Frictional response of a thick gouge sample: 2. Friction ... · empirical friction law capable of accounting for this slip-weakening behavior. We begin by providing a mathematical

Frictional response of a thick gouge sample:

2. Friction law and implications for faults

Guillaume Chambon,1,2 Jean Schmittbuhl,1,3 and Alain Corfdir4

Received 25 July 2004; revised 27 February 2006; accepted 19 May 2006; published 21 September 2006.

[1] On the basis of experimental results, we propose a new friction law aiming atdescribing the mechanical behavior of thick gouge layers. As shown in the companionpaper, the dominant effect to take into account is a significant slip-weakening processactive over decimetric slip distances. This slip weakening is strongly nonlinear and,formerly, does not involve any characteristic length scale. The decrease of the gougefriction coefficient m with imposed slip d is well modeled by a power law: m = m0 + ad�b,with b = 0.4. On this major trend are superimposed second-order velocity-weakeningand time-strengthening effects. These effects can be described using classical rate- andstate-dependent friction (RSF) laws and are associated with a small length scaledc � 100 mm. Consistent with the general RSF framework, we combine slip-weakeningand second-order effects in a slip, rate, and state (SRS) friction law with two statevariables. We then compute the fracture (or breakdown) energy Gc and the apparentweakening distance Dc

app associated with the slip-weakening process. Once extrapolated torealistic ‘‘geophysical’’ confining pressures, the obtained values are in excellent agreementwith those inferred from real earthquakes: Gc � 5 � 106 J m�2 and Dc

app � 20 cm. We alsofind that fracture energy scales with imposed slip in our experiments: Gc � d0.6.Citation: Chambon, G., J. Schmittbuhl, and A. Corfdir (2006), Frictional response of a thick gouge sample: 2. Friction law and

implications for faults, J. Geophys. Res., 111, B09309, doi:10.1029/2004JB003339.

1. Introduction

[2] Real faults generally consist of complex tridimensionalinterfaces comprising thick, metric layers of cataclastic gougeand damaged rocks [e.g., Chester et al., 1993; Chester andChester, 1998; Micarelli et al., 2003]. In modeling studies,however, these thick structures are usually treated as perfectlythin interfaces, and their mechanical properties reduced to aneffective friction law [e.g.,Campillo et al., 2001; Aochi et al.,2002; Uenishi and Rice, 2003; Lapusta and Rice, 2003]. Therole of the friction law is to prescribe the evolution of the faulteffective coefficient of friction as a function of relevantphysical parameters: slip, slip rate, asperity status, faulthistory, fault morphology, etc. In particular, the friction lawshould be able to describe the physical mechanisms respon-sible for fault weakening during initiation and development ofseismic instabilities (earthquakes).[3] Two principal forms of friction laws coexist in the

literature: rate- and state-dependent friction (RSF) laws and

slip-weakening laws. In RSF formulation, friction dependson the slip rate and on a set of variables characteristics ofthe ‘‘state’’ of the frictional interface [e.g., Marone, 1998a;Scholz, 1998]. This type of laws has been formulated on thebasis of numerous experimental results, and appears to beapplicable for a wide range of materials [Dieterich, 1979;Ruina, 1983; Dieterich and Kilgore, 1994]. On the otherhand, slip-weakening laws prescribe that the coefficient offriction essentially depends on slip displacement [e.g.,Palmer and Rice, 1973]. Slip-weakening laws are lessfrequently used than RSF laws in the context of laboratoryfriction experiments (see, nevertheless, Ohnaka and Shen[1999]). However, owing to their relatively easy numericalimplementation, slip-weakening laws are commonlyemployed for earthquake modeling.[4] RSF and slip-weakening laws are frequently consi-

dered as competing [e.g., Ohnaka, 2003]. It is true that themost classical RSF law, namely, the Dieterich-Ruina law,describes processes that are purely time- and velocity-dependent. It is thus unable to account for slip-dependentmechanisms. However, as shown by Cocco and Bizzarri[2002], Dieterich-Ruina law reduces to an effective slip-weakening law during a single rupture event [see alsoOkubo and Dieterich, 1984]. Furthermore, the generalRSF formulation is highly flexible, and can easily be castto include, for instance, true slip-weakening processes[e.g., Nakatani, 1998]. Hence, in principle, RSF andslip-weakening formulations are not incompatible.[5] In the companion paper by Chambon et al. [2006a],

we report on experiments conducted with an annular simple

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, B09309, doi:10.1029/2004JB003339, 2006ClickHere

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1Laboratoire de Geologie, Ecole Normale Superieure, Paris, France.2Now at Cemagref, Research Unit ETNA, Domaine Universitaire,

Grenoble, France.3Now at Institut de Physique du Globe de Strasbourg, Strasbourg,

France.4Centre d’Enseignement et de Recherche en Mecanique des Sols, Ecole

Nationale des Ponts et Chaussees/Laboratoire Central des Ponts etChaussees, Institut Navier, Champs sur Marne, France.

