force chains in seismogenic faults visualized with...

Force chains in seismogenic faults visualized with photoelastic

granular shear experiments

Karen E. Daniels1 and Nicholas W. Hayman2

Received 4 May 2008; revised 1 August 2008; accepted 2 September 2008; published 26 November 2008.

[1] Natural faults have many characteristics in common with granular systems, includinggranular fault rocks, shear localization, and stick-slip dynamics. We present experimentalresults which provide insight into granular behavior in natural faults. The experimentsallow us to directly image force chains within a deforming granular media through the useof photoelastic particles. The experimental apparatus consists of a spring-pulled sliderblock which deforms the photoelastic granular aggregate at a constant velocity. Particlesthat carry more of the load appear brighter when viewed through crossed polarizers,making the internal stresses optically accessible. The resulting pattern is a branched,anisotropic force chain network inclined to the shear zone boundaries. Under bothconstant volume and dilational boundary conditions, deformation occurs predominantlythrough stick-slip displacements and corresponding force drops. The particle motion andforce chain changes associated with the deformation can either be localized to the centralslip zone or span the system. The sizes of the experimental slip events are observed tohave power law (Gutenberg-Richter-like) distributions; the minimum dimensions ofevents and the behavior of force chains suggest that a particle scale controls the lowerlimits of the power law distributions. For large drops in pulling force with slip, the shapeof the size distributions is strongly affected by the choice of boundary condition, while forsmall to moderate drops the probability distributions are approximately independent ofboundary condition. These size-dependent variations in stick-slip behavior are associatedwith different spatial patterns: on average, small events typically correspond to localizedforce chain or particle rearrangements, whereas large events correspond to system-spanningchanges. Such force chain behavior may be responsible for similar size-dependent behaviorsof natural faults.

Citation: Daniels, K. E., and N. W. Hayman (2008), Force chains in seismogenic faults visualized with photoelastic granular shear

experiments, J. Geophys. Res., 113, B11411, doi:10.1029/2008JB005781.

1. Introduction

[2] Most earthquake-producing faults contain a layer ofgranular fault rocks, and fault rock structures are used toinfer the mechanical history of fault slip [e.g., Sibson,1977]. Furthermore, the mechanics of deforming fault rocksmay affect the nature of earthquakes, including seismolog-ical scaling parameters [Scholz, 1990; Sibson, 2003; Chesteret al., 2005]. Laboratory experiments focused on fault rockdeformation provide insight into the strength and stability ofnatural faults and help formulate constitutive models forfriction [Byerlee, 1978; Dieterich, 1978; Beeler et al., 1996;Marone et al., 1990; Marone and Kilgore, 1993]. Earlyslider-block and microfracturing experiments reproducedmany of the key scaling relations such as the Gutenberg-Richter (frequency-magnitude) relation [Burridge and

Knopoff, 1967; Scholz, 1968]. Yet, most rock frictionstudies do not provide direct data on the contact forceswithin a shearing media. In contrast, granular models forfault deformation explicitly address the forces within adeforming fault zone [e.g., Sammis et al., 1987]. Thegranular description of faulting is thus well suited forinvestigation of fault rock deformation, and may findapplications for a broader range of frictional phenomenain the upper crust.[3] A first approach to granular models is to use numer-

ical techniques such as the discrete element method (DEM)[Morgan and Boettcher, 1999; Morgan, 1999; Aharonovand Sparks, 2002; Mair and Hazzard, 2007]. DEMs simu-late particle-scale fault zone deformation with the benefit ofvisualizing the internal arrangement of forces during defor-mation. A second approach is the use of laboratory systemsemploying model granular materials, which reproduce suchnatural phenomena as shear localization and stick-slipbehavior [Nasuno et al., 1998; Anthony and Marone,2005; Johnson et al., 2008] and produce slip distributionsreminiscent of the Gutenberg-Richter relation for naturalearthquake populations [Bretz et al., 2006]. Of considerable

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1Department of Physics, North Carolina State University, Raleigh,North Carolina, USA.

2Institute for Geophysics, University of Texas at Austin, Austin, Texas,USA.

Copyright 2008 by the American Geophysical Union.0148-0227/08/2008JB005781$09.00

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interest, the force distributions within granular materials arequite broad and follow an approximately exponential-taileddistribution under a variety of loading circumstances [Milleret al., 1996; Mueth et al., 1998; Majmudar and Behringer,2005].[4] A particular class of laboratory granular models

utilizes photoelastic particles to investigate the spatialarrangement of internal stresses in granular materials[Dantu, 1957; Drescher and de Josselin de Jong, 1972;Howell et al., 1999; Majmudar and Behringer, 2005].Typically, the internal stresses within a granular materialarrange themselves in ‘‘force chains,’’ which are roughlycolinear groups of particles carrying more of the loadimposed on a granular system than the adjacent particles.The details of the force network change under even minuteshifts in particle positions, and are quite sensitive to theshear history of the sample [Majmudar and Behringer,2005]. Photoelastic granular experiments have not beenwidely used to see if such force chain behaviors areimportant for fault behaviors as manifested in earthquakepopulations [Yu and Behringer, 2005].[5] In this paper, we present the results of photoelastic

