A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data



[1] We propose two major revisions on the rate- and state-dependent friction (RSF) law on the basis of rigorous analysis of friction experiments. First, we find that the direct effect coefficient a, a parameter playing a central role in the RSF constitutive law, is much larger than the traditional, consensual estimate of less than about 0.01. We derive a lower bound of 0.035 for a directly from stress-velocity relations measured during carefully designed step tests, without relying on any evolution laws as traditional methods do. After correcting for state changes during the steps, inferred indirectly from observed changes in acoustic transmissivities across the interface, we obtain an estimate of a as large as 0.05. Second, we calculate values of the RSF state variable Φ by feeding the measured shear stress and slip velocity values into the constitutive law. The results showed systematic deviations from predictions of the RSF evolution law of the aging type. This leads us to propose a revised evolution law, which incorporates a previously unknown weakening effect related to the shear stress. We also present additional experiment results to corroborate the presence of this new effect. Forward simulations based on our revised evolution law, combined with the larger, revised value of a, very well explain observed variations in both the shear stress and Φ throughout different phases of experiments, including quasi-static hold, reloading after a hold, and steady state sliding at different velocities, as well as their mutual transitions, all with an identical set of parameter values.

1. Introduction

[2] It has been recognized, since the work of Bowden and Tabor [1964], that variations in frictional strength can largely be ascribed to variations in the real contact area along the interface. For instance, the linear dependence of frictional strength on the normal stress, known as the Coulomb-Amonton law, can be explained by an increase in real contact area, which is proportional to the normal stress [e.g., Dieterich and Kilgore, 1996]. Experimentally observed second-order variations in friction, such as time-dependent healing [e.g., Dieterich, 1972] and slip weakening [e.g., Dieterich, 1978], are also considered attributable to an increase and a decrease in the real contact area, respectively. Better understanding of these kinds of behavior of the frictional strength is critically important for the modeling of the dynamics of contacting surfaces such as earthquake faults.

[3] One class of empirical friction laws, called the rate- and state-dependent friction (RSF) law, incorporate the effect of these changes in the real contact area by introducing a quantity called the state variable [Dieterich, 1979; Ruina, 1983] to the constitutive relationship between applied shear stress and slip velocity:

display math

where τ and σ are the shear stress and the normal stress, respectively, V is the slip velocity, V* is an arbitrarily chosen reference velocity, a is a nondimensional, positive parameter called the direct effect coefficient, and Φ is the state variable that describes the internal physical state along the interface, which can in most cases be viewed as representing the real contact area. As Nakatani [2001] pointed out, Φ as defined in equation (1) is a natural extension of the frictional strength used in the classical friction law, because Φ represents the shear stress required to slide the interface at the reference velocity V*. The RSF constitutive law, combined with an evolution law that describes how Φ changes according to various factors, has successfully explained typical behavior of the frictional resistance observed in laboratories. Because RSF can account for a variety of slip behavior over a wide range of slip velocities—from quasi-static sliding such as earthquake nucleation [e.g., Dieterich, 1992] and afterslip [e.g., Marone et al., 1991] to high-speed sliding such as coseismic rupture [e.g., Bizzarri and Cocco, 2003]—many studies have adopted RSF in the modeling of seismic cycles [e.g., Scholz, 1998; Marone, 1998a].

[4] Despite its widespread use, however, RSF is known to have its own shortcomings. A major open question is how to describe the evolution of Φ with slip history. The physical mechanism behind the RSF constitutive law has been elucidated and attributed to thermally activated rheology of frictional junctions [Heslot et al., 1994; Nakatani, 2001]. The evolution law, however, has not been completely successful even at phenomenological descriptions [e.g., Beeler et al., 1994; Marone, 1998a]. Some candidate forms of the evolution law have been proposed, but none of them has been able to explain accurately all phases that occur in laboratory experiments, such as quasi-static hold, reloading after a hold, and velocity steps.

[5] Despite these uncertainties concerning the evolution law, RSF parameters have to be determined by fitting laboratory data to predictions based on the combination of a constitutive law and an evolution law. Use of an imperfect evolution law may lead to misestimation of important parameters of the RSF constitutive law.

[6] In this paper, we address questions of the evolution law and of the constitutive law separately. We first constrain the value of the constitutive parameter a, called the direct effect coefficient, without relying on any type of evolution law (section 4). Once the value of a is known, one can estimate Φ at any time during an experiment using the constitutive law (equation (1)), which can be rewritten as

display math

Second, we scrutinize existing evolution law models by comparing observed Φ, estimated from observed mechanical data using equation (2), and predicted Φ (section 5). On the basis of comparison, we propose a modification to one existing model of the evolution law. The revised evolution law very well explains variations in the shear stress and Φ that are recognized in various types of experiments, and provides a satisfactory solution to all shortcomings known to date.

2. Background: Existing Evolution Law Models and Their Shortcomings

[7] One popular model of the evolution law, called the slip law or the Ruina law [Ruina, 1983], is written as

display math

where L is a characteristic length scale and Φ* is a reference state variable, which is taken to be the steady state value of Φ at the reference slip velocity V*. The steady state value of Φ under a constant slip velocity V is derived by equating the left-hand side of equation (3) to zero:

display math

Combining this with equation (1), one obtains the shear stress τSS under steady state sliding:

display math

This type of velocity dependence of the shear stress under steady state sliding has been very well corroborated in many laboratory experiments [e.g., Marone, 1998a]. According to equation (3), the slip law predicts that Φ evolves toward its steady state value ΦSS with ongoing slip u, the difference Φ − ΦSS being proportional to exp(−u/L). The slip law therefore fails to explain time-dependent healing at zero or very small slip velocities [Beeler et al., 1994; Nakatani and Mochizuki, 1996], which is one of the best established features of Φ [e.g., Dieterich, 1972], although the law well explains shear stress histories observed in velocity step tests.

[8] Another popular evolution law model, called the aging law or the Dieterich law [Dieterich, 1979; Ruina, 1983; Nakatani, 2001] and written as

display math

well explains log time healing during stationary contact. The first term on the right-hand side, which represents time-dependent healing of Φ, is called the healing term. The multiplier bσ represents the increment in Φ that corresponds to an e fold increase (e: Euler's number) in the healing time [e.g., Nakatani, 2001]. The healing process is thought to result from an increase in the real contact area caused by inelastic deformation of asperities, forming real contacts, in the direction normal to the interface [e.g., Scholz and Engelder, 1976]. Brechet and Estrin [1994] showed that log time healing can be explained by a physical model along this line. Further physical interpretation was done by Nakatani and Scholz [2006]. Like the slip law, the aging law also produces equation (4) for the steady state value of the state variable Φ, which well explains experimental observations. However, slip weakening behavior predicted by the aging law is clearly at odds with most laboratory observations.

