Apparent Dependence of Rate- and State-Dependent Friction Parameters on Loading Velocity and Cumulative Displacement Inferred from Large-Scale Biaxial Friction Experiments

An earthquake cycle involves a very wide range of slip velocities, from orders of magnitude slower than a plate motion to as fast as a slip velocity at a rupture front during an earthquake. As earthquakes occur on a fault repeatedly, the internal structure of the shear zone and its mechanical properties are considered to evolve with increasing cumulative slip displacement (e.g., Beeler et al. 1996). The modeling of a sequence of earthquakes over the geologically long time scale probably requires a fault constitutive law which can comprehensively describe the mechanical properties of faults over the wide range of slip velocities and cumulative displacement.

The rate- and state-dependent friction (RSF) laws have been widely used to simulate earthquake sequences (e.g., Hori et al. 2004; Lapusta and Liu 2009; Noda and Lapusta 2013). These laws were originally proposed to model laboratory experimental data (Dieterich 1978, 1979; Ruina 1983), and the RSF parameters have been investigated using biaxial loading apparatuses at the low slip velocity from ~0.01 μm/s to ~1 cm/s, in which the cumulative displacement was of the order of cm at most (e.g., Mair and Marone 1999).

To achieve higher slip velocity and larger cumulative displacement, rotary shear apparatuses were developed (e.g., Tullis and Weeks 1986; Tsutsumi and Shimamoto 1997). Beeler et al. (1996) estimated the RSF parameters for large cumulative displacement at the slip velocity of 1–10 μm/s. Since a rotary shear apparatus is capable of producing high slip velocity up to a seismic rate, steady-state friction coefficients of various rock types have been investigated at a wide range of slip velocities, and a remarkable velocity-weakening property of rock friction was revealed (e.g., Di Toro et al. 2011). Although a rotary shear apparatus enables the investigation of rock friction properties with a wide range of slip velocities and large displacement as described above, the apparatus would not be suitable to investigate stick–slip behavior, which could be considered analogous to a sequence of earthquakes on natural faults (Brace and Byerlee 1966). To study the effect of cumulative displacement, it is important to consider the history of slip velocity in nature. In usual friction experiments at low slip rates, steady-state sliding of a fault is simulated, and the shear zone internal structure is developed under such circumstances. Natural fault hosting a sequence of earthquakes experiences quite different deformation conditions (e.g., repeated transients in the slip velocity with stress concentration at rupture fronts), and the resulting internal structure should be different from what is developed under steady-state sliding. Since the evolution of the internal structure causes evolution in the parameters in RSF (e.g., Beeler et al. 1996), it is important to study them in experiments with stick–slips to better understand behavior of seismogenic faults.

In addition to the limitations of the slip velocity and the cumulative displacement, conventional studies used small (on the order of 10 cm at most) rock specimens to estimate the constitutive friction parameters (e.g., Dieterich 1972; Marone and Cox 1994; Beeler et al. 1996). The constitutive friction parameters estimated for the small rock specimens may be different from those for the large rock specimens, as Yamashita et al. (2015) suggested that rock friction in meter-sized rock specimens starts to decrease at a work rate (the product of the shear stress and the slip rate) one order of magnitude smaller than that in centimeter-sized rock specimens.

Many previous studies stated above obtained the RSF parameters by the method of step changes in load point velocity. The RSF parameters can be estimated also from stick–slip behaviors (Mitchell et al. 2015). Mitchell et al. (2015) performed the inversions of experimental data for unstable sliding using a spring-slider model, but they ignored the inertia in their numerical simulations [see their Eq. (7)], which may lead to inaccurate estimation of the RSF parameters because in the quasi-static system, finite amplitude periodic oscillations are observed for very limited parameters and the slip velocity becomes infinite in unstable sliding regimes (Gu et al. 1984) where the inertia makes the slip and stress evolution completely different (Rice and Tse 1986).

