Simple models for granular force networks

0.41 MB·English

by  John F. Wambaugh

Simple models for granular force networks
∗
John F. Wambaugh
Departments of Physics and Computer Science and Center for
Nonlinear and Complex Systems Duke University, Durham, NC 27708†
Aremarkablefeatureofstaticgranularmatteristhedistributionofforcealongintricatenetworks.
Even regular inter-particle contact networks produce wildly inhomogeneous force networks where
8 certain “chains” of particles carry forces far larger than the mean. In this paper, we briefly review
0 past theoretical approaches to understanding the geometry of force networks. We then investigate
0 thestructureofexperimentally-obtainedgranularforcenetworksusingasimplealgorithmtoobtain
2 correspondinggraphs. Wecompareourobservationswiththeresultsofgeometricmodels,including
n random bond percolation, which show similar spatial distributions without enforcing vector force
a balance. Our findings suggest that some aspects of the mean geometry of granular force networks
J may be captured bythese simple descriptions.

4
2 PACSnumbers: 81.05.Rm,05.10.-a,89.75.Hc,89.75.Da

]
t I. INTRODUCTION materials exhibit self-organized criticality, in which the
f
o forcenetworkbehavesasanorderparameterinresponse
s to displacements of the material [2].
. Dense granular materials are composed of many par-
t For any configuration of grains (e.g., Fig. 1.1), if the
a ticles interacting through multiple, persistent contacts.
m Without knowingthe historyofaparticulargranularas- position and shape of each grain is known then a con-
tact network can be generated describing the contacts
- sembly, the mobilization of the friction of each contact
d is indeterminate [1]. Even without friction, the simplest between the grains. The equations describing a granu-
n lar material can be grouped into the known equations
granular materials, such as packings of macroscopically
o describing the behavior of the individual grains and the
uniformspheres,canstilldisplayhighlynon-uniformnet-
c unknown equations describing the contacts between the
[ works of contacts — any difference in the size of parti-
cles, including microscopic aberrations on the surface of grains as a function of material assembly and history.
4 These unknown equations can be cast as undetermined
“smooth” spheres, is sufficient to create a complicated
v relationshipbetween inter-grainforcesand the geometry
contact network [2].
4
of the contact network [8].
1 Experiments show that for any given particle within
Attempts to resolve the indeterminacy of the consti-
3 a granular material, the fluctuations in the magnitude
tutive relationshave focused upon statisticalapproaches
3 of the force carried by that particle can be extremely
0 similar to thermodynamics. Ideally, it might be possi-
large [3, 4]. This is the result of long-range force corre-
6 ble to determine aggregate properties of granular mate-
lations that can be characterized as a force network in
0 rials that do not depend upon knowing the specifics of
which some bonds of the contact network carry forces
t/ much larger than the mean force on all contacts. As in the grains to arbitrary precision. Unfortunately there
a
aretwoproblemsthat havemade suchanapproachnon-
m Fig. 1.2, forces propagate along roughly linear “chains”
trivial. First, a granular material at rest is only in a
ofcontacts. Becausetheforcenetworkdependsuponthe
- quasi-equilibrium. An arbitrarily small addition of en-
d contactnetwork,the propagationdirectionand distribu-
ergycancauseirreversible(plastic)rearrangementofthe
n tion of the chains depends strongly upon the history of
o the granular material [5, 6]. material. This is in part due to the second problem,
c the lackofananalogto temperature to“thermalize”the
The overall distribution of forces in a force network
: system. The actual energy due to temperature is far too
v is roughly exponential, indicating the possibility for ex-
i small to move grains and, in the static case, there is no
X tremely large forces. Since tangential forces are sup-
other quantity allowing the material to explore phase-
ported only by friction, their distribution falls mono-
r space for lower energy states.
a tonically, while forces along the chains are peaked at a
Despitethesecomplications,thereisarichtraditionof
mean value before falling exponentially [7]. As grains
modelstounderstandboththedistributionofmagnitude
are added, if the force networkcannot support the gran-
andthespatialcorrelationofforceswithingranularmat-
ular material, the material will rearrange until the new
ter[9,10,11]. Theapplicabilityofthesemodelstoactual
contact and force network supports the granular mate-
granular materials has been limited in part by being re-
rial. For this reason it has been suggested that granular
stricted to regular lattices of grains. Recent numerical
work [12, 13, 14], however, suggests that the spatial dis-
tributionofforcewithinirregulararrangementsofgrains
might indeed be characterized through comparison with
∗Currentaddress: National Center forComputational Toxicology,
models with regular geometries.
