Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 106, NO. B7, PAGES 13,347-13,380, JULY 10, 2001 Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology Masao Nakatani Lamont-Dohcrty Earth Observatory of Columbia University, Palisades, New York Abstract. We observed slow frictional slip occurring at a constant shear stress below the nominal friction level and compared it with the time-dependent strengthening of the frictional interface, which was also tracked experimentally. t was found that slip velocity decreases the interface strengthens duc to aging, whil' it increases with the applied shear stress. These dependencies wcrc both exponential and were of similar magnitudes, as implied by the framework law of rate- and state-dependent friction. n the spirit of the adhesion theory of friction the dependence of slip velocity on interface strength is understood be the result of the change of the shear stress acting on frictional junctions duc to the change of junction population, though the observe dependence was somewhat stronger than a simple model based on this idea predicts. By correcting the observed slip velocity for the effect of the change of the interface strength, wc could obtain a unique relationship between stress and slip velocity, which may bc readily compared with a standard theological formulation. Thus the obtained relationship between stress and slip velocity showed a reasonable agreement with the absolute rate theory over a temperature range of øC for the present experimental condition (fmc albitc powder, 20 MPa normal stress, no pore water). 1. ntroduction A class of empirical friction laws called "rate- and statedependent friction (RSF) law" has been successfully applied to fault mechanics (e.g., see $cholz [1998] for a review). The law has been developed as a phenomenological description of second-order changes in friction observed in laboratory experiments. The general framework of the RSF law is to describe these changes as correction terms to the base level friction. Because of this phenomenological presentation, the RSF law has not readily lent itself to be envisioned in terms of the underlying physics, in contrast to usual rheological laws such as Hooke's law or flow laws. The principle objective of this paper is to recasthe general framework of RSF law into a more standard rheological perspective and to examine the physical processes that are then implied. 2. Conceptual Clarification of Rate- and State- Dependent Friction 2.1. Conventional View of Rate- and State- Dependent law: Friction Let us begin with the conventional presentation of the RSF St (= '/tr) = St, +O+aln(V/Y,). (1) By convention, we take St, to be the steady state friction at a reference velocity (Y,), and hence the state variable O is zero at the steady state sliding at V,. Also, for simplicity, we Copyright 2001 by the American Geophysical Union. Paper number 2000JB /01/2000JB $09.00 consider only the case in which normal stress (or) is constant. Though dimensionless due to the normalization by or, St., O, and a have meanings as shear strength or shear stress. The frictional coefficient St is defined as the shear stress (z') applied to the frictional interface normalized by the normal stress (or). However, St is often thoughto be more a "strength" than a "stress" and is regarded as the output of the law. This follows the traditional concept of "friction law" since Amonton's law, the principle role of which is to describe the shear strength as a property of a frictional interface. (n this classical friction law the interface does not slip if the applied stress is below the frictional strength, which is static friction if the interface is not previously slipping and is dynamic friction if the surface is already slipping.) n rigorous terms, stress and strength are equivalent in modem rate and state friction because (1) assumes that the frictional interface is always slipping. However, we believe that it has been largely the case that in discussing the RSF law, we have often regarded St in (1) as strength in the classical sense, not always being aware that St is actually the applied shear stress at the same time. n section 2.2, with an alternative representation of (1), we will argue for an alternative view where St is regarded more as the applied stress, while St, +O is regarded as an extension of the "frictional strength" in the classical sense. However, let us first reiterate the conventional view of rate and state friction. Equation (1) separates the variations in St into two different classes: the effect of the current slip velocity (Y), called the direct effect (the aln(v/v,) term), and the variations due to the other reasons (the O, or state variable [e.g., Ruina, 1983]). t would be fair to say that no meaning beyond this classification is implied by (1). With this conventional view, any variations in the observed friction (st) can be incorporated in the RSF 13,347

2 13,348 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON framework (equation(1)) by making 0 evolve properly, though in practice, only limited types of variations have been incorporated. One representativevolution law for 0 is do b exp - - V (2) dt = L/- ** Z ' which describe several typical experimentally known features including the following: First, O increases with the time of stationary contact as b n(t). Second, when slid at a constant velocity, O evolves to a steady value of -b n(//g.). Third, the change of O with slip occurs over a characteristic displacement of L (equation(2)is not exactly correct about this point, but is sufficient for now, see section 4 and Appendix A). Usually, the term "RSE law" refers to the combination of equations (1) and (2) and has been always tested together by comparing the predicted and observed changes of t (output)against a variety of slip velocity histories (input). Actually, much of the effort to understand the rate and state friction has been aimed at understanding the evolution laws. However, below we argue that (1), when viewed in a different way, is a clear testable proposition in itself and has an interesting meaning beyond just a classification method as mentioned above. n this paper, the term "RSF framework (law)" exclusively refers to (1). The conventional way of viewing it discussed above is referred to as RSF view. The term "evolution law" refers to (2) or its alternatives (e.g., see Marone [1998a] for a review). The evolution law and its physics are beyond this paper's scope, and thus we adopt the main stream views such as those reviewed by Marone [1998a] in this regard. However, some new points about evolution laws will be made in section 4 and Appendix A, as required to complete the experimental test of the framework law (equation(1)) as rigorously as possible. 2.2 Alternative View of Rate and State Friction: Stress- and Strength-Dependent Slip When solved for the slip velocity, (1) becomes rift -(#. a +O) 1 /(r,e) = g. exp. (3) shown that the magnitudes of o (ln and of o (!n /) / o ( / )1o, while each changing considerably with temperature, are always similar to each other. Though there is no a priori reason that o (ln /) / o l, and o (ln V) / o ( ' / a)lo have to be related, a physical model to link them will be shown in section 5. However, first, we discuss some characteristics of the alternative view of rate- and statedependent friction represented by (3) because we believe this allows some important aspects of modern friction to be grasped more intuitively. All the conceptual clarifications made below rely on one philosophy underlying the alternative view: Strength is distinct from stress. By noticing that the separation of t. and O is rather arbitrary, it may be more appropriate to think that (3) represents the dependence of slip velocity on t. + O rather than O. At any given value of O the quantity t. +O is actually the shear stress (normalized by ) required to cause the slip at velocity V., that is, frictional strength measured at a (suddenly) imposed reference velocity. n the conventional RSF view, t. + O is the frictional strength corrected to a reference velocity for the direct effect. n any case, t. + O is a quantity that exclusively belongs to the frictional interface (which may evolve) and is a dimensionless strength. n the strict definition of the terms, stress and strength are always the same in the situation we are talking about where the interface is always slipping even if at a miniscule speed. However, following the commonplace use of the word strength as a property of the interface (which can change with time), we propose to call the t. +O the "interface strength." n addition, t. + O can be regarded as the equivalent of the frictional strength in the classical sense, as discussed later in this section. Using the concept of interface strength, we can recast the RSF friction in an alternative framework that we refer to here as the "stress- and strenl th-dependent slip (SSS)." The framework law of SSS is (3), which describes how slip occurs when a shear stress ( r ) is applied to an interface with a given interface strength (it. +O). This framework law exclusively deals with how slip occurs and is not concerned with how and why the interface strength (#. + O) changes. The latter is taken care of by an evolution law such as (2) or its alternatives. Thus the roles of the SSS framework law and the evolution law are clearly separated. The interface strength, t. + O, in principle, can always be measured by suddenly imposing the reference slip velocity V. and observing the shear stress required to do this. Therefore (3) is independently testable by measuring the slip velocity at a known shear stress and then measuring t. + O immediately afterward. Of the two factors governing how slip occurs, earlier experiments have only confirmed the direct effect ( 0( ' / o') / 0(ln Die = a or 0(ln /) / 0( ' / a)le = a-' ) but not if Here the intention is to view the frictional sliding as a shearing (output) of a frictional interface caused by the applied shear stress ' (input) [e.g., Reinen et al., 1994]. For this reason we intentionally avoid using t in place of ß/. For a given interface state O the slip velocity is uniquely determined by the applied stress. The sensitivity to applied stress is characterized by a' slip velocity increases e-fold when is raised by a. This exponential dependence of slip velocity on shear stress ( o (ln /) / o ( ' / rr)l o = a - ) is nothing new and is exactly the same thing as the direct effect 0(ln/)/o-O, =-a -. This is why the RSF framework [Dieterich, 1979] of current slip velocity on frictional (equation(1)) appeared to be such a catchall that could resistance in (1)( o ( ' / rr) / o (ln /)lo = a). incorporate any changes in t by the use of an appropriate However, (3) also says that slip velocity at a given applied evolution law. However, the SSS view highlights that this stress decreases exponentially with O. Moreover, it says that may not be the case. As mentioned earlier, there is no a priori the sensitivity to O is equal to the sensitivity to the applied reason to believe that 0(n /)/o-ol, has to coincide with shear stress, i.e., o (ln /)/ono, =-a-'. That is, an increase of -o (ln V) / o ( ' / o'){ e. f o (ln V) / a91, measured for some type of O by, say, 0.1 decreases the slip velocity by the same factor change of {9 is found to be different from -o (ln V) / o ( / (Y)le, as the decrease of the applied stress by 0.1. This latter then such change of interface strength cannot be incorporated dependence ( o (ln /)/o l,) has not been shown in (1) or (3), regardless of the evolution law employed. experimentally before. n section 3 we experimentally Another merit in regarding t. + {9 as a strength is that it demonstrate that slip velocity does have a negative and connects the modem friction concept with the classical world exponential dependence on O as in (3). Further, it will be of friction as a threshold strength. n the classical world the