Copyright 2006 by the American Geophysical Union.0148-0227/06/2004JB003339$09.00

B09309 1 of 12

http://dx.doi.org/10.1029/2004JB003339

shear apparatus (ACSA) that allows us to shear 10-cm-thickgranular samples over plurimetric slip displacements. Theresponse of synthetic gouge samples (0.6 or 1 mm angularquartz sand) in this setup is dominated by a significant slip-weakening behavior active over decimetric slip distances.Surprisingly, this slip weakening proved independent ofgouge grain size (at least in the diameter range that weinvestigated). We checked carefully that it does effectivelyconstitute a rheological property of the gouge, and is notaffected by the particular geometry of our setup. Micro-structural observations reveal that this slip weakening isprobably caused by a slow mechanical decoupling betweenthe shear band and the rest of the material. Hence the use ofthick gouge samples appears as a critical condition for thisslip weakening to occur (at least for the ranges of confiningstress and slip velocity used in our experiments).[6] In the present paper, our objective is to derive an

empirical friction law capable of accounting for this slip-weakening behavior. We begin by providing a mathematicalmodeling of the slip-induced friction decrease. We thendiscuss the existence of two distinct second-order effectsinfluencing sample strength. All these properties are thenintegrated into a generic friction law, formulated accordingto the RSF framework. Finally, we compare the predictionsof this law to seismological data (in terms of dissipatedfracture energy and apparent weakening distance). Note thatall the experimental results discussed below have beenobtained with 1 mm sand.

2. Power Law Slip Weakening

[7] We have shown in the companion paper [Chambon etal., 2006a] that the slip-weakening process active in ourexperiments is reversible. It can be virtually reset by so-called ‘‘restrengthening events’’ (namely, sense changes orshear stress releases), and reproducibly repeats during shearphases following these events. Accordingly, the relevantvariable to describe this slip weakening is partial slip dp,defined as the amount of slip undergone by the sample sincethe last restrengthening event. The connection between realfaults and these notions of restrengthening events andpartial slip is discussed in section 6.[8] Figure 1 shows that the postpeak decrease of shear

stress t with partial slip dp follows a linear path in log-logcoordinates. This property appears particularly evidentduring initial shear phases (IS): the linear decrease can thenbe observed over more than 2 slip decades. Though gener-ally over a smaller range of scales, the same feature isexhibited by all the shear phases that we conducted,regardless of the preceding shear history applied to thesample. The slip-weakening process can thus be modeled bya power law of the form

t dp� �

¼ tr þ a dp � d0� ��b

; ð1Þ

where tr represents the asymptotic shear stress value, d0 isthe particular displacement for which the law is singular,and a and b are two parameters. Remarkably, the exponentb appears to be systematically equal to 0.4 ± 0.05 for all thestudied shear phases (Figure 1). Hence this exponentappears characteristic of the slip-weakening process.

Currently, we have not found any satisfactory micromecha-nical explanation for the particular value of b (nor for thevalues of the other parameters involved in expression (1)).[9] The asymptotic shear stress tr does also remain

approximately constant for all the studied shear phases (atconstant confinement se): tr � 0.2 MPa for se = 0.5 MPa[see also Chambon et al., 2006a, Figure 7]. On the contrary,nonlinear fitting of our data show that the parameters a andd0 do vary with the considered phase. In particular, d0,which is systematically negative, tends to decrease when thetotal cumulative slip dcum undergone by the sampleincreases. (However, no clear trend is observed, probablydue to experimental noise). For potential users of our law, inFigure 2 we quote typical values of a and d0 applicable tothe case of an initial shear phase. Note finally that thesingularity for dp = d0 in expression (1) does not constitute alimitation, since it always remains outside the validitydomain of the power law (d0 < 0 while the law only appliesafter stress peak, i.e., for dp > 5–20 mm).[10] As shown in Figure 2, power law (1) provides a very

accurate representation of the observed slip-weakeningprocess, including in the immediate vicinity of the shearstress peak. Comparison between Figures 2a and 2b alsoillustrates the multiscale feature of the slip-weakeningprocess. The progressive decrease of shear stress is ex-tremely slow and prolongs over very large quantities of slip.From a formal standpoint, a power law such as (1) is scaleinvariant, i.e., it does not involve any characteristic slip

Figure 1. Logarithmic plot of shear stress t versus partialslip dp during five different shear phases (confinement se =0.5 MPa, slip velocity v = 83 mm s�1). The type of thephases is labeled as in the companion paper [Chambon etal., 2006a]: IS, initial shear phase; SD, shear phasefollowing a prescribed shear stress release; SR, shear phasefollowing a change of the shear sense. For the sake ofclarity, the three types of shear phases have been artificiallyseparated on the plot (vertical translations). Cumulative slipdcum undergone by the sample at the beginning of thedifferent phases is given in legend. The values of tr and d0used for each phase are different, and were obtained from apower law fit using expression (1). Goodness of this fit isillustrated by the dashed line with slope �0.4.

B09309 CHAMBON ET AL.: FRICTION OF THICK GOUGE SAMPLES, 2

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scale. Such a scale invariance is consistent with theobservation that slip weakening is independent of gougegrain size [see Chambon et al., 2006a]. As we explain insection 5, this property of scale invariance also has funda-mental implications regarding in particular the apparentfracture energy dissipated by the slip-weakening process.

3. Rate and Time Effects

[11] In terms of magnitude, the major slip-weakeningprocess represents by far the dominant effect in our experi-ments. Nevertheless, the frictional strength of our syntheticgouge samples appeared to be also sensitive to otherparameters such as slip speed and hold time. Severalconventional velocity-stepping and slip-hold-slip tests[e.g., Marone, 1998a] have been conducted, whose resultsare presented in Appendix A. Here, we only recall thatthese experiments allowed us to detect two second-orderprocesses, namely, a velocity-weakening behavior and atime-strengthening behavior.[12] Such rate and time dependence of frictional strength

is a common observation when investigating the response ofrock-rock interfaces or of thin gouge layers sandwichedbetween rock blocks [e.g., Dieterich, 1979; Marone et al.,1990; Beeler et al., 1996; Mair and Marone, 1999]. It isgenerally interpreted in the framework of classical rate- andstate-dependent friction (RSF) laws [Marone, 1998a]. How-ever, rate- and state-dependent behaviors are rarely reportedin experiments involving thicker granular samples, either inthe soil mechanics or in the granular physics communities[Hungr and Morgenstern, 1984; Groupement de RechercheMilieux Divises, 2004]. Let us nevertheless mention therecent studies of Tika et al. [1996], Karner [2002], and

Coste [2004]. These authors report on the shearing of thickgranular samples in rotary and triaxial setups, and doevidence a shear rate influence comparable to our results.[13] In our case, and despite the large thickness of our

gouge samples, it appears that both the velocity and the timedependences that we observe are consistent with the pre-dictions of classical RSF friction laws (see Appendix A).Even the values of the RSF constitutive parameters appearin good agreement with previous gouge friction studies[Marone, 1998a]. We find that the RSF characteristic slipscale dc is of the order of 100 mm, and that the velocity-weakening parameter B � A is of the order of 10�2.