USEPA,ResearchTrianglePark,NC27711
†Electronicaddress: [email protected] Typically the force network is determined using a
2
β
γ
q 1−q α
FIG.2: LeftIn thetwo-dimensional q-Model, a fraction q of
thetotalweightonaparticleonaregularlatticeistransferred
toonelower neighbor, and theremaining 1−q istransferred
to the other. As originally formulated by Coppersmith, et
al., the values of q can be drawn randomly from different
distributions. In the “critical” case, the value of q is limited
to either 0 or 1. Right To make rough analogy to the q-
Model for more complicated geometries, we use a geometry-
basedq-Modelinwhich thefraction ofavertex’sweightthat
isdistributedalongadownwardpointingedgeisproportional
to thesine of the angle that edge forms with the vertex.
ryingonlysmallamountsofforce,providerigiditytothe
network.
Therehavebeenmanyattemptstomodelthebehavior
of force networks, most famously the “q-Model” of Cop-
persmith (1996). In the two-dimensional q-Model, de-
picted in Fig. 2, granular materials are represented as a
latticeofsiteswithtwoupwardneighborsandtwodown-
wardneighbors. Eachsite has an intrinsic weight that is
addedtoanyweightitreceivesfromtheneighborsabove
it. The total weight on a site is distributed between the
two downward neighbors with one receiving a fraction q
FIG.1: Analysisofthetwo-dimensionalgranularassemblyof of the total weight and the other receiving 1−q. The
bi-dispersepolymerdisksdepictedin1. Thecontactnetwork q-Model allows the determination of the distribution of
for the disks can be thought as a series of vertices centered weight by working downwards from the top of the lat-
at each disk connected by edges wherever two disks are in tice. This process can be shown to be diffusive-like in
contact. 2 Because the disks are photoelastic, when they are the continuum limit [9].
illuminated between crossed polarizers bright fringes appear
Diffusivesystems,however,donotallowforlong-range
in disks understress allowing force chains to be clearly seen.
propagatingforces like those seen in force networks. Ex-
A load has been applied to the top of the pile causing the
perimentsonsmallsystemshaveshownthatevenaverag-
creation of additional contacts and force chains supporting
the load. 3 We examine the gradient of the stress image as ingovermultipleforcechainnetworksdoesnotsupporta
arough measureof thenumberof fringes ineach disk,which diffusive descriptionofgranularmaterials[16]. However,
is proportional to the applied force. 4 Finally, we apply an inthecasewhereq isallowedtoonlytakethevaluesof1
algorithmtothegradientimagetofindagraphdescribingthe or 0, the q-Model does show chain-like clusters of bonds.
force network. Thiscasewillbereferredtoasthe“criticalq-Model”[9].
In all cases the q-model is a scalar model in that the
distribution of weight,not force,is being described. The
threshold — for instance, only bonds carrying forces forces exerted upon a particular lattice site do not need
larger than some value, often the mean force, are con- to balance, and this in part explains the lack of chain-
sidered to be part of the force network. In numerical like structures in all but the critical case. Vector models
simulations, critical-like transitions from a sparsely dis- that attempt to include force balance also exist. One of
tributednetworkofforcechainstoapercolatingnetwork the simplest lattice models that includes force balance is
have been observed as the threshold criteria for includ- the“tripod”modelthat,intwo-dimensions,assignseach
ing bonds is varied. In one case, the magnitude of the site three downward neighbors, one directly below and
inter-grain mobilization of friction [15] was used and in two offset to either side by a fixed angle. In the con-
another that magnitude of inter-grain forces [12]. It is tinuum limit this model can give wave-like equations for
important to note that these thresholded force networks propagating stress [10]. Additionally, other models have
will not be rigid, since they lack bonds that, while car- included torque-balance on each site, giving meaning to
3
of edges representing the connections between the lat-
tice sites with the restriction that each lattice site has
only one load-bearing downward bond. The set of load-
bearingbondscanbethoughtofasasubsetofedgesthat,
with the vertices as the lattice sites, forms a sub-graph
of the contact network. Since weight at a given vertex
is equal to one plus the size of the tree rooted at that
vertex, the distribution of cluster sizes gives bounds on
the distribution of weights
In disordered granular materials the network of inter-
grain contacts may perhaps serve in place of a fixed lat-
tice for percolation. Taking the contact network to be
a graph, then the force network may be thought of as
thesub-graphconsistingofedgeswithgreaterthansome
threshold force.