3 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,349 (a) Classical Friction threshold strength, below which slip stops and above which slip occurs (Figure l a). n fact, this had been the case until the a ln(v/'v,) term (equation (1)) was introduced by Dieterich [ 1979]. The detailed facts such as the time-dependent increase of static friction during stationary contact [Dokos, 1946; Dieterich, 1972], the continuous reduction of friction from static friction to dynamic friction with ongoing slip displacement [Rabinowicz, 1951 ], the velocity dependence (particularly velocity weakening) of steady state friction, and the continuous change of friction with slip when velocity is changed, all had been discovered and incorporated in the friction law [e.g., Dieterich, 1978] before the work by Dieterich [1979]. Even the concept of memory of the interface, which would be generalized as state variable later [Ruina, 1983], had already been recognized [Rabinowicz, 1958] by then. However, they were all about how and why friction, in the sense of threshold strength, changes. n the (threshold strength) modem framework they were all about the evolution of state, which is (2) (or its alternatives). static or dynamic friction, depending The a ln(v / V,) term was originally introduced as a on the prior slip history. phenomenological correction term to describe the small offset in /, proportional to the log of the current slip velocity (b) Modern Friction (RSF = SSS) [Dieterich, 1979]. As we argued earlier, even now / is still perceived largely as strength in the classical sense, and the a ln(v / V,) term is perceived as merely a correction term in // in this sense. However, when we adopt the SSS view, it becomes clear that the introduction of the aln(v/v,) term into (1) has a more fundamental significance. As illustrated i n Figure lb, by noting that a takes a small value (typically ), one can see that (3) says that slip velocity changes very steeply with the applied stress. f we set the reference velocity V, at a level of slip velocity under which slip is slow enough to be regarded as negligible for some application, the value /J, + O is the threshold level of stress to cause meaningful slip velocity, and hence /J, +O is analogous to the "strength as a threshold," which is the classical sense of the term "friction." This analogy becomes more exact as a approaches zero. Conversely, we may say o f that the introduction of the a ln(v/v,) term has, in fact,.t,+o /{J extended the classic concept of friction as a simple threshold (interface strength) to one in which the slip velocity continuously (but sharply) varies, depending on the prior slip history. changes with the applied stress around the interface strength (#. +O). What happens in the situation described before, where a Figure 1. Slip velocity as a function of the applied shear stress shear stress of 0.4 (r is applied to the surface of the strength (schematic) (a) for classical friction before the wrok by Dieterich [1979] and (b) for modem friction after the work by Dieterich (/z. + 19) of 0.7, is now (in SSS view) as follows. Because the applied stress is significantly smaller than the strength, the [1979]. surface slips only at a miniscule speed given by V(r = 0.4(:r, 19 = 0.7) = V. exp[( ) / a]. n the conventerm friction, either static or dynamic, had a meaning as strength, not stress. This is well illustrated by the following situation. Suppose an frictional interface whose static friction coefficient is 0.7 and dynamic friction coefficient is 0.6. When a shear stress of 0.4 (r is applied on this interface which has been stationary, no slip occurs because the stress (0.4 (r) tional RSF view the exact same situation is stated differently: Because of the large negative direct effect of-0.3 due to the miniscule current slipping speed of V. exp[( )/a], the friction (/z)is only 0.4. We believe the SSS view is much clearer, particularly when the word friction (#) is often perceived as strength rather than stress, following the is less than the strength (0.7 r). When a stress at or above tradition of classical friction. this threshold is applied, slip at any speed can occur. Dynamic friction is also a threshold strength as illustrated by the following situation: a frictional interface sliding under a shear stress of 0.6 rr which is equal to the dynamic friction of the sliding interface. f the applied stress is slightly reduced, slip stops. n summary, in classical friction, friction was a The interface strength, /z. +19, works also as a state variable in a standard rheological sense, which represents the variable property of the object and makes the deformation rate a unique function of the applied stress at a given value of the state variable [e.g., Frost and Ashby, 1982]. Thanks to the particular functional form of (3), we could use a

4 13,350 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON macroscopically measurable value (st, +0), even with an with #60 abrasive. The normal stress was kept constant at 20 intuitive meaning, as a state variable. This may be rather MPa throughout the experiments. The sample assembly was fortunate. However, as we show later, this luck enclosed by an oversized (35 x 35 x 35 crn 3) external furnace (approximately) holds within our present experiments. Finally, we want to make two remarks to avoid possible confusion. First, because (3) is mathematically equivalent to (1), the feature that 3(ln V) / oqol = -a -s is implicit in (1), though not addressed before. Hence, what follows will not conflict with (1) or applications that have been made using it. Rather, this study affirms them more strongly by an to keep the temperature of the sample and the loading piston constant within a few degrees. Runs were carried out at fixed temperatures over the range øC. All signals (shear load, normal load, axial displacement) were measured with sensors with optimized ranges for the present experiments and then antialias filtered at 300 Hz, 14-bit digitized at 1 khz, and then digitally averaged and stored. n addition to 2-Hz recording experimental demonstration (section 3)of o (ln V)/o ] and covering the entire experimental time, quick changes upon by proposing a physical model suggesting that this stands to reason (section 5). t should be noted that o (ln V)/o Olr, as imposed sliding was recorded at 100 Hz, which will be used in section 4.1. The frequency response of the sensors are much well as other well-known features of the framework and better than 100 Hz. DC accuracy and stability of the evolution laws, has played important roles in many of geophysical applications of RSF, particularly in those for which low-velocity slip is important. Examples include earthquake preslip and afterslip [e.g., Tse and Rice, 1986], instrumentation system comparable to 14-bit accuracy has been secured with the combination of analogue electronics that consists of precision op-amps with built-in registers and a precision AD converter. The instrumentation error is spontaneous earthquake nucleation [Dieterich, 1992], negligible in the present study. aftershocks [Dieterich, 1994], and delayed earthquake triggering by stress change [e.g. Gomberg et al., 1998; Lockner and Beeler, 1999]. An example of "slide-hold-slide" tests is shown in Figure 2. Following sliding at a sliding friction level (St.) at a constant reference velocity ( V. = 2 m/s), we observed slow slip during The second remark is about the notation. Though (1) is one a "hold" under a constant shear stress ho d below the sliding of the original notations of the RSF [e.g., Ruina, 1983], the friction level, analogous to creep tests for intact samples. The following expression using 0 defined as -= b ln(0 / 0,) has been used by many authors in place of (1): ram displacement was measured using an eddy current noncontacting sensor with a catalogue resolution of 0.2 St (= v l tr) = st, + b ln(o / O, ) + a ln( V / V, ). (4) (ts range was 2 ram, and the sensor was reset during the slide period, where accurate displacement measurement was not Here 0, is a reference value of 0, usually taken at the value at necessary. Servocontrol was based on a separate displacement the steady state sliding at the reference velocity. Evolution sensor with a 20 mm range.) The resolution of displacement law (equation(2)) is modified in this case, and in fact, the measurement was further improved by averaging, and the constant b does not appear there. t appears in the framework actual limitation of the slip velocity measurement came from law (equation(4)), instead. With this notation the interface thermal expansion and contraction of the apparatus due to strength St, + is represented as St, + b ln(o / 0,). The small fluctuation of room temperature as discussed in section symmetric dependence of slip velocity on stress and strength 3.2. The measured displacement includes elastic deformation discussed earlier might be felt less dramatic because one may of the apparatus and the sample, but the change in intuitively try to look at o (lnv)/oqolz rather than displacement during the hold period does not include an elastic o (!n V) / o (b n 0)] because this notation treats 0 as an component because the applied load was kept constant. independent variable. Nonetheless, all that we observe is the Figure 2 shows that the slip gradually slows down while the change of St by = b ln(o / 0,). The feature that o (ln V) / oqo]z applied shear stress is kept constant at a designated value ~ -o (ln V)/o ( '/tr)] o of the framework law (equation (1) or 'hoid, indicating that the state of the frictional interface is equation (3)) therefore should not be discounted by arguing somehow changing. t should be noted that the slowing down that it appears less dramatic when we choose the notation of slip observed in conventional slide-hold-slide tests where based on O. n addition, when recognizing the very different the loading ram is held stationary, while similar, can not be logical roles of the framework law and evolution law necessarily attributed to the change of state because shear mentioned earlier, it is logically more consistent to have the stress in conventional slide-hold-slide tests decreases with time. constant b in the evolution law because it is the magnitude of the change of the interface strength, which should be taken t is known that the static friction (Sts)required to resume care of by the evolution law. Though understanding evolution sliding after a hold period increases with hold time, thoid, a laws is beyond this paper's scope, the representations of phenomenon known as "healing." We use the amount of representative evolution laws are given in Appendix A, healing p k ----Sts- St* (Fig. 2) as a measure of the net showing that the use of large 0 notation does not cause any change of the interface state during the hold and compare this difficulty. n fact, physical insights for the evolution laws are quantity with the slip velocity V=d measured at the end of often grasped more easily in notation. hold. Strictly speaking, because O evolves during the reloading period (from the end of hold to peak stress), the 3. Experiment and Results value of p k is somewhat different from the value of at the end of hold (m Oe,d), which V, d should be theoretically compared with. However, as shown in section 4, the difference 3.1. Experimental Technique between Op k and O, d is not sufficient to seriously affecthe Experiments were done in a double-direct shear apparatus conclusions of the present study. n section 3.3, will use [e.g., Dieterich, 1981]. A thin (~0.5 mm) layer of fine Na- p k and discuss its relation with V, d. All the points made in feldspar powder (<100 xm) was held and sheared between section 3.3 will also be seen when estimated O=d values are ceramic loading blocks having 2x4 cm 2 surfaces roughened used (section 4).

5 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,351 Opeak -,, hold time' t hold,,? F,l:hold/( E E i..i _o o ) ,.., (/,j - -i-.i... i i i i i i i ij i loo time since the start of hold (s) -i- -- loo s Figure 2. Time record of (top) the shear stress and (middle) slip displacement for a cycle of slide-hold-slide for an example at 800øC. Slip displacement was obtained from the measured axial displacement after the correction for the elasticomponent using the shear load record. (bottom) Time record of slip velocity. After sliding at a sliding friction level (#.) at a constant reference velocity ( V gm/s), shear stress was reduced by A Z' and held constant at Z'hola for a hold period ( tho d ), during which slow slip continued. Servocontrol of the axial ram during this hold period was done by modifying the command voltage for displacement control referring to the shear load signal (cascade control), which enabled smooth and quick transition between hold period at a constant shear stress and the slide period at a constant displacement rate. After the hold period the ram was advanced again at V., and the amount of healing during the hold period was measured Ore --/ s-/ *, where #s is the peak friction. The slip velocity measured at th end of hold (V d) will be analyzed as a function of Az' and Ov. Cycles of hold and slide were repeated with various experimental parameters ( :hold, tho d) over a total displacement of 18 mm in a l'un. Table lists all the experimental parameters. We employed 3.2. Measurement Errors three different Z'ho a levels, which are A Z' = #. - Z'ho d ~0.006, 0.032, 0.063, while the actual AZ' for each hold period varied a little. These levels will be referred to as errors. The only serious error occurred in reading V a, which the high, middle, and low shear stress levels, respectively. was measured by manually drawing a tangent to the last 10o The data for the low shear stress level were obtained only for the 800øC experiment because V a was too low to measure at other temperatures. Before showing the results, we discuss the measurement 25% of time slip record for each hold. (For 3162 s and 10,000 s hold, the last 20-25% were used, while the last 10% were used for other shorter hold times.) This method resulted in two

6 13,352 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON Table 1. Experimental Data T, Displace- p, Hold Time, AT / o' At / o'a øc ment, mm s Oend Oena /.a (O k+ At) / ha(. (O +A,) / ha( b b b c c c c c c b b b b b ½ ½ ½ ½ b b c c c b b c c O b b b b b ½ ½ ½ ½ OO b b b c c c b b b d d d c c c b b athe value of ' before correction (section 4) is used. bmiddle nominal shear stress level. ½High nominal shear stress level. dlow nominal shear stress level.

