Force networks and elasticity in granular silos

0.41 MB·English

by  John F. Wambaugh

Force networks and elasticity in granular silos
∗
John F. Wambaugh , Robert R. Hartley, and Robert P. Behringer
Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708†
(Dated: February 2, 2008)
Wehavemadeexperimentalobservationsoftheforcenetworkswithinatwo-dimensionalgranular
silosimilartotheclassicalsystemofJanssen. ModelslikethatofJanssenpredictthatpressurewithin
a silo saturates with depth as the result of vertical forces being redirected to the walls of the silo
wheretheycanthenbecarriedbyfriction. Byaveragingensemblesofexperimentally-obtainedforce
networksin differentways, wecomparetheobserved behaviorwith variouspredictionsfor granular
silos. We identify several differences between the mean behavior in our system and that predicted
8
by Janssen-like models: We find that the redirection parameter describing how the force network
0
transfers vertical forces to the walls varies with depth. We find that changes in the preparation
0
of the material can cause the pressure within the silo to either saturate or to continue building
2
with depth. Most strikingly, we observe a non-linear response to overloads applied to the top of

n the material in the silo. For larger overloads we observe the previously reported “giant overshoot”
a effect where overload pressure decaysonly after an initial increase [G. Ovarlezet al., Phys. Rev. E
J 67, 060302(R) (2003)]. For smaller overloads we find that additional pressure propagates to great

2 depth. Thiseffectdependsontheparticlestiffness,asgivenforinstancebytheYoung’smodulus,E,
2 of the material from which the particles are made. Important measures include E, the unscreened
hydrostatic pressure, and the applied load. These experiments suggest that when the load and

] the particle weight are comparable, particle elasticity acts to stabilize the force network, allowing
t
f non-linearnetwork effects to beseen in themean behavior.
o
s PACSnumbers: 45.70.-n,45.70.Cc,83.80.Fg,45.05.+x
.
t
a
m
I. INTRODUCTION to be in these positions [1]. Determining a statisticalde-
- scriptionofthesecontactsisatthecoreofunderstanding
d
dense granular matter.
n Thephysicsofgranularmatterisofgreatinterestboth
o Grain-scale statistical descriptions are further compli-
because of the enormous breadth of physical systems in
c cated by the presence of force networks [2, 3]. An ex-
the granular regime as well as the lack of a characteri-
[ traordinary feature of granular contact networks is that
zation of the granular state in fundamental terms. Al-
the distributionofforcesonthe contactscanbe strongly
1 though there has been some success understanding ener-
v inhomogeneous [4]. Certain sequences of contacts carry
getic granular gases in terms of kinematic models, dense
7 forceswithmagnitudemanytimesthemeanforlongdis-
granular materials continue to pose difficult problems.
8 tances along “force chains”. As shown in Fig. 1, force
3 We consider dry granular matter at low humidity where networkscan developto support the weight of the parti-
3 there are no attractive forces between the constituent
cles as well as any additional applied load.
. particles and the particles do not significantly interact
1 Despitethedifficultiesinherenttounderstandinggran-
with the interstitial fluid (air). Contact forces dominate
0 ular materials, phenomenologicalengineering models ex-
8 dense granular materials in this regime, with each parti-
istthat describe the mean behaviorwellenoughto allow
0 cle having several persistent contacts.
for the design of granular storage facilities and handling
:
v Coulomb’s law of friction is an inequality relating the processes. These models can provide useful tools for in-
Xi magnitude of the friction force, Ff to the normal force, vestigating granular materials. In particular, Janssen’s
Fn, depending upon the mobilization of the friction. If model of granular materials in vertical silos is interest-
r
a µs is the static friction coefficient, then the mobilization ing because it has been thoroughly tested, is realizable
is given by Ff/(Fnµs). For a non-moving contact the inthe laboratory,andultimately is viewedto be a work-
mobilization ranges from −1 to 1 and reflects both the ing qualitative and even quantitative description of the
direction in which the force of friction acts to resist mo- mean behavior of granular matter [5, 6].
tion and how close the forces acting on the contact are
to causingmotionin the opposite direction. Becausethe
forceoffrictionisdependentuponhowthecontactarose,
II. THE JANSSEN MODEL
the granularstate is determined bothby the positionsof
the particles and the history of how the particles came
Janssen treats the material in a silo as a continuum
and considers horizontal slices, for which the difference
of stress, the weight of the material in the slice, and
the force of friction at the wall must be balanced ver-
∗Currentaddress: National Center forComputational Toxicology,
USEPA,ResearchTrianglePark,NC27711 tically. Here we consider a quasi-two-dimensional silo,
†Electronicaddress: [email protected] such as the one we use in the experiments reported be-
2
1
0.8
e
ur 0.6
s
s
0.4
e
r
P
0.2
0.5 1 1.5 2 2.5 3
Depth Harbitrary unitsL
FIG. 2: Janssen’s model predicts that stress exponentially
approachesasaturationvaluewithdepthinasilo. Overloads
decay exponentially and if an overload equalto thethesatu-
ration stress is placed on thesilo thestress is constant. Here
we plot predictions in a system that saturates at 0.5 in arbi-
traryunitsfornoload, asaturation overload (=0.5), andan
abovesaturation overload.
