## 4 Correlation between The Fractal Dimension and Compaction Characteristics  
Graded crushed stone is a special mixture of crushed grains and its fractal dimension is related to the degree of crushing. [PERSON] [35] proposed the following expression between the fractal dimension of particle breakage and the degree of breakage:  
\[D=3-\frac{\log f}{\log d} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Sample** \\ **No.** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Measured** \\ **Water** \\ **Content/\%** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Measured DRY** \\ **Density/(g\(\cdot\)cm\({}^{-3}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Theoretical** \\ **Fractal** \\ **Dimension** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Measured** \\ **Fractal** \\ **Dimension** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Deviation of** \\ **Fractal** \\ **Dimension** \\ \end{tabular} } \\ \hline
6 & 1.82 & 2.153 & 2.4290 & 2.4024 & \(-\)0.0066 \\
7 & 3.68 & 2.219 & 2.4727 & 2.4831 & 0.0154 \\
8 & 5.44 & 2.267 & 2.5045 & 2.5103 & 0.0158 \\
9 & 6.32 & 2.242 & 2.4879 & 2.4805 & \(-\)0.0114 \\
10 & 7.66 & 2.207 & 2.4648 & 2.4616 & \(-\)0.0032 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Correlation validation results of fractal dimension with dry density.  
Figure 10: Correlation curve of fractal dimension and dry density.  
where \(f=N\alpha^{3}\) is the probability of particle breakage; \(N\) is the number of particles after breakage; \(\alpha\) is the reciprocal of the ratio of the particle size before and after particle breakage; \(d\) is the particle size after breakage.  
Equation (8) indicates that as the fragmentation probability of the particles increases, the fractal dimension of the particle distribution after fragmentation also increases. When parameters are set to \(f=\alpha^{3}\) and \(D=0\), the whole particle is void so these values are not physical. When \(f=1\) and \(D=3\), there is no void in the whole particle and the parameters are not physical again. Accordingly, the fractal dimension of the particle size distribution of the graded crushed stone varies in the range of 0 to 3.  
When the solid particles are completely broken, the fractional dimension of particle size distribution approaches a constant value, which is generally 2.6. Moreover, the index of stone grading could be evaluated using the fractal dimension. The corresponding fractal dimension in an appropriate gradation varies in the range of 2.22-2.63, and the fractal dimension corresponding to the optimal grading is generally 2.55-2.60. In this study, the fractal dimension of the particle size distribution of graded crushed stone was 2.33-2.57, and the fractal dimension increased with an increase in repeated compaction times, which was consistent with the foregoing analyses.  
Based on the performed experimental analyses, the maximum fractal dimension of the particle size distribution of graded crushed stone was 2.60. The theoretical maximum dry density of graded crushed stone samples was 2.411 g\(\cdot\)cm\({}^{-3}\) and the corresponding optimized water content was 6.02%. When the graded crushed stone reaches the theoretical maximum dry density, the ratio of the particle size of crushed stone can be calculated using Equation (3).  
In summary, the fractal dimension of particle size distribution effectively reflects the compaction characteristics of graded crushed stone. As the fractal dimension of particle size distribution increases, the theoretical maximum dry density of graded crushed stone increases, and graded crushed stone becomes more compact under optimized water content, thereby improving the working conditions.