## 1 Introduction  
Extensive research on earthquake damage has shown that earthquake-induced slope instability and landslides are among the major geological hazards in mountainous and hilly regions [1, 2, 3]. These landslides, triggered by earthquakes, are typically characterized by their widespread distribution, high frequency, and considerable destructiveness, leading to substantial casualties and economic losses [4, 5, 6, 7, 8]. The spatial distribution of earthquake-induced landslides is closely correlated with the main seismogenic faults and their surface ruptures [9, 10]. [PERSON] et al. [11] studied thousands of landslides triggered by the Mw 7.9 Denali earthquake in Alaska and found them to be concentrated within a 15 km range on either side of a surface rupture zone. Scholars have investigated the aftermath of the Mw 7.9 Wenchuan earthquake in 2008 and found that approximately 80% of large landslides were distributed within approximately 5 km on either side of the seismogenic fault surface rupture zone, with a significantly higher density of landslides on the hanging wall compared to the footwall [12; 13; 14; 15; 16; 17; 18; 19]. Experts and scholars have analyzed landslides triggered by the Mw 6.9 Yushu earthquake in 2010 and found that 80% of the landslides were concentrated within a 2 km width along a 95 km-long surface rupture zone, illustrating the typical characteristics of surface fault control [20; 21; 22]. The ground motion near strong seismic fault zones is associated with high energy, leading to a dense development of landslides. In particular, the damage caused by the landslide being directly \"cut\" by the relative dislocation of the fault or the relative motion of the two walls is more serious. For instance, [PERSON] et al. [23] and [PERSON] et al. [24] studied the Wenchuan earthquake-induced Guantan landslide (Anxian County), which was traversed by the Longmenshan fault zone. They found that the relative displacement between the two walls of the fault causes the slope to be fractured and loosened, with cracks extending deeper to form tensile fractures. The landslide experienced overall instability eight hours after the earthquake, resulting in the formation of a landslide dam. The formation mechanism and development process of this landslide could be divided into three stages: compression-toppling deformation of the natural slope, post-earthquake through-going slip surface stage, and overall failure stage of the landslide. [PERSON] et al. [25], [PERSON] et al. [26], [PERSON] et al. [27], and [PERSON] et al. [28] investigated and analyzed the Donghekou landslide. The upper wall of the Shikan fault across the slope moves upward and the surface rupture zone is found at the landslide front. The rock mass at the rear of the rupture zone developed tensile cracks, causing the outer rock mass to be ejected as a whole, moving forward approximately 2 km in the form of a debris flow on impacting with the ground. [PERSON] et al. [29] and [PERSON] and [PERSON] [30] investigated the 1994 Northridge Mw 6.7 earthquake in the United States, and found that the Loma Verde landslide, composed of sandstone, had multiple steep scarps at its top due to the thrust fault, leading to a complete fragmentation of a ridge hundreds of meters high. [PERSON] et al. [31] investigated the 2022 Menyuan Mw 6.7 earthquake and found that under the action of strong fault dislocation, significant displacement movement occurred in the fault zone where the tunnel was located (the subgrade was raised by 2 m), resulting in a large number of rockfall and collapse of the tunnel slope. Therefore, the instability of the cross-fault landslide is caused by the relative movement of the two sides of the bedrock fault. However, due to the different positions of the main rupture zone (tensile surface) caused by the fault dislocation, the formation mechanism of the cross-fault landslide is not clear. At present, many researchers have explored the failure mechanism and evolution process of cross-fault slopes through physical model tests.