4. An Extended Rate- and State-DependentFriction Law: SRS Friction Law

[14] Though consistent with the second-order velocity-weakening and time-strengthening effects that we report,classical RSF laws are unable to account for the dominanteffect in our experiments, that is for the major slip-weakening process. In what follows, we thus propose anextended rate- and state-dependent formulation, the slip,rate, and state (SRS) friction law, that is appropriate todescribe all our results.[15] When applied to granular gouges, the general RSF

theory developed by Ruina [1983] states that internalfriction m depends on slip velocity v and on a set of statevariables. The classical Dieterich [1979] law involves asmall number of such state variables Qi, typically one ortwo. These are usually considered as representative of theintergranular contact state or of the consolidation state of thegouge [e.g., Sleep, 1997]. In Dieterich law, all the statevariables Qi generally play similar roles and are governedby similar evolution equations [see also Gu et al., 1984;Tullis and Weeks, 1986]. Several authors, however,proposed to complement Dieterich law with supplementary,specific state variables in order to account for additionaleffects like thermal weakening, slip weakening, etc. [e.g.,Blanpied et al., 1998; Nakatani, 1998].[16] Here we follow this framework and introduce a

specific state variable L to describe the slip-weakeningprocess observed in our experiments. We propose thefollowing SRS friction law:

m ¼ m? þ A lnv

v?

� �þQþ L: ð2Þ

In the above expression, the logarithmic velocity depen-dence and the state variable Q are the usual Dieterich[1979] terms. They account in particular for the second-order velocity-weakening and time-strengthening effects.Among the various expressions proposed in the literature togovern the evolution of Q, the most widely used is Ruina’s[1983] aging law:

dQdt

¼ Bv?

dce�Q=B � v

v?

� �: ð3Þ

As already mentioned (see Appendix A), we checked thatthis law indeed provides an accurate fit of our velocity-stepping results.

Figure 2. (a) Illustration of the data fit provided byexpression (1). The gray curve represents the measured shearstress weakening during an initial shear phase (confinementse = 0.5 MPa and slip velocity v = 83 mm s�1). The dashedcurve corresponds to the power law (1) with b = 0.4 and thefollowing fitting parameters: tr = 0.16 MPa, d0 = �2.6 �10�3 m, and a = 5.6 � 10�2 MPa m0.4. (b) Zoomed-inportion of the plot for low values of partial slip dp.


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[17] Regarding the new state variable L, its evolutionequation directly derives from power law (1): L = C[(dp �d0)/d?]

�b, and hence

dLdt

¼ � bjvjd?C1=b L

1þ1=b; ð4Þ

where C is a new constitutive parameter complementing thetwo classical RSF parameters A and B, and d? is anormalization factor. The proportionality between the timederivative of L and the absolute velocity jvj is the hallmarkof the slip-induced character of the ongoing process. Wealso note that since b > 0, equation (4) has an evident fixedpoint: Lss = 0. However, unlike for the classical variable Q[Ruina, 1983; Gu et al., 1984], linearization is not possiblearound this fixed point. This reflects the absence of acharacteristic distance in the way L approaches its steadystate.[18] To render its physical meaning more straightforward,

the state variable Q in expression (2) is frequently replacedby an auxiliary variable q according to Q = Bln(q/q?). Thisnew variable evolves following

dqdt

¼ 1� qvdc

ð5Þ

and corresponds to an average contact time between twoasperities of typical size dc [Ruina, 1983]. Similarly, wemay define, instead of L, a new variable l = dp � d0 whichis directly representative of the partial slip undergone by theinterface. It is governed by the evolution equation:

dldt

¼ jvj: ð6Þ

268 Finally, the SRS friction law (2) can be rewritten in terms ofthese new variables:

m ¼ m? þ A lnv

v?

� �þ B ln

qq?

� �þ C

ll?

� ��b

; ð7Þ

with l? = d?.[19] Both equation sets (2)–(3)–(4), and (7)–(5)–(6)

constitute comprehensive friction laws. They can be used

to compute the postpeak evolution of the friction coefficientm during a given shear phase. In its current form, our SRSfriction law does not directly account for the restrengtheningmechanisms induced by sense changes or shear stressreleases. As a consequence, one should ‘‘manually’’ pre-scribe the initial value li of the state variable l at thebeginning of each considered shear phase. Typically, li is ofthe order of 10 mm for an initial shear phase (Figure 2), andof a few hundreds of mm for phases following restrengthen-ing events (it increases with cumulative slip dcum). Thevalues of the other constitutive parameters are given inTable 1. Recall that C does depend on the considered shearphase (and hence on the whole sample history). Neverthe-less, this slip-weakening coefficient remains systematicallymuch larger than the velocity-weakening coefficient B � A.

5. Weakening Parameters

[20] In seismological studies, the weakening of frictionduring earthquakes is usually quantified in terms of twolinked parameters: the fracture energy Gc and the charac-teristic weakening displacement Dc. To examine whetherour laboratory results could be extrapolated to real faults,we compute below the values of Gc and Dc associated withthe dominant slip-weakening process of our experiments.