In this study, we estimated the RSF constitutive parameters for the data obtained in experiment data by large-scale (on the order of meters) biaxial rock friction experiments conducted by Fukuyama et al. (2014) to investigate the dependence of the parameters on the loading velocity and cumulative displacement. For the estimation, we performed fully dynamic simulations of a single-degree-of-freedom spring-slider model. For efficient calculation, we developed a new algorithm of numerical simulations which tremendously reduces the calculation time relative to conventional methods such as embedded Runge–Kutta method.

We applied a spring-slider model with mass with one-degree-of-freedom (Fig. 7) to explain the stick–slips because we found that the displacement of the lower specimen is approx. 10% of that of the upper specimen during dominant slip (16.653–16.664 s in Fig. 5c) and can be ignored. The equation of motion is

$$\frac{{{\text{d}}v}}{{{\text{d}}t}} = {{\left( {F_{\text{s}} - F_{\text{t}} } \right)} \mathord{\left/ {\vphantom {{\left( {F_{\text{s}} - F_{\text{t}} } \right)} m}} \right. \kern-0pt} m},$$

(4)

where v is the slip velocity, t is time, \(F_{\text{s}}\) is spring force, \(F_{\text{t}}\) is the shear force on the fault, and m is mass. We perform dynamic simulations (i.e., accounting for the inertial effects), in contrast to the previous studies which ignored the inertia at low slip velocity (e.g., Rice and Tse 1986; Bizzarri 2011). \(F_{\text{s}}\) is

$$F_{\text{s}} = F_{0} + k\left( {v_{\text{L}} t - u} \right),$$

(5)

where \(F_{0}\) is the steady-state shear force at a reference slip velocity of the friction law \(v_{0}\), k is the spring stiffness, \(v_{\text{L}}\) is the load point velocity, and u is the fault displacement. \(F_{\rm{t}}\) is assumed to obey the RSF law. We examine both Slip law (Ruina law)

$$F_{\text{t}} = F_{0} + aF_{\text{n}} \ln \left( {v/v_{0} } \right) + \theta$$

(6)

$$\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - \frac{v}{{L_{\text{c}} }}\left( {\theta + bF_{\text{n}} \ln \left( {{v \mathord{\left/ {\vphantom {v {v_{0} }}} \right. \kern-0pt} {v_{0} }}} \right)} \right)$$

(7)

and Aging law (Slowness law)

$$F_{\text{t}} = F_{0} + aF_{\text{n}} \ln \left( {{v \mathord{\left/ {\vphantom {v {v_{0} }}} \right. \kern-0pt} {v_{0} }}} \right) + bF_{\text{n}} \ln \left( {{{v_{0} \theta } \mathord{\left/ {\vphantom {{v_{0} \theta } {L_{\text{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}} \right)$$

(8)

$$\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 1 - \frac{v\theta }{{L_{\text{c}} }},$$

(9)

where a and b are parameters representing the direct and evolution effects, respectively, \(\theta\) is the state variable, and \(L_{\text{c}}\) is the critical slip distance (e.g., Marone 1998a). We calculated m from rock density and dimensions (Table 4). We used the estimated values of \(k_{1}\) (Table 3) as k, which was constant in our simulations because \(k_{1}\) was almost constant through the experiments, as stated in Sect. 2.2.2. The parameters used in the simulations are shown in Table 4.

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Table 4

Simulation parameters

Property	Symbol	Value
Mass	m	1.1 × 103 kg
Direct effect parameter	a	0.008
Reference velocity	\(v_{0}\)	\(10^{ - 5} {\text{ m/s}}\)

If we solve Eqs. (4)–(7) for the Slip law and Eqs. (4), (5), (8), and (9) for the Aging law by the Runge–Kutta method with adaptive step-size control (Press et al. 1992), it takes too long to integrate in time domain during interseismic periods for some of the parameter sets. Therefore, we newly developed an efficient algorithm for the numerical time integration which is an example of the exponential time differencing method (e.g., Cox and Matthews 2002) similar to that used by Noda and Lapusta (2010), as described in “Appendices 1 and 2”.