|σ |≤ µ |σ | for wall friction µ . Janssen further as-
xy w xx w
sumes that the vertical and horizontal normal stresses
are linearly related to each other to write σ =kµ σ
xy w yy
where k represents both the mobilization of interparti-
cle contact friction (with friction coefficient δ) and the
mobilization of friction at the walls. By considering the
extremes of maximum upward and downward mobiliza-
tion, it has been shown that 1−δ <k < 1+δ [7].
1+δ 1−δ
Using Janssen’s relation for the shear stress in Eq. 1
allows us to rewrite it as a differential equation for σ :
yy
dσ 2kµ
yy w
=− σ +ρg (2)
yy
dy L
IfJanssen’sparameterk istakenasaconstant,thenthis
equation permits exponential solutions of the form:
−y ρgL −y
σyy =Φ0e λ + 2kµ (cid:16)1−e λ(cid:17) (3)
w
FIG.1: Aforcechainnetworkinasilofilledwithphotoelastic
where Φ is any load applied to the top of the system
disks undera 56 g load 0
and
L
λ≡ (4)
low. Janssen assumes that the stress does not vary hor-
2kµ
w
izontally, allowing one to write for the case of a two-
dimensional silo: sets a length-scale for the evolution of stress with depth.
AsindicatedbyFig.2,theJanssensolutionpredictsthat
L(σ (y+∆)−σ (y))=ρgL∆−2σ ∆ (1) anyoverloadwilldecayexponentially,andthatthestress
yy yy xy
withinthematerialwillsaturatewithdepth. Thisscreen-
where x and y are the horizontal and vertical directions ingeffectresultsfromfrictionatthewallsactingtocarry
respectively, ∆ is an infinitesimal change in y, σ is the the weight of the material and any overload. Although
stress tensor, ρ is the density of material (mass per unit we have presented the two-dimensional version here, the
area), g is the acceleration of gravity and L is the width Janssen model is qualitatively similar in three dimen-
of the silo. The second term on the right-hand-side cor- sions.
responds to the two frictional wall contacts — one at Janssen’s prediction of stress saturating with depth
each side of the silo. Coulomb’s law of friction gives the has been validated qualitatively in numerous three-
shear stress at the wall in terms of the horizontal stress, dimensional experiments that examine the apparent
3
weight of a confined granular pile. Though these experi- ent from predictions. Instead of a monotonic, exponen-
mentsineffectonlymeasuredtheforcesatthebottomof tialdecaytothe saturatedstresssupportedbythe walls,
thepile,byrepeatedlyvaryingtheamountofmaterial— a non-monotonic, “giant overshoot” has been observed
and thus the height of the pile — a mean description of experimentally [8, 21]. Even an overload equal to the
stresswithdepthwasobtained[2,8,9,10,11,12]. Large saturated stress deep in the silo produces a local rise in
scale simulations of three-dimensional systems have also thestressinsteadoftheflatstressprofilepredictedbythe
observedstresssaturation[13]. Inquasi-two-dimensional Janssen model. This can in part be explained by local
experimental systems that are similar to the analysis changes in the mobilization of friction and packing frac-
above,thesaturationlengthscalehasbeenfoundtoscale tion of granular matter induced by placing an overload
with system size as predicted, λ∝L [14]. on the material.
While qualitatively accurate, Janssen’s predictions To account for the overshoot effect and better fit ex-
have been shown to underestimate the actual value of perimentally observed stress profiles, a piece-wise model
saturated stress in a silo [10]. Careful analysis of stress has been proposed that describes stress in a silo as hy-
profiles in experiments and simulations have shown that drostatic - increasing linearly with depth - until a tun-
for small depths the profile is more linear than exponen- able depth where the stress then becomes exponential
tial [10, 15]. These deviations from the Janssen model [10, 15]. This piece-wise behavior arises out of the anal-
are not surprising, since key assumptions of the analysis ysis for horizontally varying vertical stress and indicates
can be shown to be incorrect. the depth to which the stress at the top boundary pene-
FortheJanssenmodelaboveweassumedthattheredi- trates[20]. Fittingexperimentalresultstothepiece-wise
rection parameter k was constant. For a cohesionless pressure profile corrects experiments that find unphysi-
granularmaterial,however,the ratioofthe horizontalto cal values of the redirection parameter when the basic
vertical stress is constant only if the horizontal and ver- Janssen model is used [8, 10].
tical directions are the frame in which the stress tensor In some experiments, a threshold for overloads has
is diagonal [7]. This is inconsistent, since Janssen as- been observed that, when passed, causes the redirection
sumes that there is a non-zero σxy at the walls. If σxx parameterk to increase[19]. Whenthe presence offorce
and σyy are not the principal stresses, then we expect networks within granular silos is considered, the mech-
k ∼σxx/σyy to vary horizontally and with depth. anism accounting for different overload profiles becomes
Several factors contribute to variation in the redirec- clearer. Granular matter has been described as being in
tion parameter with depth. In particular, it is difficult a “fragile”state. Along the directionsin whichthere are
toachieveauniformmobilizationoffrictionatthewalls. pre-existing force chains, large loads can be supported.