7 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, T 5 m 0.000õ m/s + 55 % Shear Stress ß -, 500 s time Figure 3. Reading, for a long hold. The worst case example, which happened for a hold at the middle shear StTCSS level at 600øC, is shown. types of errors. The first comes from the limitation of the accuracy of slip measurements, while the second comes from the fact that Ven d was not read at exactly the end of hold. The assessment of the first type of the error (vertical error bars in Figures 4, 5, 7, and 8) was done differently for short holds up to 316 s and long holds for 3162 and 10,000 s, constant for the long term. A longer portion of the hold (last 20-25%) was necessary to identify the rhythm of the air conditioning on-off. n this case, we estimated the error by minimum and maximum slope that we determined by drawing tangents manually. For long holds, we decided that the gm/s (Figure 3) is the lowest meaningful Ven d reading. When considering the nature of the inaccuracy slip measurement the Ven a reading was lower than the gm/s, the data in the present series of experiments. The biggest source of inaccuracy in slip measurement was a change in room point was discarded. Again, such discarde data points always happened when the expected value for Ven d from the A r and temperature by air conditioning. The air conditioner was an Optik were lower than gm/s. on-off control type. n displacement records at low 'hold where The second type of error (left horizontal error bars in slip velocity could not be distinguished from zero (such Figures 4 and 5) was estimated by thinking by how much 19 at experiment is not used in this paper, of course), a triangular the time of Ven d reading differs from that at the true end of the wave was seen. Typically, the displacement record increased hold. For example, in the case of thold -' 3162 s where we for 300 s by 1 gm and then decreased by gm for 300 s, both approximately linear. n terms of velocity this resulted in the errors of V^c = gm/s. V^c varied between and needed the last 20% of hold time for Ven d reading, Ven d was read approximately when 0.9 x thold has passed since the start of hold. Assuming that 19 roughly increases logarithmically gm/s, depending on the experiment. with hold time [e.g., Dieterich, 1972], which was For short holds up to 316 s, we estimate the error of Ven d to be the bigger of V^c and the uncertainty of manually fitting the tangent. V^c was estimated to be 0.003, , , approximately true in the present series of experiments [Nakatani, 1997], the 19 at 0.9 x 3162 s is smaller than!9 at 3162 s by (193162s s)1øg10(1/0'9) ß The value of s is 0.005, and gm/s for experiments at 25, 200, 400, 600, known from the 19,=a, measured right after this hold, and and 800øC, respectively. For short holds Ven d less than V^c 193 6s is known from the 316 s hold at the most similar was hard to read, and the readings of Ven d less than V^c were condition (in terms of A, and cumulative displacement). Such discarded. This always happened when the expected value of data were always available because holds at the same shear Ven d from the value of A r and O,,a k based on (3) were less stress level but for different tho d were done consecutively. than V^c. Conversely, this data selection did not result in This second type error turned out to be very small. exclusion of possible outliers that could show unexpectedly Besides the errors associated with Fen d reading discussed low Ven a for the condition. above, the values of 19p, and Ar can have an error due to the For long holds (3162 s and 10,000 s), we could read Ven d uncertainty of/t., which was determined for each cycle of hold lower than V^c. The worst cas example is shown in Figure 3. Though the effect of air conditioning is apparent, we can still and slide based on the long-term position-dependent trend of steady state friction. Since more than 700 gm of slip was see the increasing trend in slip for the last section of this employed for each slide period, this trend was easily hold, and velocity can be measured from one peak to the next determined, and the uncertainty in #. was too small to aft ct of the triangle air conditioning wave. t should be noted that A : and 19,=a, significantly. One thing to note here is, while the air conditioning assured that the room temperature was we discussed (see section 2) thinking of p as a single

8 13,354 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 1 o.1 ( ) hold time 32 s 100 s ( 316s O 3162 s s epeak ß ß ß (b) 200øC i i i i ' ' " ' Opeak Figure 4. Slip velocity ( 'end ) at the end of eac hold period with various ( 'hold, thold ) conditions. Plotted against O peak. White, light shaded, and dark shaded symbols denote the three different shear stress levels employed (Table ). See section 3.2 for discussion of error bars. The average of the actual AT values (Table 1) from the holds at each shear stress level are indicated on the lines. The lines are drawn for each stress level, using the indicated values and a r and a o values listed in Table 2. entity, in the actual data treatment wc only examine the effects 3.3. Results of the change of interface strength from the reference level //. and the change of the stress from the reference level z. = //.( Behavior at each temperature. n Figure4, The effect of long-term change in //. is simply excluded by Ve. d is plotted against O p k. At all temperatures it is clearly detrending [e.g., Blanpied et al., 1998]. On the other hand, the seen that slip velocity at the same shear stress level decreases sum of Op = k and A ', another important value in the present exponentially with increasing O, in agreement with (3). n study, does not depend on /.. addition, (ln V)/o Ol at different stress levels are similar,

9 .. ß NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,355 1 O.Ol ) o.ool o.oool i i i ' ' ( )peak 1 o.1 O.Ol ß o.oool (d) 600øC i i i i o.oo 0.05 i O.lO 0.15 i 0.20 ) ak Figure 4. (continued) while slip velocity for the same 9 is lower at lower shear slip velocity at a constant shear stress decreases stress, also consistent with (3). The only notable outliers are logarithmically with hold time. However, in the present one of the two 316 s holds at the middle shear stress level at experiments, O for the same thold and 'hold sometimes varied 400øC and the one 3162 s hold at the middle shear stress level substantially from test to test, but Ven d scaled with 6) rather at 600øC. With these data, we would conclude that the negative than with thold itself; slip slows down as the interface exponential dependence of slip velocity on 6), the previously strengthens. The examples include the pairs for 32, 316, and untested implication of the rate and state framework law, has 3162 s holds at the high shear stress level at 25øC, the pair of been shown to be true within the present experimental 316 s holds at the middle shear stress level at 400øC, and the conditions. pair of 316 s holds at the high shear stress level at 800øC. Since (5 generally increases logarithmically with hold Although we do not claim that we have proven that Ven d scales time, one might think that the present data only show that with 6) rather than hold time with only this many examples,

10 13,356 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON (e) 8ooøc [,,,,,! [, i,,!, [ [ [ { )peak Figure 4. (continued) 0.20 thinking so would be natural because the frictional interface However, at least at 800øC, where tests at both middle and should know the time lapsed since the start of hold only by the high shear stress levels were repeated different cumulative resulted change in the physical states of the interface, not by a displacements (Table 1), the variation of a was clearly stressstopwatch. t is quite likely that the physical states that dependent rather than displacement-dependent. The same thing govern the current slip velocity also affecthe strength may be seen from 400øC experiments, though less clearly. measured at the reference velocity, which is the definition of Variations of a at 200øC and 600øC can be interpreted as O. either stress-dependent or displacement-dependent. At 25øC, Next, we check the further implication of the framework displacement dependence of a was not recognized. law: s o (ln V)/o )l equal with -o (ln V)/o ( ' / o') e? f this is n the above, we did not check if the dependence of velocity true, plotting log(y, a / V-) versus Ar / o' + O will result in a on shear stress is really exponential or not. n fact, a rigorous data collapse. The result is shown in Figure 5. The data check of this point is difficult in these experiments because we collapse is not perfect. The lower shear stress data could get data only from a narrow range of shear stress levels systematically appear to the right of higher stress datat all (Table 1). However, as discussed in section 2, this exponential temperatures. This means that O affect slip velocity more dependence on shear stress is exactly the same thing as the sostrongly than r/o' does, in which case, the framework law (3) called direct effect, which has been well continned in previous must be rewritten as studies [e.g., Dieterich, 1979, 1981; Tullis and Weeks, 1986; Kato et al., 1992]. n addition, the lines in Figure 4 drawn Y( ', O) = V- exp exp. (5) using (5) well fit the data at different shear stress levels, a L oj though good fitting for this small range of shear stress cannot Here, in order to express what is actually examined the data be taken as evidence. analysis, stress and strength are written in terms of their Temperature dependence of a. Figure 6 deviations from the reference level, as mentioned in the last shows how a, a,, and a o change with temperature. The paragraph of section 3.2. values are also listed in Table 2. The value of a was obtained Equation (5) is exactly the same as (3) if a = a e. However, by two different methods. The first method is calculating if a a e, as suggested by Figure 5, the RSF framework a = -(A,/rr + Op ak)/ln(v a / V-) for each data point (Table would be distorted to some extent, depending on how different 1) and taking its average. Figure 6a shows the values thus they are. Though (3) says a r and a e are equal, the original obtained with the error bar of 1 standar deviation. The lines intention of the rate and state friction law does not include the in Figure 5 correspond to these values. The other method was effect of O on slip velocity, and there is no logical or least squares fitting to the function that -ln(vma/v-) physical reason to think that they have to be equal. = (A,/rr +Opeak)/a. The result is shown in Figure 6b. The At temperatures except for 400 and 800øC, tests at the high error bars are 1 sigma of the estimated a by fitting, which is shear stress level were done at larger cumulative displacement much smaller by the first method. Physical meaning of the than the tests at the middle shear stress level (Table 1). Hence temperature dependence of a will be discussed in section 6. it might be possible to argue that the apparent stress effect on Here we simply point out that a systematic increase with a (imperfect data collapse in Figure 5) does not come from the temperature is seen. difference in shear 'stress level but comes from the The values of a, and a o were obtained by the least squares displacement dependence of a [Mair and Marone, 1999]. fitting to the function that - n( Vma / V,) =

11 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, ' Aq; + ( )peak A'; + ( )peak Figure 5. Slip velocity ( Ven a ) at the end of eachold period with various ( Thold, thold ) conditions. Plotted against At + O peak. For the meaning of different symbols, see Figure 4. The solid line is drawn with the ' value listed in Table 2. The dashed lines correspond to +1 standard deviation of a = -(At/(x + Or ak )/ ln(k'en a / k',) determined from each test (Table 1). (At/ or)/a r + Op k / a e. The result is shown in Figures 6c and (Figures 6a and 6b) are much closer to a o (Figure 6d) than to 6d. The error bars show sigma of the estimated coefficients ar (Figure 6c) presumably because we had data for much wider a r and a e. The lines drawn in Figure 4 correspond to the mean range of O than A r/ (Table 1). This may also be the reason values of a r and a e (Table 2). Physical meaning of the for the large error bars for at. temperature dependence will be discussed in sections 6 and 7.2. We just point out some features here. First, a r > a O at all temperatures. Second, they both increase systematically with 4. Loss of nterface Strength During Reloading temperature in a similar fashion, indicating that a r and a e So far, we did our analysis assuming that O measured at the may be related quantities. A possible physical model to link peak stress (Opeak) is a good approximation of O at the end of them will be discussed in section 5. Third, the single a values hold (Oend). n this section, we assess this assumption

12 13,358 NAKATANh PHYSCAL UL,,,d<FCATON OF RATE AND STATE FRCTON O.Ol, (c! 4,00, ' ' ' Aq + )peak,, O.Ol ,, (d) 600,," A q + )peak Figure 5. (continued) because this could produce a systematic error. n section 4.2, To infer O, a from the experiment, we have to use an we will show revised plots corresponding to Figures 4, 5, and evolution law, for which there are three representative 6, using the calculated O,a value instead of Op ak, with the candidates. The first one is (2)(referred to here as the Dieterich resulthat all the conclusions discussed in section 3.3 are law following Marone [1998a]). The other two are preserved. We are not claiming that the revised plots are more correct, considering uncertainties inferring O,,d values. do=- (O-bln(-- -) 1 (6) Rather, the purpose of this section is to show that the dt ' estimatedifference of Op ak from O d, although which is referred to as the Ruina law following Marone nonnegligible, is unlikely to alter the conclusions based on [ 1998a], and further directly conclusions measured that values are of made Op ak. in section This also 6, applies where the to do dt = 2L/V. b xp ) by2 ( ) e - exp, (7) experimental results are compared with the absolute rate 2LK theory. which is referred to as the Perrin law following Marone

