## 3 Results
### Comparing the Effect of Barrier with Various Friction Laws
#### 3.1.2 Barrier with Rate-State Friction Law  
In Section 3.1.1, the classical slip-weakening friction law was used to simulate the effect of the barrier on the rupture process. It was found that the barrier could promote the fault to produce a supershear rupture. Is this conclusion was based on the characters of the classical slip-weakening friction law? According to Equation (1), when the slip reaches the characteristic slip distance, the friction coefficient on the fault remains unchanged, the dynamic friction coefficient state is maintained, the slip is difficult to stop, and the rupture cannot heal. Next, we used the rate-state friction law to  
Figure 3: The slip and slip rate of the center point of the barrier on the SW model fault change with time. Figure 2: (a) Distribution diagram of the rupture speed on the fault surface of [PERSON]’s model (SW for short). (b) Distribution diagram of the slip along the interface over time. simulate the effect of a barrier on rupture propagation; this is called RS model for short. According to Equation (7), the characteristic slip distance \(D_{c}\) in the rate-state friction law is approximately equal to the product of the characteristic velocity \(V_{c}\) and the characteristic time scale \(t_{c}\). When the characteristic slip distance \(D_{c}=0.4\)m, a spontaneous rupture cannot be produced by the fault when the parameters of the other parametric models are consistent with those of Section 3.1.1, which indicates that there is a great difference when using the two different friction constitutive relations to simulate the spontaneous rupture process of the fault. For this section, \(V_{c}=0.1\)m/s, \(t_{c}=0.1\) s were set up to enable the fault to spontaneously rupture and to study the effect of the barrier on the rupture process. Figure 4a shows the rupture speed distribution on the fault surface. It can be seen that the rupture propagation process of the fault was divided into six stages that were substantially the same as those in Figure 2a. However, the propagation distance of each stage was different: (1) The front of the rupture spontaneously propagated along the rupture direction from the nucleation zone, and the propagation velocity gradually increased and then tended to be stable; (2) when the rupture propagated to the barrier zone, its velocity gradually decreased; (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 (3333m/s); (4) the rupture propagates at the supershear wave velocity for about 3 km; (5) the rupture speed gradually decreased and then tended to be stable; and (6) the rupture propagated to the end of fault at the subshear wave velocity. This shows that the barrier can induce supershear rupture when the rate-state friction law is used to simulate the spontaneous rupture process of faults. Figure 4b shows the changes of the slip along the interface over time. It can be seen from the figure that the propagation distance of the rupture increased and the rupture speed exceeded the S-wave speed when the rupture passed through the barrier zone. Through the above analysis, we were able to verify that a barrier can lead to the supershear rupture of faults by using various friction laws. Figure 4: (**a**) Distribution diagram of the rupture speed of the fault surface of the RS model (a model that uses the rate-state friction law to simulate the effect of a barrier on rupture propagation). (**b**) Distribution diagram of the slip along the interface over time. It can be seen from the analysis above that regardless of the slip-weakening friction law or rate-the state friction law used in a simulation, a barrier is able to induce supershear rupture on a fault.