Experimentsandsimulationshaveshownthatagreement In other directions, only slight deformation of the inter-
with Janssen predictions can be obtained if the granular particle contacts are allowed before the material must
material is slowly lowered relative to the walls to make rearrange,forming new force chains [22]. To understand
uniform the direction and extent of mobilization for all how a granular silo responds to overloads, we need to
wall contacts [9, 16, 17]. The global packing fraction of considerhowthe forcenetworkfacilitatesthe redirection
the material has also been shown to alter the redirec- of stress to the walls, which is the goal of the present
tion parameter and it is reasonable to expect that local experiments.
variations in packing fraction would produce variations Since the Janssen model is a continuum description
with depth [8]. Finally, experiments studying the stress that at least moderately successfully predicts stress sat-
within a silo of deformable beans, instead of the more uration with depth, it is interesting to examine in what
rigid glass or metal balls typically used, have found that ways it gives an appropriate mean description of the
the redirection parameter varies with depth due to the forceswithinagranularmaterial. Itisequallyinteresting
elasticity of the particles [18]. to determine how and why it fails [1, 23]. For instance,
More elaborate analysis of the Janssen model using various models of granular force propagation give differ-
stochastic differential equations to account for random- ent predictions ofmean silo behavior[2, 24, 25]. Further
ness in the mobilization of friction has shown that the experimentalanalysisofthe silo systemcanserveto test
Janssen model only correctly predicts the stress on the these and future models of granular matter.
axis of the silo [16]. In general, the vertical stress at In our experiments, we use photoelastic particles to
a given depth has been found to vary horizontally, fur- investigate how the preparation and properties of the
ther complicating the force balance analysis [7, 19]. In granular material and the network of contact forces in-
three dimensions, attempts have been made to account teract to redirect stress to the walls and determine the
for radial stress balance by using vertical force balance development of the stress profile with depth. We con-
onconcentricringsabouttheaxiswiththegeneralresult ductexperimentsinaquasi-two-dimensionalsystemthat
thatforcesaturationduetoscreeningbythewallsisstill is described by the Janssen analysis above and has been
expected, though the approach to the saturation value shown to be similar to three dimensional systems in ex-
may not be exponential [20]. perimentsandsimulations[14,26,27]. Similarquasi-two-
Although screening-induced saturation has been vali- dimensionalexperimentshaveusedphotoelasticityinthe
dated, the behavior of overloads can be strikingly differ- pasttoexaminethepropagationofstressbutdidnottest
4
10 cm
the larger disks approximately 20% larger. The silo is
constructed so that there are movable pistons at the top
and bottom of the silo. The pile rests upon the bottom
piston, while loads can be applied to the top piston. A
10g load applied to the top of the pile corresponds to a
∼ 1000 dyne/cm stress at the top. Both pistons can be
pinned to prevent motion or unintended loading.
We measure the local disk-scale pressure, which we
refer to as “stress” on our photoelastic disks using a
technique similar to previous granularresearchfrom our
group [3]. The bi-refringent disks display bright fringes
in response to applied stress when illuminated between
m
crossed polarizers. The density of fringes within a given
c disk is proportional to the stress within the disk. We

8 determine the stress on a given disk by first calculat-
3 ing the gradient of an image of fringes. We take the
~ square of the gradient averaged over multiple directions
to be proportional to the number of fringes. We use
an experimentally-obtained calibration to convert the
squared gradient of the intensity into an equivalent load
on a monolayer of disks and hence a single disk.
We performed experiments with two different
silo/camera arrangements. In the first silo we use
approximately 650 disks — one quarter with diameter
0.9 cm and the rest 0.75 cm — resting upon a piston
that is affixed to a stepper motor. A 640×480 digital
videocamerais usedto imagethe columnata rateof30
frames/s while the piston supporting the pile is slowly
piston
lowered at a rate of hundredths of a grain diameter per
second. We examine the sequence of images to find
the first frame where an abrupt rearrangement of the
force networkis observedandtake the frame before that
FIG.3: Weexaminephotoelasticdisksconstrainedtoaquasi- as the moment that the friction of the grains at the
two dimensional geometry in a vertical ‘silo’ made from two walls is maximally mobilized upwards. We perform our
Plexiglas sheetsseparatedbyslightly morethanadiskthick- analysis on this maximally mobilized frame. To prepare
ness. Thebi-dispersedisksareconfinedbyverticalaluminum for each observation, a new force network is created
sidewallstoachannelL=10cmwideandrestuponapiston
by quickly forcing the lower piston upwards and then
that can be used to control the mobilization of friction. The
letting the entire pile raindownwardsas the piston falls.
aspect ratio of the pile is typically 4:1.