13 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,359 1,, 0.01 (e) , /k + { )peak Figure 5. (continued) [1998a]. Note that (2), (6), and (7) are expressed in different because the slip-weakening rate in the Dieterich law is forms from those in Marone [1998a] due to the different constant, independent of the current {9 (see Appendix A2.2). notation. We take {9-- b ln(0/0,) as an independent variable The reason why the {9 decrease eases off when it has become rather than 0. See Appendix A. small is that the healing rate becomes higher as {9 decreases n fact, as will be shown below, we have found a clear (see Appendix A2.1). When we start from a small {9s, the reason to use the Ruina law rather than the Dieterich law and Dieterich law thus predicts that slip weakening is exponentialthe Perfin law for the present purpose. Therefore we will like as is often observed in experiments [e.g., Ruina, 1983]. discuss the evolution laws in section 4.1, though here we However, when {9stm is large, the majority of the slip emphasize that it has nothing to do with the main point of the weakening occurs linearly with a constant slope of b/œ present paper. regardless of {9stm. As a result, apparent critical slip displacement increases almost proportionally to 4.1. Slip Weakening and the Evolution Laws However, no such phenomenon has ever been reported, even for slide-hold-slide tests [e.g., Dieterich, 1981] and velocity As will be shown in section 4.2, {9p ak is thought to be step tests involving a high ratio (100) of new and old smaller than {9e,d because significant slip occurs during the velocities [e.g., Ruina, 1983; Tullis and Weeks, 1986], where reloading period from the end of hold to peak stress. So how the evolution occurs from the {9 value considerably higher {9 evolves (particularly decreases)with slip is the most than the following steady state value. (See Appendix A2.2 for important feature of an evolution law for the present purpose. discussions of some confusing cases, which are irrelevant to The purpose of this section is to show that the Dieterich law and the Perrin law have serious flaws in this regard, while the the present study.) As shown later, the flaw of the Dieterich law in slip weakening was very clear in the present Ruina law does a good job. As discussed in Appendix A, slip weakening by these three experiments that involved a wide range of {9stm. Note that evolution laws differs in the dependence of the slip-weakening similar strange slip weakening predicted by the Dieterich law rate on the current value of {9. Let us first look at the was earlier discussed to some extent by Pert n et al. [1995], prediction of slip weakening by the three evolution laws in an Blanpied et al. [ 1998], and Marone [ 1998a]. ideal experiment, where the slip velocity (not the load point On the other hand, the Ruina law, where slip-weakening velocity) is suddenly broughto V, and maintained, starting rate depends linearly on the current {9 (see Appendix A1), from a given initial value of {9 = {9stm. The given level of does not have this problem, as seen in Plate lb. The slip {9s is prepared beforehand by some method (e.g., for larger weakening is exponential with slip over a constant {9, longer holds or steady state slip at lower velocity.) The displacement, independent of {9start' result is shown in Plate 1, using the nondimensionalizing n the Perrin law (Plate l c), slip-weakening rate depends scheme of Gu et al. [1984](see Appendix A4). exponentially on the current {9 (see Appendix A3). Hence it With the Dieterich law (Plate l a), {9 and ' (which are predicts extremely rapid weakening in the earlier stage of slip always equal in this ideal experiment because V = V,.) weakening starting from large {9, t. As shown below, this is decrease linearly with slip until {9 approaches zero. As a too extreme compared with the present observations. result, longer slip displacement is required to complete the The superiority of the Ruina law in correctly predicting the slip weakening when slip starts from larger {9 tart' This is slip weakening is seen from the comparison of the present

14 13,360 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 0.0 m (a) (b) 0, m i ß T (K) T (K) (c) (d) ao T (K) o T (K) Figure 6. Values of a, a, and a e plotted against the linear temperature. They were obtained using the directly measured values of p,. The value of a was obtained by two different methods and separately shown in Figures 6a and and 6b. See text. They are almost the samexcept for the error bars. All the values, including the error bar, are listed in Table 2. The line drawn in Figure 6a corresponds to g2 = (0.43 nm) 3, and the line drawn in Figure 6c corresponds to g2 = (0.38 nm) 3, when o'½ = 8 GPa is assumed (section 6). experiments and simulations (Plates 2, 3, and 4).-n the per second. n the experiments, stress was measured on the simulations the load point velocity is suddenly raised to g,. central block of the double-direct shear, which stayed put. The sliding block is connected with the load point with a Hence the : measured in the experiment is the (friction) stress spring of finite stiffness [e.g., Dieterich, 1981]. nertia is exerted on the sliding interface, not the spring stress pulling neglected in the simulation of these experiments, where the the block, though the difference is negligible in the lowmaximum slip velocity is at most a few tens of micrometers velocity regime of the present experiments. The starting value Table 2. The a Values From the Present Experiments o C -- a From Fitting to (ln(v. / V. ' t) V d = V. exp - O+A '/o' a and a o From Fitting to V a = V. exp - at J, ao J ar a e a e / a ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ W/th p k ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ [ rith O md ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ

15 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,361 t' -.. (a) o-' Slip Displacement/L 4 6 Slip Displacement/L Dieterich law micrometers) (b) Ruina law i 8 10 (c) Perrin law! Slip Displacement/L type healing. The long L evolution is not considered in the present simulation because such evolution only slightly affects the change of G during the short slip (several that occurs during reloading n addition to well fitting the observed slip weakening of stress, the simulations using the Ruina law well predict the slip velocity during slip weakening. This means that the Ruina evolution law predicts G itself correctly ( O'= :/ cr - a ln(v / V,) - t,), not only that its combination with the framework law predicts the shear stress correctly. Note that agreement in stress alone does not guarantee the correctness of an evolution law itself because it is possible that the mispredicted velocity (through direct effect) and mispredicted O cancels out to give the correct stress value. On the other hand, simulations with the Dieterich law (Plates 2c, 3c, and 4c) show the flaw noted above in the discussion of the ideal experiments. n the simulation, we used a smaller L for the Dieterich law in an attempt to make the simulations look closer to the data. f the same value of œ as for the Ruina law is used, the discrepancy becomes more conspicuous, but there is no reason that the same œ value has to be used for different evolution laws. Finally, the simulations with the Perfin law (Plates 2d, 3d, and 4d) misfit the experiments badly. For slip weakening after a longer hold, very fast slip is predicted because O decreases very rapidly (dashed line), while the decrease of stress is limited by the finite stiffness. Actually, the misfit by the Perfin law is exaggerated in the simulations because part of the rapid evolution comes _from the ove {edicted slip velocity (see equation (A31)). Even if the Perfin law were appropriate, such an extreme velocity might not happen because inertia becomes important at higher velocity. However, the experiments did not show any signs of very fast slip. The unrealisticness of the Perfin law can be understood by the comparison with the Ruina law, which has been shown to agree well with the observed slip weakening for a wider range of starting O. When we compare the initial slip-weakening rate from two different starting O values, for example, 2x b Plate 1. Slip weakening predicted by different evolution laws for ideal experiments where slip velocity was suddenly brought to V, and maintained constant. The blue, green, red lines correspond to and 6x b, the initial slip-weakening rate of O differs only by initial conditions of Ostar t = la, 3a, and 5a, respectively. Shear 6/2 = 3 times in the Ruina law (see equation (A11)), while it stress (solid line) measured from the reference level t,cr is equal to Ocr (dashed line) because slip velocity is assumed to be V, in differs by e4= 55 times in the Perfin law (see equation these ideal experiments. The value of b was assumed to be equal (A31)). Therefore, when the Ruina law shows a good to a. All three of these evolution laws are adequately agreement with slip weakening for a wide range of starting O, nondimensionalized by Gu et al.'s [1984] scheme (see Appendix as shown, the Perfin law cannot do so. Note that the range of A); therefore the results are shown following their scheme. O used here is realistic. Ost m for each simulation is chosen so that the simulated curve has the peak height of the measured G. Thus the chosen Using the simulation shown above, we obtained Ge, d Ost m value is the inferred O,n d value (Plate 2b). values for each hold. For the reasons noted above, we used the First, the Ruina law (Plates 2b, 3b, and 4b) fits the Ruina law as the evolution law. For stiffness, we used a value 2 experimental results (Plates 2a, 3a, and 4a) well at all x l0 s N/m throughout, a value which was obtained from the temperatures, where slip weakening is completed in the same slip displacement independent of the O t m value (larger for longer hold). Some experimental examples (316 s and 3162 s holds at 400øC, 3162 s hold at 800øC) showed additional slip weakening after the completion of the evolution over the short L. This slip weakening lasting over a very long slip displacement ( !.tm) is the erasure of another type of time-dependent healing that occurred for the long time hold at high temperatures, whose phenomenologies are clearly different [Nakatani, 1997] from those of the usualog time healing [e.g., Dieterich, 1972] that is erased within a short critical displacement. The latter is referred to here as Dieterich Correction and Results With Oen d lower stress portions of the reloading curve, where slip is negligible compared with the elastic deformation. The effective stiffness k (frictional coefficient/length) corresponding to our machine stiffness is calculated to be 6.25 (frictional coefficient/mm), using the present conditions of normal stress and the sum of the areas of the two sliding surfaces. This value is indicated in Plates 2-4. For a, we used the ' values listed in Table 2 for each temperature. For b, we assume that b = a for all temperatures. The reasoning for this is as follows. Exploratory velocity step tests at 25, 400, and 800øC have shown that the velocity dependence of steady state friction aitss / a(ln V) was between 0.1a and -0.3a, which

16 13,362 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON slip velocity (l n/s) slip veloaty (ta /s)...,......,... _o.o slip velocity ( m/s) slip velocity (izm/s), L... h,,,,,,, L... l.,. ß. '-< o, '. E o.o.o oo o o o ø.. o O oo o -o *'rt - / q q q o.. o O '*! -

17 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,363 slip velocity (t m/s) slip velocity ( m/s),...,,,!...,. h...,! , C..o o oo o ooo ooooo oo oo oøøøøøø3 øø L O '*ri - 0/ oooo....oo -.oo...%z- ½ %T.& ß ß ß o o i,' '"'J' '"' ' ''"'!... ' ''"'' q q q q. q o- slip velocity ( m/s) slip velocity (pro/s) ,.,,,.,,...,,, , _o, h,, , 0,v o o o o o o o o o _ o o o ø 0 o... t t m... li'''i"'''l... '"' 00 (,!- N o N q o. o. q q *fi - o/z... " t... '!",i' m... o. o. q o. q O, O '* t -

18 13,364 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON slip velocity (wn/s) ß...! h o o o 0 ' _o _o o 0 u 0 u 0 u'.,-:,-: q q o. 0 o o 0 o øø ooo ooø0øø f / /e O...., 0 i ' ' ' i % " ' "l'" ' ' 0 u 0 u 0 u.,-:,-: o. o. o....,. r.n O O O slip velocity ( m/s) 3 slip velocity (um/s),!...!... L o o!... L...! o -o.o o i o < ' ß 0, o o o 00 % %0 o o o o u o u o u.,..:. o. q q