granular experiments designed to test the extent to whichforce chain behavior plays a role in the dynamics of faults.As shown in Figure 1, the experimental apparatus we usedis geometrically analogous to a natural, gouge-filled strike-slip fault. By pulling a slider block at a constant velocity wedeformed an aggregate of �104 photoelastic particles andrecorded plate displacements and drops in the pulling forceduring stick-slip events. During these events we also ob-served a range of behaviors in the force chain network,including both local and system-spanning force chain reor-ganizations, and we analyzed these events with imagedifferencing techniques. The events exhibited statisticalproperties similar to those observed in natural earthquakepopulations. Notably, we observed differences in the scalingrelations for force drops and related work terms for higherand lower ranges of event size. Following the presentationof our experimental methods and results, we suggest that thebehaviors we observe in force drop data and force chaingeometries are consistent with earlier explanations ofchanges in scaling relations that depend on the size of therupture area within the seismogenic crust [Pacheco et al.,1992; Heimpel and Malin, 1998; Romanowicz and Ruff,2002]. Variations in scaling relations and deficits in radiatedenergy both suggest that the mechanics within a deformingfault zone can directly influence larger-scale slip behaviors[Kanamori and Brodsky, 2004]. We also suggest that thepatterns of force chain rearrangements are consistent withdescriptions of fault zone deformation that include bothlocalized slip on narrow zones, and ongoing distributeddeformation in adjacent, wider gouge zones. The twoimplications of our experiments are related and imply thatgranular processes underlie both the geological and seismo-logical expressions of fault slip.

2. Experiment

[6] Our experiments take place in a split-bottomed shearcell shown schematically in Figure 1. Circular (60%) andelliptical (40%) disks form a granular layer on the trans-parent split bottom, and are confined to a single layer by a

transparent cover. The disks are photoelastic (3 mm thickPhotoStress Plus PS-3 polymer from the Vishay Measure-ments Group) and have a bulk elastic modulus of 0.21 GPa.The circular disks have a diameter of 5.6 mm and theelliptical disks have major and minor axes 6.8 mm and4.7 mm, respectively. The mixture of two shapes preventscrystallization of the granular medium into a closely packedconfiguration. The total granular aggregate comprises9360 particles in a region 125 cm long and 22.8 to24.2 cm wide.[7] An imposed slip plane separates the two sides of the

cell. The plate with the movable confining wall is fixed andthe pulled side, consisting of a bottom plate with an attachedconfining wall, is driven as a unit by a stepper motorattached to a linear feed screw which moves at a constantvelocity of 0.30 mm/s. We record the pulling force which iscoupled to the slider block by a linear spring using aChatillon DFS piezoelectric force sensor and the positionusing a Celesco cable transducer, each measured at afrequency of 100 Hz. The split in the center of the cell isrequired for relative displacement of the two plates, andcreates a granular-on-granular shear zone [Fenistein andvan Hecke, 2003] rather than localizing the shear to theboundary. At their most basic level, natural faults are drivenfrom the sides and have three-dimensional particle inter-actions within the shear zone. In the experiment, theparticles deform because of the forces imposed from theside boundaries, with traction between the moving bound-ary and the particles enhanced by protrusions separated by15 mm (visible on the images in Figure 2). However, theparticles are also in frictional contact with the bottom plateand thus the experiment is a quasi-two-dimensional approx-imation of an idealized fault.[8] The photoelasticity of the disks allows us to spatio-

temporally resolve the internal stress state of the system[Frocht, 1941]. As shown in Figure 1a, the shearing of thegranular aggregate causes chains of force to develop inopposition to shear and loading. The brighter the particle is,the stronger the forces at its contacts, while particles thatappear dark are carrying negligible amounts of the total loadacting on the shear zone. We recorded the spatiotemporalemergence and evolution of force chains for the middlethird of the length and the full width of the experimental cellusing a digital video camera operating at 4 Hz and with950 � 550 pixel resolution.[9] An important feature of the apparatus is the ability to

set the boundary conditions on the stationary side of the faultto a fixed volume V, or, alternatively, to confine theparticles with a compressed spring allowing dilation. Thedilational boundary condition provides constant changes inpressure P relative to changes in volume (constant dP/dV).We refer to these boundary conditions as ‘‘constant vol-ume’’ and ‘‘dilational,’’ respectively. The advantage of thespring boundary conditions is that it is simple to realizeexperimentally and still allows us to evaluate the effects ofdeformation within a nonconstant volume. The significanceof this dilation is to stand in contrast to the constant volumecase, rather than modeling a particular constant pressure.We measure the pulling force Fk for all runs, and F? for thedilation runs via the position of the confining wall. From theposition of the spring-confined boundary, we find theconfining pressure P � 7 kPa; a typical change in loading

B11411 DANIELS AND HAYMAN: FORCE CHAINS AND GRANULAR FAULTS

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pressure over the course of the run is 20% (see Figure 2d forthe time dependence). The constant volume runs have apacking fraction of f = Vparticles/Vcell � 0.81, while over thecourse of the dilational runs f typically decreases from�0.83 to 0.79.

