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

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## Abstract

We investigated the constitutive parameters in the rate- and state-dependent friction (RSF) law by conducting numerical simulations, using the friction data from large-scale biaxial rock friction experiments for Indian metagabbro. The sliding surface area was 1.5 m long and 0.5 m wide, slid for 400 s under a normal stress of 1.33 MPa at a loading velocity of either 0.1 or 1.0 mm/s. During the experiments, many stick–slips were observed and those features were as follows. (1) The friction drop and recurrence time of the stick–slip events increased with cumulative slip displacement in an experiment before which the gouges on the surface were removed, but they became almost constant throughout an experiment conducted after several experiments without gouge removal. (2) The friction drop was larger and the recurrence time was shorter in the experiments with faster loading velocity. We applied a one-degree-of-freedom spring-slider model with mass to estimate the RSF parameters by fitting the stick–slip intervals and slip-weakening curves measured based on spring force and acceleration of the specimens. We developed an efficient algorithm for the numerical time integration, and we conducted forward modeling for evolution parameters (*b*) and the state-evolution distances (\(L_{\text{c}}\)), keeping the direct effect parameter (*a*) constant. We then identified the confident range of *b* and \(L_{\text{c}}\) values. Comparison between the results of the experiments and our simulations suggests that both *b* and \(L_{\text{c}}\) increase as the cumulative slip displacement increases, and *b* increases and \(L_{\text{c}}\) decreases as the loading velocity increases. Conventional RSF laws could not explain the large-scale friction data, and more complex state evolution laws are needed.

## Keywords

Rate-and-state friction large-scale experiment stick–slips numerical simulation spring-slider model## 1 Introduction

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.

## 2 Large-Scale Biaxial Rock Friction Experiments

### 2.1 Experimental Procedure

Conditions of analyzed experiments

Experiment ID | \(\bar{v}_{\text{L}}\) (mm/s) | Slip displacement (mm) | Gouge removal before experiment |
---|---|---|---|

LB01-127 | 0.1 | 40 | Yes |

LB01-134 | 0.1 | 40 | No |

LB01-142 | 1.0 | 400 | No |

The relative displacement of the sliding surfaces was measured by two laser displacement transducers with different measurement ranges: a long-range transducer (LDT-L) and a short-range transducer (LDT). The LDT-L was installed on the side plate attached to the end of the lower specimen, and its target was attached at the edge of the upper specimen (magenta squares in Fig. 1a). The LDT and its target were attached at the center of the lower and upper specimens, respectively (green squares in Fig. 1a). Acceleration was measured by two accelerometers installed in the upper and lower specimens at 20 mm from the slip interfaces (yellow circles in Fig. 1a). The force applied by the reaction force bar \(F_{\text{s1}}\) was measured by a load cell (Fig. 1a).

*m*is mass of the rock specimens,

*a*is the acceleration of the specimens,

*k*is the spring stiffness,

*u*is the displacement of the specimens, subscripts 1 and 2 stand for the upper and lower specimens/springs, respectively,

*t*is time, and \(F_{t}^{{ 2 {\text{d}}}}\) is the shear force between the two specimens. It should be noted that LDT and LDT-L measured

*u*(\(=\!\! u_{1} - u_{2}\)), and that \(F_{t}^{{ 2 {\text{d}}}}\) could not be measured directly.