FIG. 3: In our experimental silo for observing granular force Properly selecting the edges that comprise the force
networksbi-dispersedisksmadeofaphotoelasticpolymerare networkrequiresconsiderationofvectorforcebalancefor
confined vertically to two-dimensions between clear plastic each particular network. However, recent work has sug-
sheetsandaluminumspacersservingassidewalls. Thegrains
gested that granular force networks belong to a specific
rest upon a piston that is lowered slightly to make uniform
universalityclassthatcanbecharacterizedthroughanal-
the mobilization of friction at the walls. The entire silo is
ogy to random bond percolation, allowing the distribu-
illuminated between crossed polarizers, which allows fringes
tionsofforcesinawiderangeofisotropicandanisotropic
induced by applied stress on each grain to be imaged with a
granular systems to be scaled onto a single distribution
CCD camera.
[12, 13, 14].
In this paper we make experimental observations of
the finite extent of the grains [11]. Including both force the force networks within a granular system and use a
and torque balance allows derivation in simple cases for simple approach to extract the statistical distributions
equationsdescribingforceequilibriumasfunctionsofthe of graph properties for these force networks. We charac-
contact distribution [17]. terizeforcenetworksbothintermsofthegraph-theoretic
properties of degree distribution and cluster-size,as well
as q-model inspired properties, including the general-
II. GEOMETRIC APPROACHES TO ized q-model weight and the size of the cluster resting
GRANULAR MATTER on a given grain. We then compare our experimentally-
observedgraphswith those generatedby simple geomet-
ric models to determine in what ways simple statistical
Research into the general behavior of force networks
approaches do and do not describe the magnitude and
in granular matter has shown that certain properties,
spatial distribution of forces within dense granular mat-
including the exponential distribution of large forces,
ter.
should be expected to arise regardless of the grain-scale
model. In this case, some properties of granular materi-
alsmaybephenomenaofthestatisticsofforcenetworks,
and not the micro-mechanics of the grains [18, 19]. III. METHODOLOGY
This insight has led to approaches to granular mate-
rials from the mesoscopic level. By assuming the pres- We use the experimental setup depicted in Fig. 3 to
enceofforcechainsanddescribingtheirbehavior,itmay observe force networks within actual granular matter.
be possible to avoid some of the complexity of granular We confine 650 bi-disperse (one quarter diameter 0.9
materialsandstill achievemeaningfulresults[20,21]. In cm and the rest 0.75 cm) photoelastic polymer disks
particular,networkapproachestogranularmaterialsmay within a vertical, two-dimensional silo. When illumi-
bear fruit because the statistical mechanics of networks nated between crossed polarizers, photoelastic materials
in general are beginning to be understood [22]. show bright fringes of light in proportion to the stress
The critical q-model is similar to random bond perco- applied to the material. By taking the gradient of an
lation on a lattice. Percolation theory provides a frame- imageof aphotoelastic disk alongmultiple directions we
work for understanding how small clusters of connected effectively “count” the number of fringes and, using an
sites aggregate into larger clusters as a function of the empirical, linear calibration, can obtain the force distri-
probabilitythatabondisselected. Therearewellknown bution throughout the granular material (see Fig. 1.3).
values, depending upon the geometry of the lattice, for Using the gradient image of a particle force network
transitions from loose aggregates of clusters to system- we convolvewith an image representing a single stressed
spanning, “infinite” clusters [23]. In the case of the crit- grain to locate the positions of all stressed grains in the
ical q-model, bonds are selected at random from a set image. We then assign an edge between all vertices sep-
4
1
0.35
0.3
0.25
2 1
0.2
)
k
2 (
3 1.1 P0.15
0.1
0.05
1.6
3.5
0
0 1 2 3 4 5
Vertex Degree k
3.8
10 FIG.5: Thedistributionofvertexdegreepeaksatk=2. The
lack of asharp uptickat k=5indicates that truncatingat 5
has not significantly affected the distribution. These results
FIG. 4: Illustration of the graph quantities measured. The
arefromanalyzingonehundredexperimentally-obtainedforce
cluster size s = 10 includes all the vertices. The size of the
networks under56g load.
upward tree rooted at thewhite circle (indicated bydirected
edges), t=4includesall vertices connectedbyupward edges
only. Thedegreeofthewhitenodeisk=3. Thenumbersin-
wemeasurethesizeoftheupwardtreerootedateachver-
dicateapproximategeometricq-Modelweightsw,determined
foreachvertexbyassigningeachaninherentunitweightand tex. The upward tree is a graph made by treating each
distributingtheweightamongalldownwardedgesinpropor- edgeasadirectededgeorientedagainstgravity. Thesize
tiontothesineoftheangleeachedgemakeswiththevertex. ofthe upwardtreecorrespondstothe number ofvertices
thatareatleastpartiallysupportedbyagivenvertex. As
illustratedinFig.2,weightscanbe assignedto eachver-
arated by a distance in the range of l/4 to l where l is tex and propagated along each downward pointing edge
a manually-adjusted parameter. The lower range is nec- in proportionto the cosine of the angle each edge makes
essary because some fringe patterns within a single disk with the vertex. This allows for a q-Model-like weight
produce multiple points when convolved, which has the distribution to be generated for non-lattice geometries.
effect of artificially increases clustering of the graph. For all of these quantities an aggregate probability den-
sity function is calculated.