19 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, ' ' ' ' Oend 1, (b) øC,,,, Oend Figure 7. Slip velocity ( Ven d ) at the end of eachold period with various ( 'hol,, thol, ) conditions plotted against =a. For the meaning of different symbols, see Figure 4. The vertical error bars are the same as those in Figure 4. The horizontal bars are not error bars. They connect (,, V, ) with ( p k, V, ) denoted by the small dots, showing how much correction was made. The lines are drawn for each stress level, using the indicated values and ar and a o values listed in Table 2. suggests that b is between 0.9a and 1.3a. However, in many weakening curves. These parameters (a, b, L)were fixed for cases this steady state friction was achieved after additional each temperature. We are not sure about how constant was b evolution over a long critical slip displacement following the between different hold and slide tests at each temperature. evolution over a short critical slip displacement. This means However, for the current purpose of seeing how the systematic that the value of b to be used to estimate the evolution with difference of Op and O, d affects the plots in Figures 4 and short critical slip displacement should be smaller than 5, fixed b at each temperature would not be too crude. The O. tss/o(ln V) suggests. Considering this, we used b = a. The fixed L well fitted the slip-weakening curves at each critical slip displacement (for the short evolution, of course) temperature, except that those after the initial one or two was chosen so that the simulations fitted the observed slip- holds (at cumulative slip displacement of 1-2 mm) were fit

20 ß 13,366 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON O.Ol o.ool - o.oool (c)!.ooøc, i i i ' ' Oend o.1 O.Ol o.ool - o.oool. 0.00?o,ooc,,...,,--,%,%,...,,, Oend Figure 7. (continued) better by assuming L up to 1.5 times larger than in the other the calculated O,. To see how much the correction was, holds, qualitatively consistent with Marone and Kilgore uncorrected O p k values are plotted as dots and connected with [1993]. Two initial conditions must be assumed for each hold. O=a values using a horizontal bar. There is no significant One is the stress level upon the start of reloading, which is a difference compared with Figure 4. One hope that we had in known value A ' (Table 1). The other initial condition, the doing this correction was that the difference between different starting O value, was chosen so that the predicted peak stress shear stress levels in the 1og(V=a/V,) versus agrees with the measured value of Owa, (Plate 2b). The plots (Figure 5)may disappear. However, this hope was not starting O value thus estimated is the inferred value of O d realized, as shown by the lack of perfect data collapse still for eac hold (Table 1), as mentioned before. Note that O + t, seen in the plots against Az / o + O=a (Figure 8). and '/ o' coincide at the peak stress because the slip velocity The a values obtained from the data in Figures 7 and 8 are and the load point velocity (set at V, in the present shown in Figure 9 and Table 2. There is no significant change experiments) are then exactly the same. in trend from Figure 6, but the values are generally larger than The result of the correction is shown in Figure 7. This plot those in Figure 6 because the correction resulted in O=a is same as Figure 4 except that Op, k has been replaced with systematically larger than O

21 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, }end Figure 7. (continued) n this correction, we not only used the evolution law but also the flamework law. One may note that this is a tautology because we are attempting to test the framework law. Theoretically speaking, however, the framework law is not shows such a tendency, but it is far weaker than that seen in the data. The reasons for this discrepancy are not clear. One interesting possibility would be a, > a o noted in section 3.3. n the simulation assuming a, = a o, the slip occurs only as a necessary for this correction because we can use the measured function of the difference between :/ ry and O +/t,. As a slip history during reloading instead of the slip given by the framework law. Plugging the measured slip history into the evolution law gives the evolution of O. The measured slip history during reloading (Plates 2a, 3a, and 4a) may be of good enough quality to do this. However, we did not take this approach mainly because it would be too much labor, result, reloading curves with different Ostar t are almost the same when they are traced back from the peak stress (Plates 2b, 3b, and 4b). However, if a, > a o, low O contributes more to slip than high ß / ry. Hence it is expected that a larger preslip occurs in reloading from a small Ostar t (low O, low :/ry) than in reloading from a large Ostar t (high O, high considering that the error in Ven d measurements (see section :/ ry ). Of course, other reasons may be the case. 3.2) is the main difficulty in discussing the present data. The above discrepancy suggests that the actualoss of O Another reason why we avoided this approach was that the validity of evolution law had been previously checked only by comparing the experimental history with the history predicted by the combination of the evolution and framework laws; The validity of using the evolution law alone had not during reloading after a smaller Oen a, because of the larger preslip, may be larger than the estimation by the simulation using the slip calculated with the framework law. n other words, datapoints with the small O en a values may be undercorrected and may have to be shifted farther right in been established, although the present study (section 4.1) has Figures 7 and 8. Some of small Oen a data (Figure 7, leftmost shown that the Ruina evolution law well predicts the evolution white marker at 25øC, data with Oea < 0.02 at 200øC, of O itself during slip weakening. Furthermore, it is known that the Ruina law is not correct about healing (see Appendix A for further discussion), which may produce some error in calculating the evolution during reloading. This error may be as large as the error caused by the use of calculated reloading slip history, which is possibly incorrect if the framework law is incorrect. leftmost white markers at 800øC) are slightly to the left of the trend. However, just as many data points with small O en a are on the trend. Such a discussion is obviously beyond the data limitation of the present experiments. Caution needs to be taken that the correction may not be enough for small Oen a data points. Finally, we summarize how large is the estimatedifference To assess the possible bias or/the inferred values of Oen d between O peak and O en d. This is an assessment of the quality due to the use of reloading slip-history calculated with the of the present experimental method. Plate 5 shows O a, as a framework law instead of the actual one, we go back to Plates function of starting O (Oend) of the reloading for some 2-4. Since the difference of Opeak from Oend mainly comes representative system parameters. The O values are shown from the loss of O by the small but significant (compared scaled with a, according to Gu et al. [1984] (see Appendix with L)slip during reloading, we look at how much slip occurreduring reloading. n the experiments, preslip before the peak stress was often larger in reloading after shortest 32 s holds than in reloading after longer holds (e.g., Plates. 2a and A4). Using the nondimensionalization of the system, the system parameter has been reduced to only two free parameters. One is the ratio of b to a, which we assume to be 1 =1. The other is the ratio tc of the effective system 4a). This tendency was not exceptional but was seen in the stiffness k to a / L. majority of the data. The simulations using the Ruina law also Plate 5a shows the case where the system stiffness is only

22 13,368 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON i ' Z 'C + ( )end (b) 200øC ' ' ' Z 'C + ( )end Figure 8. Slip velocity ( ze d ) at the end of each hold period with various ( rho a, tho a) conditions plotted against, r + Oe. For the meaning of different symbols, see Figure 4. The vertical error bars are the same as those in Figure 4. The horizontal bars are not error bars. They connect (, r + Oen a, ze a) with (, r + Ope, ze d) denoted by the small dots, showing how much correction was made. The solid line is drawn with the mean E value listed in Table 2. The dashed lines correspond to +1 standard deviation of a = -(, r/cy + Oena)/ln(Fen a / g.) determined for each test (Table 1). twice a/l. This is close to the experiment at 800øC. (System from the end of hold to the peak stress. The maximum O, d at stiffness was independent of temperature, but a/l increased the 800øC experiment was 7.04a. The effect of A ' on the with temperature.) An additional initial condition is the correction is generally small in the present experimental applied stress level at the end of hold. Two cases of range of O cnd' One may wonder why is calculated to, r = 0.2ao' and, r = $ao' are shown, which cover the range increase from the end of hold to the peak stress when Oe d is of the experiments at $00øC ( 0.24ao' <, r < 2.44ao', Table very small. This is because apparent healing occurs when is 1). When ( a is greater than 0.$a, ( is smaller than smaller than O,,(g), which is the target value of evolution by (,,a. The lowest datapoint corresponds to ( d = 0.77a, the Ruina law (see Appendix A l). which occurred for the 32 s hold at, r = 0.4a y, and thus all n order to determine a accurately, what matters most is the the data points fall in the regime where ( decreases somewhat ratio of Opeak to Oe d because a is determined as

23 ..,., NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, ', (c) 400øC i i i i ' 0.20 Aq + )end '= o.oool 0.00 (d) 600øC i i i i , i 0.20 Aq + )end Figure 8. (continued) a = -(az'/cr + O)/ln(Ven d / V.) and a e is determined as has been confirmed in the present experiments. The model is a o = -O / n{fen d exp(a : / a,cr) / }. Plate 5a shows that based on the adhesion theory of friction (ATF)[Bowden and O ak preserves 70-80% of O end, quite stable over a wide Tabor, 1964] summarized below. range of relevant O en a values. Things are only improved at Rough surfaces under normaload are in contact only at lower temperatures because of improved tc. Plate 5b is for tc = "junctions" at the tips of contacting asperities, whose total 5, close to the condition of the 25øC experiments area 1 r is proportional to the normal stress cr [Bowden and (0.83act < Az' < 4.53act, 1.14a < Oen a < 6.06a, Table 1). Tabor, 1964; Greenwood and Williamson, 1966]. n the classical adhesion theory of friction (ATF) the proportionality 5. Junction Shear Stress Model for Stress factor depends on a bulk matedhal constant called the and Strength Dependence of Slip Velocity indentation hardness cr c' n this section, a model is proposed to explain the dependence of slip velocity on the interface strength, which is implied by the framework law of the rate and state friction and A r O', (8) where 1 is the nominal area of an interface. Frictional

24 =..,. 13,370 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON '= - (e) 800øC o.oool 0.00 i!! 0.05 i i i i O.lO o i i Aq; + )end Figure 8. (continued) (a) (b) 0,04 '"= 0.03 "' a ! ' ' ' T (K) ' ' o T (K) (c) (d) ae T (K) ' ' T (K) Figure 9. Values of a, a, and a e plotted against the linear temperature. They were obtained using the inferred values of O=a. The value of a was obtained by two different methods and is shown in Figures 9a and 9b. See text. They are almost the samexcept for the error bars. All the values, including the error bar, are listed in Table 2. The line drawn in (a) corresponds to = (0.41 nm) 3, and the line drawn in (c) corresponds to = (0.37 nm) 3, when cr c = 8 GPa is assumed (section 6).

