# Study of the Internal Rebreaking Characteristics of Crushed Gangue in Mine Goal during Compression
## 3 Results
### Confined Compression Deformation Characteristics of Fractured Gangues with Different Lithologies
#### 3.1.3 Fractal Dimension  
Many studies have shown that the particle size distribution of crushed gangue mass has a fractal structure. Fractal theory is widely used in the study of particle size distribution and the grading evolution characteristics of crushed gangue, which can comprehensively reflect the particle size distribution and fragmentation of crushed gangue before and after loading [49]. The fractal dimension, expressed by \(D\), is the representation of the complexity of the composition of a material and how it is self-organized. Under the action of a certain stress level, the particles in the crushed gangue sample move and slide, which changes its spatial arrangement and pore distribution. Because the cylinder used in the test had a large stiffness, the radial deformation of crushed particles was not considered in the loading process [50]. According to fractal theory in gangue crushing, the total mass of the crushed gangue is set as \(M_{t}\), where the mass of particles smaller than the particle size di is \(M_{1}(d_{i})\), and the mass of particles larger than the particle size \(d_{i}\) is \(M_{2}(d_{i})\), where  
\[M_{t}=M_{1}(d_{i})+M_{2}(d_{i}) \tag{1}\]  
The relationship between the volume \(V\) and the fractal dimension \(D\) of the fractured gangue in mine goaf is as follows:  
\[V=C_{v}\left[1-\left(\frac{d_{i}}{\lambda_{v}}\right)^{3-D}\right] \tag{2}\]  
where \(C_{v}\) and \(\lambda_{V}\) are constants, and their size depends on the shape and distribution of the crushed gangue particle size. According to [PERSON]'s assumptions [51], if the density of each particle in the crushed gangue is the same, then  
\[M_{2}(d_{i})=\rho C_{v}\left[1-\left(\frac{d_{i}}{\lambda_{v}}\right)^{3-D}\right] \tag{3}\]  
From the above formula, when \(d_{i}\) = 0, there is  
\[M_{t}=M_{2}(d_{i})=\rho C_{v} \tag{4}\]  
When \(d_{i}\) = \(d_{\max}\) and \(M_{2}(d_{\max})\) = 0, substituting Equation (3), we can obtain  
\[\lambda_{v}=d_{\max} \tag{5}\]  
From Formula (2) to Formula (6), the fractal relationship between the particle quality and particle size of crushed gangue can be obtained:  
\[M_{1}(d_{i})/M_{t}=[1-M_{2}(d_{i})]/M_{t}=(d_{i}/d_{\max})^{3-D} \tag{6}\]  
The logarithm of the two sides of the equation can also be obtained:  
\[\lg[M_{1}(d_{i})/M_{t}]=(3-D)\lg(d_{i}/d_{\max})=\lg(d_{i}/d_{\max}) \tag{7}\]  
The relationship between the linear slope and fractal dimension is seen from Equation (8). The crushed gangues before and after compaction were screened and weighed. The logarithmic relationship between the mass ratio and particle size ratio was calculated according to Equation (8). The different combinations of crushed gangue were graded by particle size. All gangue samples were sieved to the particle size ranges of 0-2, 2-5, 5-10, 10-15, 15-20, 20-25, 25-30, 30-35, and 35-40. The gangue samples in each particle size range were weighed and the mass proportion was calculated. Each particle size range contained sandstone and sandy mudstone. The gangue samples of single lithology and combined lithology after crushing were compared. The proportion of sandy mudstone in the large range (35-40 mm) gradually decreased after compression. It further showed that the high-strength gangue block was mainly skeleton-bearing, while the low-intensity gangue block was mainly pore-filling for the combined gangue sample. According to the gradient of the linear equation and the fractal dimension, the fractal dimension of single and combined gangues under  
Figure 7 shows that the fractal dimension increases exponentially with the variation in different lithologies. The fitted result is  
\[ \begin{array}{l}\frac{y=-0.06767\exp(-x/0.40344)+2.34467}{R^{2}=0.99885}\end{array} \]  
In the process of loading and compression, the breakup of the composite gangue samples increased with the increase in sandy mudstone, with the rate gradually decreasing.