Simulations have shown that granular piles created by
‘raining’particlesfromaboveareindependentoftherate
at which grains are added [13]. To apply an overloadwe
the Janssen model [3, 26, 28]. Rather than measure the
gently lower the upper piston onto the top of the pile
apparent weight at the bottom of a three-dimensional
before the pile is lowered by the stepper motor. The
pile, these photoelastic techniques allow us to examine
top layers of grains rearrange slightly as the overload is
the actual force network responsible for Janssen screen-
appliedbutthe remainderofthe particlesmaintaintheir
ing. Weinvestigatetheroleofparticleelasticityandforce
relative positions.
networkswithinsilosbyexaminingthemeanresponseto
In the second silo arrangement, approximately 1200
overloads, by characterizing the properties of the force
slightlysmallerdisksareused(anapproximately3:1mix-
network, and by varying the elasticity of the particles.
tureof0.6cmand0.75cmdisks). Thelowerpistonrests
upon a base that can be manually lowered a small dis-
tance. Withoutafixedconnectiontoasteppermotor,we
III. METHODOLOGY
areabletocreatenewforcenetworksbyphysicallylifting
the entire silo and flipping end-over-end. After the pile
Westudyaquasi-twodimensionalarrangementofpho- is lowered slightly to mobilize friction more uniformly, a
toelastic disks constrained vertically by two aluminum high-resolution(3264×2468) digital still camera is used
walls and onthe faces by two transparentsheets to form totaketwopictures,onewithoutpolarizationthatcanbe
a channelL=10cmwide, as depictedin Fig.3. To pre- usedtofindparticlecentersandonewithpolarizationfor
ventordering,weuseabi-dispersemixtureofdisks,with obtaining force from photoelasticity. We can use this in-
5
8000
)
m
c6000
/
e
n
y
d
(

e4000
r
u
s
s
e
r
P
2000

Low Stress High Stress 0 0.5 1 1.5 2 2.5 3
Depth from Surface (L)
FIG.4: (ColorOnline)Thesecompositeimages,generatedby FIG. 5: By binning the images of force networks horizon-
merging sets of seventy to one hundred gradient squared im- tally, averaging sets of seventy to one hundred images and
agesofforcenetworks,displaytheonsetofJanssen screening then using a calibration to convert intensity gradient into
astheoverload isincreased from 0to106ginapproximately pressure, we can generate pressure profiles. The dashed line
25 g increments. The same intensity scale is used for all im- indicates the unloaded case, while the increasing solid lines
ages, indicating greater pressure deep in the silo for smaller indicateoverloadsappliedatthetopincreasinginincrements
loads. of roughly 25 g: no load (•), 31 g ((cid:4)), 56 g (N), 81 g ((cid:7))
and 106 g (H). Error bars indicate the standard deviation
of means calculated for an ensemble of subsets of the data.
formationto characterizethe contactandforcenetworks DepthisgivenintermsofthewidthofthesystemL=10cm
withinthematerial. Toinvestigatetheinfluenceofparti- ≈13 smaller disk diameters.
cle elasticity on these networks, we use disks made from
two photoelastic polymers of different stiffness (Young’s
modulus E = 4 MPa for softer particles, E = 210 MPa bottom of the silo. This increase is also observed for all
for harder particles). Both soft and hard disks are 0.635 overloads applied to the top of the pile. This effect is at
cmthick,andthematerialfromwhichtheyaremadehas least partly due to the way that the preparation history
a density of 1.2g/cm3. affects the unloaded state. As seen in the left-most im-
For all figures, the standard error indicated by bars age of Figure 4, the mean stress is not constant across
for mean quantities is determined using a ‘bootstrap’ al- thelayeratagivenheight,particularlynearthe bottom.
gorithm where the mean of the quantity is calculated This effect is presumably associated with the fact that
for each of 200 randomized data sets constructed by se- whenthepistonpushesupwards,notalltheparticlesare
lecting elements of the original data set at random with easily thrown upward, or that there is wall drag as the
replacement[29]. Thestandarddeviationofthesemeans particles fall which affects their packing.