3. Results

3.1. Particle and Force Chain Deformation

[10] At the beginning of each experimental run we startfrom a randomized configuration with low internal stressand few force chains. When the pulling starts, particlesrearrange, and the aggregate dilates if the boundary con-ditions permit, in response to the shear; force chainsdevelop in opposition to the motion. Animations 1 and 21

are samples of these dynamics. As can be seen in Figures 2band 2e, the force chains intersect at junctions betweenwhich they follow arched paths surrounding relativelyunstressed ‘‘spectator’’ particles or regions. The result isan anisotropic network which balances the forces within thegranular media, carries the load applied from the bound-aries, and evolves through changes in the geometric con-figuration of the branched force chains. From the beginningof the experimental runs, displacement proceeds dominantlythrough stick-slip motion which corresponds to both drops

in the pulling force and steps forward in the displacementprofile (see Figures 2a and 2c). Initially, these steps are largeand periodic until a fully developed stress state is formedafter approximately 200 s, after which the stick-slip eventsare aperiodic and irregular in size.[11] On first inspection, the images of the force chains

before and after stick-slip events reveal little informationsince they are quite similar, as shown in Figure 3. In fact, ascan be seen through image differencing, each event iscomposed of particle displacements and/or changes in thestrengths of force chains which provide adjustments to thenetwork. The image differencing technique uses a pixel-wise subtraction of the initial image from the final image foreach event. Thus, we can visually inspect the granularmechanisms associated with each event. In thesedifferenced images, the image is white where a force chainstrengthened, black where a force chain weakened, and graywhere no change occurred. A chain that is lined with a whitestrip on the left side and dark on the right slipped sidewaysas a unit; a chain structure that is completely white formedduring the slip event; a chain structure that is completelydark disappeared during the slip event. Finally, particles thatslipped sideways appear as faint rings.[12] Slip events all involve a macroscopic displacement

along the imposed fault. The slip events comprise bothparticle motions and force chain changes, both of which areremarkably varied in their spatial extent. This variability isshown by the examples in Figure 4, drawn from both

Figure 1. (a) Closeup of force chains in photoelastic particles. Brighter particles are carrying moreforce. Schematic of laboratory fault apparatus (not to scale). (b) Cross-sectional view of apparatusshowing arrangement of polariscope used to visualize force chains. Pulled side bottom and stationaryback wall move as a single unit. (c) Top view. Stationary side can have either constant V (not shown) ordilational (shown, constant dP/dV) boundary conditions. The central region is filled with densely packedphotoelastic disks. The pulled side is driven at constant velocity via a spring coupling (shown at left side).Boundaries parallel to the shear (dark gray bars) are rough on the scale of the particles; end boundaries(light gray bars) are smooth.

1Animation 1 and 2 are available in the HTML.


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boundary conditions and a variety of event sizes. The twomajor classes of behavior, (1) particle kinematics, referredto here as ‘‘particle slips’’ and (2) rearrangements of theforce chain network, correspond to two different mecha-

nisms for dissipating energy during the slip event. Either ofthese types of changes can take place on either a local or aglobal scale. Thus, we observe events in which the entirenetwork of force chains slid forward through slip at the far

Figure 3. Example of image differencing technique. Images before and after a slip event at the fault,subtracted to produce a differenced image.

Figure 2. Sample time series of position (red/gray) and force (black) for the slider block with(a) constant volume and (c) dilational boundary conditions. The thick bar along the time axis indicates theinterval used for calculating event statistics. (b and e) From the central region of the shear cell (and thefull width) at the end of the run and show force chains. (top) The fixed side of the experiment, and(bottom) the slider block side. Particles are visible with out polarizing filters in the lower left corners ofthe images. (d) The friction coefficient (black, left axis) normal pressure (red/gray, right axis) for the samedilational run. For animations of Figures 2b and 2e, see Animations 1 and 2, respectively.


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boundary, as shown in Figure 4a but also large-scalechanges in the force chain network that were accompaniedby only local particle slip events (Figure 4b). We refer tosuch penetrative changes as system-spanning events. Theforce chain changes can also span the shear zone in onearea, but be ‘‘blocked’’ in an adjacent region, as shown inexample Figure 4c.

[13] Another common pattern we observed was for theparticle slips and force chain releases to be localized towithin a few particle diameters of the full length of theimposed fault plane (Figures 4d and 4e) or to even just afinite patch along the fault (Figures 4f and 4g). In occa-sional circumstances, isolated changes were observed totake place far from the imposed fault (Figure 4h). The

Figure 4. Differenced images from dilational runs (via the method given in Figure 3), showing (a) asystem-spanning event dominated by particle slip; (b) a system-spanning event dominated by changes inthe force network; (c) large-scale slip adjacent to a ‘‘blocked’’ region without changes; (d) particle slipand (e) force chain changes localized to within a few particle diameters of the imposed fault plane;(f) particle slip and (g) force chain changes within a highly localized patch near the boundary; (h) forcechain changes within a localized patch far from the boundary. (i– l) Differenced images of similarexamples drawn from constant volume runs. All images have an imposed fault plane indicated by theblack dashed line at the bottom edge. Scale bars represent the change in the intensity value of the pixelover the span of the event, out of a possible 8 bits (256 units) of intensity. The force drop DFk and eventenergy W values are calculated using equations (1) and (2), respectively.