[9] The second term on the right-hand side of equation (6), where L is not a characteristic slip distance but simply serves as a coefficient of slip weakening as /L, represents slip weakening in Φ at a constant weakening rate per unit slip, as long as the normal stress σ remains constant [Nakatani, 2001]. This means that the slip distance has to be larger when the change in the strength (∼Φ) is larger. In contrast, laboratory experiments show that slip weakening proceeds exponentially, rather than linearly, with slip displacement and ends at a characteristic length that is independent of the magnitude of the strength change [e.g., Ruina, 1980, 1983; Tullis and Weeks, 1986]. Another well-known misprediction of the aging law is that the shear stress responds differently to a step increase and a step decrease in the slip velocity. In experiments, they show near-symmetric patterns when plotted against slip displacement, but the aging law predicts clearly asymmetric evolutions [e.g., Marone, 1998a; Kato and Tullis, 2001].

[10] A modified evolution law, presented by Perrin et al. [1995], is written as

display math

The first term, the same as in the aging law, represents purely time-dependent healing. The second term proposes that the rate of slip weakening depends also on the current slip velocity and the current value of Φ. This type of slip weakening is free from the question of asymmetry. It has been pointed out, however, that the slip-weakening rate predicted by equation (7) is much larger than what is observed in laboratory experiments [Nakatani, 2001].

[11] Slight differences in the evolution law can sometimes have important consequences not only for explaining experimental observations but also for forecasting earthquake cycles. It is known that numerical simulations of seismic cycles based on the RSF constitutive law vary significantly depending on which sort of evolution law is used [e.g., Rice and Ben-Zion, 1996]. A recent work by Ampuero and Rubin [2008] shows that different evolution laws may lead to qualitatively different predictions on earthquake nucleation patterns, indicating that the problem has crucial consequences for earthquake prediction.

3. Experiment Setup

[12] All experiments described in this paper were conducted in a double direct shear apparatus (Figure 1) under room temperature and humidity. The sliding interfaces were created between bare surfaces of fine-grained Aji granite, prepared with #600 abrasive. The area of each sliding surface remained constant at 50 × 50 mm during all stages of slip. The normal stress was retained at 10 MPa in all experiments. The shear displacement was measured at a point in the loading medium, not directly on slip interfaces, whereas the normal and shear stresses applied on the two sliding surfaces were measured directly with load cells. The slip displacement (u) and the slip velocity (V) were estimated from the loading point displacement (uL) by subtracting elastic deformation between the frictional interface and the loading point:

display math
display math

where k is the system stiffness, or the coefficient of elastic interaction between the slip interface and the loading point, and is estimated from the measurement of loading point displacement caused by a change in the shear stress under a negligible slip velocity. The displacement used for the estimation of k included elastic deformation of the interface itself. In all experiments shown below, k was estimated at 0.2 MPa/μm.

Figure 1.

Experiment setup and an example of received acoustic waves.

[13] During all experiments, P wave transmissivity of the friction interface was measured to monitor the state of contact [Nagata et al., 2008]. A pair of PZT ultrasonic transducers, with a central frequency of 1 MHz (Panametrics V103RM), were attached on the outer faces of the two outside blocks. This configuration allowed the relative transmitter-receiver positions to remain unchanged by the sliding offset. One cycle of 1 MHz sine waves, with a repetition rate of 1 kHz, was used to excite the transmitter. The other transducer received waves that had traveled through the three blocks and across their interfaces. The received waves were amplified and digitized at 100 megasamples per second over a time window of 8192 samples by using a high-speed digitizer (Gage CompuScope 14100). The digitized waveforms were averaged over 10 shots and the peak-to-peak amplitudes of the averaged waveforms were recorded. The acoustic transmissivity of each interface was calculated by ∣T∣ = inline image, where A is the transmitted wave's amplitude and Ai is the amplitude of the pulse that has traveled through the same length of an intact rock. Nagata et al. [2008] showed that acoustic transmissivities measured in this way very well reflected changes in the contact state variable Φ. Although the acoustic data supported all conclusions of the present paper, we base our main arguments, for prudence, on mechanical data alone.

4. Estimation of the Direct Effect Coefficient a Without Recourse to an Evolution Law

4.1. Traditional Method to Estimate a

[14] Under the RSF constitutive law (equation (1)), the shear stress is a function of the slip velocity and the state variable:

display math

The first term represents the direct effect. This relation suggests that the parameter a can be estimated from changes in the shear stress and the slip velocity under a fixed state variable.

[15] Typically, velocity step tests, where abrupt changes are imposed on the velocity of the loading point (VL), are used to ensure that Φ is unchanged. Under ideal test conditions (the slip velocity V always equal to the time-varying VL to add to very fast servocontrol), Φ does not change, because no aging or slip weakening takes place during the step. Hence the corresponding change in the shear stress, Δτpeakideal, should only stand for the direct effect and have a magnitude of aσΔlnV (Figure 2a). The direct effect coefficient a should therefore be estimated by

display math

where V1 and V2 are the prestep and poststep slip velocities, respectively.

Figure 2.

Schematic illustration of (a) an ideal step test and (b) a step test in reality in which the load point velocity (VL) is servocontrolled.

[16] In reality, however, the slip velocity cannot be changed instantaneously because of elastic deformation of the medium between the slip interfaces and the loading point. Hence Φ changes to some extent before the shear stress reaches its peak (Figure 2b). Equation (10) shows that the amount of the direct effect has to be estimated by subtracting the change in Φ at the peak stress (ΔΦpeak) from Δτpeak:

display math

Note that, when the stress is peaked, the slip velocity (V) is equal to the controlled loading velocity (VL).

[17] Usually, ΔΦpeak is inferred by assuming a certain sort of evolution law. This is what is essentially done in determining RSF parameters by fitting laboratory shear stress data to predictions based on a model that combines a constitutive law, an evolution law, and a system stiffness. As noted before, however, every existing evolution law model has its own deficiencies. The estimate of a may therefore be subject to serious errors, reflecting imperfections in the evolution law assumed.