25 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, Hence the dependence of re on the applied shear stress (r) and O is with r Ocr r(r,e) = r c. +, (9) re, - r,-- (= St,cr ; (10) 2'-, O- Ax/o-02a idea1 Ax/o-5a o_/,, 0.0 Z Oe /a fix/o - 0.2a... Ax /o - 5a... ideal...! i Z /a L Z.O Plate 5. Estimated change of during reloading from the end of hold to peak stress. Estimation was done using the combination of framework law and the Ruina law. The value of b was assumed to be equal to a. resistance is given by the adhesive shear yield strength of junctions multiplied by A r. The theory [Bowden and Tabor, 1964] explains the most important feature of friction: the macroscopic friction coefficient St is independent of or. t also explains why St is insensitive to the surface material: Lower shear yield strength expected of junctions of softer material is compensated by their larger A r due to lower cr½. For the same reason, St is also insensitive to temperature. n the context of ATF, the increase of O due to log time healing is due to the time-dependent increment of A r [e.g., Dieterich, 1979; Dieterich and Kilgore, 1994] from the value instantaneously attained upon normaloading [Greenwood and Williamson, 1966]. On the other hand, the shear deformation rate of the junction should be governed by the average shear stress re acting on them, which is obtained by multiplying the applied nominal shear stress r by A/A r. At a given applied stress r, re should scale inversely with A r and hence with St. + O, which we assume to be proportional to A r. 0.0 re, can be regarded as the junction shear stress at the steady state sliding athe reference velocity. From (9), we obtain Comp,,ring,,na o r½ = St,(; re. (l+ocr/-' St,(; ) (ll) O rc = St.cr rc* St crl / 1 + St- -j Ocr/-2 o3(ocr). (12) noting th,,t lel <<,u,, we ot, t,,in o3r c ø r = -(r/st,a) (13) a(ea) Because 0.9 < rhold/st,cr < 1.0 in the present experiments, the effects of Ocr and r on re, and in turn on V, are expected - to be of similar magnitudes but of the opposite signs, as observed. Thus ATF theory suggests that the dependencies of slip velocity on r and O share the same physical root: shearing rheology whose rate is governed by the junction shear stress r c. Except at 600øC where a r estimation yielded a very large error bar, the ratio ao/a, was in a range of for values obtained directly from experiments and for the values obtained using the corrected data (Table 2). f we further exclude the 400øC result, which showed the second largest error bar, the ratio is almost constant at all temperatures (25, 200, and 800øC), whereas both a o and a, changes by ~4 times over this temperature range. However, we must note that the above simple model does not completely explain the observed strength dependence of slip velocity. The observedifference between a and a o was of the opposite sense than the prediction by (13): The experiments show that Ocr affects the slip velocity more strongly than r does (ao-s>a -'), while (13) says the opposite (ao-'/a,-' = rhold / St,cr < 1). Though a, was not well constrained in the present experiments, if the observed difference is true, what would be the reason? One possibility is that the healing increases the shear strength (per unit real contact area)of the junctions as well as it increases the net area of real contact. n other words, the physical state that is reflected by macroscopic interface strength O may involve the quality of contact (T. Tullis, personal communication, 2000) in addition to the quantity of contact [Dieterich and Kilgore, 1994]. 6. Rheological Analysis of Frictional Sliding as a Thermally Activated Process We now apply the classical rheological analysis to frictional sliding by correcting the measured slip velocity for the effect of O. The correction can be done by defining Ve e ( r, T) -- V / exp(-o / a e ). (14) Following the model proposed in section 5, this corrects for the change of r e due to the evolution of the junction population. However, as mentioned in section 5, this model '

26 13,372 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON does not explain the observed effect of O on slip velocity completely. Hence, strictly speaking, the correction by (14) is empirical. Applying (14) to (5), we obtain Vofr (*, T) = V, exp (-/x,) exp (,/rr \ a, /' (15) t may look strange that a, appears as the denominator of /,, which is a part of the interface strength (section 2). However, this makes sense if we note that the /2, in (15) actually comes from A ' -- '- t,cy in (5) and hence is the normalized (by or) reference stress level ',, which has been taken to be t,cy in the convention of the present study. The following representation using ', instead of t,cy may be clearer: /oeec(*, T) - Y, exp. The analysis given here is not seriously affected by the convention of setting the reference levels of interface strength (/,), applied stress (,,), and slip velocity (Y,). We may compare YOfree (equation (15)) with the general expression for exponential creep based on absolute rate theory [e.g., Poirier, 1985]: V = V o exp expl J, (! 7) -5 which generally describes thermally activated creep in which the deformation rate is limited by the thermally activated jump -10 in a pinning potential field with a valley depth Q (activation energy) and spacing of valleys of the order of gl /3(activation volume). T is the absolute temperature, k is the Boltzman 10 ' s constant, and Y0 is the product of the attempt frequency and -2O the jump distance. The exponential dependence of the deformation rate on the applied stress is typically observed in the very high stress regime [e.g., Frost and 4shby, 1982] and =L ' i v -30 is called anelasticreep, the specific mechanisms for which lo x include stress corrosion cracking and dislocation glide. Considering the very high level of junction shear stress of the order of % (equation(10)), it is natural to assume anelastic -4O creep in the shearing of junctions. Comparison of the stress-dependent parts of (15) and (17) -45 leads to the relation k suggesting that a, increases proportionally with the absolute Figure 10. Arrhenius plot of slip velocity corrected for the temperature [e.g., Heslot et al., 1994; Sleep, 1997]. n effects of and r. Triangles and circles were calculated for the obtaining (18), the estimation of the level of :½(equation lsigma uncertainty of a, (Table 2) with the average of/, from (10)) was used. The present data (Figures 6c and 9c) show a tests (Table 1) at each temperature. Horizontal bars were rough agreement with (18), though the assessment is seriously calculated for/j, from each test with the mean of a,. (a) The ar limited by the large uncertainties a, at 673 and 873øK. values obtained from the direct measurements (Table 2) are used. Assuming a value of rye = 8 GPa [Bowden and Tabor, 1964] for (b) The a, values obtained from the corrected (section 4) for feldspar, the lines drawn in Figures 6c and 9c correspond to its change during reloading (Table 2) are used. gl = (0.38 nm) 3 and gl = (0.37 nm), respectively, indicating that the junction shearing is ratelimited by atomic-scale obstacles. (Typical atomic spacing between oxygen atoms in The Arrhenius plot compares the slip velocity at 0=0 (i.e. tekto-silicates is 0.26 nm.) Y,) at different temperatures, taking it into account that the On the other hand, from the comparison of the stress- slip velocity occurs by the application of a shear stress of independent parts of (15) and (17), we obtain Q (Figure 10). ', = ]/,Cr. Figure 10 is the Arrhenius plot of the slip velocity corrected One assumption behind making the Arrhenius plot of (19) for the effect of and : as is that the physical state corresponding to O = 0 is equivalent at different temperatures. This assumption cannot be true. However, if we adopt the idea that the physical state underlying O is the real contact area, this assumption may [/'O'lfreefT) --[/'Ofree( " r)/exp =. lo v rz'/o'/-- k a' -j exp/ -**/. (19) lo i v x 10'35- -4O 'C 400'C 25'C i (a) x10 1/T (K) 800'C 400'C 25'C (b) o.s s 2.o 2.s 3.o 4.0x10 1/T (K)

27 ... NAKATAN: PHYS1CAL CLARFCATON OF RATE AND STATE FRCTON 13,373 not be so bad as long as ere does not strongly depend on the temperature. The horizontal bars in Figure 10 correspond to V, exp(-,u, / a,) calculated using #, from each test (Table 1 ) and the mean value of a, determined for each temperature (Table 2). While #, varies considerably at tach temperature (by ), we can see the effect of,u, is not so strong. The uncertainty a, affects the Arrhenius plot more strongly (see triangles). The affect of the assumption of the equivalence of the O = 0 state at different temperatures may be assessed from these horizontal bars. f the difference in physical state at different temperatures can be adjusted by shifting the definition of O= 0 by, say, _+0.05, this difference does not seriously affect the Arrhenius plot. Figure 10 may suggest a single mechanism operating over the entire temperature range (25-800øC). Note that the large uncertainty of a, at 600øC does not affecthe Arrhenius plot as seriously as it affects the linear temperature plot (Figures 6c and 9c) because the Arrhenius plot becomes less sensitive to a, as its value increases. As discussed above, the theoretically correct parameter to study for the temperature dependence is a,. However, since the present experiments could not confidently prove that a, an,d a o take different values, we may also look at the results of the analysis under the assumption of a, = a o, as implied by the standard formulation of the framework law (equations (1) or (3)). The value, obtained under this assumption, is plotted againsthe linear temperature in Figures 6a and 9a, and the Arrhenius plots using in place of a, in (19) are shown in Figure 11. n these plots, there is a change in trend at ~400øC. f a reason were found to believe that a t has to be equal to ao, this trend change in the plots using might be interpreted as showing the change of the mechanisms accommodating frictional sliding, overriding the the plot using a, that suggests the operation of a single mechanism as mentioned earlier, because was determined more confidently than a from our data. However, unless we assume a, = a o, the presently obtained ff is essentially a o as noted in section The temperature dependence of, therefore, can not be necessarily related to the deformation mechanism of sliding. We below discussome possible specific mechanisms for frictional sliding (equal to junction shearing). The value Q kj/mol, obtained from the analysis at high temperatures may be argued to be in the range of dislocation 1/T (K) creep, butwice as large as its typical value for silicates [e.g., Figure 11. Same as Figure 10 except that ' (Table 2) was used Kirby and Kronenberg, 1987]. The value of gl is also twice as instead of a,. large as typical values. The low-temperature activation energy Q-90 kj/mol from ff analysis is in the range of stress corrosion [e.g., Atkinson, 198'4]. Although there have been 1981 ]. However, we must note that Shelton's value is for some suggestions [e.g., Dieterich, 1979; Lockner, 1998] to power-law creep with exponent 3-5. relate the direct effect in friction to the strain rate dependence Though some specific deformation mechanisms have been of intact rocks coming from stress corrosion cracking [e.g., raised above, it should be noted that we do not necessarily Lockner, 1998; Masuda et al., 1988], it is not convincing expect an exact match of the deformation mechanism of because experiments at extremely dry conditions [Dieterich junction shearing with one of the mechanisms identified for and Conrad, 1984] have shown that the direct effect is materials in a crystalline state because the frictional junctions observed as significantly as in the experiments at room should be in a highly disordered state [e.g., ¾und et al., 1990]. humidity conditions, while time-dependent healing, for which t may be necessary to consider a special mechanism for stress corrosion cracking is considered to play an essential amorphous solids such as "thermally activated premature role [e.g., Scholz and Engelder, 1976], disappears under the depinning," to which the traditional direct effect observed for same dry condition. On the other hand, the value of Q ~200 polymer surfaces has been attributed [Baurnberger et al., kj/mol obtained from a, analysis for the entire temperature 1999]. range is close to the value for the dislocation creep of the mportant conclusion of this section is that the framework present experimental material (albite, 234 kj/mol) [Shelton, law of modem friction is compatible with the idea that -5 ' 0 ' s lo, ! Q- x 10'3ø- 10'3s- -4O O i '3ø x (i) ' øC 400øC 25oc i (a) i i i i i x10 1/T (K) 800øC 400øC 25øC (b) x10