is taken to be the standard error. Wenextexaminedtheresponsetofouroverloadsinin-
crementsofroughly25g(i.e. atwo-dimensionalstressof
roughly 2500 dyne/cm). As shown in Fig. 5, for the two
IV. RESULTS largestoverloadsa non-monotonicovershooteffect is ob-
served. This behavior is similar to previous experiments
A. Fixed Silo Method–Softer Particles [8, 21] – after an initial increase in pressure at the top
of the silo, the largest overloads are screened by arching
Wefirstusedthefixedsiloandstepper-motorarrange- and the pressure converges with the unloaded pressure
menttoobserve∼100unloadedforcenetworksand∼75 profile with depth.
networkseachforfourdifferentoverloads. Figure4shows The behaviorofthe two smalleroverloadsis strikingly
the merged squared gradient images of these force net- different from expected and previously-observed behav-
works, roughly depicting mean response. If we bin the ior. The additional pressure of small overloads persists
responsesacrossthewidthofthesilointohorizontalslices with depth and is not screened by wall friction. The
theheightofoneparticle,wecanobtainapressureprofile difference between large and small overloads is clear in
with depth as in Fig. 5. Fig.6,wherethe unloadedpressureprofilehasbeensub-
Considering first the case of the silo without an over- tracted from the four loaded pressure profiles. Deep
load, we see distinctly non-Janssen behavior. Although withinthepile,theadditionalpressureoftwolargerover-
the pressure initially seems to saturate with moderate loads decaysto zero,while the pressure fromthe smaller
depth, the pressure starts increasing again toward the two loads persists.
6
8000
) 30
m
c 20
/
ne 6000 m) 10
y
d c 0 0.5 1 1.5 2 2.5 3
ure ( 4000 λ (1/ 30
s 20
s n
e e 10
Pr 2000 ss 0 0.5 1 1.5 2 2.5 3
d n
a a
o 0 J 30
rl
e 20
v
O 10
−2000
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Depth from Surface (L) Depth from Surface (L)
FIG. 6: Plotting the difference between the pressure profile FIG. 8: The Janssen screening length-scale λ as a function
resulting from an overload and the unloaded profile shows of depth for, from top to bottom, no load (•), a 31 g load
how the overload is distributed. Four different overloads, 31 (N)and an 81 g ((cid:7)) load for both the initial (dashed) force
g ((cid:4)), 56 g (N), 81 g ((cid:7)) and 106 g (H), were applied across network and themaximally mobilized (solid) force network.
thetopofthesystem. Smallerloadsseemtopenetratedeeper
into the pile, while larger loads seem to show the previously
observed overshoot effect [8] followed by theexpected decay. we estimate λ numerically as:
P(z)
λ(z)= (5)
ρ φg− dP(z)
ps dz
25
whereρ isthedensityofthe photoelasticmaterial,φis
ps
the packing fraction in the silo, P(z) is the horizontally-
m) 20 binned pressure and dPd(zz) is estimated numerically us-
ing local quadratic fits to P(z). Since λ varies as the
c
( inverse of the redirection parameter k, this result indi-

λ
cates that the assumption that k is constant with depth
n 15
e within our system is incorrect. Though the scatter in
s
s the pressure profile for 31 g introduces substantial noise
n
a into the calculation of λ, hence this data is omitted, we
J 10
see that the variation of λ with depth is roughly similar
for both unloaded and loaded silos. This indicates that
thepreviously-observedload-inducedvariationinkisnot
5
responsible for variation in load propagation[19].
0 0.5 1 1.5 2 2.5 3
Depth from Surface (L) Though Janssen’s k varies with depth, we can still in-
vestigate if k(z) behaves as a depth-dependent redirec-
FIG. 7: The Janssen length-scale λ can be calculated nu- tion parameter might be expected to behave. In Fig. 8,
we calculate λ as a function of depth for the ensemble of
merically using the density of the granular material and the
pressureprofilesfornoload (•),56g(N),81g((cid:7))and106 g unloaded networks,and the ensembles for overloads just
(H). We observe that λ varies as a function of depth instead above and below the transition from deep-propagation
ofremainingconstantasintheJanssenmodel. Sincethecal- to screening. We compare the network in the first im-
culated length-scale varies with the inverse of the derivative ageofourloweringsequence,correspondingtotheinitial
of pressure, variations in the measured pressure profile can networkbeforethepilehasbeenloweredtochangemobi-
be greatly amplified, as with the for 31 g results at shallow lization,tothenetworkinthesubsequentimagewherewe
depth.