4.2. New Method and Test Design

[18] We hereby propose a new method to constrain the value of the direct effect coefficient without recourse to an evolution law. Our strategy simply consists in performing a step test with as little change in Φ as possible. Generally, in step tests starting from a steady state, Φ changes in the opposite direction to the change in velocity. The ratio between Δτ and Δ ln V, obtained directly from actual step tests, therefore sets a lower bound on the value of a:

display math

By minimizing the change in Φ, alb can be brought closer to the true value of a. And the change in Φ can in fact be minimized by servocontrolling the shear stress rather than the load point velocity.

[19] As mentioned above, elastic deformation in the loading system makes it difficult to realize sharp steps in velocity step tests, where the load point velocity is controlled. By contrast, the shear stress applied on the slip interface can be servocontrolled directly by monitoring the shear load measured at the loading point, which is equal to the shear load on the frictional interface as long as the slip velocities remain small (i.e., negligible inertia).

[20] In addition, unlike the slip velocity, the shear stress can be measured instantaneously. This makes servocontrol much more effective in the first place. A stress step test can therefore be much closer to ideal than a velocity step test.

[21] Unfortunately, even in an ideal “stress step” test, it would still not be possible to get measurements done with no change in Φ, because measurement of the poststep velocity requires a certain amount of slip displacement anyway. In addition, changing the stress takes a finite length of time, depending on the performance of the servocontrol system (about 1 s in our tests). To minimize the change in Φ during the step and during the measurement of the poststep velocity (ΔΦ1−2), we took additional care.

[22] First, the slip velocity under steady state sliding before the step was chosen to be small. The rate of increase in Φ by time-dependent healing [e.g., Dieterich, 1972] is written as

display math

[e.g., Brechet and Estrin, 1994; Nakatani and Scholz, 2006]. Equation (14) shows that the time-dependent increase in Φ is slower for larger Φ, reflecting a lower driving local normal stress due to a larger contact area [Brechet and Estrin, 1994; Nakatani and Scholz, 2006]. During steady state sliding, Φ is known to be larger for smaller slip velocities [Dieterich, 1978; Dieterich and Kilgore, 1994]. The same observation was made by acoustic monitoring of the contact state [Nagata et al., 2008]. Figure 3 shows acoustic transmissivities of the frictional interface measured after a step decrease in the shear stress following steady state sliding at different velocities. At large hold times, all curves asymptotically approach a linear log time increase. At earlier stages, however, the growth rate is smaller than this asymptote. As shown by Nakatani and Scholz [2006], the duration of this phase of slower growth is inversely proportional to the prior slip velocity. Similar results were obtained for the static friction coefficient (μ = τ/σ = (Φ + ln (V/V*))/σ) [Marone, 1998b]. Therefore, the poststep change in Φ due to healing should be smaller if the test starts at a smaller slip velocity. In the experiments, we set the steady state sliding velocity before the step at 0.01 μm/s, which is the lowest velocity of steady state sliding achievable by our apparatus.

Figure 3.

Increase in transmissivity during quasi-stationary contact following steady state sliding at different velocities.

[23] Second, the step in the shear stress was chosen to be negative (Δτ < 0), because an increase in the shear stress could result in runaway slip acceleration.

[24] Third, the magnitude of the negative step in the shear stress was chosen to be small. This is because a larger negative step, or a smaller poststep slip velocity, would require longer time for the velocity measurement to be completed, involving a larger change in Φ due to healing.

4.3. Results and Analysis

4.3.1. Lower Bound Estimate for a

[25] Figure 4 shows an example of a stress step test, where the shear stress was reduced by 0.05 MPa from steady state sliding at 0.01 μm/s. The slip displacement (u) was estimated with equation (8).

Figure 4.

An example of a stress step test. The dark-colored lines that run through the centers of the slip displacement and transmissivity curves denote 1 s moving averages.

[26] The prestep slip velocity could be estimated with confidence from a record of long-sustained steady state sliding before the step (thick yellow line in Figure 4). To measure the poststep slip velocity just after the stress step, we used data corresponding to a 0.1 μm slip distance after a new constant stress was achieved within a few seconds of the step operation. In both cases, the slip velocity was determined by least squares line fitting.

[27] Figure 5 shows the values of alb obtained by equation (13). The estimates of alb were larger for the smaller value of the stress step magnitude ∣Δτ∣. This is probably due to a difference in ΔΦ1−2, or the change in Φ before and after the step. Figure 6 shows the changes in acoustic transmissivity, Δ∣T1−2, between the prestep steady state and the point where the poststep velocity was measured, which we believe reflect ΔΦ1−2. Because the measurement of the poststep velocity required a certain duration of time, Δ∣T1−2 was evaluated using ∣T∣ averaged over that time duration. In Figure 6, as expected, Δ∣T1−2 was smaller in experiments with the smaller ∣Δτ∣, which suggests that the alb estimates obtained for the smaller ∣Δτ∣ were closer to the true a value. Presumably, the larger poststep slip velocity in experiments with the smaller ∣Δτ∣ led to a larger slip distance, which helped to cancel the healing effect on Φ to a larger extent during measurements of the poststep velocity. We therefore refer to the results for the smaller ∣Δτ∣ = 0.05 in Figure 5 (alb = 0.035 ± 0.002 in mean and standard deviation) and use 0.035 as the lower bound estimate for a.

Figure 5.

Estimates of the lower bound on the value of a. The ∣Δτ∣ is the magnitude of the stress step used in the test. The error bars are derived from uncertainties in poststep velocity measurements.

Figure 6.

The change in acoustic transmissivity before and after the step, or Δ∣T1−2, for the stress step tests shown in Figure 5.

4.3.2. Exact Value Estimate for a

[28] In the method described above, a lower bound on the value of a was estimated with equation (13) by ignoring ΔΦ1−2. Equation (10) shows that, when ΔΦ1−2 is not zero, the true value of a has to be estimated by

display math

One should therefore know the amount of ΔΦ1−2 to estimate the true value of a from stress step test data. This idea is not very different from that of the traditional method. We chose, however, to estimate the value of ΔΦ1−2 on the basis of acoustic data instead of relying on an evolution law.