5.1. Apparent Fracture Energy

[21] The fracture (or breakdown, see Tinti et al. [2005])energy Gc corresponds to the specific energy dissipated inexcess of the frictional work during a weakening process. Inour case, it can be easily computed for each shear phase byintegrating the t versus dp relationship [Rice, 1980; Ohnaka,2003]:

Gc ¼Z df

dit dð Þ � t df

� �� dd; ð8Þ

where df represents the partial slip at the end of theconsidered shear phase and di is defined by t(di) = t(df)(Figure 3b).[22] Figure 3a shows that the fracture energy Gc dissi-

pated during a shear phase increases linearly with theapplied confining pressure se. This behavior was expectedsince shear stress t itself does vary linearly with se for allvalues of partial slip dp [Chambon et al., 2002, 2006a]. Aswe demonstrate in section 5.3, Gc also depends on theamount of slip imposed during the considered shear phase.For a 0.6-m-long shear phase with se = 0.5 MPa, we findGc � 2 � 104 J m�2 (Figure 3a). With the same confiningpressure but for a 1.5-m shear phase, we find Gc � 5 �104 J m�2.

5.2. Apparent Characteristic Weakening Displacement

[23] The characteristic displacement Dc represents thetypical amount of slip required by a weakening process tocomplete. As already mentioned, the power law slip weak-ening evidenced in our experiments does not involve anycharacteristic slip scale. Hence only apparent weakeningdisplacements can be defined. For instance, from plainobservation of Figure 2a, we can estimate that most of thestrength drop during a 1.5-m shear phase occurs over Dc �50 cm. More quantitatively, an apparent weakening dis-

Table 1. Typical Values of the Parameters Involved in Our SRS

Friction Lawa

Parameter Value

B � A 10�2

C 0.7 (0.7–1.0)dc, mm 100b 0.4

aValues of m?, v?, and q? are not given since these are normalization

factors. Because of limited data resolution, only the difference B � A is

given. For practical purposes, the reader may choose values of A and B

consistent with the literature (such as those cited in Figure A2). The

coefficient C has been computed from a (see section 2) by taking l? =

10 mm and se = 0.5 MPa. We give both the value appropriate for initial

shear phases and the range of values obtained for phases which follow

restrengthening events (in parentheses).


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placement Dcapp can be derived, for each shear phase, from

the dissipated fracture energy [Rice, 1980]:

Dappc ¼ G0

c

t dpeak� �

� t df� � ; ð9Þ

where dpeak represents the partial slip at the peak of the tversus dp curve. Fracture energy is denoted G0

c in (9)because, rigorously, it has to be computed between dpeak anddf, and not between di and df as in expression (8).[24] Figure 4 shows that unlike the fracture energy, the

apparent weakening displacement Dcapp is roughly indepen-

dent of the applied confining stress se. This feature mightbreak down at very low confining stress (se < 0.2 MPa), butadditional data would be needed to confirm this point. Onthe contrary, and as expected, the apparent weakeningdisplacement does depend on the imposed amount of slip:We find Dc

app � 14 cm for 0.6-m-long shear phases(Figure 4), and Dc

app � 25 cm for 1.5-m-long shear phases.

5.3. Comparison of Our Results WithSeismological Data

5.3.1. Values of Gc and Dc

[25] A recurrent concern in earthquake mechanics is themajor quantitative disagreement between the values of Gc

and Dc found in the lab, and those inferred for real faults.The process of friction weakening active on faults duringearthquakes can be inverted from seismological recordsusing various methods. Owing to resolution issues, theseinversions generally yield better constraints on the fractureenergy Gc than on the characteristic weakening displace-ment Dc [Guatteri and Spudich, 2000; Spudich andGuatteri, 2004; Piatanesi et al., 2004]. For large earth-quakes (slip > 1 m), however, all the existing studies seemto converge to values of Gc in the range 10

6–108 J m�2, andto values of Dc in the range 0.5–1 m [Aki, 1979; Ide andTakeo, 1997; Bouchon et al., 1998; Peyrat et al., 2001;Ampuero, 2002; Ohnaka, 2003;Mikumo et al., 2003; Rice etal., 2005].[26] In contrast, the characteristic length scales generally

found in experimental friction studies lie in the range 10�3–1 mm [Marone et al., 1990; Beeler et al., 1996; Ohnaka andShen, 1999]. Care should be taken since the values reportedin most of these studies actually correspond to the charac-teristic distance dc estimated in the framework of classicalRSF laws. The associated weakening displacement Dc canbe up to 10–20 times larger, and even more in the presenceof fluid and thermal effects [Cocco and Bizzarri, 2002;Bizzari and Cocco, 2003, 2006a, 2006b]. Still, severalorders of magnitude systematically separate experimentaland seismological estimates of Dc. Similarly, the dissipatedfracture energy Gc reported in previous friction experimentsnever exceeds 101–102 J m�2 (for studies conducted underhigh confining stresses), far smaller than seismological data[Li, 1987; Abercrombie and Rice, 2005].[27] On the contrary, the new slip-weakening process

evidenced in our experiments compares fairly well with

Figure 3. (a) Evolution of fracture energy Gc as a functionof applied confining stress se. Each point corresponds to a0.6-m-long shear phase occurring after a prescribed sensechange (SR phases). Error bars are estimated using twoindependent realizations of each shear phase (except for thelowest value of se, for which we only have one exploitablerealization). The dashed line, whose equation is indicated,represents a linear regression of the data set. (b) Frictionalresponse during one of the 0.6-m shear phases used in themain panel (case se = 0.5 MPa). The fracture energy Gc

corresponds to the grayed area under the curve. Note that inactual computations, the term t(df) in expression (8) isreplaced by the average shear stress over the last 50 mm ofthe shear phase.