Adding an edge to any grain that already has the
threshold value k edges causes the longest edge to
max
that vertex to be removed. We use the smallest k
max
IV. RESULTS
that does not significantly alter P(k).
The specific values of the network-finding parameters
A. Experimental Results
useddependsuponmanypropertiesoftheimagesofforce
networks, including the image resolution and dispersion
ofthedisks. Theseparameters,inadditiontothecorrela- We analyzed one hundred force networks obtained
tionthresholdforlocatingvertices,mustbetunedtopro- fromexperimentalimagesofgrainsinaverticalsilosub-
duce force networkgraphs that accuratelydescribe force ject to a 56 g overload that created an approximately
networks. We assigned edges within the range l = 13 uniform distribution of force chains, as in Fig. 1.2. Fig-
pixelsandweallowedamaximumofk =5edges. As ure5showsthe distributionofvertexdegreesand,asex-
max
discussed below, the distributions reported here do not pected for chain-like structures,vertices most often have
changeunexpectedlywithvariationofcorrelationthresh- twoedges. Fromgeometricconsiderations,themaximum
old. The set ofverticesandedges foundthis wayarethe number ofcontactsfor asingle disk ina bi-dispersemix-
experimentally obtained force network (see Fig. 1.4). ture of disks with radii rsmall and rlarge is:
Using ensembles of force networks we can determine
k =π/2/arcsin[r /2/(r +r )] (1)
max small small large
theprobabilitydistributionfunctionfortheaveragenum-
ber of contacts per vertex and number distribution of In this case k = 6.85 and there is no probability
max
cluster sizes. We can also record the observed force at of seeing a vertex with degree higher than six. Thus,
each vertex and create an ensemble probability distribu- as Fig. 5 shows, the imposed cutoff of k = 5 used
max
tion for force. in the edge finding routine does not significantly affect
Figure 4 illustrates several graph quantities that we the distribution of P(k) since fewer than a tenth of all
measure. In addition to cluster size and vertex degree, vertices have degree larger than 4.
5
−1 −1
10 10
ns nt
−3 −3
10 10
−5 −5
10 10
0 1 2 3 0 1 2 3
10 10 10 10 10 10 10 10
Cluster Size s Upward Tree Size t
FIG.6: Thedistributionofns,thenumberofclustersofsizes FIG. 7: The logarithmically-binned distribution of nt, the
dividedbythetotalnumberofclusters,appearstoberoughly numberofupwardclustersofsizetdividedbythetotalnum-
linear over many decades of probability and two decades of ber of such clusters, appears linear on a log-log plot, similar
cluster-size (corresponding to the system size) when binned totheclusterdistributioninFig.6butwithashallowerslope.
logarithmically. A power-law distribution would support the
ideaofRouxetal. [2]thattheforcenetworkmayperhapsbe
considered a critical phenomenon.
0
10
The distribution of the number of observed clusters
of size s (as a fraction of the total number of clusters)
appears to follow a power law, as indicated by Fig. 6. −2
)10
This behavior lends support to the suggestion of Roux, F
(
et al. [2] that the force network may be an example of P
a self-organized critical phenomena. Fits were obtained
−4
forregressionstothedata(neglectingtheendpoints)for 10
a power law (n = a sa2), the product of a power law
s 1
and an exponential (n = b sb2eb3s), and a log-normal
s 1
(ns = cs1ec2(s−c3)2) distribution. In all cases, the ra- 10−6 −2 0 2
tio of the variance predicted by the regression to the 10 10 10
observed variance (R2) is greater than 0.95, indicating Force F/<F>
likelyfits[24]. Thefitsforthepowerlaw-exponentialand
log-normaldistributions,however,bothproducepositive FIG. 8: The force observed at each vertex computed from
curvatureindicatingincreasinglikelihoodwithverylarge thegradient isnon-monotonicandthegreatest probabilityis
clustersize. Webelievesuchdistributionsareunphysical for forces below the mean force in the network (indicated by
sinceincreasingprobabilitywithincreasingclustersizeis F/hFi=100).
not normalizable, so that a power law with an exponent
a =−1.6 seems most likely.
