General stick–slip behaviors
This paper reports preliminary results from a series of initial tests at loading rates of 1.0, 10, and 100 mm/s, at normal stresses to 1.32 MP, and with displacements of about 0.39 m, using a pair of ground specimens of Indian gabbro. Violent stick–slip events occurred in most experiments with shear force drops reaching 8 × 105 N. The overall stick–slip behaviors observed in our experiments exhibited the same features as recognized previously, i.e., nearly constant rise time of slip events or a constant duration of slip event (17.7 ms, Fig. 10b), and a linear relationship between the average velocity and shear force drop or shear stress drop (Fig. 11a, c; cf. Ohnaka 1973, 1978; Johnson and Scholz 1976; Shimamoto et al. 1980). Frequent stick–slip events must have been caused by rather soft loading system with stiffness of 1.15 × 108 N/m (153 MPa/m for our specimens), a value determined from a force drop ΔF and an amount of slip Δd during stick–slip events at loading rates of 1.0 and 10 mm/s (Fig. 10a). This stiffness is much smaller than the stiffness of a large-scale biaxial friction apparatus at the U. S. Geological Survey (2.6 × 109 N/m or 3.3 GPa/m; Lockner and Okubo 1983), and this is a reason for the occurrence of violent stick–slip in our experiments. Some events were so large that the shear stress went on to show negative values (Fig. 4) due to overshooting. Such overshooting of slip was suggested to have occurred during the 2011 Tohoku-Oki earthquake, resulting in the inversion from compressive stress before the earthquake to the tensile stress field over wide areas in subduction zone after the earthquake (e.g., Ide et al. 2011; Sibson 2013).
We conducted a simple analysis of a spring–slider block model with one degree of freedom assuming a constant static friction coefficient μ
s (= 0.8) and a constant kinetic friction coefficient μ
k in Appendix 1 (μ
s was assumed to drop instantly to μ
k upon the initiation of slip). The overshooting of the shear stress to a negative value begins to occur when μ
k is less than 0.4 and the overall stick–slip behaviors in Fig. 4c are similar to such a behavior. Thus, μ
s must have dropped to slightly less than half of μ
s in violent stick–slip events in our experiments. Oscillatory slip with Coulomb damping begins to occur for μ
k < 0.8/3, but such slip oscillations did not occur in our experiments.
The stick–slip amplitude appears to increase with the increasing loading rate from 1 to 100 mm/s, as seen from the results in Fig. 4. This is an opposite trend from that recognized in previous experiments (e.g., Teufel and Logan 1978, Fig. 5). In our experiments, however, the same specimens were used repeatedly, and damage accumulated on the sliding surface with the increasing number of experiments (Fig. 5a, c). Gouge was removed after each run, but the amount of generated gouge during each run was different from one run to another. Thus, the effects of loading rate and the damage/gouge accumulation cannot be separated in our experiments. The conspicuous abrasive grooves in Fig. 5 are similar to the wear grooves associated with stick–slip, described in Engelder (1974). He reported that the grooves were carrot shaped, and their lengths rarely exceeded the amount of slip during stick–slip, but the grooves were much longer than the amount of abrupt slip in our experiments. The wear grooves will be described in more detail elsewhere, along with the partition of frictional work in the gouge generation.
The observed stick–slip behaviors are complex, and we now consider how the friction apparatus behaved during stick–slip. The velocity fluctuated during and following slip events (Fig. 6b, d, f), and small oscillatory fault motion occurred during the loading or the so-called stick periods of stick–slip events (Fig. 7). Initially, we interpreted the decaying oscillatory motion during the stick periods as derived from oscillatory fault motion due to the oscillatory movement of the very heavy shaking table with low stiffness and huge mass (M
st = 1.8 × 105 kg). However, Nick Beeler (personal communication) pointed out that such a Coulomb damping could not occur unless a shear stress dropped to the negative side by a large amount and suggested that the oscillations in the displacement records might have come from vibration of the sensor holder. We measured the characteristic frequencies of the target bar of the displacement transducers and confirmed that the decaying oscillations during the stick-periods in Fig. 8 are due primarily to the oscillations of the target bar, although the oscillation in the X direction (or sliding direction) is more complex than the simple oscillation of the target bar (Appendix 2).
