– Inthisreport,westudiedthecreepre-
sponse of a granular packing below the Coulomb fluidiza-
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p-6