In Fig. 4, temporal variation of coefficient of friction during an experiment is shown. In the inset of Fig. 4, which is a magnified plot between 211 and 217 s, we can see many stick slip events whose stress drop is about 0.1 in coefficient of friction (or about 0.3 MPa in shear stress drop). We analyzed the shear strain array data at each stick slip event. It should be noted that for the analysis of the l
c
estimation, we used the data of the third experiment in LB04 specimen series, called “LB04-003.” The reason why we chose this data set is simply because there are many supershear rupture events included. Actually, it was quite difficult to control the rupture velocity of the stick slip events because the propagation velocity depends on the fault surface roughness as well as the amount of off-fault damages, which we could not control yet.
Figure 1 shows a schematic illustration of stress and slip behavior at the rupture front based on the theoretical formulation. This will help for the explanation of the method to estimate l
c
here. Cohesive zone is defined as the distance between the peak stress and residual stress along the fault at the rupture front. This zone is considered as inelastic behavior so that the stress does not diverge at the rupture front and gradually decreases behind the rupture front toward the frictional stress level. If we have a spatial variation of shear strain along the fault as shown in Fig. 1, we will be able to measure the l
c
value by picking the position for the peak strain (A in Fig. 1) and that for the residual strain (C in Fig. 1). However, since the strain gauge was installed slightly off the sliding surface, the strain waveforms were contaminated by elastic waves. This makes the detection of the residual strain location (C in Fig. 1) practically difficult. Thus, we picked the position where the shear strain drops to the initial strain value (B in Fig. 1) instead of picking the residual position. Therefore, in this paper, we call the distance between A and B the cohesive zone length (l
c
). It should be reminded that this l
c
value is an approximation to directly measure from the observation data. We will discuss the accuracy of this approximation later using theoretical strain waveforms.
It should be noted, however, that what we measured is the temporal variation of shear strain, which can be converted to shear stress by multiplying the rigidity of the medium. And, what we need to estimate l
c
is the spatial variation of shear stress. Since we could measure the propagation velocity of the rupture front, we are able to estimate the spatial distribution of shear strain by converting the spatial array of temporal strain change. When the rupture velocity (v) is constant, x-vt becomes the variable of the array data, where x and t are spatial and temporal coordinate parameters, respectively. Therefore, the temporal strain data s(x
0, t) at position x
0 can be considered as a spatial snapshot of s(vt
0
, -x/v) at time t
0 if t∼t
0 and x∼x
0.
In Fig. 5, 32 shear strain waveforms are shown for the stick slip event E0049, as an example of the observed data. This event was initiated between ST31 and ST32 at 213.91015 s after the loading started and the rupture propagated mostly unilaterally at a velocity of 5.0 km/s. We estimated v by a least square fitting of the rupture time data. The rupture time was measured by picking the time when the strain waveform suddenly decreased and crossed the initial strain level before the stick slip (B in Fig. 1). The estimated v is much faster than the v
s
of this material (3.63 km/s) and is close to \( \sqrt{2}{v}_s \).
In Fig. 6, snapshots of the spatial distribution of the strain ahead of and around the rupture front for the stick slip event E0049 are plotted, which were converted from the temporal strain array data along the fault (Fig. 5) assuming a constant v of 5.0 km/s. In this plot, we assume that the reference time for the space-time conversion is set at the rupture front arrivals at an observation point. Then, we consider the data recorded before this time as the data recorded in space ahead of the rupture propagation at this reference time by adjusting the space-time coordinate using the assumed rupture velocity. In Appendix A, we show the comparison between converted spatial strain distribution with the snapshot of the strain along the fault, which was the measured strain value at a particular time at each position. We can see that the conversion works reasonably well. Using these plots, we estimated l
c
values by measuring the distance between A and B in Fig. 1.
It should be noted that, as can be seen in Fig. 6b, at some temporal snapshots, the peak strain location was not clear, which could be due to the local rupture velocity perturbation or due to the station location at finite distance off the fault. The local velocity variation might be related to heterogeneous stress drop along the fault. And, when the observation point is not on the fault, near-fault waves might contaminate the strain waveforms. Since we did not take into account these effects here, these could result in estimation errors of l
c
values in the present study.