## 4 Numerical Simulation Analysis
### Effective Stress Analysis of Specific Element  
In the numerical simulation analysis, the stress experienced by a particular element in the numerical model is primarily described by analyzing its von Mises equivalent stress, which is used to analyze the failure characteristics of rock blasting. When the peak effective stress experienced by the selected element exceeds the dynamic tensile strength of the rock, the rock element is considered damaged [39]. Based on the numerical simulation model presented in the previous section, the effective stresses of the selected characteristic elements were output and analyzed using the LS-PREPOST 4.10 post-processing software to investigate the influence of different inter-hole delay times during pre-splitting blasting on stress propagation. The model was cut in half along the Z-axis, and four characteristic elements that were symmetrically distributed along the centerline connecting the blast holes were selected. During sequential detonation, the rightmost hole (Hole 1) was the first to detonate. The selected characteristic elements were A (H709667), B (H709399), C (H512737), and D (H512469), which were chosen sequentially from right to left. The locations of the selected characteristic elements are shown in Figure 12, and the time-history curves of the effective stresses for these elements are presented in Figure 13.  
Figure 12: Position diagram of feature unit selection. Figure 13: Effective stress time–history curve of characteristic elements. As shown in Figure 13, under the simultaneous detonation method, since the four selected characteristic elements are located near the midpoint of the line connecting the blast holes and are symmetrically distributed, the detonation wave propagates outward from the center of the explosive package after detonation, acting on the rock wall near the blast holes and forming an initial stress wave. The time taken for the explosion stress wave to reach these four characteristic elements is almost the same, resulting in similar time-history curves for the effective stress of the four selected elements. After the explosive detonates, the explosion shock wave propagates rapidly, and stress begins to appear in elements A, B, C, and D at around 0.2 ms. The effective stress quickly reaches a peak at around 0.4 ms, with peak values of approximately 50 MPa for all four elements. This peak value is less than the dynamic compressive strength of the rock but greater than its dynamic tensile strength, indicating that the rock elements are damaged under tensile stress. Subsequently, the effective stress decreases and levels off. As shown in Figure 13b, under the sequential detonation method, the time-history curves of the effective stress for the four selected characteristic elements exhibit a \"step-like\" pattern, with similar curves for adjacent elements. After the detonation of the first hole (rightmost), elements A and B, being closer to the explosion source compared to elements C and D, reach their first peak effective stress at around 0.4 ms, while elements C and D reach their first peak at around 0.9 ms. Due to their greater distance from the explosion source at this time, the change in the effective stress for elements C and D is smaller. As the explosion stress wave propagates, its stress intensity decreases as the distance increases. Therefore, the peak effective stress of the selected elements is inversely proportional to their distance from the explosion source, with larger peaks closer to the source and smaller peaks farther away. The peak effective stresses are as follows: element A (29 MPa) > element B (24.3 MPa) > element C (5.91 MPa) > element D (3.94 MPa). At this point, the peak effective stresses of elements A and B exceed the dynamic tensile strength of the rock, causing damage to these elements; meanwhile, elements C and D remain undamaged as their peak stresses are below the dynamic tensile strength. Subsequently, the effective stress decreases and levels off. At 12 ms, the middle blast hole detonates.