## 1 Introduction  
The so-called supershear rupture refers to a phenomenon in which the rupture propagation velocity \(\mathrm{V_{r}}\) on a fault exceeds the propagation velocity \(\mathrm{V_{s}}\) of the shear wave (S-wave) in the medium around the earthquake source area. Earthquake observations have shown that the rupture speed of the fault in most earthquakes is lower than the S-wave speed, but many strike-slip earthquakes produce a supershear rupture, such as the 1979 Imperial Valley earthquake in the United States (Mw = 6.4) [1], the 1999 Kocaeli (Izmit) earthquake in Turkey (Mw = 7.4) [2], the 1999 Duzce earthquake in Turkey (Mw = 7.2) [2], the 2001 West Kunlun earthquake in China (Mw = 7.8) [3], the 2002 Denali earthquake in Alaska of the United States (Mw = 7.9) [4], and the 2013 Craig earthquake in the United States (Mw = 7.5) [5]. In addition, it has been proven by theoretical analysis [6; 7; 8; 9], numerical simulations [10; 11; 12; 13; 14; 15; 16; 17], and laboratory experiments [18; 19; 20; 21; 22; 23; 24; 25; 26; 27] that fault rupture speed can exceed the S-wave speed and produce a supershear rupture. When a fault ruptures at supershear wave velocity, the wavefronts that are formed by the new and old ruptures interfere with each other, and the superposition effect greatly increases the amplitude of ground motion and generate Mach waves, which greatly intensify the seismic hazard. Therefore, it is important to study the formation mechanism of supershear rupture earthquakes in order to understand earthquake sources and to evaluate seismic hazards. A barrier is a strong section of a fault that requires more energy to break [28; 29]. Recent studies have shown that a barrier may induce the supershear rupture of faults [30; 31]. According to a dimensionless stress ratio \(S~{}=~{}(\tau_{\rm s}-\tau_{0})/(\tau_{0}-\tau_{\rm d})\) that was defined by [PERSON] [10; 11] (wherein \(\tau_{s}\), \(\tau_{0}\), and \(\tau_{d}\) represent the fault static shear strength, initial shear stress, and dynamic shear stress, respectively), in the rupture process, by using two-dimensional numerical simulation and a slip-weakening model, a sub-Rayleigh wave velocity rupture occurs when \(S\) is large enough, a supershear rupture occurs when \(S\) is small enough, and the critical value of \(S\) ranges from 1.7 to 1.8. However, due to the existence of a barrier, in certain conditions, the rupture speed of faults changes from a subshear rupture speed to a supershear rupture speed. When simulating the 1992 Landers earthquake, [PERSON] et al. [32] found that the rupture speed of fault exceeds the S-wave speed when there is a high stress zone on the fault that is close to the fault width. Since then, [PERSON] and [PERSON] [9] and [PERSON] and [PERSON] [33] found the same phenomenon in numerical simulation experiments. [PERSON] et al. [30] found a peculiar phenomenon in their study that a barrier may induce a supershear rupture in certain situations. A supershear rupture that is induced by a barrier eventually transforms into a subshear rupture. The duration of a rupture is related to the size of the barrier and the distance between the nucleation zone and the barrier. [PERSON] et al. [31] studied the effect of barriers on the propagation process of supershear ruptures in a three-dimensional semi-infinite space, and they demonstrated that the effect of a barrier on the rupture process is related to the size, strength and location of the barrier. A fault is large in spatial scale, and its rupture process cannot be directly detected in the deep underground. Dynamic studies of the fault rupture propagation process are mainly carried out by numerical simulations. This difficulty is faced by researchers when they establish a numerical model of earthquake source dynamics in the setting of friction relation. At present, we are incapable of directly obtaining the true friction constitutive relation on fault surfaces.