[29] For this purpose, we need to know the quantitative relationship between Φ and ∣T∣. Figure 7 shows relations between observed ∣T∣ and Φ, the latter calculated by equation (2) with different values of a assumed. The data in Figure 7 were taken from the work of Nagata et al. [2008], who conducted experiments on various phases, including hold at a constant shear stress, reloading, steady state sliding at different velocities and their mutual transition, using the same blocks that were used in this study. The ∣T∣ − Φ relations were almost linear and unique when a was assumed to be greater than 0.03 (Figure 7). Fortunately, we already know that a is greater than 0.035. We therefore assume a linear relationship:

display math

where r represents the sensitivity of the acoustic transmissivity to changes in Φ.

Figure 7.

The relationship of ∣T∣ − Φ for different assumed values of a to calculate Φ using equation (2). The data were taken from an experiment, described by Nagata et al. [2008], on various phases including quasi-static hold, reloading after a hold, steady state sliding at a number of different velocities (0.01, 0.1, 1, 10 μm/s), and their mutual transition under a normal stress of 10 MPa.

[30] The sensitivity r can be determined from the slope of the ∣T∣ − Φ plot in Figure 7. To draw such a plot, however, we first need a, the very value that we want to evaluate. To escape from this logical circle, we introduce an independent constraint from acoustic data taken during steady state sliding at different velocities (Figure 8). Let us make just one assumption here that the velocity dependence of Φ under steady state sliding is given by equation (4). Certainly, this equation is part of the evolution law concept, but there is a broad consensus on this type of velocity dependence of ΦSS (the effective contact time concept [Dieterich, 1978; Marone, 1998a]), although the applicability of evolution laws to nonsteady states remains a controversial issue. Using the data shown in Figure 8, where Δ∣T∣ corresponding to an e fold increase in VSS was −0.0068, one can express r−1 as a function of b:

display math

because the change in ΦSS corresponding to an e fold increase in VSS is − (equation (4)).

Figure 8.

P wave transmissivity under steady state sliding at different slip velocities. The dotted line shows the slope of the ∣T∣ − lnV relationship.

[31] Further, the difference between the parameters a and b is available from the dependence of the shear stress on the slip velocity under steady state sliding according to equation (5). The τSS values observed in our experiments have suggested ab = −0.0056. One can thus rewrite equation (17) to represent the sensitivity r as a function of a:

display math

Substituting equation (18) into equation (16) and then substituting the output into equation (15), one obtains a as follows:

display math

This means that a can be derived from only three quantities measured during step tests. Δτ/σ and ln(V1/V2) were used to obtain alb (equation (13)). Δ∣T1−2 is the difference between the transmissivity under steady state sliding before the stress step and the average transmissivity during velocity measurement after the step (Figure 6). The a values estimated by equation (19) are shown in Figure 9. They range between 0.037 and 0.064. They are scattered over different ranges for the different values of Δτ, but the ranges overlap. We take a ∼0.05, in the overlap region, as our best estimate.

Figure 9.

Estimates for the exact value of a. The error bars are derived from uncertainties in poststep velocity measurements.

[32] This value of a, substituted into equation (18), gives an estimated sensitivity of r(≡Δ∣T∣/ΔΦ) = 0.012 /MPa. Both values agree with those obtained by Nagata et al. [2008] using a different method, which we think was less reliable, because the assumptions in their arguments included a controversial part of the evolution law, although they tried to use the least affected data. Despite uncertainties in the value of a obtained here, all conclusions of this paper remain valid as long as a is greater than 0.03. And the discussions in section 4.3.1 preclude all values of a smaller than 0.035.

[33] The above estimate of a is based on the results of negative shear stress step tests because, as mentioned earlier, an increase in the shear stress could have resulted in runaway slip acceleration. We did not perform positive stress step tests. As shown in section 5, however, numerical simulations based on our estimate of a and a modified evolution law we propose here fit very well the observations of both positive and negative velocity step tests, which suggests that a similar value of a value would have been derived from positive step tests.

5. Proposal of a Modified Evolution Law

[34] The question of evolution laws has been left largely unsolved partly because their predictions concerning the state variable are very hard to corroborate. RSF has always been described in terms of a pair of an evolution law and a constitutive law, and comparison has customarily been made only on the shear stress. In this paper, however, we have evaluated the direct effect coefficient a on the basis of a constitutive law alone. We now fix the value of a, and analyze Φ obtained by the constitutive law (equation (1)) using mechanical measurement data.

[35] The a value of 0.05, obtained with our method, is much larger than typical values known before. With this value of a, existing evolution law models totally fail to explain observed shear stress histories. Figure 10 shows examples of velocity step test simulations with an aging law (equation (6); Figures 10a and 10b) and a slip law (equation (3); Figures 10c and 10d), overlaid on experimental data. Since the value of ab has been estimated with confidence from the velocity dependence of the shear stress under steady state sliding according to equation (5), the value of b is automatically fixed when the value of a is fixed. For both types of the evolution law, then, the only one parameter adjustable is the length parameter L. When we use a larger value of L, which means a smaller weakening rate per slip, the predicted peak in the shear stress after the step becomes too large because of the large value of a (Figures 10a and 10c). On the other hand, when we use a smaller value for L to suppress the peak stress following the step, very unstable behavior follows, unlike in the observation (Figures 10b and 10d). In short, as long as we accept the large value of a as indicated by the present study, both the aging law and the slip law perform very poorly.

Figure 10.

Shear stress and Φ simulated using an aging law with a fixed at 0.05 for a positive velocity step test (from 0.01 μm/s to 0.1 μm/s). In all plots, b = 0.056, k = 0.2 MPa/μm, σ = 10 MPa, and V* = 1 μm/s. (a) Results for an aging law with L = 1 μm. (b) Results for an aging law with L = 0.4 μm. (c) Results for a slip law with L = 1 μm. (d) Results for a slip law with L = 0.4 μm. The observed shear stresses and acoustic transmissivities are also shown in all plots. The acoustic transmissivity ∣T∣ is so scaled that variations in ∣T∣ and Φ overlap on the charts during steady state sliding.

[36] We therefore have a stronger motivation to look for an evolution law that works better. Using the obtained a value, Φ can be calculated at any time during an experiment from the instantaneous values of V, τ, and σ according to equation (2). In addition, when V is too small to be measured accurately, the acoustic technique of Nagata et al. [2008] is available to monitor the contact state. We can therefore infer real values of Φ directly from the observations and compare them with predictions of existing evolution laws.