28 13,374 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON frictional sliding is entirely thermally activated. Except for Stesky's [1978] pioneering work done before the rate and state feature was recognized, frictional sliding of rocks has been thought to be typical "brittle" deformation and the roles of thermally activated process for sliding have been limited to "small" direct effect [e.g., Chester, 1994; Sleep, 1997; Baumberger et al., 1999], except for special minerals [Reinen et al., 1994] and special conditions [e.g., Chester and Higgs, 1992] where the evolution of state are not significant. However, with the help of SSS view (section 2.2, equation (14)), the present study emphasizes that the frictional sliding can be entirely thermally activated even for the conditoins where the evolution effect with short..l, a typically brittle behavior, is conspicuous. Note that the present experiments were done at a low normal stress of 20 MPa. Hence even the experiment at 800øC is in the brittle regime, where the nominal strength (a#.)is proportional to the normal stress and insensitive to the temperature [e.g., Scholz, 1990]. 7. Discussion over a wide range of temperature. They provided a consistent data set up to 845øC for nominally dry tests and 600øC for wet tests. Figure 12a plots their a value from the dry experiment against the linear temperature. An increasing trend with temperature may be seen, but the data scatter is very large. The data scatter is also large for wet case (Figure 12b), whereas the change around 600 K [Blanpied et al., 1998] seems still significant. n conclusion, these plots do not conflict or support the a result from the present study and its interpretation. Considering the present observation (e.g., Table 1) and the theory (sections 5 and 6) both suggesting that a is a well reproducible quantity, we think that the large data scatter in Figure 12 is simply due to the technical difficulty discussed below. The difficulty in Blanpied et al.'s [1998] data is that O changed a lot during the time from the end of steady state sliding at the prior velocity to the peak stress at the new velocity. This O change is taken into account in fitting the experimental data with the combination of the framework law and an evolution law. However, as they cautioned, the experimental data constrain a very poorly. This difficulty seems to have occurred because what they could see directly 7.1. Velocity Step Tests by Blanpied et al from the experimental data (Az'step) was often much smaller than what should have been seen upon the ideal velocity step As discussed in section 6, a is the theoretically correct (Figure 13, inset). f A rstep in the experimental stress parameter to be studied for the temperature dependence. displacement records of ten-fold velocity step tests of However, the present experiments could not determine a with great accuracy because of the limited range of AT employed. Since a is exactly the same thing as the direct effect, one Blanpied et al. [1998] is compared with aln10 using their inferred a values listed with the record, A s /6 is shown to be 20-90% of a ln10. n other words, the experiments lost 80- might expect that velocity-stepping tests, which should 10% of the information that should have been obtained in ideally give a, is a better approach. By ideally, O does not change during the time from the end of steady state sliding at the prior velocity to the peak stress at the new velocity. n ideal experiments. When the experiment loses this much information, its constraint becomes weak, and estimation is problematic, despite the careful modeling mehotd employed in this case, a is directly determined from the amount of stress their study. Also note that the value of a obtained by the change upon velocity step (Arstep) and is not affected by a o. n reality, however, a from velocity stepping tests is not so straightforward. To our knowledge, Blanpied et al., [1998] give the only example where a was obtained from velocity stepping tests fitting involving the framework law cannot be regarded as pure a but is affected by a o to some extent. The above problem becomes more severe as : decreases [Reinen et al., 1994; Blanpied et al., 1998]. Figure 13 shows the ratio of A 'step/aa n 10 as a function of '. The A 'step Was (a) (b) Dry Granite Gouge [Blanpied et al., 1998] ß Wet Granite Gouge [Blanpiecl et al., 1998] normal stress ~400MPa O effective stiffness -0.3 mm 'l ß effective stiffness -0,8 mm 'l ß ß normal stress =400MPa pore pressure =100MPa effective stiffness -0.8 mm ' a a o o o o o o o o o! T (K) Figure 12, Values of a obtained by fitting the 10 times velocity stepping tests with the combination of the framework law and the Ruina law (Table 2 of Blanpicd et al. [1998]) plotted against the linear temperature. Note that vertical scales arc different for these two figures ß ß ee ee T(K) ß ß ß ß ß

29 , NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13, ß r.oaln lo Figure 13. Quality of information obtained from velocity step tests as a function of nondimensionalized effective stiffness The Ruina law with b = a was assumed in simulating 10 times velocity stepup. calculated with the combination of the Ruina law with,/ = 1 and the framework law. The effective stiffness of Blanpied et al. [1998] was k mm -!. Assuming typical values of a = 0.01 and L = 10 gm, this value yields a range of tc = Figure 13 shows that velocity step tests (step up by 10 times) done under such values of tc preserve only 40-60% of the information. This estimate does not depend on the velocity range employed, as understood from Gu et al.'s [1984] nondimensionalization scheme but weakly depends on the magnitude of the velocity step ( ln(vn w / Vo d)) and its sign. Many pairs of a, b, and L values listed by Blanpied et al. 1998] lead to a further poorer assessment of the quality of the information that the experiments provided. This is not the fault of the velocity step method itself, as Az' t p/act n 10 (Figure 13) is improved to a level comparable to the Opeak/Oend(Plate 5) in our slide-hold-slide tests when tc is increased to a comparable level. Rather, the low effective stiffness due to the high normal stress (400 MPa)was responsible. The present experiments' small loss of O during reloading (section 4.2) largely benefited from the low normal stress employed (20 MPa). Unfortunately, we did not do systematic velocity stepping tests because we were not aware of the physical significance of a T at the time of experiment. Note that the values of b and L associated with the evolution over the long critical displacement of Blanpied et al. [ 1998] do not suffer the difficulty discussed above. in that the RSF framework law treats multiple evolution mechanisms by replacing O with YjOj [e.g., Gu et al., 1984; Ruina, 1983]. n the present series of experiments, however, except for the long 3162 s and 10,000 s holds at 600 and 800øC, O was dominated by Dieterich-type healing, and hence the present experiments can be essentially regarded to have observed the effect of this particular type of healing. The 3162 s and 10,000 s holds at 800øC (Figures 4e and Figure 7e), where significant contribution (---20% of O p k) occurred from another mechanism of healing [Nakatani, 1997], still lie on the trend from the shorter hold times. At face value, this result may suggest that healing from this second mechanism contributed to suppression of the slip velocity in the same manner as Dieterich-type healing. On the other hand, the temperature dependence of a o shows a significant trend change at 400øC (Figures 6d and 9d), whereas such a trend change is not clear in at. This might indicate that the second type of healing mechanism, which becomes significant above --400øC [Nakatani, 1997] affects the slip velocity with different efficiency from Dieterich-type healing. n any case, however, we emphasize that the data points involving significant healing by the second mechanism is too few to conclude anything about its effect on slip velocity. 8. Conclusions 1. The framework law of RSF (equation(1)) implies that slip velocity has a negative exponential dependence on the change of interface strength in addition to a positive exponential dependence on the change of applied shear stress (equal to direct effect). The former was not recognized previously. Here the interface strength is defined by the stress necessary to cause slip at a reference velocity (section 2). 2. The quantity #. + in the modem framework, not # (= r / tr), is regarded as an extension of the classical frictional strength as a threshold (section 2). 3. The present experiments (on an albite gouge layer, under 20 MPa normal stress, no pore water) have shown a negative exponential dependence of slip velocity on the change of interface strength throughout a temperature range øC (section 3.3). This affirms the qualitative validity of the use of the RSF framework law in many applications for which the prediction of slow creep plays a crucial role. 4. The present experiments indicated that the dependence of the slip velocity on the change of interface strength was stronger than the dependence on the applied stress (section 3), whereas the standard RSF formulation implies that they are the same. Still, the experiments suggest that both dependencies share the same root, consistent with a model based on the adhesion theory of friction (section 5). 5. The temperature dependence of the parameter a (ideally at) roughly agrees with the idea that frictional sliding is 7.2. Types of Healing As mentioned in section 4.2, the increase of value thermally activated shearing of frictional junctions by a mechanism whose rate has a strong exponential dependence on the junction shear stress (section 6). While the analysis of during the hold in the present experiments were not necessarily achieved by a single healing mechanism, as indicated by the existence of long-term slip weakening some cases (Plates 3a and 4a). Strictly speaking, the slip a suggests a change of specific mechanisms around 400øC, a logically more correct analysis based on a T suggests that a single mechanismay be operating throughout the wide range of temperature the nominally dry conditions of the present velocity was compared against the sum of the contributions experiments. However, a firm conclusion has not been from different strengthening mechanisms. This is consistent obtainedue to the low quality of the a T estimation.

30 13,376 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 6. n evaluating the stress and strength dependence of slip velocity (a or a r and ao), the present method, where the slip presented in the following is, of course, just a mathematical variable conversion, and hence the two notation systems are velocity at the end of hold at a given stress level is compared completely equivalent. However, better physical insights are with the interface strength measured at the following peak stress, shows promise because only 20-30% of the strength sometimes obtained from evolution laws written in large O notation. Note that some of pioneering works on RSF have change earned in the hold was lost during reloading (section used O notation [e.g., Ruina, 1983; Gu et al, 1984], though 4). Though this method's full value was not realized in the 0 has been favored in later studies. The following two present experiments, this is only because the lowest relations directly obtained from (A)will be useful in this measurable velocity for short hold times were limited by short period room temperature fluctuations (section 3.2). 7. Velocity stepping tests, if ideally done, can give a, not affected by a o. n reality, the quality of data strongly depends variable conversion: on the effective stiffness. The only available data from do b velocity step tests over a wide range of temperature [Blanpied do 0 et al., 1998] is not of enough quality for the present purpose (A3) of examining the temperature dependence of ar because of the low effective stiffness employed (section 7.1). 8. Strictly speaking, the change of interface strength in the present experiments did not come from a single healing mechanism. However, contributions fros,i the mechanisms other than Dieterich-type healing were minor except for the The framework law is written as + a ln(v / V.) #(= r/a) = #, + blnll in 0 notation, while it is written as (A4) long holds at high temperatures. Therefore, the presently # (=r/co=#, +O+aln(V/g,) (A5) confirmed dependence of slip velocity on interface strength in O notation. At the steady state sliding at an arbitrarily can be essentially regarded as the dependence on the change by chosen reference velocity V,, O is zero (by definition) and 0 Dieterich-type healing. (section 7.2). is L/V, as given by the evolution laws discussed below. The n addition to the above conclusions about the framework level of friction (or applied shear stress/normal stress) in this law, the following things have been found about the evolution situation is tt, in both notations. laws. Appendix A: Representation of the Evolution Laws in Large O Notation Using (A9), (A7) is rewritten as Here, the representation of the rate- and state-dependent friction law is shown in large O notation instead of conventional small 0 notation. The relationship between dt L them is O -- b n(l%). (A) Probable physical interpretation of O is that it represents the change of the interface strength from the reference value, (present paper) and that of 0 is the effective contact time [e.g., Dieterich, 1978; Beeler and Tullis, 1997]. What will be 0 =L exp/- ) (A2) 9. nspection of evolution laws in large O notation highlights that the three representative evolution laws differ A1. Ruina Law in the dependence of slip-weakening rate on the current O value (Appendix A). The Dieterich law assumes a constant rate Regarding the Ruina law, everything discussed here has already been known [e.g., Ruina, 1983]. t is repeated here for independent of O, the Ruina law assumes a linear dependence reader's convenience. The Ruina law [e.g., Marone, 1998a] in on O, and the Perrin law assumes an exponential dependence 0 notation is on O. When a moderately wide range of O (e.g., 1-7 times of a in the present experiments) is considered, the difference between these three laws is huge. Only the Ruina law can dt L adequately fit the experimentally observed slip-weakening while the same equation becomes evolution (section 4). 10. True time-dependent healing is described in the first term of the Dieterich law, as was already known. nspection of dt L the Dieterich law in large O notation highlights that the logarithmic nature of the healing comes from the fact that the underlying process's rate has a negative exponential dependence on the current O value (Appendix A2.1), consistent with recent physical models [e.g., Brechet and in O notation. Equations (A2) and (A3) were used to convert (A6) to (A7). By equating the left-hand sides of (A6) and (A7) with zero, steady state value of the state variables are given as a function of the velocity as Estrin, 1994; Berthoud et al., 1999]. Further, an apparent L cutoff time of healing depending on the initial state is O (r) = -- v predicted. (^8) ao = r [o_ %(v)]. (^10) For an evolution at a constant V, (A0) can be further rewritten as do = - [o- Oss(V)]dx (Al l) L where dx is the small increment of slip. This can be interpreted as the slip-dependent evolution toward a target '