We confirmed for the Slip law that the time-integration method provided results that were identical to those obtained by the Runge–Kutta method, but required only for 1/10,000 of the calculation time for \(a = 0.008\), \(b = 0.0092\), \(L_\text{c} = 0.5 \, \upmu {\text{m}}\), and the constant \(v_{\text{L}}\) of 0.1 mm/s (Fig. 18).

The loading velocity slightly fluctuated in our experiments, as stated in Sect. 2.2.2. Therefore, we assumed in our simulations that \(v_{\rm{L}}\) is the faster loading velocity \(v_{\text{Lf}}\) for the duration of \(d_{\text{f}}\) after a stick–slip event finishes and \(v_{\text{L}}\) is the slightly slower loading velocity \(v_{\text{Ls}}\) at other times. We defined that the event occurs when \(v > 10v_{\text{Ls}}\) and it finishes when \(v \le v_{\text{Ls}}\). Table 2 lists the values of \(v_{\text{Lf}}\), \(d_{\text{f}}\), and \(v_{\text{Ls}}\), which were decided from the observed movement of the shaking table (Sect. 2.2.2).

Using the new time-integration method, we conducted many numerical simulations with various combinations of the evolution parameter b and the state-evolution distance \(L_{\text{c}}\) while keeping the direct effect parameter a constant. We then estimated the combinations of b and \(L_{\text{c}}\) which reproduce the recurrence time and the friction drop consistent with \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\). We also calculated the slip-weakening curve with each combination, because we could not determine the optimal parameters only by the recurrence time and the friction drop as described in the next section.

The results of this study suggest that when a friction experiment starts without gouges on the fault, both the b and \(L_{\text{c}}\) values increase as the cumulative slip increases, as stated in Sect. 4.1.1. Beeler et al. (1996) also reported the decrease in \(a - b\) in some initially bare surface experiments. On the other hand, Leeman et al. (2016) suggested that decreases in both \(a - b\) and \(L_{\text{c}}\) with the cumulative slip. They constructed 3-mm-thick layers of powdered silica to simulate granular fault gauges, which might cause the contradiction between their results and ours on \(L_{\text{c}}\). Our results on the slip dependence of the RSF parameters can be partly explained by the production of gouges, since the state-evolution distance \(L_{\text{c}}\) increases with increasing gouge layer thickness as suggested by Marone and Kilgore (1993). In fact, many gouge particles were produced during the present friction experiments as described by Fukuyama et al. (2014). From the estimated gouge production rate, we calculated 5.013 × 10⁻⁷ and 1.303 × 10⁻⁶ m as averaged thicknesses of the gouge layers for W1 and W2 in LB01-127, respectively. See “Appendix 3” for detail of this estimation. Note that we estimated these thicknesses assuming that the produced gouge materials are uniformly distributed over the fault surface. Actually, the gouge materials were locally produced in and around the generated grooves as revealed by Yamashita et al. (2015). Therefore, these thicknesses could be minimum estimates. The estimated b and \(L_{\text{c}}\) values were larger in experiment LB01-134 (open squares in Figs. 12, 15) than in experiment LB01-127 (triangles and circles in Figs. 12, 15), which is consistent with the slip dependence described in Sect. 4.1.1, because LB01-134 was conducted after several experiments following LB01-127 without the removal of gouges. The conventional RSF laws with a single set of the RSF parameters were not sufficient to explain the results of the long cumulative displacement experiments, and more complex state evolution laws accounting for gouge production are needed to comprehensively describe the large-scale experimental data.

It is important to note that most earthquake cycle simulations (e.g., Hori et al. 2004; Lapusta and Liu 2009; Noda and Lapusta 2013) used the conventional RSF law with a single state-variable. Based on the present findings, however, the evolution of friction as a function of slip during evolution of the internal structure of the shear zone (e.g., accumulation of wear material) could not be well expressed by the conventional RSF law. To account for the large-scale behavior in which the apparent RSF parameters evolve with changes in the internal structures of the shear zone, a different framework is required.