2
As shown in Fig. 7, the number distribution of up- mate force probability distribution. This non-monotonic
ward clusters is shallower (a = −1.0) than for whole distribution peaks below the mean force and then falls
2
clusters but still appears to be linear. We expect the off faster than a power law.
distribution of upward trees to generally be shallower Wethencomparetheactualforcedistributiontowhat
than the distribution of clusters since each large cluster wouldbe predictedusingourgeometricq-Model,obtain-
contains many upward trees. Regressions to power law, ing the distribution in Fig. 9. The q-Model weights are
power law-exponential, and log-normal distributions all sub-power law but nearly monotonic, unlike the actual
have R2 > 0.85, but only the fits for the power law- force data.
exponential distribution give the observed drop-off with Thegeometriclimitof6possiblecontactsperdiskcor-
large upward tree size. responds to 6 potential edges per vertex, however the
Figure 8 shows the probability distribution of inten- average number of contacts (coordination number) for
sity of the gradient squared image at the pixel location a disordered granular material is 4 [2]. Assuming that
of each vertex. The intensity corresponds with the force there are four contacts per vertex, we can calculate a
on the disk at that vertex and thus Fig. 8 is an approxi- chance p that a bond carrying a large force is present.
6
0.4
0
10
0.3
−2
)10 )
P(w P(k0.2
−4
10
0.1
10−6 0
−2 0 2 0 1 2 3 4 5
10 10 10
Vertex Degree k
"q−Model" Weight w/<w>
FIG. 9: The probability density function of weight assigned
toverticesintheexperimentally-obtainedforcenetworksbya
−1
method similar totheq-Model. Unliketheactual force data, 10
thecurveis monotonic except for a slight dip.
s
n Increasing
−3
10 Threshold
Foraparticularvertexi,p =k /4andtheaveragevalue
i i
p≡PNi=1pi/650=PNi=1ki/4/650, where N is the total
numberofverticesintheforcenetworkgraph,and650is
thetotalnumberofparticles. Fortheobservednetworks,
−5
p=0.2214. 10
0 1 2 3
10 10 10 10
Additionally, since we expect 4 contacts on average, Cluster Size s
Fig. 5 indicates that grains in the force network are typ-
ically in contact with one or two grains not in the force FIG. 10: Top Distribution of vertex degree for experimen-
network. tally obtained force networks as a function of acceptability
threshold for convolution with our fringe kernel. The dashed
Foreachvertexwecanalsocomputetheclusteringco- line indicates the distribution for the acceptability threshold
efficient—howmanyedgesaresharedbytheneighboring of0.5usedfortheanalysisinthispaper. Loweringthethresh-
vertices(cid:0)4(cid:1)possibleedges[25]. Unlikearandomnetwork oldfrom0.6(thickestline)to0.4(thinnestline)inincrements
2
whereanyvertexmaybecontactwithanunlimitednum- of0.05increasesthenumberofvertices“found”andskewsthe
ber of other vertices at any position, the choice of our distribution away from chain-like structures towards higher
vertex degree. Bottom Despite the change in vertex degree
radii and the quasi-two-dimensionalgeometryof our silo
distribution, the distribution of cluster size still indicates a
restricts the number of neighbor-neighbor contacts be-
power law with the same slope for all thresholds considered.
tween n neighbors to n−1. We calculate c=0.1219 for
The dashed line again indicates 0.5.
the observed force network. This amount of clustering
is similar to that seen in scale-free networksthat exhibit
power law behavior, such as the World-Wide Web and
B. Comparison with Spatial Models
power grids [26], although since we have restricted the
number of neighbor-neighbor contacts this is not a per-
Havingcharacterizedactualforcenetworks,itispossi-
fect analogy.
ble to make comparisonsto simple models. Using anen-
Finally, we examine the robustness of our results. To sembleofrealizationsofthecriticalq-Modelona10×40
obtain force network graphs from our experimental data vertex triangular grid (chosen to reflect the aspect ratio
we convolved our images with a single fringe image ker- and roughly the number of vertices in our experiments)
nel that does not necessarily match all arrangements of as in Fig. 11.1, it was possible to determine distribution
fringes. We used an acceptability threshold of 0.5, on a functions for vertexdegree,cluster size, upwardtree size
scale where 1.0 is a perfect match and 0.0 is the oppo- and weights assigned by the q-Model.
