## 3 Fracture and Damage Analysis of Model Experiments
### Damage Characteristic Analysis  
Based on the binarized image of the blast-induced crack in the PMMA plate, the fractal dimension of the crack distribution of each layer is calculated using the box counting dimension method. According to the basic principle of the box counting dimension, the fractal dimension of any non-empty bounded target set can be expressed as follows [22]:  
\[D=\underset{k\rightarrow\infty}{lim}\frac{\lg N(\delta_{k})}{-\lg\delta_{k}} \tag{1}\]  
where \(D\) is the fractal dimension; \(\delta_{k}\) is a decreasing sequence with the square grid side length as the element; and \(N(\delta_{k})\) is the minimum number of grids required to cover the target geometry with a square grid with a side length \(\delta_{k}\).  
Figure 7 shows the fractal dimension calculation of the crack distribution of PMMA plates at partial layers. There are pronounced differences in the fractal dimension of cracks in different layers. In the quantitative analysis of rock blasting damage, the fractal dimension can be used to characterize the damage degree. The corresponding relationship between fractal damage \(\omega\) and crack fractal dimension \(D\) under the blasting load can be expressed as follows [23]:  
\[\omega=\frac{D-D_{0}}{D_{max}-D_{0}} \tag{2}\]  
where \(\omega\) is the fractal damage under blast loading; \(D\) is the fractal dimension of blast-induced cracks after blasting; \(D_{0}\) is the fractal dimension before blasting; and \(D_{\max}\) is the  
Figure 6: Changing curve of the average length of directional cracks with the layer number.  
fractal dimension under complete damage and destruction. For two-dimensional plane problems, \(D_{\text{max}}=2\); for three-dimensional problems, \(D_{\text{max}}=3\).  
It should be noted that for this model experiment, the PMMA plate is homogeneous and non-damaged before blasting, so \(D_{0}=0\), \(\omega=0\). According to the above calculation results, the fractal dimension and the damage value of each layer of the specimen shown in Table 2 are obtained, and the changing curve of the damage value with the layer number shown in Figure 8 is further drawn. It can be observed in the figure that the damage values of the first and fourteenth layers (non-charging, no slit) are the smallest, and the damage value of the second layer (detonation position) is the largest. The damage value gradually decreases with the increase in the layer number. That is, in the layered rock mass, the blasting damage shows a gradually decreasing variation along the detonation direction. This is basically consistent with the distribution law of the directional crack propagation length along the detonation direction obtained by the previous analysis. The crack distribution and damage fracture in the layered rock mass are significantly different from that in the unlayered rock mass.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Layer Number** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **6 th** & **7 th** \\ \hline Fractal dimension & 1.049 & 1.369 & 1.324 & 1.294 & 1.245 & 1.245 & 1.274 \\ Damage value & 0.525 & 0.685 & 0.662 & 0.647 & 0.623 & 0.623 & 0.637 \\
**Layer number** & **8 th** & **9 th** & **10 th** & **11 th** & **12 th** & **13 th** & **14 th** \\ Fractal dimension & 1.262 & 1.225 & 1.252 & 1.251 & 1.193 & 1.206 & 1.012 \\ Damage value & 0.631 & 0.613 & 0.626 & 0.626 & 0.597 & 0.603 & 0.506 \\ \hline \end{tabular}
\end{table}
Table 2: Fractal dimension and damage value of each layer of the specimen.  
Figure 7: Fractal dimension calculation of the crack distribution of PMMA plates at partial layers: (**a**) second layer; (**b**) seventh layer; (**c**) eighth layer; (**d**) thirteenth layer.