### 2.2 Experimental Results

#### 2.2.1 Behavior of Stick–Slips

In addition to the dependence on the cumulative slip, those characteristics of stick–slips depended on \(\bar{v}_{\text{L}}\); the slip amount during an event and \(\Delta \mu^{\prime}_{\text{ob}}\) were larger and \(\Delta T_{\text{ob}}\) was shorter for the experiments with faster \(\bar{v}_{\text{L}}\). The average \(\Delta T_{\text{ob}}\) was approx. 3.6 s for slow \(\bar{v}_{\text{L}}\) (\(\bar{v}_{\text{L}} = 0.1{\text{ mm/s}}\), LB01-134, window W3 in Fig. 3a) and approx. 0.44 s for fast \(\bar{v}_{\text{L}}\) (\(\bar{v}_{\text{L}} = 1.0{\text{ mm/s}}\), LB01-142, window W4 in Fig. 3b). The average \(\Delta \mu_{\text{ob}}^{\prime }\) was approx. 0.052 for slow \(\bar{v}_{\text{L}}\) (window W3) and approx. 0.060 for fast \(\bar{v}_{\text{L}}\) (window W4). The slip amounts per event were approx. 0.35 mm for slow \(\bar{v}_{\text{L}}\) (window W3) and 0.39 mm for fast \(\bar{v}_{\text{L}}\) (window W4) (blue triangles and red crosses in Fig. 4).

^{3}) and mass volume, \(a_{1}\) measured by the accelerometer (Fig. 5a), and \(F_{\text{s1}} = - k_{1} u_{1}\) measured by the load cell (Fig. 5b).

For the estimation of \(u_{1} - u_{2}\), we conducted a double time-integration of \(a_{1} - a_{2}\) because LDT-L did not have enough resolution. Figure 5c shows a comparison of \(u_{1} - u_{2}\) obtained by the double time-integration of \(a_{1} - a_{2}\) and \(u\) measured by LDT, indicating that we can estimate the short-term slip displacement from the accelerograms. Examples of estimated slip-weakening curves are shown in Fig. 5d. In this estimation, we corrected the timing of the recording system as pointed out by Fukuyama et al. (2014). We applied a 400-Hz Butterworth-type low-pass filter to \(F_{\text{s1}}\). We examined the contribution from high-frequency waves by applying a 500-, 750-Hz, and 1-kHz low-pass Butterworth filter to the acceleration data, and we computed the slip-weakening curves (black, blue, light blue curves, respectively, in Fig. 5). Since we did not observe any significant differences in the slip-weakening curves, we confirmed that the high-frequency waves did not contribute to the estimated slip-weakening curves and used a 400-Hz cutoff. We will show the results using the acceleration data in which the 750-Hz low-pass filter was applied below.

We estimated the peak slip velocity during stick–slip events by time-integration of \(a_{1} - a_{2}\). The peak slip velocity was approx. 0.02 m/s for time window W1, 0.04 m/s for time windows W2 and W3, and 0.05 m/s for time window W4.

#### 2.2.2 Behavior of Experimental Apparatus

*l*), as follows. We divided each time window from W1 to W4 (Figs. 2, 3) into the shorter time windows for large and small slopes of

*l*, and we estimated

*l*for each time window by a straight line, as shown in Fig. 6c. The average values of the large and small slopes and the duration of the large slopes correspond to \(v_{\text{Lf}}\), \(v_{\text{Ls}}\), and \(d_{\text{f}}\), respectively. Table 2 lists the values for each time window from W1 to W4.

Estimated loading velocity

Experiment ID | Time window | \(v_{\text{Ls}}\) (mm/s) | \(v_{\text{Lf}}\) (mm/s) | \(d_{\rm{f}}\) (s) |
---|---|---|---|---|

LB01-127 (short cumulative disp.) | W1 | 0.0753 | 0.9622 | 0.039 |

LB01-127 (long cumulative disp.) | W2 | 0.0750 | 1.9717 | 0.042 |

LB01-134 | W3 | 0.0740 | 2.3608 | 0.036 |

LB01-142 | W4 | 0.7356 | 3.3027 | 0.046 |

Estimated stiffness

Experiment ID | Time window | Stiffness \(k_{1}\) (\(10^{8} {\text{ N/m}}\)) | Normalized critical stiffness \(\kappa\) | |
---|---|---|---|---|