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remaining four images in Figure 4 show similar behaviors inexamples drawn from runs with constant volume instead ofdilational boundary conditions. Finally, it is important tonote that many events resembled more than one of thecategories described above, and further experimental workis needed to determine under what specific conditions eachof these behaviors is observed. In section 3.4, we willexamine the correlations between the source parametersof the events and the spatial extent of the associateddeformation.

3.2. Effects of Boundary Conditions

[14] Our second approach focuses on the source param-eters under the two different boundary conditions, constantvolume and dilational, while leaving other system parame-ters as constant as possible. We performed 15 experimentalruns with dilational boundary conditions, and 24 runs withconstant volume. A side-by-side comparison of pullingforce Fk(t) and plate position x(t) is shown in Figures 2aand 2c and elucidates the similarities and differencesbetween the two boundary conditions. For each forwardslip event, we see a corresponding drop in the pulling force.Under both boundary conditions, the shear zone strength-ened and developed a force chain network throughapproximately periodic stick-slip motion during an initialperiod of �200 s. Once the internal stresses were wellestablished, the experiments entered a regime of aperiodicbehavior, marked by the thick bar along the time axis inFigures 2a and 2c.[15] Both constant volume and dilational experiments

generate a strengthening in the pulling force over time,resulting from the development of the force chain network.The dilational boundary condition permits a partial relievingof the stress and thus the pulling force increases moreslowly. The images of force chains in Figure 2 provide anoptical indication of these contrasting strength evolutions.The dilational boundary conditions (Figure 2e) show fewerand weaker force chains than the constant volume case

(Figure 2b), where the chains are well developed andstrongly anisotropic. These differences are similar to tran-sitions in force chain behavior which have been previouslyreported for changes in packing fraction [Howell et al.,1999].[16] Variation in the behavior of Fk(t) for individual runs

can be seen in the examples shown in Figure 5. Thisvariation is notable because the runs use the same materialstarting from nominally the same initial conditions, differingonly in their rearrangement (by hand) prior to the start ofeach run. On average, the constant volume runs strengthensignificantly over time, while the dilational runs do not.Furthermore, constant volume deformation includes inter-mittent events with larger force drops than the largest eventsfor the dilational runs. However, for any short time interval(for instance, the interval from 200 to 300 s in the fourthcurve of Figure 5b), differences between the two boundaryconditions are not as readily apparent. Therefore, it ishelpful to examine the probability distributions of eventstatistics rather than single runs or events in order todetermine the effect of changing the boundary conditions.[17] Arguably, none of the experiments, and particularly

those at constant volume, achieved a true steady state in thepulling force Fk(t) as all runs exhibit an upward trend overthe course of the run. This lack of steady state behavior is inpart a consequence of the finite strain accessible to a systemof this geometry. While it would be possible to obtain truesteady state behavior through the use of an annular cell (asin the work by Howell et al. [1999]), in such a geometry theshear localizes to the region of highest curvature, which isan undesirable effect for modeling natural faults. None-theless, we observe that the source parameters of the eventsare drawn from approximately stationary distributions, asdiscussed in section 3.3 below. For the case of the dilationalboundary conditions, steady state was approximatelyachieved in both the pulling force and the coefficient offriction (Figure 2d). We determined the coefficient offriction, m � 0.2 to 0.3, from the ratio of the pulling force

Figure 5. Five representative time series of the pulling force, taken under (a) dilational and (b) constantvolume boundary conditions. Curves are offset by 15 N in order to provide visual separation but are plottedon the same scale. Thick bar along the time axis indicates the interval used for calculating event statistics.


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Fk(t) to the compressive (normal) force F?(t) supplied bythe confining springs.

3.3. Event Parameters

[18] In order to further evaluate the character of the stick-slip deformation, we extract two event parameters to de-scribe the size of each event. Because every stick slip eventdisplaces the entire shear zone boundary, we consider bothparameters to describe the rupture of the full length of thefault, L. To detect individual events, we both smoothed the100 Hz time series using a window of 0.6 s and obtained theFourier space derivative to alleviate issues with triggeringevents on the basis of noise. To obtain accurate times t1 andt2 marking the beginning and end of each event, we locallyfit a parabola to minimally smoothed data (the bottom twocurves in Figure 6). For each such event, we directly extracttwo parameters describing the size. The pulling force dropDFk (measured in newtons) corresponds to the (positive)drop in the pulling force during the event:

DFk ¼ Fk t1ð Þ Fk t2ð Þ: ð1Þ

Additionally, we define a work term (measured in N mm)

W ¼Xt2

i¼t1

Fk;i xiþ1 xið Þ; ð2Þ

which we refer to below as the event energy and resultsfrom various loss processes. These two values (DFk, W)were extracted for each event and problematic events werediscarded from the analysis to avoid small events of thewrong sign. We also discarded events from the initial 200 s.This procedure resulted in �1500 stick-slip events for eachboundary condition.[19] We first examine the statistical distributions of the

two event sizes DFk and W, shown in Figure 7. The firsttwo panels of each row plots the cumulative probabilityfunction for each size, but with different axes to show thedifferent scaling. In Figures 7a and 7d, we see that bothDFkand W for the constant volume data set exhibit power law-like tails. We fit each using the method of Clauset-Shalizi-Newman (A. Clauset et al., Power law distributions inempirical data, 2007, available at http://arxiv.org/abs/

