## 1 Introduction  
As the world's population and economy continue to expand, the demand for energy and raw materials is on the rise [1], leading to the further development of various mineral mining operations. Rock rupture is an inevitable problem during mineral mining, which poses a potential threat to construction safety. Therefore, understanding the propagation of cracks in rock masses is of great significance to ensure the stability of mining structures. There are many initial cracks of different scales inside the rock mass. These cracks not only alter the mechanical characteristics of the rock but also, under external forces, the propagation and connection of cracks lead to rock failure and ultimately result in engineering accidents [2; 3; 4]. The propagation and interconnection of cracks within the rock mass account for the majority of failure occurrences in rock and soil engineering, such as the landslide that occurred in Yanchihe Phosphorus Mine, Yichang City, Hubei Province, China, in 1980, and the water seepage accident during the construction of Foshan Subway Line 2 in 2018. In recent years, experts have conducted theoretical analysis, numerical simulation experiments, similar material, and mechanical tests on single-cracked and multi-cracked rock specimens, and have achieved rich research results. [PERSON] et al. [5] carried out Brazilian splitting tests on shale disks incorporating pre-existing cracks that oriented at different angles. The findings revealed that the pre-existing cracks have a certain weakening influence on the crack resistance strength of the rock disks. [PERSON] et al. [6] performed uniaxial compression load tests on pre-cracked marble specimens. The results indicated that the cracks not only changed the mechanical properties of the rock but also the size of the rock bridge between the cracks, affecting the morphology of the fracture plane and the failure mode of the rock. Studies such as references [7; 8; 9] have investigated the influence of pre-existing cracks on the mechanical properties of rocks under compression conditions. [PERSON] et al. [10] established a rock fracture stable propagation model based on crack strain, providing a comprehensive depiction of the evolution patterns of crack strain and wing crack length. [PERSON] et al. [11] used acoustic emission (AE), digital image correlation (DIC), and scanning electron microscopy (SEM) techniques to explore the effects of creep loading on the fatigue behavior and acoustic emission characteristics of pre-cracked sandstone during fatigue loading. Due to the difficulty of conducting in situ loading tests on rock specimens containing cracks, similar material tests have become one of the main research methods for studying the mechanism of crack propagation and connection in rocks. [PERSON] et al. [12] used theoretical analysis and similar material methods to study and found that the connection between multiple cracks is an important cause of rock failure. [PERSON] et al. [13] studied the influence of multiple cracks on the compressive strength of rock-like materials by using the method of embedding inserts in cement mortar specimens. [PERSON] [14] used a certain proportion of cement mortar to make disc specimens containing pre-existing cracks for Brazilian splitting tests. By combining particle flow theory and numerical simulation, the study analyzed the influence of a solitary crack on the tensile properties and fracture patterns of the rock. Given the intricate nature of crack propagation and interconnection in rocks, theoretical analysis alone faces significant challenges. Consequently, numerical simulation has become a commonly used research method among scholars [15; 16]. [PERSON] et al. [17] conducted uniaxial compression tests on rock samples containing fissures using Particle Flow Code (PFC) 2D software, and systematically investigated the effects of different fissure lengths and inclinations on the failure modes and crack initiation and propagation of rock-like structures. [PERSON] et al. [18] and [PERSON] et al.