site of the kernel. As indicated by Fig. 10, varying the In addition to the q-Model results, graphs were also
acceptability threshold between 0.4 and 0.6 only slightly constructed via bond percolation. Two regular trian-
affects properties like vertex degree distribution without gular lattices were considered, one where in addition
significantly changing the cluster size power law. to downward-angled edges there were horizontal edges
7
0.5
0.4
0.3
)
k
(
P
0.2
0.1
0
0 1 2 3 4 5 6
Vertex Degree k
FIG. 12: The probability density function for degree vertex
for the critical q-Model and random triangular graphs. The
critical q-Modelisindicated byadashedline. Thebond per-
colation results for the experimentally obtained p = 0.2214
are indicated by (cid:3). Other bond probabilities, p = 0.15, 0.3
and0.5,arerespectivelyindicatedbylinesofincreasingthick-
ness. While the q-Model by design gives the expected peak
at k = 2 — indicating chain-like structures — since in the
critical case of the q-Model k ≤ 3, the best match to Fig. 5
FIG. 11: From left to right: 1 An instance of the q-Modelin appears to be p=3.
thecriticalcase. 2Arandomnetwork(p=0.25)composedof
edgesfromatriangularlatticeorientedhorizontally. 3Aran-
dom network(p=0.25) composed ofedges from atriangular
peak appropriately at k = 2, but instead at k = 1 in-
lattice oriented vertically.
dicating many pairs of bonds but no clusters. Instead,
the peak of the distribution for p = 0.3 is most like the
experimentally-observed distribution. The distribution
(see Fig. 11.2), and one where there were vertical edges
for p = 0.3 over-represents single bonds when compared
(Fig. 11.3). For each realization a subset of these edges
with experiments, but the distribution of larger k values
were chosen with a probability p to create a randomized
matches well.
graph. Withsixpossibleedgespervertex,thetriangular
latticeissimilartothebi-dispersesystemthoughangular Also plotted inFig.12 is the distributionfor p=0.15,
distributions are severely restricted. below the measured p and not agreeing with the exper-
A range of different p values were examined for both imentally measured distributions. Finally, the distribu-
triangular orientations. In addition to the p = 0.2214 tion for p=0.5,shows too much skew towardlarge clus-
thatwasdeterminedexperimentally,weexaminedvalues ters.
of p = 0.15 and p = 0.3 above and below the estimated As indicated in Fig. 13, the q-Model gives a distribu-
p. Finally,weexaminedp=0.5,avaluegreaterthanthe tion (with logarithmic binning) of cluster size that falls
percolationthreshold,pc =0.3473,forasystem-spanning linearly(a2 =−1.53)onalog-logplotuntilbroadlypeak-
cluster in an infinite triangular lattice. ingatroughlyaquarterofthesystemsize(100vertices).
We generated 5000 realizations each for the q-model Because we have chosen the critical case of the q-model
andthe twotriangularlattice orientations. Forthe bond toensurelongchainsofcontacts,itisnotsurprisingthat
percolationgraphsbothvertexdegreeandclustersizeare we see more large clusters. In contrast, the cluster size
independentofwhetherthe basisisorientedhorizontally distributions for graphs constructed from triangular lat-
or vertically and so the joint ensemble of 10000 realiza- ticesshowninFig.14aremonotonicandroughlylinearly,
tions was analyzed. with decreasing slope as p is increased until the critical
The distributions of vertex degree for the critical q- thresholdis crossed,after which the slope for small clus-
Model and triangular lattices with different bond prob- ter sizes increases and a sharp peak at the size of the
abilities are shown in Fig. 12. Since the q-Model has system appears, indicating infinite clusters (similar “bi-
to give chain like structures, it is not surprising to see modal”distributionswereobservedpreviouslybondper-
that P(k) peaks at k =2, indicating that most elements colation simulations of much larger (105) systems [27]).
are members of a single chain. For a random graph con- The slope for p=0.3 is found to be most similar to that
structedwiththebondprobabilityp=0.2214calculated of the experimentally observed networks: a = −1.77
2
from our experimental data, the distribution does not with R2 ≈0.97 compared to a =−1.6 for experiment.
2
8
−1 −1
10 10
s s
n n
−3 −3
10 10
−5 −5
10 10
0 1 2 3 0 1 2 3
10 10 10 10 10 10 10 10
Cluster Size s Cluster Size s
FIG. 13: The distribution of number of clusters of size s for FIG. 14: The distribution of the number of clusters of size s
the critical q-Model. On a log-log plot the distribution of for random triangular graphs. The bond percolation results
smaller clusters is roughly linear (dashed line indicates slope for the experimentally obtained p = 0.2214 are indicated by
−1.53 found by fitting to linear region), but there is a peak (cid:3). Other bond probabilities, p = 0.15, 0.3 and 0.5, are re-
at large cluster sizes since the critical q-Model requires that spectively indicated by lines of increasing thickness. For all
each site has a downward neighbor. cases there is roughly linear behavior for small to moderate
(∼ 100) cluster sizes. The transition value to universal clus-
ters is pc = 0.3473 for an infinite triangular lattice and as
expected the slope varies non-monotonically, decreasing as p
Figure 15 shows the distribution of “weight” assigned
isincreaseduntilabruptlydroppingatp=0.5whereasecond
via the q-model for the critical q-Model graphs. The
peak at large cluster size indicates that infinite clusters are
model distribution is more skewed toward large weights
forming.