Behavior of the friction apparatus during stick–slip
Measuring only relative displacement between the two specimens is not enough to understand the behaviors of both stationary side with upper specimen and the shaking table with lower specimen. Thus, we refer to a supplementary experiment that measured the relative displacement d at a different position with reduced vibration problem of the target bar, a displacement of the shaking table l, and the accelerations of the upper and lower specimens, a
1 and a
2, with two accelerometers (Fig. 12a; LB01-127, slip rate v = 0.1 mm/s, normal stress σ
n = 1.3 MPa; to be reported elsewhere by Y. Urata and others). Table 3 of Fukuyama et al. (2014) gave the history of experiments with the specimens. The laser–displacement transducer was set to a wooden bar that was fixed to the lower moving specimen, and its target plate was glued to the middle part of the upper stationary specimen (see Fig. 12a for their positions). Integrations of a
1 and a
2 twice yield displacements of the upper and lower blocks u
1 and u
2, respectively. An inset diagram in Fig. 12b exhibits seven stick–slip events starting from 299 s after the onset of experiment (time = 0 corresponds to 299 s in the diagram), with vertical axis showing shear force F (black curve), relative displacement u
2 − u
1 (pink curve), and displacement of the shaking table l (green curve). The displacement u
2 − u
1 is the same as the displacement d = d
2 − d
1 as measured by a laser-displacement transducer in the X direction (Figs. 2, 3b), but we use different symbols because u
1 and u
2 were measured differently. Figure 12b shows temporal changes in F, d, l, –u
1,
u
2, and u
2 − u
1 with different colors as shown in the diagram, for the sixth stick–slip event marked with a dashed rectangle in the inset diagram. Note that (u
2 − u
1) shown by a pink curve coincides with d in orange curve, but u
1 and u
2 in red and blue curves were plotted until the numerical integration of a
1 and a
2 was properly done (the numerical integration error accumulated, and the low-frequency response of the accelerometers was not sufficient to determine −u
1 and u
2 beyond those plotted in Fig. 12b).
Shear force F began to drop abruptly at time t of 16.395 s and reached a minimum value in 15 ms as shown by two dashed black lines in Fig. 12b. This time interval is plotted as the stick–slip rise time in Fig. 10b. The movement of the upper specimen stopped in 11.8 ms after the onset of shear-force drop (red curve), after which −u
1 decreased by 0.033 mm when F reached minimum (u
1 is defined positive leftward, and a decrease in (−u
1) corresponds to the movement of the upper block in the loading direction). During the same period, the displacement of the lower specimen u
2 increased by 0.037 mm (blue curve), slightly larger than the drop in (−u
1), and the relative displacement (u
2 − u
1) slightly increased as shown by the pink curve. However, this much of difference between −u
1 and u
2 can be due to the cumulative integration errors of a
1 and a
2. At about the same time, fault displacement d stopped increasing (orange curve) at around 11.8 ms, but it is unclear where exactly the fault motion stopped due to the oscillation in the d record (the oscillation was reduced with the new target plate, but was not eliminated completely). Thus, the time for the complete stop of fault motion could not be determined accurately from the current data. However, the movement of the upper block was reversed sharply at 11.8 ms from the onset of shear-force drop and after this point, the upper and lower specimens moved in the same direction by almost the same amounts as shown by −u
1 and u
2 curves. We thus consider that the 11.8 ms is the most likely time for the stop of the fault motion. Our stiffness data in Fig. 10 can be improved by conducting similar experiments as those shown in Fig. 12.
An interesting result is that a rapid movement of the shaking table started in about 12 ms after the onset of shear-force drop (green curve in Fig. 12b) and increased for 50–70 ms. The onset and stop of this rapid movement of the shaking table could not be determined clearly, but the displacement during or following an abrupt slip is seen as small steps in l–t record (green curve in the inset diagram of Fig. 12b). This displacement of the shaking table, which is in the loading direction, increased the shear force because the fault was already locked. This process corresponds to an increase in the shear force F from J to K in Fig. 8a, b. There are oscillations in l causing oscillations in F in Fig. 12b, and similar oscillations in F can be recognized in Fig. 8b as well.
The result in Fig. 12b gives an insight on the cause of the bimodal distribution of the stick–slip rise time t
r in Figs. 10b, 11b and e. We determined the stop of abrupt slip during stick–slip from the point of minimum shear force F or minimum shear stress τ (e.g., dashed vertical line on the right side after 15 ms since the onset of slip in Fig. 12b). However, the fault motion is likely to have stopped after 11.8 ms from −u
1 record in this case (red vertical dashed line), as discussed above. Notable oscillation in F is overlapped on the F versus time record, and this oscillation probably caused the minimum shear force slightly after the stop of fault motion. There are 14 oscillations in a time interval of 90 ms after 16.41 s (black curve in Fig. 12b), and this gives an average duration of 6.4 ms for one oscillation. This is consistent with the time interval of 7 ms (= 24–17 ms) between the two peaks of t
r in Fig. 10b, strongly suggesting that large values of t
r were due to the effect of oscillation in F. A similar situation is recognized in an example shown in Fig. 7a. Shear force F or shear stress τ reached local minimums at J′ and M′, but the real minimums of F and τ were achieved at J and M which were taken as the stops of abrupt slip. In view of the results in Fig. 12b, however, J′ and M′ are probably close to the stop of fault motion. Thus, the data on t
r in Figs. 10b, 11b and e are not reliable, and t
r should be determined based on the improved measurements of fault displacement d and on the measurements of acceleration −u
1 of the stationary block in the future. The displacement d, used to determine Δd in Fig. 10a, remains about the same after 11.8 ms since the onset of slip, although d oscillates for several tens of ms (the orange curve in Fig. 12b). However, ΔF would have been overestimated slightly by taking the minimum point of F as the stop of fault motion, owing to the overlapped oscillation in F, and hence the stiffness k (= ΔF/Δd) determined from the data in Fig. 10a might have been overestimated slightly as well.