Figure 4. Evolution of the apparent weakening displace-ment Dc

app (calculated from the fracture energy usingexpression (9)) as a function of the applied confining stress.The points correspond to the same 0.6-m-long shear phasesas those presented in Figure 3. Error bars are estimatedusing two independent realizations of each shear phase.Unfortunately, we only have one exploitable shear phase forthe atypical point corresponding to se = 0.2 MPa. As aconsequence, the significance of this point cannot beassessed (question mark).


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seismological data (Figure 5). The apparent characteristicweakening displacement Dc � 20 cm is much larger than inprevious experimental studies, and fully consistent withvalues inferred for large earthquakes. The value of fractureenergy, Gc � 104 J m�2, is also much larger than in previousstudies, but yet a little lower than seismological data. Recallhowever that our experiments were conducted at very lowconfining stress se, while Gc proved to be strongly depen-dent on this parameter. We thus performed the exercise ofextending the linear relationship displayed in Figure 3atoward a realistic confinement level se = 100 MPa. Inter-

estingly, this yields a value of Gc = 5 � 106 J m�2 which isfully consistent with seismological inferences (Figure 5).Obviously, such an extrapolation of our results is highlyspeculative, and further work would be needed to check thatthe power law slip-weakening mechanism effectivelyremains active at high values of se (see also section 6).Nevertheless, we believe that the agreement found at thisstage between our laboratory results and seismologicalfriction data is remarkable [see also McGarr et al., 2004].5.3.2. Scaling Laws[28] Also interesting is the comparison between the

scaling properties predicted by our friction law and thoseinferred from seismological data. Because of the multiscalefeature of the slip-weakening process observed in ourexperiments, both the fracture energy Gc and the character-istic weakening displacement Dc continuously evolve withpartial slip dp. Integrating power law (1), one finds thefollowing scaling relationship between Gc and dp (in thelimit dp ! 1):

Gc � dpz; ð10Þ

where z = 1 � b = 0.6. The same relationship also holds forthe apparent weakening displacement Dc

app:

Dappc � d0:6p : ð11Þ

Accordingly, neither Gc nor Dc should be regarded asintrinsic material parameters in our experiments. The valuesreported in Figure 5 only represent apparent weakeningparameters appropriate for the case dp � 1 m.[29] Similarly, various seismological studies have indi-

cated that fracture energy Gc on real faults does not behaveas a material parameter. From a wide data compilation, Ideand Beroza [2001] showed that the energy radiated byearthquakes is roughly proportional to the seismic momentM0 over more than 14 orders of magnitude in M0. Togetherwith the classical results that earthquake stress drop andapparent stress are independent of M0 [Abercrombie, 1995;Ide et al., 2003], this observation implies that the fractureenergy should be roughly proportional to earthquake slip s:Gc � s. In an independent study, Abercrombie and Rice[2005] directly evaluated the fracture energy of variousearthquakes, and ended up with a similar, though slightlydifferent, scaling relationship: Gc � s1.3. This value of 1.3for the scaling exponent is also supported by the results ofRice et al. [2005]. A recent study by Tinti et al. [2005]suggests that this exponent might even be larger (close to 2).[30] Hence, just as in our experiments, the fracture energy

inferred from real earthquakes displays multiscale propertiesand increases significantly with total slip. This resultstrongly contrasts with the predictions of classical RSFlaws, for which Gc and Dc essentially constitute materialparameters [e.g., Marone, 1998a; Madariaga and Olsen,2002]. Furthermore, we note that the scaling relationshipsdeduced from seismological data are similar to that derivedfrom our experiments (see equation (10)). (We implicitlyassume here that partial slip dp is analogous to fault slip s,which is discussed in section 6.) One could argue that ourexperimental scaling exponent z = 0.6 is lower than theseismological values, generally comprised in the range

Figure 5. Relationship between fracture energy Gc andcharacteristic weakening displacement Dc. This plot,extracted from Figure 7 of Ohnaka [2003], is a compilationof both experimental data (coming from various friction andfracture studies) and seismological inversions. We addedtwo points to the original figure: the raw data pointrepresenting our experimental data with se = 0.5 MPa(‘‘ACSA raw result’’), and the data point obtained whenextrapolating our results to a realistic geophysical confine-ment level se = 100 MPa (‘‘ACSA extrapolated result’’).


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1.0–1.3. It is not clear to us whether this difference reallyentails a physical meaning, or whether it could be attrib-uted to the insufficient resolution of earthquake inversions.The precise value of the error bar attached to seismologicalestimates of z is difficult to assess. We have checked,however, that a scaling exponent of 0.6 would be reason-ably compatible with the data presented by Abercrombieand Rice [2005] [see also Chambon et al., 2005].

6. Discussion: Extrapolation of Our Results toReal Faults

[31] Section 5 shows that our results represent a signifi-cant breakthrough in the prospect of bridging the quantita-tive gap between laboratory and seismological friction data.Not only our experiments give rise to fracture energies andcharacteristic weakening displacements consistent with seis-mological inferences, but they also reproduce the scalingrelationships observed between these parameters and earth-quake size. Can we conclude from this spectacular agree-ment that the friction weakening mechanisms along realfaults are of the same type than in our samples?