Slip law | Aging law | |||

LB01-127 (short cumulative disp.) | W1 | 1.42 | 2.6 × 10 | 2.4 × 10 |

LB01-127 (long cumulative disp.) | W2 | 1.48 | 3.2 × 10 | 1.6 × 10 |

LB01-134 | W3 | 1.48 | 7.4 × 10 | 7.4 × 10 |

LB01-142 | W4 | 1.48 | 6.5 × 10 | 7.7 × 10 |

The fast recovery of the friction immediately after the sharp friction drop observed at a stick–slip event would be related to the oscillation of the shaking table. This is because the ratio of \(\dot{F}_{\text{s1}}\) during the fast recovery of the friction and during stick was similar to the ratio of \(v_{\text{Lf}}\) and \(v_{\text{Ls}}\), that is, the stiffness \(k_{1}\) was almost constant through our experiments.

## 3 Method for Numerical Simulations

*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

*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)

*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.

Simulation parameters

Property | Symbol | Value |
---|---|---|

Mass | | 1.1 × 10 |

Direct effect parameter | | 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.

## 4 Results

### 4.1 Slip Law

#### 4.1.1 Dependence of Constitutive Parameters on Cumulative Displacement

We estimated the combinations of constitutive parameters *b* and \(L_{\text{c}}\) to reproduce the stick–slip events that occurred at short and long cumulative displacement in the experiment LB01-127 with \(\bar{v}_{\text{L}} = 0.1{\text{ mm/s}}\) (windows W1 and W2 in Fig. 2).

*b*and \(L_{\text{c}}\) values for the long cumulative displacements (window W2). The colored and gray circles indicate the parameters with which the system reached the limit cycle and the stable sliding, respectively. The crosses indicate parameters which provide the combination of plural recurrence time. The multiple recurrence time does not appear in the simulations for the constant \(v_{\text{L}}\); it comes from the changes in \(v_{\text{L}}\) in our simulations. If \(\Delta T_{\text{sy}}\) for \(v_{\text{Lf}}\) is shorter than \(d_{\text{f}}\) (duration of \(v_{\text{Lf}}\)), the stick–slip events occur during \(v_{\text{L}}\) of \(v_{\text{Ls}}\) and \(v_{\text{Lf}}\), and the multiple recurrence time arises. Our simulations are not suitable in these cases because \(v_{\text{L}}\) decreases from \(v_{\rm{Lf}}\) to \(v_{\rm {Ls}}\) after duration \(d_{\text{f}}\), whether or not a stick–slip event occurs during \(v_{\text{Lf}}\). However, we need not consider these cases because \(\Delta T_{\text{ob}}\) is much larger than \(d_{\text{f}}\). The diamonds show the parameter sets (

*b*, \(L_{\text{c}}\)) that provide \(\Delta T_{\text{sy}}\) within \(\Delta \bar{T} \pm \sigma_{\text{T}}\) and \(\Delta \mu_{\text{sy}}^{\prime }\) within \(\Delta \bar{\mu }^{\prime } \pm \sigma_{\mu }\), where \(\Delta \bar{T}\) and \(\Delta \bar{\mu }^{\prime }\) are the average \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) for window W2, respectively, and \(\sigma_{\text{T}}\) and \(\sigma_{\mu }\) are the standard deviation of \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) for window W2, respectively. The parameter sets providing \(\Delta T_{\text{sy}}\) within \(\Delta \bar{T} \pm \sigma_{\text{T}}\) are similar to those providing \(\Delta \mu_{\text{sy}}^{\prime }\) within \(\Delta \bar{\mu }^{\prime } \pm \sigma_{\mu }\). To model \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) simultaneously, we calculated the following evaluation function

*b*and \(L_{\text{c}}\) values shown by the diamonds in Fig. 8c can reasonably reproduce \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\). Figure 8c shows that \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) have little information on \(L_{\text{c}}\), and thus we cannot constrain the parameter set of

*b*and \(L_{\text{c}}\) uniquely only from the recurrence time and friction drop data.