31 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,377 value Oss(V), as Ruina [1983] pointed out. t is either slip weakening (when O > Oss(V)) or slip strengthening (when O < Os (V)) at a rate (per unit slip) proportional to the difference between the current O and the target value Oss(V), resulting in an exponential evolution with slip over L. driving stress at the contact should decrease as the contact area grows. Along this line, Brechet and Estrin [1994] has attributed the logarithmic healing to asperity creep whose rate depends exponentially on the junction normal stress. Berthoud et al. [1999] has shown the quantitative agreement of Though the Ruina law predicts apparentime-dependent this model with experiments on polymer surfaces. Moreover, healing during hold tests if O < Oss(V) [e.g., Ruina, 1983], it has been shown that slip is not as critical as the Ruina law says at least for nominally bare rock surfaces [Beeler et al., 1994; Nakatani and Mochizuki, 1996]. Though Perrin et al. [ 1995] have pointed out that the prediction of healing by the Ruina law deviates from Beeler et al.'s [1994] experiments only moderately, Nakatani and Mochizuki's [ 1996] stress corrosion, which has been proposed to be the underlying mechanism of time-dependent healing of silicate rocks [e.g., Dieterich and Conrad, 1984; Scholz and Engelder, 1976], has an exponential dependence [e.g., $cholz, 1972] on the drying stress. With a general initial condition of O = Oi i at t = 0, the solution of (A4) is experiments have shown that clear log time healing occurs even when shear stress during hold was almost zero, showing that truly time-dependent healing definitely occurs, t, (AS) disagreeing with the Ruina law. Cautio needs to be taken that this is not established as well for the surfaces separated by an O=O i+blnl+ -,½x L ;( ) artificial gouge layer. No time-dependent healing was which is a log time growth with the rate do / d(ln t) = b after a observed when :hold is low [Nakatani, 1998; Karner and cutoff time Marone, 1998; Nakatani, 1997] on such surfaces, which might be apparently consistent with the Ruina law [Nakatani, 1998]. However, Nakatani [1997] argued that the Ruina law's prediction of :hold effect on time-dependent healing differs as has long been known [e.g., Ruina, 1983], except for the from their detailed experimental results on the simulated gouge dependence of the cutoff time on the initial condition Oi,, i. layer. The meaning of this cutoff time can be illustrated by thinking A2. Dieterich Law of a situation where true stationary contact holding occurs immediately after a steady state slip at V = V. mi. Since the The Dieterich law in 0 notation [e.g., Marone, 1998a] is do steady state value of O by the Dieterich law is same as that by the Ruina law, using (A9), the initial t9 value at the start of this hold is dt n notation it becomes VO 1 -. (A2) L do b exp - ---V. (A13) dt L/V, L t½ = - -. exp, (A6) Oin i = bin/v' ). (A17) k, Vini J Plugging (A 17) into (A 16), we obtain Equations (A2) and (A3) were used to convert (A12) to (A13). t½ = L / Vi,,i. (A8) The first and second terms in these equations are discussed When we define the effective contact time at a steady state separately in sections A2.1 and A2.2. sliding at V as A2.1. Healing term of the Dieterich law (the first term). As easily seen from inspection of the first term tenco,, L'/V, (A9) of (A12) [e.g., Ruina, 1983], the first term of the Dieterich law represents true time-dependent healing. n O notation (A13) the healing term is with some representative length scale L' of the interface [e.g., Dieterich, 1978], we can see that the cutoff time in (A5) means that the log time growth does not occur conspicuously (in log time axis)until the hold time reaches the effective contact time at the beginning of the hold. This do dt L/V. b exp(_ bo_..) (A14) prediction has not been tested experimentally but sounds The solution of this differential equation is the log time growth of O [Dewers and Hajash, 1995; Nakatani, 1997]. Before showing the solution of (A4), the physics it implies reasonable. n (A8)this length scale L' is taken to be L, which is experimentally determined from the slip weakening of friction resulting from the second term of (A3), as will be discussed briefly here, namely, that the instantaneous discussed in section A2.2. However, there has not been any healing rate is a function of the current O [Nakatani, 1997]. This point is difficult to notice in 0 notation because the first experimental or physical reason to believe that L' has to be equal to L. t may be more reasonable to set a L' in the first term of (A2) does not include 0. The dependence is negative, term of (A13) as different from L in the second term. and this is the reason why healing slows down with time. The Note that the cutoff time discussed above is an "apparent" particular dependence of the exponential in (A4) merely cutoff time resulting from the initial condition of hold and is results from the fact that time-dependent growth of O (healing) is observed to be logarithmic. However, the idea obtained from (A4)that a stronger interface heals less efficiently sounds reasonable. For example, if we believe that distinct from the "true" cutoff time which is expected for more fundamental reasons [e.g., Okubo and Dieterich, 1986]. The fundamental cutoff time is not included in the formulation of the evolution laws discussed in this paper. f it exists, it would the healing is due to the increase of real area of contact [e.g., be observed as the lack of evolution effect following velocity Dieterich, 1972; Dieterich and Kilgore, 1994], the actual step in a very high velocity regime. Also note that the

32 13,378 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON apparent cutoff time (A 6) result solely from the first term of (A2) or (A3) and is distinct from another type of apparent cutoff time resulting from the second term [Marone, 1998b]. A2.2. Slip-weakening term of the Dieterich law (the second term). Slip weakening by the Dieterich law is contained in the second term of (A2) and (A3), that is, in O notation (A3), ao v. dt L Marone, 1998a] were aware of this problem, and Perrin et al. [ 1995] proposed another evolution law. n small 0 notation it is [e.g., Marone, 1998a] do = 1 -(VO) 2 (A22) Basically, they squared'the second term of (A2) for "faster" slip weakening. To be consistent in the steady state friction value, the framework law is slightly modified to This is an explicit slip weakening at a constant rate independent of the state, as more readily seen by rewriting (A4) as do = b dx. (A21) L This point is difficult to notice in 0 notation because the second term of (A2) includes 0. However, as pointed out in section 4.1, (A21) disagrees with most experimental results, even though it is not physically unreasonable [e.g., Beelet and Tullis, 1997]. Although there are a few experiments reporting linear slip weakening with a constant slope independent of peak friction height, none of those examples have been shown to be related with the erasure of the strength increment due to time-dependent healing, as discussed below. The first example is the erasure of the time-independent strengthening reported by Nakatani [ 1998], which he interpreted to be due to the mechanical consolidation of gouge caused by lowered shear stress during the hold. The slip weakening was clearly linear, and the slope was independent of the magnitude of the strengthening. As a result, the slip displacement necessary for the erasure increased linearly with the magnitude of the strengthening. The slope was far below the machine stiffness of the experiment. The strengthening occurred by lowering :ho a, with the magnitude proportional to A : but independent of hold time. (This time-independent consolidation strengthening occurred also in the present series of the experiments, but they were limited for low cases that are not used in the present paper.) A similar phenomenon has been observed by Karner and Marone [1998] in similar experiments. Another example is Richardson and Marone's [ 1999] experiment where normal stress was vibrated during the hold. This experiment also showed the linear gentle slip weakening with a constant slope independent of the peak height, strikingly similar to that of Nakatani [1998]. Though Sleep et al. [2000] interpreted Richardson and Marone's strengthening as time-dependent healing, we think it is due to mechanical consolidation caused by the normal stress vibration instead of by lowered shear stress [Nakatani, 1998], as Sleep et al. [2000] admit as an undeniable possibility that they do not favor. The minor increase of Richardson and Marone's strengthening with hold time would be due to the added mechanical consolidation by repetition of vibration [e.g., Lambe and Whitman, 1969]. Slightly modifying (A1), we redefine O as O--bln/ 2L/V. 0 (A24) so that we can keep the same framework law (A5) in O notation. From (A24), two useful relationships for conversion are obtained: do b d0 0 A3. Perrin Law This is similar to the Ruina law (All) in that the slipweakening rate is a function of the difference ( O - 0,, ) of the As discussed above, the Ruina law has a flaw in its inability current state from the target value, but the dependence is much to predictruly time-dependent healing (section A l), while the stronger in (A31), which is actually way too strong as shown Dieterich law has a flaw in correctly predicting slip-dependent in section 4.1. Also, different from the Ruina law, (A31) only evolution (section A2.2). Though the latter point has not been recognized as acutely as in the present study (section 4.1), som earlier works [Perrin et al., 1995; Blanpied et al., 1998; describeslip weakening. Steady state is achieved by the balance between the slip-weakening term and the healing term, as is the case in the Dieterich law. (A26) Using these relations, (A22) converted into large O notation is The first term do = b exp - - exp. (A27) dt 2L/V. 2LV, dt 2L/V, represents the truly time-dependent healing as in the Dieterich law. The only difference fi'om (A14) is that L in (A14) is now 2L. However, this difference does not have a significant meaning as seen from the earlier discussion about L and L'. The meaning of the second term of (A27), that is, do_ dt 2LV. bv 2 exp/ / (A29) becomes clearer when focusing on the evolution at a constant slip velocity. Then, (A29) becomes 2L V. which suggestslip weakening at a rate (per unit slip) that depends positively both on the current slip velocity and current O. Since the steady state value of O from (A27) coincides with (A9), (A30) can be further rewritten as do= b exp dx. (A31) 2L b '

33 NAKATAN: PHYSCAL CLARFCATON OF RATE AND STATE FRCTON 13,379 A4. Nondimensionalized Equations velocity-dependent volumetric strain of fault zones, J. Geophys. Res., 102, 22,595-22,609, The framework law (A5), and all the three evolution laws Beeler, N.M., T.E. Tullis, and J.D. Weeks, The roles of time and discussed above can be non-dimensionalized by Gu et al.'s displacement in the evolution effect in rock friction, Geophys. Res. [1984] scheme. We show the nondimensionalized equations Lett.,21, , below because they are used in sections 4 and 7. Following Gu Brechet, Y., and Y. Estrin, The effect of strain rate sensitivity on et al. [1984], we scale the terms with the meaning of shear dynamic friction of metals, Scr. Metall. Mater., 30, , stress or strength by a: Berthoud, P., T., Baumber er, C. G'Sell, analysis of the state- and rate-dependent friction law: Static friction, (9' (9/ a (A32) Phys, Rev. B, 59, 14,313-14,327, r' -- (r / rr -/,)/a. (A33) Blanpied, M.L., C.J. Marone, D.A. Locknet, J.D. Byedee, and D.P. King, Quantitative measure of the variation in fault rheology due to When choosing L and L/V. for the natural units of length fluid-rock interactions, J. Geophys. Res., 103, , and time, velocity should be scaled with Bowden, F.P., and D. Tabor, The Friction and Lubrication of Solids, Part e 2, 544 pp., Oxford Univ. Press, New York, t' = t/(l/ Y.) (A34) Chester, F.M., Effects of temperature on friction: Constitutive equations and experiments with quartz gouge, J. Geophys. Res., 99, , (A35) Chester, F.M., and N.G. Higgs, Multimechanism friction constitutive Then the framework law (A5) becomes model for ultrafine quartz gouge at hypocentral conditions, d. Geophys. Res., 97, , r' = O' + n V'. (A36) Dewers, T., and A. Hajash, Rate laws for water-assisted compaction The Ruina law (A7) becomes and stress-induced water-rock interaction sandstones, J. Geophys. 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