5.1. Misprediction Analysis

[37] We start from the aging law, because it describes time-dependent healing more properly. Although it is known that slip weakening is not correctly described by an aging law [Nakatani, 2001], the slip-weakening term (second term of equation (6)) is necessary to explain the observed velocity dependence of the shear stress under steady state sliding (equation (5)) when the healing term (first term of equation (6)) is present. One approach to find a way out of this paradox would be to look for mispredictions that emerge only under a nonsteady state.

[38] For misprediction analysis, we used experimental data, part of which was shown by Nagata et al. [2008]. The data include various phases of slip experiments, such as quasi-stationary hold, reloading after a hold, steady state sliding at velocities of 0.01, 0.1, 1, 10 μm/s, and transitions between them. A sequence of slip experiments, including these phases, was repeated three times under a constant normal stress of 10 MPa. We inferred real values of the state variable, using the τ and V values observed during the experiment (Φobs), according to equation (2). Here we compare inline imageobs, the time derivative of the observed state variable, with predictions for the same quantity ( inline imagepredict), calculated by substituting observed V and Φobs into the aging law (6). As explained in the introductory part of section 5, the b parameter is virtually fixed, so only one parameter, L, is adjustable in equation (6) for calculating inline imagepredict.

[39] Since we were interested in finding out evolution effects that emerge only in a nonsteady state, we plotted misprediction in the time derivative of the state variable ( inline imageobs − Φpredict) against the time derivative of the observed shear stress ( inline image) and looked for possible systematic mispredictions. Figure 11 shows results for different values of L assumed. If an aging law with an appropriate L correctly predicted variations in Φ, all data points would lie on inline imageobs − Φpredict = 0. This, however, was not the case, whatever value we might assume for L. This is only natural, because it is known that the aging law can explain only limited aspects of experiments. On the other hand, we found a systematic dependence on inline image when L = 0.33 μm was assumed (Figure 11b). This suggests that an increase in the shear stress has an additional evolution effect on the weakening of Φ.

Figure 11.

Difference between inline imageobs and inline imagepredict plotted against inline image. The predictions were made with the aging law (equation (6)) with different values of L assumed. In addition, a = 0.05 and b = 0.056 were assumed throughout Figure 11.

[40] Assuming, for simplicity, that the difference between the real and predicted values of inline image is linearly proportional to inline image as in Figure 11b, the aging law can be modified to

display math

where c is a positive coefficient that determines the magnitude of the newly introduced shear stress effect. We set c at 2 from the average slope in Figure 11b. The modified evolution law (equation (20)), with c = 2 and L = 0.33 μm, should therefore be able to explain the evolution of Φ much better than the original aging law (equation (6)) throughout different phases of experiments.

[41] The third term in equation (20) may be rewritten as

display math

which we would call a stress-weakening effect. To confirm the presence of this effect, we designed an experiment to look at the relationship between ΔΦ and Δτ under a condition where the two other terms (healing and slip weakening) had minimal effects. Because the effect of healing is smaller for larger Φ [e.g., Nakatani and Scholz, 2006] and the effect of slip weakening is expected to be small when the slip velocity is small, the tests were performed after a hold sustained over some time (hold time). Following a quasi-static hold under a constant shear stress (τhold), the shear stress was increased by Δτ and was held constant at τafter thereafter (Figure 12). Measurement of the slip velocity, a necessary procedure to infer Φ with equation (2), is difficult under such extremely small velocity conditions. Therefore, we used instead the acoustic transmissivities, measured continuously during the tests, as a proxy for ΔΦ. A linear relationship was found out earlier between ΔΦ and the change in acoustic transmissivity, Δ∣T∣, under the same experiment conditions, with a sensitivity of r(≡Δ∣T∣/ΔΦ) = 0.012 /MP (section 4.3.2). Figure 12 clearly shows an instantaneous decrease in acoustic transmissivity upon a shear stress increase, providing evidence for an instantaneous stress weakening effect on Φ. In more detail, Figure 12 suggests that a further decrease in acoustic transmissivity may follow the instantaneous decrease, but we include only the instantaneous effect in the evolution model proposed presently.

Figure 12.

An example of a stress step test that demonstrates the weakening effect of an increase in τ on Φ (the acoustic transmissivity T is used as a proxy for Φ).

[42] Figure 13 is a plot of Δ∣T∣ caused by different shear stress increases Δτ in the stress step tests, including the case shown in Figure 12. The decrease in acoustic transmissivity is approximately linearly related to the increase in shear stress. Using the sensitivity of r = 0.012 /MPa obtained earlier, Figure 13 yields c ≈ 2. This agrees with the independent estimate of c = 2 derived from Figure 11b.

Figure 13.

Relationship between the change in acoustic transmissivity and the change in shear stress from an initial state of τhold = 8.2 MPa and for a hold time of thold = 300 s. The broken line shows a reference slope for c = 2 with an assumed sensitivity of r(≡Δ∣T∣/ΔΦ) = 0.012 /MPa.

[43] The declining steepness of the row of dots in Figure 13 for smaller Δτ suggests a declining sensitivity of the acoustic transmissivity to the shear stress in low shear stress ranges. This observation is corroborated by another test, where the shear stress varies over a broad range from 0.3 MPa to the peak stress (Figure 14). The solid curve in Figure 14 shows a direct comparison between the acoustic transmissivity and the shear stress, not their respective changes. The sliding velocity stayed closed to zero in most parts of Figure 14 except near the peak stress, under which the sliding velocity was equal to the controlled loading velocity (1 μm/s). To account for the reduced sensitivities at low shear stresses, c needs to be defined as a function of τ. Figure 14 shows that c varies continuously from a small value at τ ∼ 0 to approximately 2 at τσ. From Figure 14, an approximate form may be given by

display math

where c* is a reference value under the reference shear stress τ*, and τc is a phenomenological, characteristic stress scale. The τc may vary in proportion to the normal stress (10 MPa in our experiments), as is generally the case with all macroscopic friction law parameters that have the dimension of the shear stress.

Figure 14.