*i*, where \(Nj(i)\) is the number of data for the event

*i*, and \(\mu_{\text{sy}}\) and \(\mu_{\text{ob}}\) are the synthetic and observed shear force divided by the normal force, respectively. For this calculation, we applied the linear interpolation to the numerical result. Note that the initial friction coefficient \(\mu_{0} = {{F_{0} } \mathord{\left/ {\vphantom {{F_{0} } {F_{n} }}} \right. \kern-0pt} {F_{\rm{n}} }}\) affects only the absolute level of \(\mu_{\text{sy}}\). We analyzed all of the events for window W2. Figure 9a shows the average \(\bar{J}_{2} = \sum\nolimits_{i = 1}^{{N_{i} }} {J_{2} (i)/N_{i} }\), where \(N_{i}\) is the number of the events, for many computations with parameter sets (

*b*, \(L_{\text{c}}\)). The diamonds show \(\bar{J}_{2}\) less than the minimum value of \(\bar{J}_{2} + \sigma_{\rm{J}}\) in all simulations, where \(\sigma_{\text{J}}\) is the standard deviation of \(J_{2} (i)\) for each simulation. The parameter set with small value of \(\bar{J}_{2}\) can reproduce the observed slip-weakening curves, but the parameter set with large value of \(\bar{J}_{2}\) cannot, as demonstrated in Fig. 9b.

*J*(star in Fig. 10a) gives the optimum parameter set \((b , { }L_{\rm {c}} ) = (0.0094 , { }0.3 \, \upmu {\text{m}})\), and \(J \le 2\alpha_{1}\) (diamonds in Fig. 10a) can be a possible range of

*b*and \(L_{\text{c}}\). The comparison of the observed and synthetic time histories for 20 s of displacement and of the spring force is shown in Fig. 10b, c. Our simulation could reproduce the very sharp friction drop and the subsequent fast recovery of the friction observed at a stick–slip event. These behaviors in our simulation result from the variable \(v_{\text{L}}\) (Table 2) because the features do not appear in the simulations with the constant \(v_{\text{L}}\). In addition, the synthetic recurrence time, slip amount during an event, cumulative slip for 20 s, friction drop, and the slip-weakening curves for the optimum parameter set are similar to the observations, as demonstrated in Fig. 10b, c and by the red solid line in Fig. 9b.

*b*and \(L_{\text{c}}\) to reproduce the stick–slip events that occurred at the short cumulative slip displacement (window W1), in the same manner as at the long cumulative slip displacement (window W2). By a joint inversion of the recurrence time, the friction drop, and the slip-weakening curves (Eq. (12)), we obtained the optimum parameter set \((b, \, L_{\text{c}} ) = (0.0085, \, 0.09 \, \upmu {\text{m}})\), as shown in Fig. 11. Our simulations cannot reproduce the observed fluctuations with events of \(\Delta T_{\text{ob}}\), \(\Delta \mu_{\text{ob}}^{\prime }\), slip amount during an event, and the slip-weakening curves, which are larger than those at the long cumulative slip displacement. However, \(\Delta T_{\text{sy}}\), \(\Delta \mu_{\text{sy}}^{\prime }\), slip amount during an event, and the slip-weakening curves for a large value of

*J*are out of the observed fluctuations. Therefore, our estimation is reasonable.

*b*, \(L_{\text{c}}\)) are summarized in Fig. 12. For the short cumulative slip displacement case (window W1), \((b , { }L_{\text{c}} ) = (0.0085 , { }0.09 \, \upmu {\text{m}})\) was the best of the examined cases (i.e.,

*J*was the minimum, Fig. 11a), and the possible ranges for

*b*and \(L_{\text{c}}\) were \(b = 0.0085\) and \(0.04 \le L_{\text{c}} \le 0.1 \, \upmu {\text{m}}\), respectively (triangles in Fig. 12). For the long cumulative slip displacement case (window W2), \((b , { }L_{\text{c}} ) = (0.0094 , { }0.3 \, \upmu {\text{m}})\) was the best (Fig. 10a), and the possible ranges of

*b*and \(L_{\text{c}}\) were \(0.0092 \le b \le 0.0096\) and \(0.04 \le L_{\text{c}} \le 0.8 \, \upmu {\text{m}}\), respectively (circles in Fig. 12).

