## 1 Introduction
[19] investigated theeffects of low-temperature nitrogen thermal shock on the porosity, permeability, and mechanical properties of shale, and they proposed a multi-cycle thermal shock experimental method. By subjecting the rock to three cycles of heating followed by nitrogen injection, they observed significant enhancements in porosity, permeability, brittleness index, and fracturing index. Furthermore, a number of studies have investigated the physical and mechanical properties of granite following exposure to high-temperature liquid nitrogen cooling. For instance, [PERSON] et al. [20] conducted a study on the properties of three types of rocks following exposure to high-temperature and liquid nitrogen cold-shock treatment. Through mechanical strength and permeability analysis, it was determined that liquid nitrogen cold shock significantly increased the permeability of the rocks. Sandstone, with its larger pores and mineral composition, exhibited the least thermal shock effect. [PERSON] et al. [21] conducted liquid nitrogen jet experiments on granite at temperatures ranging from 200 \({}^{\circ}\)C to 300 \({}^{\circ}\)C and analyzed the effects of liquid nitrogen fracturing on the fracture characteristics of dry hot rock reservoirs. The results showed that liquid nitrogen cold shock significantly increased the number and complexity of fractures as well as the permeability of the rock. In a related study, [PERSON] et al. [22] investigated the effects of liquid nitrogen, water, and air cooling on the physical and mechanical properties and microstructural changes of high-temperature granite. Their findings indicated that liquid nitrogen cooling resulted in more substantial damage to granite compared to other cooling methods. In a related study, [PERSON] et al. [23] investigated the effects of liquid nitrogen treatment on the damage characteristics of high-temperature granite. This included the effects of liquid nitrogen cold shock on P-wave velocity, tensile strength, energy evolution, and acoustic emission parameters. The study concluded that liquid nitrogen cold shock significantly increased the damage level of high-temperature granite. [PERSON] et al. [24] investigated the fracturing effects of liquid nitrogen (LN\({}_{2}\)) on coal rock from the Karaganda Basin in Kazakhstan and proposed two novel treatment methods: freeze time (FT) and freeze-thaw cycles (FTCs). Their study demonstrated that liquid nitrogen fracturing can weaken the mechanical properties of coal by regulating freeze-thaw conditions, thereby enhancing permeability. In the domain of practical rock engineering, the rock body is typically in a two- or three-dimensional stress state, with compressive shear damage being the most prevalent mechanical response of the rock body [25]. [PERSON] et al. [26] investigated the impact of elevated temperatures on the shear characteristics through conducting variable-angle shear tests on granite samples subjected to different temperatures. The experimental findings demonstrated that both shear strength and cohesion decreased with increasing temperature, while the friction angle increased. [PERSON] et al. [27] investigated the mechanical characteristics of rock shear damage at high temperatures and determined that an increase in temperature led to a decrease in shear strength and modulus. The friction angle was found to have a significant effect on the behavior of shear damage. [PERSON] et al. [28] analyzed the shear strength and acoustic emission characteristics of rocks in the variable-angle shear test and the direct shear test. The authors found that, although the main types of microcracks generated by shear damage and compressive damage were the same, the shear damage generated by large-scale microcrack extension activity was more intense. [PERSON] et al. [29] conducted numerical simulations of rock mechanical behavior under triaxial compression and proposed a finite element method based on the Mohr-Coulomb criterion.