We found an increase in b as \(\bar{v}_{\text{L}}\) increased, as stated in Sect. 4.1.2. This indicates a positive correlation between \(\bar{v}_{\text{L}}\) and the friction drop. Some previous studies, however, suggested that the friction drops of the stick–slip events decreased with the increase of the loading velocity (Karner and Marone 2000; Mair et al. 2002; Anthony and Marone 2005; McLaskey et al. 2012). The negative correlation between \(\bar{v}_{\text{L}}\) and the friction drop shown by the previous studies can be interpreted to result from contact aging associated with frictional healing during the inter-seismic period of the seismic cycle. Our results do not deny the effect of the frictional healing because peak friction was slightly higher at lower \(\bar{v}_{\text{L}}\) in our experiments as shown in Fig. 3. Instead, our results may suggest that the velocity-weakening effect is stronger than that expected from the conventional RSF law with a single set of the RSF parameters. The estimated peak slip velocity was approx. 0.02–0.05 m/s and higher for the faster \(\bar{v}_{\text{L}}\) in our experiments (Sect. 2.2.1). In this velocity range, friction weakens as a function of slip velocity (e.g., Di Toro et al. 2011); therefore, larger friction drop would occur for higher slip velocity. Kato et al. (1991) obtained similar data in experiments with a granite specimen as well as a composite specimen of granite and marble. The correlation between the friction drop and the loading velocity might relate to the characteristics of the apparatus used in the experiments.

The estimated \(L_{\text{c}}\) values except for time window W3 are smaller than those obtained by the previous studies (0.7 μm or longer, e.g., Dieterich 1979; Marone et al. 1990). The small \(L_{\text{c}}\) could result from the thin gouge layer. As described above, Marone and Kilgore (1993) proposed a scaling relation that \(L_{\text{c}}\) is proportional to the gouge thickness. The gouge layer thicknesses estimated in this study are two orders of magnitude smaller than those in the previous experiments (e.g., Marone et al. 1990; Marone and Kilgore 1993); therefore, \(L_{\text{c}}\) could be smaller in our experiments than in the previous studies. The small \(L_{\text{c}}\) might be also related to the velocity weakening processes. This is because the previous studies obtained \(L_{\text{c}}\) by velocity step change tests, while we obtained \(L_{\text{c}}\) from stick–slips which involve high slip velocity (0.05 m/s at most).

Unstable (seismic) slip may occur for spring stiffness smaller than a critical value, as theoretically shown by Ruina (1983). Leeman et al. (2016) showed that the behaviors of stick–slips and stable sliding are related to normalized critical stiffness \(\kappa = {k \mathord{\left/ {\vphantom {k {k_{\text{c}} }}} \right. \kern-0pt} {k_{\text{c}} }}\) where \(k\) is loading system stiffness, \(k_{\text{c}} = {{\sigma_{\rm{n}} \left( {b - a} \right)} \mathord{\left/ {\vphantom {{\sigma_{n} \left( {b - a} \right)} {L_{\rm{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}\) is critical stiffness of a fault, and \(\sigma_{\text{n}}\) is the normal stress. From the stiffness and the RSF parameters obtained in Sects. 2.2.2 and 4, we estimated \(\kappa\) as shown in Table 3. Many stick–slip events were observed for \(\kappa \ll 1\), which are consistent with Leeman et al. (2016) and the theoretical works (e.g., Ruina 1983). However, stick–slip behaviors cannot be explained only by \(\kappa\) in our experiments. For example, \(\Delta \mu_{\text{ob}}^{\prime }\) of stick–slip events and slip amount per event were high in the order of W4, W3, W2, and W1, but \(\kappa\) was small in the different order.

