## 1 Introduction
established statistical models and general expressions for the failure probability, and on this basis, they determined the SE of quasi-brittle materials through the Weibull and Poisson distribution models, revealing that the defect probability and rock strength increase monotonously with the volume [10]. [PERSON] et al. proposed a numerical simulation method for studying the SE of a macroscopic cross-dimension rock mass [11]. In addition, under different loading conditions, the mechanical properties of rock vary greatly. In a laboratory, different values for the LR can be used to simulate the loading conditions of specimens subjected to different stress environments. The LR is likewise one of the factors affecting rock's mechanical parameters. By conducting experiments on marble with different LRs, [PERSON] et al. found that with an increase in the LR, the rock damage morphology experienced changed the localized shear to complete shear-tension damage and to conical damage [12]. By analyzing the energy evolution mechanism of marble at nine LRs, [PERSON] et al. proposed that the degree of fracturing is positively correlated with the LR [13]. [PERSON] et al. studied the mechanical properties of deep sandstone under different loading rates and revealed the damage mechanism of deep sandstone under different strain rates [14]. They analyzed the relationship between the SE of rock and the different confining stress conditions through numerical simulations, and the strength of the rock increased significantly with increasing confining pressure [15; 16; 17; 18]. Through statistical analysis of previous research results and experimental verification, [PERSON] et al. concluded that the SE for a rock's strength depends on the type and structure of the rock [19]. Most rock formations in coal mines are sedimentary rock. There are still differences of opinion about the SE and LR of sedimentary rocks. [PERSON] proposed that sandstone specimens reach their peak strength in the range of 40-60 mm in height, and the strength will decrease when the height is beyond this range [20]. [PERSON] et al. proposed that the long-term strength values of rock will gradually decrease and plateau as the specimen size increased [15]. [PERSON] et al. used RFPA2D to study the mechanical behavior of carbonaceous mudstone specimen with a HDR of 2:1, and founded that \(\varphi\)150 mm \(\times\) 300 mm was the boundary size of rock strength [21]. [PERSON] et al. conducted mechanical testing experiments on red sandstone and found that the peak strength of the specimen is negatively correlated with the HDR and positively correlated with the LR [22]. [PERSON] et al. experimentally challenged the assumption underlying the Weibull method that the critical crack size increased with specimen size [23]. [PERSON] proposed a brittle dynamic failure model and predicted that the observed dynamic compressive strength would be enhanced with the increase in the LR and specimen size [24]. The surrounding rocks of deep coal mines have undergone extensive geological structural deposition, resulting in a highly complex stress environment. Additionally, the coal and rock mass are relatively incomplete, which complicates the successful drilling of rock and coal specimens with a HDR of 2:1 on site [16; 25; 26; 27; 28]. Furthermore, the mechanical properties of rock and coal specimens drilled from different locations exhibit significant discrepancies. Consequently, recent experimental studies have increasingly focused on the development of rock-like materials in the laboratory to investigate the associated mechanical properties of rock. [PERSON] et al. combined experiments and numerical simulations to investigate the mechanism of micro-crack propagation and coalescence in brittle rock-like materials. They discovered that crack propagation in rock-like materials may resemble that of natural micro-cracks [29]. This was achieved through orthogonal experiments, [PERSON] et al.