than the q-Model-like weight assigned to the experimen-
tallyobtainedgraphs. Thisisunsurprisingsince,bycon-
struction,thecriticalq-Modelisaconnectedgraph. The when there are vertical edges the distribution of cluster
distribution of upward trees in Fig. 16 is roughly linear, sizes is broader. This is reasonable since the addition of
with nearly the same slope (a2 =−1.52) as cluster size. vertical edges makes large trees more likely.
InFig.17,wehavethedistributionofweightsassigned
by the geometric q-Model. None of the models showed
the experimentally observed peak for actual force found V. CONCLUSION
in Fig. 8. Graphs composed from edges in vertically-
aligned lattices — where an additional downward edge Wehavecharacterizedthegraphsgeneratedfromana-
is possible — showed slightly broader distributions than lyzing experimentally-obtained force networks in granu-
those that were composed from horizontally-aligned tri- lar material. Assuming that there are an averageof four
angular lattices. In the vertically-aligned lattices, the contactspergrain,wecalculatethattheprobabilitythat
verticeswithweightbelowthemeanaredistributedmore a contact carries a large force is p=0.2214. We observe
broadlythaninthehorizontalcase. Forbothalignments, evidenceforapowerlawscalingofclustersofbondswith
the distribution for p = 0.5 was the most similar to the largeforce,possiblyindicatingasimply-characterizedun-
distribution for graphs obtained from experimental data derlying geometry.
— this values is much largerthan the p measuredexper- When we compare ourexperimentalnetworksto bond
imentally. None of these distributions can be said to be percolation using random sets of edges from triangular
a good match for the force or q-Model distributions for lattices, we find that geometric properties of force net-
experimentally obtained graphs. workscanindeedbematcheddependingupontheproba-
As Fig. 18 shows, the distributions of upward trees bilityofthepresenceofabondp. Wefindthatwecannot
for graphs below the percolation threshold are simi- match all properties of the experimental networks using
lar. For p = 0.3 and below, the distributions are well- any one bond probability, including p = 0.2214. How-
describedbytheproductofapowerlawandanexponen- ever,the valueofp=0.3workedwellforboththevertex
tial(R2 >0.98)howeverthecoefficientsareverydifferent degree and cluster size distribution.
fromthe experimentally-observeddistributionofupward Aslightlylargervalueofpmaybeanartifactofunder-
trees. Forp=0.5thedistributionisqualitativelysimilar samplingthenumberofminimallystressedparticlesthat
totheexperimentaldistributioninFig.7,butisnotwell- are hard to detect visually but do contribute stressed
described by any of the distributions considered (power bonds. Itispossiblethatfurtherrefinementofthegraph
law, log-normal and power law-exponential). Generally, finding algorithm is needed to get the correct value of p
9
distributions require more elaborate, vector analysis.
100 100 a 100 b
) 10−2 10−2
w
(
P
)10−2 10−4 10−4
w
(
P 10−6 10−6
10−2 100 102 10−2 100 102
−4 " q−Model" Weight w/<w>
10
FIG.17: Probabilitydistributionofgeometricq-Modelweight
for graphs assembled from horizontally- (a) and vertically-
−6
10 (b) aligned triangular lattices. The results for the experi-
−2 0 2
10 10 10 mentally obtained p = 0.2214 are indicated by (cid:3). Other
q−Model Weight w/<w>
bond probabilities, p = 0.15, 0.3 and 0.5, are respectively
indicated by increasing line thickness. The two alignments
FIG. 15: Probability distribution of weight assigned by the differinthedistributionoflowerweights. Inbothcases,only
q-Modelforthecriticalq-Modelgraphs. Becausetherearefar thelargest, p=0.5, approaches thedistribution observed for
more vertices with weights below the mean, the distribution experimentally-obtained networks (Fig. 9).
ofq-Modelweightsisdifferentfromboththeactualforceand
geometricq-Modelweightdistributionsforexperimentallyob-
tained force networks.
a b
10−1 10−1
nt10−3 10−3
−1
10
10−5 10−5
100 102 100 102

Upward Tree Size t
nt
−3
10 FIG. 18: Distribution of numberof upward trees of size t for
bondpercolationgraphsassembledfromhorizontally-(a)and
vertically- (b) aligned triangular lattices. The results for the
experimentallyobtainedp=0.2214areindicatedby(cid:3). Other
bond probabilities, p = 0.15, 0.3 and 0.5, are respectively
10−5 indicated by increasing line thickness. Only when infinite
100 101 102 103 clusters are present (p = 0.5) do the distributions change
Upward Tree Size t significantly,in both cases giving adistribution that islinear
overthe same ranges as observed experimentally (Fig. 7).