These results suggest that the evolution-related constitutive parameters (*b* and \(L_{\text{c}}\)) increased as the cumulative displacement increased, even in a single experiment. By comparing the triangles and the circles in Fig. 12, we can say that the *b* value is significantly different. The \(L_{\text{c}}\) value might be different between these two time windows, but the error range was too large to judge the differences.

We compared \(\Delta T_{\text{sy}}\) at limit cycles with \(\Delta T_{\text{ob}}\) not at limit cycles. The synthetic stick–slips reach a limit cycle after several stick–slips in the simulations with the possible parameters. However, the observed stick–slips did not reach a limit cycle even after many stick–slips occurred. Therefore, other processes such as changes in the state of the fault surfaces may have occurred in the experiment, which were not taken into account in synthetic model. This will be discussed in Sect. 5.

#### 4.1.2 Dependence of Constitutive Parameters on Loading Velocity

*b*and \(L_{\text{c}}\) to reproduce the stick–slip events observed in experiments LB01-134 with the slow \(\bar{v}_{\text{L}}\) (\(\bar{v}_{\text{L}} = 0.1{\text{ mm/s}}\)) and LB01-142 with the fast \(\bar{v}_{\text{L}}\) (\(\bar{v}_{\text{L}} = 1.0{\text{ mm/s}}\)) (Fig. 3). Throughout each of these experiments, \(\Delta T_{\text{ob}}\) and \(\Delta \mu_{\text{ob}}^{\prime }\) were almost constant (blue triangles and red crosses in Fig. 4). We estimated the constitutive parameters

*b*and \(L_{\text{c}}\) in the same manner as that described in Sect. 4.1.1. Figures 13 and 14 show the results of the joint inversion of the recurrence time, the friction drop, and the slip-weakening curves for the slow and fast \(\bar{v}_{\text{L}}\), respectively. The observed fluctuations of \(\Delta T_{\text{ob}}\), \(\Delta \mu_{\text{ob}}^{\prime }\), and the slip-weakening curves with events are smaller than those in LB01-127 (Sect. 4.1.1), and thus the synthetic slip-weakening curves fitted the observations better and we constrained the \(L_{\text{c}}\) value better.

For the case of the slow \(\bar{v}_{\text{L}}\), the best parameter set was \((b , { }L_\text{c} ) = (0.0098 , { }0. 9 { }\,\,\upmu {\text{m}})\), and the possible *b* and \(L_{\text{c}}\) values were \(0.0097 \le b \le 0.0098\) and \(0.7 \le L_{\text{c}} \le 1.0\,\, \upmu {\text{m}}\), respectively (open squares in Fig. 12). For the case of the fast \(\bar{v}_{\text{L}}\), the best parameter set was \((b , { }L_{\text{c}} ) = (0.0103 , { 0} . 1 \,\,{ }\upmu {\text{m}})\), and the possible *b* and \(L_{\text{c}}\) values were \(0.0103 \le b \le 0.0105\) and \(0.09 \le L_{\text{c}} \le 0.2 \, \upmu {\text{m}}\), respectively (solid squares in Fig. 12). Therefore, the constitutive parameters show clear dependence on \(\bar{v}_{\text{L}}\); *b* increases and \(L_{\text{c}}\) decreases as \(\bar{v}_{\text{L}}\) increases. By comparing the open and solid squares in Fig. 12, we can say that the *b* and \(L_{\text{c}}\) values are significantly different.