assumefrictionismaximallymobilized. Sinceλvariesas
theinverseofthemobilizationk,theobserveddecreasein
λafterloweringthepile indicatesthatwehaveincreased
the mobilization. The mobilization does not change sig-
Given the non-monotonic, and sometimes non- nificantly at the top of the pile, indicating that our low-
saturating behavior of the observed pressure profiles, it ering technique does not uniformly effect the pile. The
is not surprising that the Janssen length-scale λ= L difference between initial and mobilized frames is simi-
2kµw
varieswithdepth,asindicatedinFig. 7. Usingourdata, larforbothsmall,deep-propagatingoverloadsandlarge,
7
10000 60
50
) 8000
m
c m)
e/ c 40
n 6000 (
y λ
(d n 30
e e
ur 4000 ss
s n 20
s a
e J
r
P 2000
10
0 0
1 2 3 4 5 6 0 1 2 3 4 5 6
Depth from Surface (L) Depth from Surface (L)
FIG. 9: By changing our procedure for creating new force FIG.10: ThenumericallyevaluatedJanssenscreeninglength-
networksweobtained pressure profilesthat indicatepressure scale for the pressure profiles for an unloaded silo (•) and
saturation with depth for an unloaded silo (•) and overloads overloads of 20 g ((cid:4)), 40 g (N), 60 g ((cid:7)) and 80 g (H) shows
of 20 g ((cid:4)), 40 g (N), 60 g ((cid:7)) and 80 g (H). Despite the convergence with depth to a nearly constant value of λ.
change in pile creation procedure the overshoot effect is still
observed for large loads and small loads are still observed to
6
x 10
penetrate to a considerable depth. 15
)
m
c
/
e
decaying overloads. n
y10
We note four important differences between the pres- d
(
sure profiles in Fig. 5 and those predicted by Janssen- e
r
like models, e.g. Fig. 2: λ, and hence k, is not constant; u
s 5
pressure from larger overloads first increases with depth s
e
before decaying; overall pressure does not seem to satu- Pr
ratewithdepth;andmoststrikingly,pressurefromsmall d
a 0
overloads does not decay. The first two differences have o
been observed previously and are not surprising, given erl
v
the previously discussed drawbacks of Janssen’s analysis O
[8,18,21]. Tobetter understandthe lackofoverallpres-
0 1 2 3 4 5 6
suresaturation,weconductedexperimentsinoursecond Depth from Surface (L)
apparatus that allowed the particles to be rained from
above, rather than pushed from the bottom. As shown FIG.11: Forhard(E=210MPa) photoelasticdisksweonly
inFig.9wefoundthatthesecondpreparationprocedure seetheoverloadprofileduetotherelativeinsensitivityofthe
didcreatepressureprofilesthatsaturatewithdepth. The harder material to stress. We examined overloads of 1837 g
largestoverloadis observedto overshootbefore decaying (•), 2267 g ((cid:4)) and 3402 g (N). All seem well-described by
and the smallest load propagates to great depth. exponential fits(dashed lines).
Numerically determining the Janssen screening-length
λ for the saturating pressure profiles via Eq. 5 indicates
that after an initial transient, the length-scale actually a combination of both preparation — the act of placing
converges to a constant value of roughly 20 cm with an overload at the top — and non-uniform response to
depth, as in Fig. 10. This implies a nominal value of loweringthe pilewithdepth. Thesetwofactorsmayalso
µk =L/2λ=0.25. Interestingly,theλ(z)forthemoder- account for the giant overshoot effect seen for the large
ate overloads are the flattest, while the screening length overload.
convergesfromalargervalueforthelargeroverloadsand
from a smaller value for the silo with no overload and
the smaller deep-penetrating 20 g overload. The non- B. Silo Flipping Method–Softer and Harder
constant value for the unloaded case may be due to sev- Particles
eralcauses. Onemaybe anunderestimationofthesmall
forces at the top of our pile when using our gradient We then used the silo flipping preparation to examine
method to calibrate photoelastic fringes. Alternatively, both the effect of preparation and of changing the elas-
the variation at the top of the pile may be explained as ticityoftheparticles,fromYoung’smodulusE =4MPa
8
60
)
50 > −1
P10
) <
m
c 40 >)/
(
λ P
n 30 <
e − −2
s P10
s
n 20 ((
a b
J o
r
10 P
0 −2 0 2 4
0 0.5 1 1.5 2
Depth from Surface (L) (P−<P>)/<P>
FIG. 12: Due to the insensitivity of the harder photoelas-
ticparticles, determiningthelocalJanssen screeninglength-
)
scaleλnumerically isonlypossibleforthelargest 3402 g(N) > −1
P10
overload. The observed screening length is roughly half as
<
long as for thesoft (E =4 MPa) particles. /
)
>
P
<
to E = 210 MPa. For the harder photoelastic material, − −2
P10
the sensitivity to applied stress was lower, with the re-
(
sult that the stress due to gravity was not visible even b(
atgreatdepth in the silo. Inthis case,any pressure pro- o
r
file is effectively an overload decay profile, since there is P
no unloaded profile to subtract. As shown in Fig. 11,
the additional pressure due to the overloadsdecays with −2 0 2 4
depth after overshootingfor allthree cases. For allthree (P−<P>)/<P>
overloads,the mean pressure eventually drops below the
range of observable sensitivity, though large fluctuations
FIG. 13: Probability distributions for the difference in pres-
aboutthe meanwerestillpresent. Allthree casescanbe
sure from the mean value at the center (solid) and sides
fit with exponentials if the initial overshootis neglected. (open) of thesilo forsoft (top) particles in an unloaded silo
Since the non-decaying, deep-penetrating overloads for (•)andwithoverloadsof20g((cid:4))and80g(N). Wealsoshow
the softer particles were close to the saturationpressure, distributions for hard (bottom) particles with overloads of
itisnotpossibletodeterminewhetherornotsucheffects 1837 g (•), 2267 g ((cid:4)) and 3402 g (N). The curves differ for
are present in the harder particles using our method. low pressures duetothe differences in sensitivity of thehard
Determining the Janssen screening length-scale λ nu- andsoft photoelastic polymers, butgenerally thereisagreat
deal of overlap and we cannot distinguish the sides from the
mericallyforthehardparticleoverloadswasproblematic
center.