0706.1062) and find exponents of 2.61 ± 0.14 and3.42 ± 0.12, respectively. For the dilational boundaryconditions, both DFk and W exhibit exponentiallike tails(see Figures 7b and 7e) with decay constants of 0.41 N and7.2 N mm, respectively. The heavier tail of the constantvolume case can be seen qualitatively in Figure 5, where theFk(t) plots show larger DFk events.[20] In Figures 7c and 7f, the data are binned in unequal

width bins, properly normalized to provide probabilitydensity, and plotted on log-log axes to allow examinationof the full range of DFk and W values. In each case, wehave also plotted the data sets drawn from only the first half(dotted lines) or second half (dashed lines) of the data sets toillustrate that these distributions are stationary in spite of thestrengthening which took place. We also observe that the Wvalues for the two boundary conditions are quite similar toeach other, with the exception of the large-size tail asdescribed above for the case of the cumulative distributions.For the two boundary conditions, the median DFk value isapproximately 0.2 N, and decays away for both higher andlower values. On the small-size side of these distributions,the curves have quite similar slopes down to 103 N, withsmall events being less common for the dilational boundarycondition. The center of the distribution is both slightlyhigher for the dilational boundary conditions and thedistribution is more skewed, with both effects leading to aproportionally greater number of larger DFk events.[21] In Figure 8, we provide a comparison of these two

measures of the size of events for each boundary condition.For DFk ^ 0.2 N (the median value from Figure 7), weobserve an approximately linear relationship between DFkand W. To understand the origin of this scaling, we considerthat DFk is proportional to the slip Dx of our events,because of the spring coupling. Therefore, it is alsoproportional to the stress drop

Ds ¼ EDx

L¼ E

k

DFk

Lð3Þ

for an approximately constant elastic modulus E, constantpulling spring constant k, and constant rupture length Lcorresponding to the length of the experimentally imposedfault. Alternatively, we can view DFk as a measure of the

Figure 6. Example of event-finding technique. Bold line at top is smoothed data used for initialdetection, along with Dx(t) and DFk(t) used for finding start and stop times (marked by vertical dashedlines) and event statistics.


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size since the torque, or moment, on this fixed rupturelength is

M0 ¼ DF?L �DFk

mL ð4Þ

for an approximately constant friction coefficient m. Thus,the approximately linear relationship between DFk and Wobserved in Figure 8 stems from the fact that both areapproximately proportional to the slip. The relationshipbetween this result and similar scaling in natural faults willbe explored further in section 4.[22] We therefore consider the observed deviations from

this simple linear scaling: the constant of proportionality(offset on a log-log plot) depends on the choice of boundarycondition for DFk ^ 0.2 N, and there is increased scatter forDFk ] 0.2 N. This observed variation among events of thesame ‘‘size’’ is, in fact, to be expected because of the spatialvariations shown in Figure 4: the relationship between DFkandDF? is clearly not simply proportional to m as would berequired for the approximation given in equation (4). Theutility of this relationship, therefore, should be thought of inqualitative terms only. Note that the constant of proportion-ality has units of length and ranges from 3 to 100 mm forthe dashed lines shown in Figure 8. Therefore, one possibleinterpretation is that different amounts or numbers of

particle-scale slips occur in different events. Since thedilational boundary conditions have fewer strong forcechains (see Figure 2), it is possible that fewer slips arepresent on average in the dilational events.

Figure 8. Relationship between two size variables pullingforce drop DFk and event energy W. Dashed lines are forlinearly proportional behavior. Dashed line at 0.2 N is thesame division used in Figures 7 and 9.

Figure 7. Probability distribution functions of DFk and event energy W, plotted as cumulativedistributions on (a and d) log-log and (b and e) log linear axes; and (c and f) as probability densitydistributions for increasing-spaced bins on log-log axes, to allow examination of the full range of data.The thick dashed lines in Figures 7a, 7b, 7d, and 7e) are fits to power law (Figures 7a and 7d for constantvolume data) and exponential (Figures 7b and 7e for dilational data) distributions. Figures 7c and 7f, thedata sets are into early times (less than the median) and late times to check for stationarity. The verticaldashed line in Figure 7c is value 0.2 N used to divide large- and small-DFk events in Figure 9.


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[23] Because the median or peak of the force drop DFkdivides the data set into two size regimes, we further explorethe scaling of events either above or below DFk = 0.2 N. Todo this, we reexamine the frequency-size distribution of Woriginally plotted in Figure 7, but we now divide each dataset into a large-DFk portion and a small-DFk portion.Figure 9 shows the cumulative probability distribution ofthe event energy W for these two subpopulations under eachboundary condition. The power law scaling range is, in fact,only a property of the large-DFk events, and extends from� 5 N mm to 100 N mm. The lower limit of power lawscaling, what we refer to here as a ‘‘rollover’’ in thedistribution, is around W � 5 N mm for both boundaryconditions. The rollover is therefore not observed to besensitive to such variables as overall stress, packingfraction, or boundary condition.