6.1. Analogy With Faults

[32] Let us first examine the conditions leading to theappearance of large-scale slip weakening in our experi-ments. As explained in the companion paper [Chambon etal., 2006a], the most critical ingredient is the use of thickgouge samples. Slip weakening is strongly related to theexistence of a thick bulk zone surrounding the shear band,and to the mechanical coupling between these two struc-tures. The second important condition is that the gougelayer should have underwent a perturbation sufficientlystrong to erase any influence of previous shear episodes.In our experiments, such perturbations are achieved throughso-called restrengthening events, namely, changes of theshear sense or relaxations of the shear stress.[33] Turning now to real faults, we recall that they

generally comprise cataclastic and damage layers whosethickness can reach hundreds of meters [Chester et al.,1993; Sibson, 2003]. Furthermore, we show in anotherpaper [Chambon et al., 2006b] that the structuring of oursheared samples strikingly resembles that of real faultzones. The restrengthening events imposed in our experi-ments do not have evident analogues on the field. Never-theless, numerous processes active during the stoppingphase of earthquakes as well as during the interseismicperiod probably contribute to disturb and ‘‘reset’’ the gougestructure. We think for instance to dynamic shaking due tostill-slipping patches, to chemical cementation, or to inter-seismic creep. Accordingly, postulating that the slip-weak-ening process active in our experiments might be active alsoon real faults appears quite reasonable. One could argue thatour sand samples are far from being representative of realgouge (monodisperse size distribution, no cohesion, nofluid). To us, this material constitutes nevertheless a goodfirst ‘‘approximation’’ to account for the complexity andheterogeneity of natural materials.

6.2. Open Questions

[34] In spite of the above considerations, a direct extrap-olation of our results to real faults would still remain

debatable. The main questions arise from the relatively‘‘gentle’’ conditions at which our experiments are con-ducted. The confining pressure, first, is very low comparedto realistic, geophysical values (in the range 10–100 MPa).We may argue that since the slip-weakening mechanism inour experiments primarily depends on the structuring of thethick gouge layer, it should remain active at higher con-finement levels. However, this point clearly deserves to beproperly investigated. We are aware, in particular, that somestudies reported a transition from strain-softening to strain-hardening behavior with increasing confining pressure insandstones and sand samples [Wong et al., 1997; Karner etal., 2003]. Such results could suggest, on the contrary, aprogressive disappearance of our slip-weakening process athigh values of se. None of these studies, however, pro-longed over sufficiently large slips to be compared to ours.[35] Second, the slip velocities imposed in our experi-

ments are orders of magnitude below those characteristic ofseismic rupture (around 1 m s�1). As frequently pointed out,it is likely that high slip velocities initiate various specificweakening processes involving, for instance, rock melting,off-fault damage, or thermal pressurization [e.g., Tsutsumiand Shimamoto, 1997; Lapusta and Rice, 2003; Andrews,2005; Bizzari and Cocco, 2006a, 2006b]. We note howeverthat a significant slip-weakening effect, apparently verysimilar to ours, has recently been reported at higher butstill subseismic slip speeds (up to 100 mm s�1) in rock-rockfriction experiments [Goldsby and Tullis, 2002; Di Toro etal., 2004]. This was interpreted as resulting from theformation of a lubricating layer of ultracomminuted gelalong the frictional interface. We do wonder, however,whether a mechanism similar to what happens in ourexperiments, namely, the formation of a relatively thickdamaged layer around the frictional interface, could also berelevant. In this case, the similarity between the afore-mentioned results and ours would suggest that the slip-weakening process of our experiments may remain active,and presumably dominant, during at least the wholenucleation phase of earthquakes.[36] Finally, let us also mention the particular, annular

geometry of our experimental setup. In the companionpaper [Chambon et al., 2006a], we explicitly checked thatthe mechanical response of the samples was unaffected bypotential artifacts such as development of hoop stresses.We also showed that the slip-weakening process wasindependent of gouge grain size. However, we cannotexclude that our results are dependent on some internallength scale of the setup, like the inner cylinder radius forinstance. This possible scale dependence would need to beinvestigated in detail prior to extrapolating our friction lawto fault scale. Recall however that our slip-weakeningprocess does not involve any characteristic scale. Henceany first-order scale dependence would at most concernthe prefactors of our friction law (typically, the parameterC in equation (7)).

6.3. Use of Nonlinear Slip-Weakening Laws

[37] Because of all these open questions, it would cer-tainly be premature to conclude that friction weakeningalong real faults is caused, as in our experiments, by aprocess of mechanical decoupling inside the gouge layer.Let us just say that given the excellent agreement between


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our results and seismological data, this mechanism repre-sents a good candidate. In addition, and regardless of theunderlying physical mechanisms, we also believe that theform of friction law derived from our experiments (namely,a strongly nonlinear law devoid of any characteristic slipscale) appears very promising for earthquake modeling. Theuse of such slip-weakening laws has already been advocatedby a few authors on the base of purely seismologicalconsiderations [Abercrombie and Rice, 2005; Ampuero,2002; Rice and Cocco, 2006]. Subtle differences existbetween the various formulations already proposed in theliterature and ours, regarding for instance the values of thepower law exponents (see section 5.3) or the existence of anasymptotic shear stress level for large slip. From a generalperspective, however, all these laws share common prop-erties which could instigate deep changes in our under-standing of earthquake mechanics. For instance, since theyinvolve a whole hierarchy of length scales, these lawscould provide a framework to understand the complexityof earthquake distributions on faults, a longstanding diffi-culty with classical friction laws [Ben-Zion and Rice,1997; Shaw and Rice, 2000]. Because of the absence ofa characteristic frictional length scale, these laws couldalso have important effects on the modeling of the nucleationphase [Campillo and Ionescu, 1997; Ampuero et al., 2002;Uenishi and Rice, 2003].[38] To finish, we want to insist that accounting for a slow

slip-weakening process in the friction law is not necessarilyincompatible with more classical RSF effects. On thecontrary, both slip weakening and Dieterich-Ruina termscoexist in expression (2). Such a coexistence will undoubt-edly induce further complexity in the earthquake nucleationprocess, with presumably dominating RSF mechanismsduring the first microns of slip then relayed by slip weak-ening. Note however that the classical RSF effects appear tosignificantly evolve with cumulative slip dcum in our experi-ments. The velocity-weakening effect vanishes, on average,when the cumulative slip increases (Figure A2b), whereasthe logarithmic aging during hold time tends, on thecontrary, to be reinforced (Figure A4). The persistence ofclassical RSF effects in pervasively sheared gouge layersthus appears problematic and should be investigated further.