Relationship between the acoustic transmissivity and the shear stress during a test where the shear stress was increased from 0.3 MPa until it reached a peak, nearly equal to the normal stress of 10 MPa. The load point velocity was controlled at 1 μm/s. The slip velocity was close to zero for all shear stresses except for very high shear stresses near the peak. The dashed line shows a reference slope for c = 2 with an assumed sensitivity of r (≡Δ∣T∣/ΔΦ) = 0.012/MPa. The dotted curve shows predictions from equation (23).

[44] The corresponding drop in Φ in the experiment shown in Figure 14 can be evaluated by integrating equation (22) with respect to the shear stress, resulting in

display math

where Φ0 is the value of Φ under the initial low of the shear stress τ. The dotted line in Figure 14 shows the dependence of the acoustic transmissivity ∣T∣ on the shear stress τ, predicted by equation (23) and the assumed sensitivity of r = 0.012/MPa, with c* = 2, τ* = 9.4 MPa, and τc = 2.5 MPa.

[45] Figure 14 also shows that the sensitivity of the observed ∣T∣ to the shear stress increases rapidly toward infinity at very high shear stresses, outside the test range shown in Figure 13. This is presumably attributable to slip weakening, because high shear stresses cause considerable slip (see equation (1)), and does not necessarily indicate enhanced sensitivities at very high shear stresses. Indeed, Figure 11b, where the slip-weakening effect was offset by using an aging law, indicated a largely constant sensitivity of c ∼ 2, although some of the data were from very high shear stress ranges.

[46] Although the coefficient c varies considerably with the shear stress τ, the relationship between Φ and τ may be approximated by a linear one, assuming a constant c, when the shear stress varies fairly little (Figure 11b). Like in most experiments and applications of fault mechanics, all of our experiments were performed at high shear stresses, in excess of 8.7 MPa, where it is safe to regard c as nearly constant at around 2 (Figure 11b). In the following simulations, therefore, we use the simple, linear dependence of Φ on τ as described by equation (20).

5.2. Simulations Based on the Modified Evolution Law

[47] The proposed evolution law (equation (20)), combined with the RSF constitutive relation (equation (1)) and the elastic relationship between the measured displacement and the slip velocity (equation (9)), explains the laboratory velocity step tests very well. Figure 15 shows the results of forward calculation and their fit to the observed shear stresses both in positive and negative velocity step tests. In the calculations, a was fixed at 0.05, and b was accordingly fixed at 0.056. Only L and c were varied in search of best fit to the observed shear stress variations. The optimal values chosen, c = 2.0 and L = 0.33 μm, agreed well with the estimate of c from Figure 11b.

Figure 15.

Variations in the shear stress and Φ during (a) positive and (b) negative velocity step tests, simulated using a modified aging law (equation (20)), a constitutive law (equation (1)), and an elastic relationship between the measured displacement and the slip velocity (equation (9)), with a = 0.05, b = 0.056, c = 2, L = 0.33 μm, k = 0.2 MPa/μm, σ = 10 MPa, and V* = 1 μm/s. The observed shear stresses, acoustic transmissivities, and Φ are also shown. The acoustic transmissivity ∣T∣ is so scaled that variations in ∣T∣ and Φ overlap on the charts during steady state sliding.

[48] In the proposed evolution law (equation (20)) L is not a characteristic slip distance but is simply a coefficient of slip weakening, as was also the case with the conventional aging law (equation (6)). The characteristic slip distance was estimated at about 1 μm when an exponential type of slip weakening, like the one predicted from the slip law (equation (3)), was assumed to fit the observed shear stress in velocity step tests (Figure 15). On the other hand, Figure 15 shows that changes in the shear stress and in the contact state occurred over a few μm of slip in our experiment, which gives another estimate for the characteristic slip distance. In either case, the characteristic slip distance was not very small in our experiments.

[49] In the simulations, the shear stress variations were well reproduced by the same set of parameters for both the positive and negative velocity step tests. The variation histories of Φ were also well reproduced. The dramatic improvement in the prediction of shear stress variations is evident when compared with Figure 10, which is based on conventional evolution laws.

[50] Figure 16 compares observations with simulations for different phases of experiments, including hold under a constant shear stress, reloading, steady state sliding at different velocities, and their mutual transitions. The simulation parameters were the same as those used in Figure 15. The comparison shows that the modified evolution law (equation (20)) can explain all phases of experiments very well with an identical set of parameters.

Figure 16.

Comparison between observations and simulations for different phases of experiments, such as (left) a hold under a constant shear stress, (middle) a reloading following the hold, and (right) steady state sliding at different velocities. These tests were conducted sequentially. The simulations are based on the modified aging law with the same set of parameters used in Figure 15.

6. Discussions

6.1. Discrepancy With the Results of the Traditional Method to Estimate a

[51] The new method of estimation, described here, produced a lower bound of about 0.035 on the direct effect coefficient a. This lower bound is recognizably larger than typical ranges of a (0.005 to 0.015) reported for rocks [e.g., Dieterich and Kilgore, 1994], where exact values of a, not its lower bounds, were estimated by using the traditional method, which corrected for the changes in Φ during velocity step tests by assuming a certain type of evolution law.

[52] In order to demonstrate that this discrepancy is attributable to the different estimation methods, not to the different experimental data, we applied the traditional method to the same velocity step data obtained in this work (Figure 17). We used both available types of evolution laws: the aging law (equation (6)) and the slip law (equation (3)). Fitting of the shear stress records by simulations based on equation (1) and the aging law (Figures 17a and 17b), and based on equation (1) and the slip law (Figures 17c and 17d), both produced a estimates of about 0.017, consistent with typical values obtained with the traditional method.

Figure 17.

Fit of the shear stress and Φ variations, simulated with traditional evolution laws, to the observed shear stress variations during velocity step tests. The velocity was increased from 0.01 μm/s to 0.1 μm/s in Figures 17a and 17c and was decreased from 0.1 μm/s to 0.01 μm/s in Figures 17b and 17d. The observed shear stress curves in Figures 17a and 17c are the same as in Figure 10. (a and b) Simulation results based on an aging law with a = 0.017, b = 0.023, L = 0.62 μm, k = 0.2 MPa/μm, σ = 10 MPa, and V* = 1 μm/s. (c and d) Simulation results based on a slip law with a = 0.017, b = 0.023, L = 1 μm, k = 0.2 MPa/μm, σ = 10 MPa, and V* = 1 μm/s.