3.4. Size Dependence of Force Chain Behavior

[24] Given the observation that the energy force dropscaling relation (Figure 8) and frequency energy distribu-tions (Figure 9) differ for events of different ranges of DFk,we return to the differenced images of the experimentalshear cell to examine the behavior for stick-slip events oflarge and small event energy. For the largest events (such asFigures 4a, 4b, 4i, and 4k), the differencing techniquereveals that changes in the particle positions and/or forcechains span the whole length and width of the cell. For thevery large constant volume event shown in Figure 4i, nearlyevery force chain slipped sideways intact, while forsomewhat smaller events (such as Figures 4a, 4b, and 4k)the failure of individual force chains was more dominant.Therefore, we observed that the large events (particularlyprevalent for the constant volume boundary conditions)typically have system-spanning rearrangements with somesliding (translation) along the shear zone boundaries orwithin the granular media.

[25] Moderate to small events, such as shown in theremainder of the differenced images, can exhibit force chainrearrangements and particle slips throughout the shear zoneor only within spatially localized patches. Individualpatches can be dominated either by particle slips or byforce chain changes. The patches can be quite inhomoge-neous, with only some of the force chain changes spanningthe width of the system. The smallest events are often quitespatially localized or even consist of barely perceptiblechanges in particle positions or force chains (Figures 4fand 4l). Importantly, very small stick-slip events commonlyexhibit rearrangements (either particle slips or force chainchanges) which are isolated from the shear zone boundaries.[26] As can be seen from the images in Figure 4, there is a

general trend that the larger the event energy W, the largerthe proportion of the granular material participated in theevent. To quantify this effect, we consider the fractional areaA of these images that exhibited an above threshold changein pixel brightness. While this measure does not distinguishbetween force chain changes and particle slips, it can givean semiquantitative estimate for the spatial extent of thechanges. Figure 10 plots A against the event energy W andfinds a strong correlation between the two quantities,particularly forW ^ 5 N mm. Interestingly, this correspondsto the rollover value in Figure 9, suggesting that theminimum scaling size is set by a minimum granularrearrangement scale. Since a single particle contributes anarea fraction A � 6 � 104, this breakdown in scalingoccurs for values of A that are below a size of approximately10 particles. This value is consistent with a patch theapproximate width of the shear band. We therefore suggestthat the event energy at which the rollover in Figure 9occurs is controlled by the discontinuous nature of thegranular material; further experiments are necessary to testthis finding.

4. Discussion

[27] Inspired by earlier experimentalists’ use of photoe-lastic materials [e.g., Mandl et al., 1977], we have imple-mented current approaches used by the granular physics

Figure 9. Cumulative probability distributions for bothconstant volume (black) and dilational (red) experimentaldata, split so that high-DFk (thick, DFk > 0.2 N) and low-DFk (thin, DFk < 0.2 N) subpopulations are plottedseparately from each other. Low-W data are identical toFigures 7a and 7d and are here truncated at 101 N mm toimprove readability.

Figure 10. Plot of area fraction A (fraction of pixels whichchanged by more than a threshold value) as a function of theevent energyW. Horizontal line is A for a single-particle event.


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community such as real-time digital imaging of force chainbehavior [e.g., Howell et al., 1999] to investigate a granulardescription of natural faulting. The simplicity of a photoe-lastic granular fault has significant benefits: a first advan-tage is that because the particles remain unchanged duringruns and repeated runs are possible with the identicalmaterial, we can build up a data set in which any variationis inherent to the emergent dynamics, and not a function ofvariability in the material or fault properties. A secondadvantage is that we can assume that the majority of theenergy released by stick-slip deformation is associated withparticle slips and/or force chain rearrangements and thusisolate those effects from fluid phase and fracture effects.There are inherent limitations to such an approach, forinstance the use of nonnatural materials, the two-dimen-sionality and the difficulty of providing a wide range ofparticle characteristics. These experiments thus constitute apromising direction for investigation of granular phenome-na in fault zones and several future directions are possiblewhich will expand the utility of these techniques. Futureexperiments can, for example, monitor both the particlekinematics and the normal pressure, use the photoelasticityof grains to locate all failure events rather than relying on amacroscopic slip for event detection, and systematicallymodify particle properties such as shape and particle sizedistribution.[28] To place the experiments in a geological context, it is

useful to consider several quantitative properties of thegrains. The particles have an elastic modulus of 0.21 GPawhich scales to experimentally realized pressures by a factorof 104 to 105; natural faults scale by approximately 103

(i.e., 10 to 100 MPa imposed on a rigidity of �10 GPa,using nominal values from Turcotte and Schubert [2002]).The granular materials also achieved a roughly steady statefriction of m � 0.2 to 0.3. Though well below Byerlee’s law(0.6 to 0.85), numerical models of two-dimensional granulardeformation achieve similar steady state friction coefficientsof � 0.2 to 0.6 [e.g., Mora and Place, 1998; Morgan, 1999;Frye and Marone, 2002; Mair et al., 2002] and smoothround particles in triaxial shear tests have coefficients of

friction between 0.1 and 0.4 [Anthony and Marone, 2005].Weak natural faults, although not widespread, are nonethe-less important structures to understand [Hayman et al.,2003; Hickman and Zoback, 2004]. With this in mind, wecan make meaningful comparisons between experimentaldata and observations from natural faults.[29] We begin making connections to natural faults by