since the pressure profiles eventually decayed below the
sensitivity of the hard photoelastic material. In Fig. 12,
we show the calculated λ(z) for the largest, 3402 g over-
load in the top third of the silo. What we see is qual-
itatively similar to the behavior of the largest overload
in Fig. 10 — λ approaches a constant value from above. maximumload onthe harderparticles, ∼3400g,this ra-
Surprisingly, although the system geometry is the same, tio is ∼1.0×10−4. Thus, in appropriately scaled terms
thescreeninglengthisroughlyhalfofwhatwasfoundfor the loads are comparable in both cases, and hence on
the soft (E = 4 MPa) particles used to obtain the first this basis, elasticity cannot explain the difference in the
two pressure profiles (Fig. 5 and Fig. 9). experimentalobservations. However,the ratioof the ap-
The two sets of experiments nominally contrast soft plied load to the weight of the material is significantly
and hard particles. However, there is another aspect different in the two cases, namely about 0.5 for the soft
which may be at least as important as granular elastic- particles, and 17 for the hard particles. Thus, we con-
ity. We note that the ratio of the applied loads to the clude thatforthe softerparticles,the loadsarenotlarge
Young’s modulus, E, are comparable for the hard and enough to overcome the effects caused by gravitational
the soft material. For the maximum load on the softer loading, which necessarily reflects the preparation his-
particles, ∼ 100g, this ratio is ∼ 1.6×10−4 and for the tory.
9
C. Horizontal Variations and Fluctuations
12000
Itisalsointerestingtoinvestigatevariousotheraspects
10000
of the force profiles which differ from the usual Janssen m)
picture. In particular, we consider horizontal variabil- c
ity of the forces, and also force fluctuations. In this re- e/ 8000
n
gard, we note model studies by Pitman which exhibited y
d
horizontal dependence on fluctuations and stress within ( 6000
e
granular silos[16] due to inhomogeneous mobilization of ur
friction. ss 4000
e
As shown in Fig. 13, we measured the local deviation r
P
from the mean pressure at a given height for both the 2000
centerandsidesofthesiloforalloverloadsforbothhard
and soft particles. We did not see significant horizontal 0
1 2 3 4 5 6
variation in the distributions, and for the soft particles
Depth from Surface (L)
the distributions were nearly identical for all cases and
7
veryreminiscentofforcedistributions observedforother x 10
2
granular systems [30]. Forces below the mean fall off
quickly, while forces larger than the mean tail off slowly,
and roughly follow an exponential. m)
1.5
Recent simulations for vertical granular matter with c
/
e
periodic boundary conditions (in the horizontal direc- n
y
tions)haveindicatedthatthe averaginglength-scalesuf- d
( 1
ficienttodeterminethemeanpressureispossiblyassmall
e
as a grain diameter [31]. Though our geometry has side ur
s
walls, it is still worthwhile to determine the length over s
which we must average for the pressure profile to con- re0.5
P
verge. To do this, we determine a grid of points within
the silo andaveragethe pressurewithin atunable radius
toassignapressuretothatpoint. InFig.14,weplotthe 0
1 2 3 4 5 6
pressureinthe middle of the silo (onaxis)withdepth as
Depth from Surface (L)
determined by averaging over regions with radii of 1, 5,
10 and 15 small particle diameters. For the two larger
FIG. 14: Pressure profiles on axis converge as the length R
averagingregions,theprofilesconvergetothoseobserved
overwhichaveragingisconductedvaries,asindicatedbylines
using horizontal binning by depth in Fig. 9. This con- of increasing thickness for R=1, 5, 10 and 15 small particle
vergence indicates a correlation length-scale between 10 diametersforbothsoftparticles(top)inanunloadedsilo(•)
and 15 particle diameters — approximately the same as and with overloads of 20 g ((cid:4)) and 80 g (N) and for hard
the Janssen screening length-scale obtained numerically. particles (bottom) with overloads of 1837 g (•), 2267 g ((cid:4))
Thisisinterestingsinceinthisinstance,themacroscopic and 3402 g (N).
scale of the Janssen parameter λ is comparable to the
averagingscale,eventhoughtheJanssenmodeldoesnot
include particle size. As can be seen for the lower image between two particles must be equal to the sum of their
in Fig. 14, the the effect of averaging length is roughly radii. Thisallowsustodeterminealocalpackingfraction
the same regardless of particle stiffness. throughout the silo.
