## 1 Introduction  
With the increasing demand for mineral resources and the booming development of infrastructure projects for high-speed and convenient transportation, excavation projects such as tunnels [1], mineral mining [2], and pipelines [3] have become increasingly frequent. Such construction projects usually occur in complex rock environments, where large rock masses are the main blasting objects for easy excavation and mining [4]. In a wide range of construction projects, the drilling and blasting method has become the main construction method to effectively control post-blasting disasters and provide effective guarantees for construction safety and quality. However, due to geological activities and induced excavation, the presence of joint planes [5] and cracks in large rock masses is common [6], as shown in Figure 1. In the Figure, the red arrows represent the load generated by the impact of blasting, the black lines represent the naturally occurring joints and cracks in the rock structure, and the white round holes represent the gun holes. These pre-existing cracks have a significant impact on the stability of the overall structure. In addition, under external interference such as blasting, new cracks occur and expand around the blast holes and pre-existing cracks easily, ultimately leading to overall structural instability, seriously delaying work and production progress, threatening the safety of relevant personnel, and causing incalculable economic losses. Therefore, studying the impact of cracks in rocks on blasting has a profound significance on safe construction. In the past, most studies on excavation or mining have adopted the assumption of rock models without cracks. The advantage of this approach is that fewer factors affect the generation and propagation of cracks, making it easier to simulate. However, in recent years, more and more scholars have conducted research on this type of model with cracks. [PERSON] et al. [7] stated that the presence of faults in rocks is an important factor that leads to overall instability. [PERSON] [8; 9; 10] systematically studied the crack growth behavior under various dynamic loads and obtained some analytical results and conclusions, which confirmed that when subjected to blasting load, the cracks around the blast hole are more likely to generate new growth at the tip. However, because this process is affected by a complex environment, it is difficult to obtain a sufficiently accurate analytical expression. Moreover, under high-speed dynamic loads such as explosions, the interaction between the crack's tip and the dynamic stress field causes fluctuations in the crack propagation speed at the crack's tip [11]. Therefore, in response to the reasons that affect the initiation of tip cracks, some scholars [12] have studied the process of stress wave propagation and found that when the stress wave propagates to a static crack with two tips, it will scatter near the perturbed tip and reflect on the crack surface, forming a superimposed stress field, which is usually related to the incident angle. Additionally, based on such a model, [PERSON] gives some analytical expressions for this field by using integral transform [13]. Regarding rock material models with cracks, some experimental explosion simulation studies [14] or theoretical summaries of rock materials [15] are based on the existence of a large number of natural structural joints or faults in the rock itself. Based on an experimental analysis and numerical simulation, [PERSON] [16] completed a study on the crack diffraction process using P-wave as an example and determined the influence of joint space and other factors on wave propagation. [PERSON] [17] discussed the slope failure under the action of the Rayleigh wave, and the stress wave theory was adopted to analyze the model. [PERSON] and [PERSON] used finite element models to study the effect of stress wave pairs on crack angles in the cited reference [18]. [PERSON] et al.