## 4 Discussion
### Effect of Fissure Penetration on Rock Fragment Characteristics  
Fractal theory offers a fresh approach to studying crack propagation and damage evolution in rocks and is extensively applied in mining engineering. In this study, to examine the rock chips produced during the sidewall failure of rectangular tunnels under varying fissure penetration conditions, fractal theory was applied to investigate the fine particles and chips generated. Based on fractal theory [41], the mass ratio of cuttings with a diameter, r, smaller than R to the total mass generated during failure is given by Equation (2):  
\[\frac{\mathrm{M}(\mathrm{r}<\mathrm{R})}{\mathrm{M}_{\mathrm{T}}}=(\frac{ \mathrm{R}}{\mathrm{R}_{\mathrm{L}}})^{3-\mathrm{D}} \tag{2}\]  
Taking the natural logarithm of both sides of Equation (2) yields Equation (3):  
\[\ln(\frac{\mathrm{M}(\mathrm{r}<\mathrm{R})}{\mathrm{M}_{\mathrm{T}}})=(3- \mathrm{D})\ln(\frac{\mathrm{R}}{\mathrm{R}_{\mathrm{L}}}) \tag{3}\]  
where r represents the particle size of the rock particles, R denotes the selected sieve size, D is the fractal dimension, M is the mass of particles smaller than R, M\({}_{\mathrm{T}}\) is the total mass of  
Figure 19: Failure of rectangular holes under 62 MPa vertical stress: (**a**) S-0-0, (**b**) S-0-25, (**c**) S-0-50.  
Figure 18: Failure of rectangular holes under 50 MPa vertical stress: (**a**) S-0-0, (**b**) S-0-25, (**c**) S-0-50, (**d**) S-0-100.  
the cuttings, and R\({}_{\text{L}}\) is the diameter of the largest sieve. Initially, rock chips generated from sidewall failure in various specimens were passed through sieves of different sizes. The sieved chips are shown in Figure 20. The mass of chips in each size range was measured, and the results are provided in Table 5.  
From the figure, it can be seen that as fissure penetration increased, rock slices with particle sizes greater than 8.00 mm changed from elongated strips to block-like shapes, with a decreasing length-to-width ratio. When the particle size was less than 8.00 mm, the rock chips were primarily block-like, with little difference in length and width. According to the video monitoring, larger rock blocks mainly originated from the surface of the sidewall during the first stage of plate fissure failure. As plate fissure failure progressed, the particle size of the rock chips decreased, with chips in the 4.75 mm-to-8.00 mm range coming from  
Figure 20: Distribution of rock fragments: (**a**) S-0-0, (**b**) S-0-25, (**c**) S-0-50, (**d**) S-0-100.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\begin{tabular}{c} **R** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **R\({}_{\text{L}}\)** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **M (r < R)** \\ **(g)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **M  
deeper rock layers. The smallest chips mainly originated from the deeper sections of the V-shaped trough.  
Comparing the experimental results for various fissure penetration levels, the rectangular hole of specimen \"S-0-0\" exhibited the most severe failure, producing the largest quantity and mass of rock blocks. As fissure penetration increased, the failure degree and the quantity and quality of rock chips produced by the sidewalls gradually decreased.  
By applying Equation (3) to the data in Table 5, the fractal dimension D was calculated for each fissure-penetration condition, with the fitting results shown in Figure 21. From Equation (3), the slope of the linear fit is 3\(-\)D and the intercept is 0. The fractal dimensions for specimens \"S-0-0\", \"S-0-25\", \"S-0-50\", and \"S-0-100\" were 2.3382, 2.2405, 2.1161, and 2.2905, respectively. This indicates that fractal dimension D initially increased and then decreased with increasing fissure penetration.