The supplementary experimental data in Fig. 12 clearly demonstrated that the behaviors of different parts of the apparatus behaved differently, and we now consider the apparatus behaviors in more detail. The mass is distributed in the stationary sides of the apparatus (parts 2–4, 7, 12, and 13 in Fig. 1), and we estimate an effective mass of the stationary side (M
eff) following the procedures by Shimamoto et al. (1980, Eq. (20)):
$$ M^{\text{eff}} = \sum\limits_{i = 1}^{n} {\left( {\frac{{m_{i} }}{3}} \right)} \left( {\frac{{k_{m1} }}{{k_{i} }}} \right)^{2} + \sum\limits_{i = 2}^{n} {m_{j} } \left( {\frac{{k_{m1} }}{{k^{i - 1} }}} \right)\left( {\frac{{k_{m1} }}{{k^{i} }}} \right) $$
(3)
where the number of parts n is 8. We renumbered those parts and gave stiffness k
i
and mass m
i
of part i in Table 1, and the stiffness values give the bulk stiffness of the stationary side k
m1 of 0.159 GN/m. Shortening or elongation of part i is given by (k
m1/k
i
)d
1 assuming uniform force distribution, where d
1 is the displacement of the stationary side (Fig. 3b). Then the effective mass of part i during its dynamic deformation is given by (m
i
/3)(k
m1/k
i
)2 (the first term in Eq. (3); Table 1, sixth column). We did not include the dynamic deformation of the upper specimen, and its stiffness is set to infinite in Table 1. On the other hand, displacement of each part has to be evaluated to estimate an effective mass of part i during its rigid-body transformation. The displacement at the bottom of the reaction force base is set to zero, and the displacement at the right end of part i is given by a sum of shortening of parts 1 to i (footnote of Table 1). Then the effective mass of the rigid-body transformation is given by the second term in Eq. (3) (Table 1, the seventh column in Table 1; part 1 is fixed, and its effective mass is zero). Note also that a displacement of each part reduces, and its effective mass becomes smaller than the real mass toward the fixed end. Thus, the total effective mass of the stationary side M
eff becomes 1,689 kg if a uniform force distribution is assumed. The rise time of stick–slip of the stationary side (t
r
m1
) is given by
$$ t^{r}_{m1} = \pi \left( {M^{\text{eff}} /k_{ml} } \right)^{ 1/ 2} $$
(4)
if constant static and kinetic friction coefficients, μ
s and μ
k
, are assumed and if the friction coefficient μ is assumed to drop instantly from μ
s to μ
k upon the onset of slip [see Eq. (5) in Appendix 1]. Using Eq. (4), the above M
eff and k
m1 values give 10.2 ms for the rise time, and this is in reasonable agreement with the duration of movement of the upper specimen (11.8 ms; see red curve in Fig. 12b). The rise time of abrupt slip becomes longer when μ does not drop instantly to μ
k as in the cases of real faults (e.g., Dieterich 1978), but simple analyses with constant μ
s and μ
k are useful to consider the overall system behaviors as first approximations.
