## 3 Results  
### Comparing the Effect of Barrier with Various Friction Laws  
#### 3.1.1 Barrier with Classical Slip-Weakening Friction Law  
We first studied the effect of barriers on fault rupture process and discussed whether the barrier's ability to induce supershear rupture was related to the friction constitutive relations that were used in the simulation. Therefore, the classical slip-weakening friction law, modified slip-weakening friction laws, and the rate-state friction law are were used to simulate the effect of a barrier on the propagation process of fault rupture. In this section, the parameters of the slip-weakening friction law are the same as those of [PERSON]'s model [30]--SW model for short: The static friction coefficient \(u_{s}=0.63\), the dynamic friction coefficient \(u_{d}=0.525\), the characteristic slip distance \(D_{c}=0.4\)m, and the barrier strength \(\Delta\sigma_{\text{n}}/\sigma_{\text{n}}=16\%\). The simulation results showed that the fault first began to rupture in the nucleation zone and then spontaneously propagated along the fault. Figure 2a is a distribution diagram that shows the rupture speed distribution on the fault surface, and it can be seen that the rupture propagation process on the fault was roughly divided into six stages: (1) The front of rupture spontaneously propagated along the rupture direction from the nucleation zone, and the propagation velocity gradually increased then tended to be stable (\(-2800\) m/s) at a distance of 20 km from the nucleation zone; (2) when the rupture propagated to the barrier zone, its velocity gradually decreased and tended to stop; (3) the propagation velocity of the rupture gradually increased after the front of rupture crossed the barrier zone and exceeded the S-wave speed of the medium (3333 m/s) at a distance of about 4 km from the barrier zone; (4) the rupture propagated at a supershear wave velocity for about 10 km; (5) the rupture speed tended to be stable (\(-2800\) m/s) after gradually decreasing; and (6) the rupture propagated to the end of fault at the subshear wave velocity. The whole rupture process of the fault was basically consistent with that of the previous results, which indicates that the simulation results of the nonlinear finite element method that was adopted by us were credible, and it also verifies that when the classical slip-weakening friction constitutive relation is used to simulate the spontaneous rupture process, a barrier can induce a supershear rupture of the fault. Figure 2b shows the slip along the interface over time. It can be seen from the figure that the relative displacement between the two sides of the fault gradually increased after the rupture of the fault. After the front of rupture reached the barrier zone, the propagation distance of rupture continuously decreased in an equal time interval (1 s). When the rupture passed through the barrier zone, the propagation distance of the rupture increased, and the rupture speed exceeded the S-wave speed. Figure 3 shows the changes of the slip and slip rate over time at the intermediate position of the barrier zone on the fault. As we can see from the slip-time curve of Figure 3, when the rupture propagated to the intermediate position of the barrier zone, the slip of the point rapidly increased after the rupture began; then, it remained unchanged, as if the slip had stopped on either side of the fault for about 1 s; finally, the slip at this point began to slowly increase again. As we can see from that slip rate-time curve of Figure 3, the slip rate of the point was zero for about 1 s, followed by a sudden increase in the slip rate. According to a dimensionless stress ratio \(S\) (\(S\) = \((\tau_{\text{s}}-\tau_{0})/(\tau_{0}-\tau_{\text{d}})\)) defined by [PERSON] [10; 11], the stress ratio \(S\) = 2, which was larger than the critical value of 1.77 for the supershear rupture in the initial state (no rupture), so the rupture fell into the subshear rupture.