## 4 Discussions
### Three-Dimensional Fractal Dimension and Damage Analysis of Internal Fractures in Rock Samples  
The fractal dimension, as a geometric method for describing the complexity of objects, can be used to quantitatively characterize the complexity of internal fissures in rock samples [32]. The principle for calculating the fractal dimension in Avizo 2020.1 software is as follows:  
For an object with fractal characteristics, divide it into n equal units. The number of units, unit scale, and the number of units at different scales have relationships as described by Equation (6) [33]:  
\[N(\delta)=\delta^{-D}, \tag{6}\]  
where \(\delta\) represents the smallest scale after dividing the object n times; \(N(\delta)\) is the total number of units at the smallest scale \(\delta\); and \(D\) is the fractal dimension, which corresponds to the number of units at different scales. For the area \(A\) of the remaining fractal body after division, it is equal to the product of the area at the unit scale and the number of units at that scale, as given by Equation (7):  
\[A=N(\delta)\cdot\delta^{2}, \tag{7}\]  
By substituting Equation (6) into Equation (7), we obtain Equation (8):  
\[A=\delta^{2-D}, \tag{8}\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Sample State** & **Total Fissure Volume (nm3)** & **The Volume Proportion of** & **The Volume Proportion of** & **Maximum Through** \\  & **Volume (nm3)** & **Therough Fissure (\%)** & **Independent Fissure (\%)** & **Fissure Length (nm)** \\ \hline After impact at a confining & 39,474,100 & 95\% & 5\% & 76,101,952 \\   pressure of 5 MPa & 7,961,934 & 73\% & 27\% & 54,654,324 \\   pressure of 10 MPa & 4,052,681 & 32\% & 68\% & 51,974,100 \\   after impact at a confining & 2,947,010 & 17\% & 83\% & 36,900,516 \\   pressure of 20 MPa & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical change in parameters of rock samples in different states after impact. Equation (8) illustrates the relationship between the smallest scale \(\delta\), fractal area \(A\), and fractal dimension \(D\). Therefore, \(D\) represents the area fractal dimension. Extending this equation to three-dimensional space yields Equation (9):  
\[V=\delta^{3-D_{v}}, \tag{9}\]  
In Equation (9), \(V\) denotes the three-dimensional volume, and \(D_{v}\) is the three-dimensional fractal dimension. In [PERSON]'s geometry, straight lines and curves are one-dimensional, planes and spheres are two-dimensional, and forms whose length, width, and height all exist are three-dimensional. But for complex forms, their dimensions are not described by integers like one, two, or three. This problem was solved by the advent of the Koch curve and the Menge sponge. The Koch curve occupies a limited area with its unlimited length, which is bigger than one dimension and smaller than two, i.e., its dimension is bigger than one and smaller than two. The Menge sponge is similar in principle; the surface area of the hole inside the Menge sponge is unlimited, but it occupies a limited three-dimensional space, so its dimension is between two and three [34]. From the above analysis, it is known that the three-dimensional fractal dimension is between two and three, a higher fractal dimension indicates greater complexity in the rock's fissures, while a lower dimension suggests simpler fissure structures. The use of fractal dimensions to analyze the extent of damage within a rock sample is a common method based on [PERSON]'s definition [35] of damage:  
\[w=\frac{D_{v}-D_{0}}{D_{v}^{max}-D_{0}}, \tag{10}\]  
In Equation (10), \(w\) is the degree of damage; \(D_{v}\) is the fractal dimension of the rock sample after impact; \(D_{0}\) is the fractal dimension of the rock sample before impact; and \(D_{v}^{max}\) is the maximum value of the fractal dimension of the rock sample after impact for the three-dimensional problem \(D_{v}^{max}=3\).