### 4.2 Aging Law

*b*and \(L_{\text{c}}\) and the possible ranges are summarized in Fig. 15. The results for the short and long cumulative displacements (triangles and circles in Fig. 15) suggest that the evolution-related constitutive parameters (

*b*and \(L_{\text{c}}\)) increase as the cumulative displacement increases in a single experiment. The

*b*value is significantly different. The \(L_{\text{c}}\) value can be different between these two time windows, but the error range was too large to judge the differences. The comparison of the results for the slow and fast \(\bar{v}_{\text{L}}\) (open and solid squares) suggest that

*b*increases and \(L_{\text{c}}\) decreases as \(\bar{v}_{\text{L}}\) increases and that the

*b*and \(L_{\text{c}}\) values are significantly different.

*J*(Eqs. (10)–(12)) for the estimated parameter sets were slightly different between the Slip and Aging laws, but we did not see any superiority of one over the other (Table 5). The

*b*and \(L_{\text{c}}\) values estimated for the Aging law are smaller than for the Slip law (Fig. 12) in all of the examined cases (windows from W1 to W4). However, our results for the Aging law stated above are the same as for the Slip law in Sect. 4.1.

Values of evaluation functions for optimum parameter sets

Time window | Slip law | Aging law | ||||
---|---|---|---|---|---|---|

\(J_{1}\) | \(\bar{J}_{2}\) (\(\times 10^{ - 5}\)) | | \(J_{1}\) | \(\bar{J}_{2}\) (\(\times 10^{ - 5}\)) | | |

W1 | 0.180 | 0.932 | 2.52 | 0.108 | 0.930 | 1.97 |

W2 | 0.0710 | 5.21 | 2.00 | 0.0414 | 5.21 | 1.99 |

W3 | 0.0743 | 2.96 | 2.53 | 0.0268 | 5.68 | 4.16 |

W4 | 1.28 | 5.77 | 4.44 | 1.29 | 6.89 | 4.86 |

## 5 Discussion

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.

*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.

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

| \(J_{1}\) | \(\bar{J}_{2}\) (\(\times 10^{ - 5}\)) | |
---|---|---|---|

0.005 | 0.0287 | 4.69 | 1.56 |

0.011 | 0.0321 | 5.05 | 1.99 |

*G*is the shear modulus and \(\nu\) is the Poisson’s ratio, was estimated by Ampuero and Rubin (2008).

*G*and \(\nu\) for Indian metagabbro used in our experiments are 39.3 and 0.31 GPa, respectively. From the parameters obtained in Sect. 4.2, \(L_{\text{b}}\) is estimated to be 0.31, 0.48, 2.8, and 0.36 m for windows W1 (short cumulative slip displacement), W2 (long cumulative slip displacement), W3 (slow \(\bar{v}_{\text{L}}\)), and W4 (fast \(\bar{v}_{\text{L}}\)), respectively. The increase in \(L_{\text{b}}\) for windows from W1 to W3 could represent the gouge production. The \(L_{\text{b}}\) values are smaller than the sample size, except for window W3; therefore, a rupture can propagate dynamically as a stick–slip event. It matters whether or not the sample size is larger than the nucleation size, and the effect of dynamic rupture propagation on the overall friction deserves future experimental and theoretical investigation.

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.

## 6 Conclusions

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.

## Notes

### Acknowledgements

This research was supported by the NIED research project entitled ‘Development of Earthquake Activity Monitoring and Forecasting’ and the JSPS KAKENHI, Grant No. 23340131. Assistance for the experiments provided by Tetsuhiro Togo, Hironori Kawakata, Nana Yoshimitsu, Tadashi Mikoshiba, Makoto Sato, Chikahiro Misawa, Toshiyuki Kanezawa, Hiroshi Kurokawa, Toya Sato, and Toshihiko Shimamoto is greatly appreciated. Anonymous reviewers’ comments were quite valuable in improving our manuscript. Friction experiment data are available upon request.

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