## 1 Introduction  
Some rock engineering activities, including coal mining [1; 2; 3], underground chamber excavation [4; 5] and shale gas exploitation [6; 7], generally lead to new rock fractures. Under external loads, the rock mass often begins to crack from the defect position until failure occurs [8; 9; 10; 11]. Therefore, it is of great significance to study the crack initiation characteristics and energy evolution of fractured rock mass to ensure the stability of rock mass engineering structure. To study the crack initiation characteristics of fractured rock, researchers have carried out a large number of laboratory tests, and these tests focus on single flaw [12; 13; 14; 15; 16], parallel double flaws [17; 18; 19; 20] and non-parallel double flaws [21; 22; 23]. For example, through the high-speed photography and acoustic emission (AE) technology, [PERSON] et al. [14] studied the fracture instability process of sandstone with a single flaw and analyzed the effects of flaw inclination angle, length and rock bridge dip angle on crack initiation, crack propagation and the mechanical properties of rock mass. Through the uniaxial compression experiments on Carrara marble with flaws, [PERSON] et al. [16] observed the crack formation by a high-speed camera, analyzed the microscale-macroscale phenomena related to the fracture process of the sample, and distinguished the shear crack and tensile crack. [PERSON] et al. [19] used a high-speed camera to record the compression failure process of granite sampleswith prefabricated parallel double flaws, and analyzed the influence of the geometry of prefabricated flaws on the crack initiation mode. [PERSON] et al. [22] analyzed the influence of flaw inclination angle, flaw length and rock bridge length on crack initiation position and crack propagation with the help of the high-speed camera. Most scholars use high-speed cameras and AE technology to the study cracking mechanisms of rock material with flaws. These methods analyze the influence of crack geometry and external experimental conditions on the crack initiation mode [24; 25; 26], and summarize the crack initiation characteristics of fractured rocks (including cracking position, cracking angle and extending direction and crack types) [27; 28; 29]. However, because it is difficult to directly measure of stress field inside the sample, most of the studies on the mechanism of crack initiation remain at the macroscopic level. Compared with laboratory tests, numerical simulation can analyze the crack initiation characteristics from the meso point of view. [PERSON] et al. [30] proposed a discrete element method (DEM), and simulated rock-like or rock materials by using particles and the contact between particles, and PFC\({}^{\text{2D}}\) was developed based on particle discrete element method. Unlike the finite element method, DEM can avoid selecting sophisticated crack initiation criteria and more accurately depict the sample's fracture processes [31; 32]. Compared with laboratory experiments, the fracture processes involving crack initiation, coalescence, and propagation of rock can be clearly demonstrated by the breakage of inter-particle bonds in DEM [33], and all numerical data, including those that are difficult or impossible to get directly from laboratory experiments, such as inter-particle contact forces, can be obtained at any stage of simulation [34]. For instance, [PERSON] et al. [35] carried out uniaxial compression tests on red sandstone samples with two non-parallel flaws by using PFC\({}^{\text{2D}}\) based on laboratory tests, and revealed the crack evolution mechanism of brittle sandstone samples with two non-parallel flaws through force chain evolution analysis. However, they only performed qualitative analysis and the verification of physical experiments. In recent years, some scholars have paid more attention to the quantitative analysis of crack initiation characteristics of fractured rocks by using stress and displacement fields. [PERSON] et al.