## 5 Results and Analysis
### Fractal Behavior of Particle Size Distribution during the Crushing Process
The theoretical curve of the particle size distribution is predicted based on the value of \(D\) in Table 2 and Equation (1) and a comparison was made between the theoretical data and the measured data of the particle size. The degree of coincidence of the measured data and the strictly self-similar fractal distribution was investigated. It can be seen in Figure 7 that the particle size distribution is represented by the double logarithm axis. Since Equation (1) is a strictly self-similar fractal distribution, the theoretical curve of the particle size distribution is linear. It was found through comparing the measured data and the theoretical curve that, for each test sample at different initial status, the linear relation of the measured data with \(D\) value greater than 2.2 is fairly sound, and is quite consistent with the corresponding theoretical curve (the solid line in Figure 7), so that it has fairly strict self-similar nature. The linear relation of the measured data with \(D\) value smaller than 2.2 goes bad, gradually deviating from the corresponding theoretical curve (the imaginary line in Figure 7), and with the decrease of value \(D\), the deviation is getting serious and serious and the data points become non-linear. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Stress Level**} & \multicolumn{2}{c}{**T\({}_{1}\) Sample**} & \multicolumn{2}{c}{**T\({}_{2}\) Sample**} & \multicolumn{2}{c}{**T\({}_{3}\) Sample**} \\ \cline{2-7}  & \(D\) & \(R\) & \(D\) & \(R\) & \(D\) & \(R\) \\ \hline
0 & - & - & - & - & - & 1.430 & 0.9444 \\
0.8 & 1.219 & 0.9353 & 0.898 & 0.9333 & 1.220 & 0.9549 \\
1.6 & 1.229 & 0.9667 & 0.919 & 0.9739 & 1.298 & 0.9669 \\
3.2 & 1.430 & 0.9792 & 1.066 & 0.9742 & 1.509 & 0.9777 \\
6.4 & 2.048 & 0.9841 & 1.222 & 0.9578 & 1.931 & 0.9672 \\
12.8 & 2.158 & 0.9948 & 1.975 & 0.9766 & 2.039 & 0.9814 \\
25.6 & 2.231 & 0.9984 & 2.088 & 0.9908 & 2.101 & 0.9891 \\
51.2 & 2.305 & 0.9989 & 2.222 & 0.9976 & 2.272 & 0.9977 \\
102.4 & 2.377 & 0.9983 & 2.311 & 0.9977 & 2.331 & 0.9985 \\
204.8 & 2.432 & 0.9977 & 2.360 & 0.9983 & 2.399 & 0.9963 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of fractal dimension and regression coefficient. From the above analysis, the following can be obtained: the significance level and fractal dimension of the self-similarity of the measured data of the particle size distribution are very sensitive to the accuracy of the prediction results in Equation (1). The \(D\) value corresponding to the \(R\) value with relative low significance level will lead to a great error between the measured data of particle size distribution and theoretical distribution. Therefore, in the study of the grain breakage, when the initial uniform particle size distribution is described as fractal distribution, the significance level of the self-similarity of particle size distribution should be tested first to determine whether the particle size distribution has finished the transformation to fractal distribution. In addition, for the model prediction of particle size distribution curve, it is necessary to adopt the fractal parameters with relatively strict self-similarity to avoid large error of the prediction results.