A similar analysis with a mass of the shaking table (M
st = 1.8 × 105 kg) and a stiffness of the lower part of the system (\( k_{{m2^{{\prime }} }} = k_{\text{st}} k_{m2} /(k_{\text{st}} + k_{m2} ) \) = 4.16 × 108 N/m, see Sect. 4) gives the rise time of 65 ms for the motion of the shaking table using Eq. (4). The displacement record of the shaking table indicates that the shaking table moved for 40–50 ms (green curve in Fig. 12b) which is of the same order as those estimates. Those calculations and the result in Fig. 12b raise an important question on the meaning of the data on ΔF/Δd in Fig. 10a. Our interpretations in Sect. 4 were that the slope of ΔF versus Δd relationship gives the stiffness of the whole system k and that this k and k
m1 values, used above, give the stiffness of 4.16 × 108 N/m (555 MPa/m) for the lower part (k
m2 and k
st connected in series; Fig. 3b). However, if the shaking table did not move much before the fault motion stopped at around the minimum shear force (Fig. 12b), the stiffness given by ΔF/Δd does not include the stiffness of the shaking table k
st (Fig. 3b). Then the stiffness k
m2 (4.16 × 108 N/m) is more likely to give a stiffness of k
m2. The mass of the lower specimen (1.6 × 103 kg) and this stiffness give a rise time of 6 ms. The displacement of the lower specimen u
2 exhibits a small peak at 4.4 ms after the onset of shear-force drop (blue curve and vertical dashed blue line in Fig. 12b). This is fairly close to the above rise time and probably the frame and holders of the lower specimen reacted immediately following the onset of slip. Subsequent displacement of the lower specimen may be due to the elastic deformation of the part of the shaking table and the displacement of the shaking table itself. A complex shape of u
2 record (blue curve in Fig. 12b) should reflect a complex behavior of the loading side with the shaking table and actuators, and should be analyzed in the future, by determining the stiffness of mass of each part.
The overall shape of the shear force F versus time t curve from a period of 16.41–16.5 s is similar to the shape of shaking-table displacement l versus t curve in Fig. 12b. This is natural if the fault is locked at around t = 16.41 s (vertical dashed red line in Fig. 12b) and the amount of displacement and an increase in displacement give a stiffness of about 1.4 × 108 N/s. This is fairly close to the stiffness (1.15 × 108 N/m) determined from the slope of ΔF–Δd line in Fig. 10a, and we conclude that the rapid increase in F following an abrupt slip (e.g., J–K in Fig. 8a, b) is due primarily to the delayed movement of the shaking table. The oscillatory changes in F have frequency of about 150 Hz (14 oscillations in 90 ms) and can be correlated with the oscillations in l although the latter appears to be more complex (Fig. 12b). We have not identified the source(s) of the oscillations yet, but any changes in l can cause changes in F when the fault is locked.
A future task for measuring friction along a fault during stick–slip
The new NIED biaxial apparatus has a merit of producing high slip rates and a large displacement on a pair of large-scale specimens by using the shaking table as the loading device, allowing us to observe dynamic rupture propagation and even to produce small-scale ruptures confined within the fault as demonstrated by Fukuyama et al. (2014). However, in order to study the evolution of friction during dynamic rupture propagation, it is essential to measure the shear stress directly along the sliding surface as conducted by Dieterich (1981b), Okubo and Dieterich (1981, 1984), Lockner and Okubo (1983), Ohnaka et al. (1987), and Beeler et al. (2012). The shear force gauge between the reaction force bar and the upper specimen (12 in Fig. 1c, FG in Fig. 3b) cannot separate the axial forces due to the friction along the sliding surface and dynamic forces to accelerate or decelerate machine elements and the upper specimen. The situation is similar to a spring–slider block system in Fig. 13a. The shear stress as calculated from the restoring force in the spring divided by the fault area is used to determine the shear stress/normal stress ratio τ/σ in Fig. 13 which varies sinusoidally, and yet the real friction along the fault during slip is constant with μ
k. The situation is exactly the same as calculating the friction coefficient μ from the axial force records such as those in Fig. 4.
There are two ways to determine the friction along the simulated faults during stick–slip. One is to measure the friction directly, for instance, by using strain gauges bonded on the specimens. The other way is to restore frictional properties from observed stick–slip behaviors by modeling of the system behavior with assumed form of friction law(s). Slight deviations in behaviors from the sinusoidal behaviors for constant μ
s and μ
k (Fig. 13) are the source of information to determine the frictional properties, in the case of a simple spring–slider block system with one degree of freedom in Fig. 13a. It should be kept in mind that a real friction apparatus is more complex and the frictional properties cannot be restored unless the apparatus behaviors are understood properly, and this is the main reason why we conducted the present study. The behavior of the stationary side may be close to that of a spring–slider block system if the kinetic friction coefficient is constant, as discussed in the previous subsection. However, if friction depends on slip and slip-rate as in the case of rate-and-state friction (e.g., Dieterich 1979, 1981a), then the behavior of the loading side affects the fault motion, and the loading side cannot be separated from the analysis. The behavior of this side is more complex because of the huge mass of the shaking table as revealed by the slip histories of the lower moving specimen and the shaking table (Fig. 12b). We conventionally separated two springs k
m2 and k
st for this side in Fig. 3b, but we could not fully model the behavior of the moving side yet in this paper. Full understanding of the elastic and inertial properties of the loading side will be a key to restore the frictional properties accurately from the observed stick–slip.