6.2. Comparison With a Micromechanical Model

[53] The exponential dependence of the slip velocity on nominal shear stress, which appears as a direct effect (equation (1)), can be explained by a physical model where the shear deformation of frictional junctions obeys an exponential constitutive relation [Dieterich, 1979; Heslot et al., 1994; Nakatani, 2001]. Constitutive relations of the exponential type are often observed in the deformation of materials under very high driving stresses. The same mechanism may be at work at frictional junctions along a slip interface, because local stresses can be very high at frictional junctions even when the nominal stress is not particularly high. A general expression for exponential creep based on the absolute rate theory [e.g., Frost and Ashby, 1982; see also Nakatani, 2001], where the deformation rate is limited by thermally activated jumps in a pinning potential field, may be written as

display math

where τc is the shear stress at frictional junctions, T is the absolute temperature, kb is the Boltzmann constant, V0 is the product of the attempt frequency and the jump distance, Q is the valley depth of the potential field, called the activation energy, and Ω is the activation volume, which is on the order of the valley spacings cubed.

[54] Comparison of the stress-dependent parts of equation (1) and equation (24) leads to the relation [Nakatani, 2001]

display math

where σc is the normal stress at frictional junctions. The average junction stresses are related to the nominal stresses by τc/σc = τ/σ. Assuming a value of σc = 10 GPa for quartz [Westbrook, 1958], the estimated a = 0.05 translates to Ω = (0.20 nm)3. This value is smaller than previous estimates [Nakatani, 2001; Rice et al., 2001] because of the larger value of a used, but is in fact very close to the typical spacing between oxygen atoms (0.26 nm) and that between silicon and oxygen atoms (0.16 nm) in tektosilicates, supporting the view that frictional sliding is regulated by reaction rates on the atomic scale.

6.3. The b Value

[55] We have argued that the direct effect coefficient a is much larger than formerly believed. Revision of the a value requires revision of other features of RSF as well. One example concerns the interpretation of velocity step tests. Figure 18a shows a typical shear stress history, observed during a velocity step test involving an e fold increase, where Δτpeak is 0.013σ and the difference in τSS between the prestep and poststep steady states is −0.006σ. (These values may be compared to our observations shown in Figure 10 and Figures 17a and 17c, although these were for a 10 fold step.) The dashed curves in Figure 18b show the traditional interpretation, based on an existing evolution law and a constitutive law, with a = 0.017 obtained from fitting of our data (Figure 17). According to equation (10), the observed shear stress variations are interpreted as the sum of a direct effect, which is worth 0.017σ, and an evolution effect (ΔΦ), which should be −0.004σ at the peak stress and should be −0.023σ under the steady state that follows. If a is 0.05 as we propose, however, the same variations in the shear stress are interpreted as a sum of the direct effect, which is worth 0.05σ, and an evolution effect (ΔΦ), which should be −0.037σ at the peak stress and −0.056σ under the steady state that follows, as shown by the solid curves in Figure 18. Arguing for a larger a value is therefore tantamount to arguing that the evolution effect, and accordingly also the b value, should be much larger than previously believed.

Figure 18.

(a) Shear stress history typically observed during a velocity step test. (b) Traditional interpretation with a = 0.017 (dashed lines) and new interpretation with a = 0.05 (solid lines) of the observed shear stress.

[56] Let us look at this point from the results of healing tests, another method used to estimate the b value (an increase of bσ in Φ corresponds to an e fold increase in the contact time). The circles in Figure 19a show acoustic transmissivity ∣T∣ values at the end of hold tests with different hold times. The triangles in Figure 19a show ∣T∣ at the peak in the stress during the reloading phase following the hold. Comparison shows that more than 80% of the decrease in ∣T∣ from the end of a hold until the next steady state takes place before the shear stress peak. A similar result—much of the decrease in the contact area takes place before the shear stress peak—was also shown by Dieterich and Kilgore [1994] in their optical observation of the contact area. Figure 19b shows the same plot for the state variable Φ, inferred using equation (2) with a = 0.05 assumed. It shows that Φ also incurs large losses during reloading, and indicates that the conventional method, which uses the hold time dependence of the peak shear stress [e.g., Dieterich, 1972; Nakatani and Mochizuki, 1996] may grossly underestimate the value of b because of the loss in Φ.

Figure 19.

(a) Observed acoustic transmissivities ∣T∣ at the end of hold tests (circles) and at the peak shear stress during reloading (triangles). (b) The state variable Φ inferred by equation (2) with a = 0.05 assumed. The load point velocity was 0.1 μm/s during the reloading phase.

[57] The rate of increase in Φ under the peak stress (triangles in Figure 19b) with increasing hold time gives an apparent b value of 0.009, which falls in the range of typical b estimates derived from the traditional method. However, the rate of increase in Φ immediately following the hold (circles) with increasing hold time agrees with the estimate of b = 0.056 derived earlier from careful analysis of step test data (section 5.2). The latter value should be adopted. Although many earlier studies have attempted to correct for the loss in Φ by using some sort of evolution law [e.g., Beeler et al., 2001], such a large value of b has never been reported.

[58] We consider that this was because the loss in Φ was underestimated due to imprecise evolution laws. As long as either of the traditional evolution laws was used, slip weakening effects tended to be underestimated, because strong slip weakening (i.e., small L) failed to explain observations satisfactorily (Figures 10b and 10d). This means that the losses in Φ and ∣T∣ during reloading tended to be underestimated, resulting in underestimations of the b value. The introduction of an additional “stress-weakening” term helps to evaluate the losses more appropriately.

6.4. Solution to Shortcomings of the Original Aging Law

[59] Our revised evolution law, stated by equation (20), has pretty much resolved two well-known flaws in the aging law, although the revision was not specifically intended to fix those flaws.

[60] As mentioned earlier (section 2), one of the shortcomings of the aging law is that it predicts asymmetric curves for the shear stress plotted against slip displacement for positive and negative velocity changes (Figure 20a), at odds with experimental observations. However, the shear stress versus slip curves for velocity changes of opposite signs become nearly symmetric when the stress-weakening effect is added as in equation (20) (Figure 20b), irrespective of the velocity range covered by the test. In drawing Figures 20a and 20b, a = b was assumed so that the steady state shear stress does not vary with velocity. In the modified aging law (Figure 20b), L was set at the same value used in the simulations shown in Figures 15 and 16. In the original aging law (Figure 20a), the value of L was chosen to make the transition slip distances about the same as those in Figure 20b. The system stiffness k was set very large to suppress elastic deformation of the loading system so that we could focus on the pure evolution behavior. Even with the modified aging law, the predicted shear stress histories are not perfectly symmetric, and the extent of asymmetry is larger for larger velocity changes (Figure 20b). However, even for a 100 fold velocity change, the simulation results are barely distinguishable from near-symmetry observed in the laboratories.