Anadditionalaspectofinterestisthegeometricnature Since previous studies have shown that the packing
of the contacts. Relevant quantities include the packing fraction of a pile relates to the redirection parameter,
fraction, φ, the number of contacts per particle, Z, and weinvestigatedepth-dependentvariationsofthepacking
clustering coefficient, defined below. fraction [8]. Due to the dilation of the material, we find
With our second preparation procedure, we are able thatthe packingfractionis slightlylowerforloadedsilos
to obtain to obtain all of these quantities. We do this for the depths availablein the unloadedcase. We do not
by capturing both polarized images showing the force find systematic correlation between the packing fraction
network within the photoelastic disks and unpolarized and the screening length-scale.
images showing the locations of those particles. Particle Theparticlelocationsandcontactscompriseacontact
centers can be identified using a convolution with sep- network for each pile. Analyzing these contact networks
arate kernels for the two particle sizes. We then use a allowsthedeterminationofgraphandnetworkproperties
numerical algorithm to reject overlapping particles and such as the average number of contacts per particle, Z.
identify neighbor particles in contact with each particle We can also determine the ‘clustering coefficient’, which
baseduponthegeometricconsiderationthatthedistance gives a measure of how tightly interconnected is the net-
10
Contact Network Force Network
1
5 5
4 4
0.9
φ
n Z 3 3 S
o o
acti 0.8 cle 2 2 ft
g Fr 0.7 Parti 1 1
kin er 00 2 4 6 00 2 4 6
c
a P
P 0.6 s
ct 5 5
a
0.5 nt 4 4
0 2 4 6 o
C
3 3
Depth (L) H
n
1 Mea 2 2 ard
1 1
0.9 0 0
φ 0 2 4 6 0 2 4 6
n Depth (L)
o
cti 0.8
a
Fr FIG. 16: The average number of contacts per particle is
g 0.7 roughly constant for the contact network (left) but varies
n
ki at the top of the pile for the force network (right). This is
c true for both soft particles (above) in an unloaded silo (•)
a
P 0.6 and with overloads of 20 g ((cid:4)) and 80 g (N) as a well as for
hard particles (below) with overloads of 1837 g (•), 2267 g
((cid:4)) and 3402 g (N).
0.5
0 2 4 6
Depth (L)
material, although slightly lower due to the presence of
FIG. 15: The average packing fraction for soft (left) and the walls [34]. The average number of contacts in the
hard (right) particles varies slightly depending upon over- force network is slightly less than to two, which is not
load. Whileitisroughlyconstantwithdepthforanunloaded surprising since many particles in a chain have only two
silo (•) it varies for overloads of 20 g ((cid:4)) and 80 g (N). contacts and, in our relative small silo, many particles
make contact with the walls.
Significantly, for the force network within both the
work [32]. For each particle we consider the n neighbor- hard and soft particles, the average number of contacts
ingparticleswithwhichthatparticleisincontact. There at the top of the pile varies depending upon the over-
are n−1 possible neighbor-neighbor contacts for round load, before converging with depth, unlike the contact
particles in two dimensions, so we define the clustering network which does not vary with overload. This indi-
coefficient as the ratio of the m actual contacts present cates that the force network is quantitatively different,
between those neighbors to the n−1 possible contacts, depending uponthe size ofthe appliedoverload. For the
C ≡ m . deep-penetratingoverloadonthesoftparticles,themean
n−1
For each contact network, we can determine a subset numberofcontactsatthe topisclosesttothesaturation
ofcontactsforwhichtheparticle-scalepressureisgreater value, perhaps indicating that the structure of the force
thanthemeanforthewholesilo. Thissubsetofcontacts network is made visible by the additional pressure, but
canbethoughtofasroughlycomprisingaforcenetwork, is not disturbed as in the case of larger overloads.
for which the same graph and network properties as the Fortheclusteringcoefficient,Fig.17showsthecontact
contact network can also be calculated [33]. network to be better connected than the force network.
We first examine the dependence of the mean number Aswiththenumberofcontacts,theclusteringcoefficient
of contacts per particle with depth for both the over- is roughly constant for the contact network but varies
all contact network and the force network. As shown in with depth for the force network. For hardparticles, the
Fig.16,wefindthatforthecontactnetwork,thenumber overloads do not seem to alter the clustering coefficient
of contacts is nearly independent of overload and con- of the contact or force network.
stant with depth at a value close to k = 4, the typically Thedifferencesbetweenthecontactandforcenetworks
observed coordination number for a disordered granular may indicate an explanation for the deep propagation