We assumed that a value is constant since a is considered as the material property. However, a might depend on temperature change (e.g., Blanpied et al. 1995; Nakatani 2001) and on the loading velocity because Marone (1998b) showed that the static friction increases with the loading rate (1–10 μm/s) by double-direct shear experiments. To investigate the dependency of a on the estimation of b and \(L_{\text{c}}\), we conducted the same computation using \(a = 0.005\) and \(a = 0.011\) for window W2. We estimated the constitutive parameters (b, \(L_{\text{c}}\)) and their possible ranges, in the same manner as in Sect. 4.1.1, summarized in Fig. 16. The estimated \(L_{\text{c}}\) values were slightly different among the examined cases. On the other hand, the estimated \(b - a\) values increased as a values decrease because \(\Delta T_{\text{sy}}\) and \(\Delta \mu_{\text{sy}}^{\prime }\) depended on not only b value but also a value. The values of the evaluation functions \(J_{1}\), \(J_{2}\), and J (Eqs. (10)–(12)) for the estimated parameter sets were similar in the cases with different a values, as shown in Table 6. This means that the optimum parameter set (a, b, \(L_{\text{c}}\)) cannot be determined uniquely. Thus, the \(L_{\text{c}}\) values estimated in Sect. 4 do not depend on the choice of a value, but the dependence of b values on the cumulative displacement and on \(\bar{v}_{\text{L}}\) could be explained by the dependence of a values.

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Table 6

Values of evaluation functions for optimum parameter sets for a of 0.005 and 0.011

a values	\(J_{1}\)	\(\bar{J}_{2}\) (\(\times 10^{ - 5}\))	J
0.005	0.0287	4.69	1.56
0.011	0.0321	5.05	1.99

There is evidence that the stick–slip behavior may be related to the damage of sliding surface as described below. Unfortunately, in the series LB01 presented here, we did not conduct the stick–slip experiment at the beginning. However, we observed the dependence of the stick–slip behavior on the slip surface damage in the LB09 series, in which the width of the lower rock specimen was reduced to 0.1 m to increase the normal stress to 6.7 MPa. The rock was Indian metagabbro, same as in the present experiments described above. The fault surface was repeatedly slid for 900 s at \(\bar{v}_{\text{L}}\) of 0.01 mm/s, and the gouges were removed after each experiment. We compared the stick–slip behaviors at the same time window (600–700 s) to avoid the effect of cumulative slips from gouge removal.

We found that \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) were small in the first experiment of the LB09 series (Fig. 17a), whereas both became larger and almost constant in subsequent experiments (Fig. 17b). These observations support the idea that the friction drops and the recurrence time depend on the damage on the fault surface in addition to \(\bar{v}_{\text{L}}\) and the cumulative displacement demonstrated in this paper. The effects of the damage on the fault surface will be further investigated in future works.

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We estimated the constitutive parameters in the RSF law (for both Slip and Aging laws) by fitting numerical simulations to stick–slip experiments with a large-scale biaxial rock friction apparatus at the NIED. During the friction experiments, many stick–slip events were observed, and their features are summarized as follows. (1) The friction drop and recurrence time of the stick–slip events increased with cumulative displacement within a single experiment when gouges were removed before the experiment. (2) The friction drop and recurrence time became more or less constant throughout the experiment when the experiment was done after several experiments without removing gouges. (3) The friction drop became lager and the recurrence time was shorter for an experiment with faster loading velocity. We estimated the slip-weakening curves during the stick–slip events from measured spring force and the accelerations of the specimens. We applied a one-degree-of-freedom spring-slider model with mass to explain the observed stick–slips. We developed an efficient algorithm for numerical time integration, and we conducted many numerical simulations with various b and \(L_{\text{c}}\) values while keeping a constant. We then identified the values of b and \(L_{\text{c}}\) that provided a consistent recurrence time, friction drops, and slip-weakening curves during the stick–slip events.

The results of our analyses suggest that (1) both b and \(L_{\text{c}}\) increase as the cumulative displacement increases, and (2) b increases and \(L_{\text{c}}\) decreases as the loading velocity increases, for both Slip and Aging laws. Therefore, the conventional RSF laws with an invariable single set of the RSF parameters cannot explain the whole of the experimental data. More complex state evolution laws are needed to comprehensively describe the experimental data and to consider an earthquake cycle involving a wide range of slip velocities and a sequence of earthquakes over geologically long times during which the evolution of the internal structure of the shear zone is significant.