## 4 Discussion  
To explain a variety of observations on ruptures in the Earth [28; 29], such as the increased speed and increased slip distance of local ruptures, researchers have put forward the concepts of \"barrier\" and \"asperity.\" From a study on barriers with the slip weakening friction law and the rate-state friction law, we found that both laws can induce supershear rupture. However, we found that the two  
Figure 8: Distribution diagram of peak ground acceleration near faults with barriers of different strengths. types of friction laws generate different influences on rupture propagation velocity. [PERSON] (1958); (1960) showed that when the static and dynamic friction coefficients in the slip-weakening friction constitutive relation are small, the fault easily produces a supershear rupture. By defining a dimensionless stress ratio \(S=(\tau_{s}-\tau_{0})/(\tau_{0}-\tau_{d})\) in which \(\tau_{s},\tau_{0},\text{ and }\tau_{d}\) represent the fault static shear strength, initial shear stress and dynamic shear stress of the fault, respectively, [PERSON] concluded that by using a two-dimensional numerical simulation with a slip-weakening model, a sub-Rayleigh wave velocity rupture occurs when \(S\) is large enough, but a supershear wave velocity rupture occurs when \(S\) is small enough. The critical value of \(S\) is between 1.7 and 1.8. However, when the rate-state-dependent friction law was used to simulate the fault rupture process in our study, it was found that even if the stress ratio \(S\) was higher than the critical value, a small characteristic slip distance \(D_{c}\) led to a supershear rupture of the fault when the stress ratio \(S\) remained constant. In addition, to further investigate the effect of barriers on fault rupture process, we took an extreme case into consideration: the barrier could not be broken. We built two new models (Model 1 and Model 2). The barrier in Model 1 had a width of 5 km, and the barrier in Model 2 had a width of 10 km. The friction coefficient of the fault in the barrier zone was sufficiently maximized to ensure no slip in the barrier zone. All other model parameters (such as the initial stress field, boundary conditions, and friction law) remained the same as with the model in Section 3.1.1. Figure 9 shows the snapshots of the slip profiles along the fault interface at different times in Model 1 and Model 2. It can be seen from Figure 9a that the front of the rupture spontaneously propagated along the rupture direction from the nucleation zone; when it arrived at the barrier, no slip occurred on the fault in the barrier zone; when the stress wave propagated through the barrier, the fault re-nucleated to form a new rupture which propagated forward along the fault. However, in the whole rupture propagation process along the fault, there was no supershear rupture appearance. By combining these results with those that were achieved in Section 3.2, we found that the barrier may induce the supershear rupture of faults, and as the strength of the barrier increases, the distance of the rupture that propagates at the supershear wave velocity increases. However, when the strength of the barrier reaches up to a certain degree and the barrier zone cannot be broken, the rupture jumps the barrier and keeps propagating along the fault, but no supershear ruptures appear. Namely, the barrier neither stops the propagation of the rupture nor generates a supershear rupture. It can be seen from Figure 9b that, in the simulation, when the length of the barrier was 10 km, the existence of the barrier stopped the rupture propagation. Therefore, a supershear rupture can only be generated when both the length and width of a barrier are kept within a certain range. Figure 9: (**a**) Distribution diagram of the slip along the interface over time of Model 1. (**b**) Distribution diagram of the slip along the interface over time of Model 2. Our numerical results suggested that a barrier may induce the supershear rupture of faults.