observing that pulling force drop DFk and event energy Weach have power law tails for constant volume boundaryconditions. This is similar to what is observed (with adifferent exponent) for the most widely studied naturalscaling relation, the frequency-moment (Gutenberg-Richter)relation. As depicted in Figure 11a, we monitor Fk(t) andx(t) and the internal behavior of the shear zone analogous toplane X/X0–Y/Y0. Two associations with natural eventparameters are possible. First, because there is a fixedrupture length, equation (4) suggests that for DFk � m DF?(constant frictional behavior), the pulling force drop DFk, infact, plays a role similar to the seismic moment. Anyvariations in m over the course of the experiment will not belarge enough to change the order of magnitude of thisestimate. Second, the event energy W represents thedissipation of stored energy; in an idealized seismic event,this energy is dissipated via mechanisms involving radiatedenergy. In natural earthquakes, the radiated energy fre-quently deviates from the seismic moment, potentiallyindicating that local mechanisms within the fault areexerting a large control on how energy is released. Forexample [after Kanamori and Brodsky, 2004, and referencestherein]: (1) radiated and fracture energy are not everywhereequivalent, and thus some energy dissipates throughmechanical work within the shear zone, (2) temperaturesvary between slipping faults, and generally are lower thanexpected. This indicates slip weakening processes withinfaults that vary among faults, and (3) smaller earthquakeshave a wider range in stress drop (0.1 to 100 MPa) thanlarger ones (1 to 10 MPa) [Abercrombie, 1995], indicatingthat smaller earthquakes may obey different sourcemechanics. Future experiments can more directly tacklethe way energy is radiated from our shear cell, but we

Figure 11. (a) An oblique view into a schematic, right-lateral strike-slip fault during an earthquakeillustrating some of the source parameters we can address with our experimental analog. The plane X/X0–Y/Y0 is a cut through the fault zone such as our experiment is meant to simulate. (b) Plane X/X0–Y/Y0

described as a system of force chains within the fault zone (left), and as a geologic fault (right). Zone I ofthe fault zone is an outer damage zone which accommodates little shear strain, zone II is a zone of faultrocks that accommodates strain through the development of foliations (f) and Riedel shears (r), and zoneIII is a localized, central slip zone.


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immediately note that the size dependency of force drops(as in the work by Abercrombie [1995]) is already detectedin our experiments (see Figure 8).[30] We thus ask: what are the important departures from

scaling relations in our granular experiments? First, we notethat the behavior for both pulling force drops and eventenergies differs for the two boundary conditions in theexperiments: a steeper tail is observed for the dilationalboundary conditions. Although all of our events displace thefull the length of the fault, the internal particle slips andforce chain changes are not identical for all events. For anygiven event energy there is a range of sizes of rearrange-ments, as shown in Figure 10. Yet, larger events tend tocorrespond to a larger area affected by particle and forcechain rearrangements. Thus, for the largest events, wewould anticipate that more of the force chains are in contactwith the boundary and thus these events have an enhancedsensitivity to the boundary conditions. This force chaindescription is perhaps a granular version of the argumentof Pacheco et al. [1992] and Romanowicz and Ruff [2002]that the largest subduction zone earthquakes tended todepart from the global frequency-magnitude scaling relationbecause their ruptures encountered the edges of the seismo-genic zone. Second, for DFk < 0.2 N, there is a dramaticchange in the slip behavior, and event energy no longerscales with DFk and instead shows a great deal of scatter.These small event sizes correspond to generally local forcechain changes rather than the system-spanning changesobserved for larger size events. Perhaps such force chainrearrangements underlie the deviations from scaling rela-tions observed for small earthquake sequences in nature[Heimpel and Malin, 1998].[31] The scaling relations that stem from seismic faulting

are intimately related to the structural evolution of a faultzone over many increments of slip through geologic time.For example, the thickness of a fault develops over geologictime through the competing processes of strain localizationand shear zone widening [e.g., Scholz, 1987]. The thicknessof the fault will in turn partly govern the temperatureachieved during slip, which could exceed 1000�C if thefault is centimeters thick [Sibson, 2003]. Some faults haveindeed reached such temperatures as indicated by thepresence of glasses formed from frictional heat. However,the relative paucity of frictional melts in the geologicrecord, and lack of heat flow anomalies on major crustalfaults, lead most workers to consider alternative models forthe energetics of faulting. For example, either through shearheating, or changes in frictional stability, faults can dynam-ically weaken during slip, thereby affecting the energyreleased by the fault [e.g., Rice, 2006]. The slip weakeningand localization process are mechanically related [Beeler etal., 1996; Chester and Chester, 1998].[32] With this in mind, we consider how our experimental

results relate to fault zone thickness and strain localizationby considering the internal, geologic structure of a seismo-genic fault, shown schematically in Figure 11b. Wherefaults are exposed at the Earth’s surface, they are typicallymeter-scale composites of three zones: zone I is a widedamage zone of closely spaced fractures, far from slipplane, considered here to be ‘‘wall rock’’ and outside ofthe shear zone boundaries; zone II contains granular faultrocks which have both foliations inclined to the shear zone