7. Conclusions

[39] We performed gouge shearing experiments using anovel annular simple shear apparatus. Our setup, as well asa detailed review of our main experimental results, arepresented in the companion paper [Chambon et al.,2006a]. In this paper, we attempted to formalize the me-chanical behavior of our thick gouge samples through a newfriction law, the SRS friction law (Slip, Rate, and Statefriction law). This law conforms with the general frame-work of rate- and state-dependent friction. Its main featurescan be summarized as follows:[40] 1. Friction evolution is dominated by a significant

slip-weakening process which is devoid of any characteris-tic slip scale. This process is represented by a dedicatedstate variable which decreases with slip according to apower law.[41] 2. Slip-weakening coexists with second-order

velocity-weakening and time-strengthening effects. These

effects are well consistent with classical RSF laws and arethus described by standard Dieterich-Ruina terms. They areassociated to a characteristic slip distance dc � 100 mm.[42] 3. The fracture (or breakdown) energy Gc and

apparent weakening displacement Dcapp associated with the

slip-weakening process are in good quantitative agreementwith those inferred for real faults. Furthermore, owing to itspower law character, the slip-weakening process alsoaccounts for the seismological scaling relationships betweenfracture energy and earthquake size.[43] Our SRS friction law could thus allow for recon-

ciling the longstanding discrepancy between laboratoryand earthquake weakening parameters. More precisely,our results show that neither Gc nor Dc should beregarded as intrinsic material properties since they bothevolve with total slip. We expect that the non linear slip-weakening process observed in our experiments may helpsolving some important seismological issues concerning,for instance, earthquake nucleation or earthquake com-plexity. However, proper extrapolation of our experimen-tal results to realistic fault conditions will require furtherwork.

Appendix A: Results From Velocity Steppingand Slip-Hold-Slip Experiments

A1. Rate Effects

[44] Figure A1 shows that prescribed velocity changesduring a shear phase induce reproducible variations in shearstress. In particular, sudden increases in slip velocity v from1.67 to 100 mm s�1 trigger sharp decreases in shear stress t(Figure A1b), which indicates a velocity-weakening behav-ior. Yet, velocity-induced variations in shear stress neverexceed a few percents, and sometimes hardly emerge from

Figure A1. (a) Evolution of shear stress t during an initialshear phase involving prescribed changes in slip velocity v(confinement se = 0.5 MPa). (b) Close-up of one portion ofthe curve. Each small-scale jump in t corresponds to avelocity change. We imposed a periodic succession of fourvelocity levels whose values are given in the plot (length ofeach velocity step is 20 mm).


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the noise level. Velocity weakening thus constitutes asecond-order process compared to the major slip-weaken-ing trend. This is particularly evident when both processesare superimposed, like in Figure A1a.[45] As mentioned by Chambon et al. [2002], the results

of our velocity-stepping experiments appear in good agree-ment with classical rate- and state-dependent friction (RSF)laws. We recall in particular the two following points. First,as predicted by RSF laws, the evolution of shear stress uponprescribed velocity changes is not instantaneous, butextends over a few hundreds of mm (Figure A2a). Second,also consistent with RSF laws, the shear stress t at steadystate was found to decrease roughly linearly with thelogarithm of velocity v, despite a relatively large variability(Figure A2b).[46] To check further the relevance of classical RSF laws

to our velocity-stepping experiments, we attempted tomodel the shear stress evolution induced by the velocityjump presented in Figure A2a. We used a very simple one-dimensional model consisting of a slider governed by a one-state variable Dieterich friction law [e.g., Tullis and Weeks,

1986; Marone et al., 1990; Marone and Kilgore, 1993]. Asan evolution law for the state variable, we chose the so-called Ruina’s aging law (see section 4) which is supportedby numerous experimental studies [e.g., Beeler et al., 1994].Values of the friction parameters A, B, and dc compatiblewith our data have been determined by trial and error, withthe constraint of keeping B � A � 10�2 consistently withFigure A2b. Slider models with and without elasticity havebeen tested, the latter yielding the best results (Figure A2a).What however turned out an essential ingredient to accountfor, is the inertia of the motor. Typically, it takes about 20 sfor the velocity to effectively stabilize at its new level after aprescribed increase.[47] As shown in Figure A2a, Dieterich-Ruina friction

law does indeed provide a good fit for our experimentalresults. Apart from the few damped oscillations whichcannot be reproduced without elasticity, the overall shapeof the shear stress evolution is well modeled. In particular,the absence of a marked direct effect is satisfactorilyreproduced (no immediate friction increase when the veloc-ity is increased). This distinctive property of our data seemsessentially due to the inertia of the motor. Looking in detail,one can actually note that a small direct peak is discerniblein the simulated curve. This peak hardly emerges from theexperimental noise level, and thus does not affect theagreement between the model and the data. However, it ispossible that an even better agreement (i.e., a smaller directeffect) could be obtained with more refined models takinginto account, for instance, velocity-dependent compactionrates [Beeler and Tullis, 1997].