Figure 20.

Predicted histories of the shear stress, shown as deviation from its steady state value and normalized by the peak, during velocity step tests. (a) Prediction based on the original aging law, equation (6), with a = b = 0.05, L = 1 μm, and k = 100 MPa/μm. (b) Prediction based on our modified aging law, equation (20), with a = b = 0.05, c = 2, L = 0.33 μm, and k = 100 MPa/μm.

[61] Another well-known shortcoming of the aging law is that it predicts a constant slip-weakening rate per unit slip, and therefore an increasing amount of slip displacement, with increasing initial Φ, necessary to complete the slip weakening (Figure 21a) [e.g., Nakatani, 2001; Guatteri et al., 2001]. As mentioned earlier (section 2), this is at odds with experimental observations, where the characteristic slip weakening distance is independent of the initial Φ. Figure 21b shows slip weakening predicted by equation (20), where the curves follow exponential, rather than linear, trends, and where the slip-weakening distance depends little on the initial Φ as observed in the laboratories.

Figure 21.

Predicted histories of the shear stress, shown as deviation from its steady state value and normalized by the peak, during slip weakening starting from three different initial values of Φ. (a) Prediction based on the original aging law, equation (6), with a = b = 0.05, L = 1 μm, and k = 100 MPa/μm. (b) Prediction based on our modified aging law, equation (20), with a = b = 0.05, c = 2, L = 0.33 μm, and k = 100 MPa/μm.

6.5. Speculation on the Physical Mechanism of Stress Weakening

[62] We only have one straightforward hypothesis for the mechanism of the newly proposed stress-weakening effect in equation (20). Since we found the effect to be pretty much instantaneous in nature, we speculate that it involves elastic deformation of the interface. Previous models of changes in the contact state [e.g., Beeler and Tullis, 1997] have assumed all asperities to be rigid and have only considered inelastic deformation of junctions, but elastic deformation can be considerably large at asperities, because they are more compliant than the surrounding bulk medium.

[63] As shown schematically in Figure 22, asperities may be distorted by elastic deformation when the shear stress increases, resulting in the partial breakage of junction bonds. Such breaking may also lower the asperity stiffness, similarly to the case where partial slip occurs along the edges of asperities [Johnson, 1985; Boitnott et al., 1992]. Such nonlinear elasticity of contacts may account for the nonlinear dependence of Φ on the shear stress as shown in Figure 14. Quantitative micromechanical modeling that would account for the effect of elastic deformation on Φ is left to future work.

Figure 22.

Conceptual diagram of elastic deformation of an asperity under an increased shear stress.

7. Summary and Conclusion

[64] We have proposed a new method to constrain the value of a, a parameter that plays a central role in the rate- and state-dependent friction (RSF) law. Unlike the traditional way to estimate a, our new method does not rely on any evolution law to correct for changes in the state variable Φ. Instead, we have designed stress step tests carefully to minimize possible changes in Φ. A lower bound on a was estimated with confidence at 0.035, much greater than estimates for the exact value of a according to the traditional method that relies on evolution laws. This lower bound estimate has been derived without recourse to the acoustic measurements that were conducted alongside mechanical measurements during the shear stress step tests. The value of a proposed here, much larger than previously thought, ensures a linear and unique relationship between the acoustic transmissivity ∣T∣ and the state variable Φ.

[65] The exact value of a was estimated by using the acoustic measurement data. With an additional assumption that Φ under steady state sliding depends log linearly on the slip velocity by a coefficient b (equation (4)), and using the estimates of ab from shear stresses under steady state sliding at different velocities (equation (5)), the sensitivity of the acoustic transmissivity to Φ can be represented as a function of a (equation (18)). This helped to estimate the exact value of a, after accounting for changes in Φ, at about 0.05, and the sensitivity at r ≡ Δ∣T∣/ΔΦ = 0.012/MPa.

[66] Under existing evolution law models, the estimate of a = 0.05 would result in wild discrepancy between observed frictional behavior and theoretical predictions. This necessitates revision of the evolution law to account more properly for the evolution of Φ.

[67] Comparison between predictions of the aging law and experimental observations has revealed that an additional term, accounting for weakening by shear stress, is necessary to explain the observed variations in Φ. Acoustic measurements have supported the presence of stress-dependent weakening under high shear stress conditions, not far from the so-called dynamic friction, which is the main target of fault mechanics and friction experiments. The stress-weakening effect seems to become fairly insignificant under low shear stress conditions. We speculate that the stress-weakening term may be attributable to the breakage of junction bonds by elastic deformation of asperities along the frictional interface.

[68] We have proposed a revised evolution law by adding a linear stress-weakening term to the traditional aging law. The proposed evolution law, combined with a larger direct effect coefficient than presumed before, can very well explain the variations in both the shear stress and Φ, during various phases of experiments, such as positive and negative velocity steps, holds at constant stress and subsequent reloading, with an identical set of parameters.

[69] Both of these changes—the addition of the stress-weakening term and the larger a—involve considerably large consequences in quantitative terms. Certainly, the traditional RSF—no stress-weakening term and the smaller a—performed fairly well in explaining shear stress histories observed in experiments. The grossly underpredicted state variable changes in the traditional law did not lead to significant misfit in the shear stress data, because their effects on shear stress predictions tended to be offset by those of the grossly underestimated a. Of course, the mutual cancellation did not work perfectly, and the traditional law had some persistent problems in predicting shear stress histories, if not very large in quantity.

[70] We emphasize that the two revisions proposed in our present paper help to better explain not only shear stress histories but also state variable histories. These revisions therefore provide a satisfactory solution to all known problems in reproducing observed shear stress histories and help to explain different observations in a much more consistent way.


[71] We would like to thank Allan Rubin and an anonymous reviewer for useful comments. A part of this work was supported by Grant-in-Aid for JSPS Fellows. This work was also partly supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan under its Observation and Research Program for Prediction of Earthquakes and Volcanic Eruptions.