FIG. 16: Number distribution of upward tree size for the
critical q-Model.
Acknowledgements
for actual force networks.
Becauseofobvioussimilarities,wehavemadecompar- I thank John Reif for suggesting studying the applica-
isons with the q-Model and attempted to assign weight tionsofgraphtheorytoforcenetworks;XiaobaiSunand
to, and determine “force” distribution of, vertices in Nikos Pitsianis for computer science advising; Richard
both q-Model and bond percolation graphs. None of Palmer,JoshSocolar,BrianTigheandTrushMajmudar
the methods studied provide good agreement with the for helpful conversations; and my Ph.D. advisor Robert
experimentally-obtained force distribution. This indi- Behringer. ThisresearchwasfundedbyNationalScience
catesthatthoughgraphtheoreticapproachesmayinfact FoundationgrantsDMR-0137119andDMS-0204677and
capture the geometry of force networks, the actual force NASA grant NNC04GB08G.
[1] T. C. Halsey and D. Erta¸s, Physical Review Letters 83, [2] S. Roux and A. Hansen, in Powders and Grains, edited
5007 (1999). by J. Biarez and R.Gourv´es (1989), pp.249–254.
10
[3] G. W. Baxter, in Powders and Grains, edited by R. P. [16] J. Geng,G. Reydellet,E.Cl´ement,and R.P.Behringer,
Behringer and J. T. Jenkins (1997), pp.345–348. Physica D 182, 274 (2003).
[4] P.Dantu,G´eotechnique 18, 50 (1968). [17] S.F.EdwardsandD.V.Grinev,PhysicalReviewLetters
[5] J.Geng, D.Howell, E.Longhi, R.P.Behringer, G. Rey- 82, 5397 (1999).
dellet, L. Vanel, E. Cl´ement, and S. Luding, Physical [18] K. Bagi, in Powders and Grains, edited by R. P.
ReviewLetters 87, 035506 (2001). Behringer and J. T. Jenkins (1997), pp.251–254.
[6] F.Radjai,D.Wolf,M.Jean,S.Roux,andJ.Moreau,in [19] J.D.Goddard,InternationalJournalofSolidsandStruc-
PowdersandGrains,editedbyR.P.BehringerandJ.T. tures 41, 5851 (2004).
Jenkins(1997), pp.211–214. [20] J. E. S. Socolar, D. G. Schaeffer, and P. Claudin, Euro-
[7] T. S. Majmudar and R. P. Behringer, Nature 435, 1079 pean Physical Journal E 7, 353 (2002).
(2005). [21] J.H.Snoeijer,T.J.H.Vlugt,M.vanHecke,andW.van
[8] S. F. Edwards and D. V. Grinev, Physica A 294, 545 Saarloos, Physical Review Letters 92, 054302 (2004).
(2001). [22] J. Park and M. E. J. Newman, Physical Review E 70,
[9] S.N.Coppersmith,C.-h.Liu,S.Majumdar,O.Narayan, 066117 (2004).
and T. A. Witten, Physical Review E 53, 4673 (1996). [23] M. F. Sykes and J. W. Essam, Journal of Mathematical
[10] P. Claudin, J.-P. Bouchaud, M. E. Cates, and J. P. Physics 5, 1117 (1964).
Wittmer, Physical Review E 57, 4441 (1998). [24] D.S.MooreandG.P.McCabe,IntroductiontothePrac-
[11] J. E. S. Socolar, Physical ReviewE 57, 3204 (1998). tice of Statistics (W.H. Freeman and Company, 2003),
[12] S.Ostojic, E. Somfai, and B. Nienhuis,Nature 439, 828 4th ed.
(2006). [25] D.J.WattsandS.H.Strogatz, Nature393,440(1998).
[13] S.OstojicandB.Nienhuis,inTrafficandGranularFlow [26] R. Albert and A.-L. Barab´asi, Reviews of Modern
(2005). Physics 74, 47 (2002).
[14] S. Ostojic, T. Vlugt, and B. Nienhuis, Physical Review [27] L. K. Gallos and P. Argyrakis, Physica A 330, 117
E 75, 030301(R) (2007). (2003).
[15] K. Maeda, N. Kuwabara, and H. Matsuoka, in Powders
and Grains, edited by Y.Kishino (2001), pp. 223–256.