Figure A2. (a) Close-up on the evolution of shear stress t(solid curve) (normalized by the constant confining pressurese) during one particular velocity increase of the initialshear phase presented in Figure A1. Modeling result(dashed curve) corresponding to a one-dimensional slidersystem without elasticity governed by a Dieterich-Ruinafriction law with the following RSF parameters: A = 5 �10�3, B = 1.5 � 10�2, dc = 30 mm. Note that the actual slipresolution of our data is 77 mm [Chambon et al., 2006a]. Forthe modeling, however, they were interpolated down to aresolution of 10 mm. (b) Plot of the normalized shear stresst/se at steady state (slip evolution is removed) versus shearvelocity v in semilogarithmic scale (reproduced fromFigure 2c of Chambon et al. [2002]). Note that datadispersion tends to increase, and the velocity-weakeningtrend tends to vanish, when the cumulative slip dcumundergone by the sample increases. The slope of therepresented straight line yields an estimate of the RSFparameter A � B � �10�2 for low values of dcum.

Figure A3. (a) Evolution of shear stress t versus elapsedtime t during a slip-hold-slip experiment (initial shear phaseof the sample, confinement se = 0.5 MPa). The grayportions correspond to the prescribed hold periods at zerovelocity. Velocity v during the shear periods is 50 mm s�1.(b) Close-up on one particular hold period lasting 1000 s.The response during the hold itself is also shown. (Forillustrating purposes, the hold displayed does not come fromthe experiment presented in Figure A3a but from another,similar run.)


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[48] Because of the limited slip resolution of our data, aswell as of experimental noise, we cannot conclude onprecise estimates for the RSF constitutive parameters A,B, and dc. Values cited in Figure A2 are only indicative oforders of magnitude, in particular for what concerns thecharacteristic slip scale dc. Accounting for experimentalerror bars, we may conclude that this parameter is of theorder of 100 mm in our experiments.

A2. Time Effects

[49] Another typical feature predicted by classical RSFlaws is a logarithmic aging of sample strength during holdtimes [Beeler et al., 1994; Marone, 1998b]. In our case, itappears at first glance that shear stress t is completelyunaffected by hold periods of various durations intercalatedduring a shear phase (Figure A3a). The major weakeningprocess simply pauses during the zero velocity stages, andthen proceeds, without significant restrengthening whenshear is resumed. Incidentally, note that the frictionalresponse to plain hold times thus strongly differs from theresponse to holds accompanied by a prescribed shear stressrelease [see Chambon et al., 2006a]. Such an observationconstitutes a clear evidence that the major weakeningprocess in our results is effectively slip-, and not time-,induced.[50] In detail, one can nevertheless observe a small

influence of the hold periods on sample frictional strength(Figure A3b). First, the shear stress notably evolves (it

generally relaxes) during the first instants of the hold itself.This behavior is due to the elasticity of both the setup andthe sample [Beeler et al., 1994]. Second, and more impor-tantly, we also notice that the restart of shear after a holdperiod is accompanied by a slight peak in shear stress t(Figure A3b). Hold periods thus induce a second-orderrestrengthening effect which, though never exceeding afew percents of the prehold stress value, can systematicallybe distinguished from the noise.[51] When looking at data plotted versus slip, we can note

that the typical slip distance required for the posthold stresspeak to fade out is of the order of the estimate of dc obtainedin section A1, i.e., approximately 100 mm. (Thecorresponding plot is not presented here. Evaluation of thefading slip directly from Figure A3 is biased due to the inertiaof the motor.) Moreover, as shown in Figure A4, theamplitude of the second-order restrengthening clearlyincreases with the duration thold of the imposed hold. Forat least two of the displayed experiments, this increase isroughly logarithmic. The effect of hold time on our samplesthus appears, at least qualitatively, in good agreement withthe predictions of classical RSF laws.

[52] Acknowledgments. The experiments were conducted at theCERMES, ENPC/LCPC, France. We acknowledge P. Lerat, S. Roux,J. Sulem, J.-P. Vilotte, J. Dieterich, J. Rice, P. Bernard, T. Tullis, andA. Cochard for very fruitful discussions. The comments of M. Cocco,S. Karner, C. Marone, and two anonymous reviewers were of great help toimprove the presentation of the manuscript. G.C. and J.S. were partlysupported by the ACI programs ‘‘Jeunes Chercheurs’’, RNCC, and ALEASof the French Ministry of Education and by the program CTT of the FrenchNational Agency for Research.

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Figure A4. Evolution of the posthold restrengtheningamplitude Dpt (normalized by confining stress se) as afunction of hold duration thold in semilogarithmic scale.The quantity Dpt is defined as the difference between theposthold peak stress and the average stress level during thelast millimeter of slip preceding the hold (see Figure A3b).Error bars represent the typical amplitude of the noiseaffecting our friction measurements. The four data setscorrespond to four independent slip-hold-slip experiments.Note that the corresponding curves seemingly steepen asthe cumulative slip dcum undergone by the sample beforethe experiment increases. The slope of the straight lineyields an estimate of the RSF parameter B � 3 � 10�3 forlarge values of dcum.


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��G. Chambon, Cemagref, Unite ETNA, Domaine Universitaire, 2 rue de la

Papeterie, B.P. 76, F-38402 Saint-Martin d’Heres Cedex, France.([email protected])A. Corfdir, CERMES, ENPC/LCPC, Institut Navier, 6 et 8 avenue Blaise

Pascal, F-77455 Champs sur Marne, France.J. Schmittbuhl, Institut de Physique du Globe de Strasbourg, 5 rue Rene

Descartes, F-67084 Strasbourg Cedex, France.


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Frictional response of a thick gouge sample: 2. Friction ... · empirical friction law capable of accounting for this slip-weakening behavior. We begin by providing a mathematical