boundaries, and Riedel shears which are inclined in theopposite direction to the foliations; and zone III is a welldefined, centimeter- or millimeter-scale principal slip zone(PSZ) within zone II [Logan et al., 1979; Chester andLogan, 1987]. Though codified at a locality along the SanAndreas fault system, faults worldwide are reported to havesome variation of these structural zones [e.g., Sibson, 2003].[33] Geologic evidence shows that shear zones evolve

over time through increments of slip, and shear strainprogresses inward from zone I into zone II and then zoneIII [e.g., Chester and Chester, 1998]. Such geologic evi-dence has been used to argue that the localization process ismostly irreversible; that is, once the PSZ of zone IIIdevelops, further deformation within the surrounding faultrocks of zone II is relatively insignificant. The implication isthat the PSZ of zone III accommodates most of the fault slipand most of the coseismic rupture. Though strain localiza-tion produces extremely narrow faults with extraordinarilylarge cumulative displacements, some geologic exposurescontain evidence for distributed deformation in zone II afterlocalization into the PSZ of zone III [Cowan et al., 2003;Hayman et al., 2004; Hayman, 2006]. Models of earthquakepropagation also hold that during an earthquake a large areaof the fault will be critically stressed and energy will beradiated away from the localized slip zone, causing damagein the surrounding fault and wall rock [e.g., Rice et al.,2005].[34] Our experiments provide insight into the situation of

a mature, gouge-filled fault subjected to shear. The experi-ments show an initial strengthening of the force chainnetwork near the boundaries of the shear zone, and asubsequent localization of particle slips at the center ofthe shear zone. The general pattern of localization is thusconsistent with a progression in natural faults from zone Iinto zone II and ultimately into zone III, with a PSZ of a fewparticle diameters. However, even after localization occurs,there are significant force chain changes and particle slipsoccurring throughout the shear zone: within the experimen-tal analogue for zone II fault rocks. Such granular defor-mation away from the central slip plane has a range ofmanifestations, from system-spanning force chain rear-rangements, to blocked regions, to local patches of forcechain changes. As described above, the different behaviorsare correlated with the size of the stick-slip event. For thelargest events, we observe not only sliding along the shearzone boundaries and central slip surface, but also system-spanning force chain responses within the granular media.[35] In summary, the force chain description of natural

shear zones holds that there are perhaps continued modifi-cations to a wider gouge-filled shear zone even afterlocalization of slip into a narrow principal slip zone. Theforce chain description also holds that scale dependence ofseismic rupture may have its origin in the behavior ofintrafault force chains. Thus, our experiments can providea conceptual link between geologic processes within faultzones, and the earthquakes that can result from fault slip.

5. Conclusions

[36] A photoelastic granular experiment shows that sheardeformation near an imposed fault is partly accommodatedat the particle scale by the rearrangement of force chains,


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the approximately colinear arrangements of particles thatcarry most of the load. There is a scale dependence to thedynamics of failure within the granular media, observed viathree fundamental trends in the statistics of pulling forcedrop, event energy (work), and images of force chains. First,the size scaling closely mimics the natural scaling for eventswith sufficiently large force drops (DFk ^ 0.2 N), whilesmaller force drops depart from that scaling. Second, theforce drop-dependent behavior is reflected in the frequencysize plots (analogous to the Gutenberg-Richter relation),where only the larger force drop events are sensitive to theboundary conditions. For those large-DFk events, the shapeof the power law tail depends on the choice of boundarycondition, suggesting that small-DFk events do not extendfar enough spatially to be affected by the boundaries. Third,the minimum scale for Gutenberg-Richter like behaviorappears to be set by aminimum granular rearrangement scale.[37] The experiments inspire a force chain description of

earthquake populations. In this description, as proposed byearlier earthquake seismology studies, changes in frequencysize statistics for natural earthquakes are caused by theextent to which the ruptures interact with the limits of theseismogenic zone. The experiments also bear on the geo-logic interpretation of natural shear zones. Namely, theexperiments demonstrate that deformation can occur withinwide gouge zones even after localized slip zones begin toaccommodate coseismic slip; such behavior is presumablyrelated to the long-range correlations induced by forcechains. Our experiments thus provide a conceptual linkbetween earthquake distributions and structures withingeologic fault zones in that scale dependence in each hasunderlying granular controls.

[38] Acknowledgments. We thank Peter Malin (University of Auck-land, New Zealand) for being instrumental in the genesis of these experi-ments and in particular for urging us to examine moment-scaling relations;Robert Behringer and Peidong Yu (Duke University) for discussionssurrounding the development of the experiments; and NCSU undergraduatestudent Greg Gibson for assistance in developing the experimental appa-ratus. Chris Marone, an anonymous referee, and Associate Editor EmilyBrodsky contributed substantial improvements during the revision of themanuscript. K.D. received support from North Carolina State Universityand NSF-DMR 0644743. N.H. received support from NSF-OCE 0222154(to Jeffrey Karson) and the University of Texas Institute for Geophysics ofthe Jackson School for Geosciences.

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K. E. Daniels, Department of Physics, North Carolina State University,

Box 90305, Raleigh, NC 27695, USA. ([email protected])N. W. Hayman, Institute for Geophysics, University of Texas at Austin,

Box 8202, Austin, TX 78758, USA. ([email protected])


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force chains in seismogenic faults visualized with...

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