## 5 Results and Analysis
### Stress Strait and Relative Breakage Rate in the Crushing Process  
According to the previous analysis, the particle size distribution when the \(D\) value is larger than 2.2 is taken as the boundary for fractal distribution and the mutual impact mechanism between the fractal distribution and \(B_{r}\), \(\sigma_{v}\) and \(\epsilon_{v}\) during the evolutionary process of particle size distribution is studied. Among them, \(B_{r}\) is analyzed separately as per Hardin index and Einav index. Figure 8 shows the impact of relative breakage rate on fractal dimension. Under the same breakage state, the breakage amount described by Hardin indexes is smaller than that described by Einav indexes. The value of \(D\) of each sample grows with the rise of \(B_{r}\). When the breakage develops to a certain degree, fractal distribution occurs (above the imaginary line in Figure 8). When the initial distribution is different, the amount of breakage needed to evolve into fractal distribution is also different. The T\({}_{1}\) sample with relatively large initial particle requires larger breakage amount compared with T\({}_{2}\) sample  
Figure 7: Comparison between results of Equation(1) and measured data of particle size distribution: (a) T\({}_{1}\) sample, (b) T\({}_{2}\) sample and (c) T\({}_{3}\) sample  
with relatively small particles, and the T\({}_{3}\) sample with sound grain size distribution requires a smaller amount of breakage amount compared with T\({}_{1}\) and T\({}_{2}\) with poor grain size distribution. When the particle distribution evolves to fractal distribution, \(D\) increases linearly with \(B_{r}\). Figure 9 indicates the impact of stress level on the relative breakage rate. Under the same stress level, \(B_{r}\) is affected by the initial distribution and is related to the critical crushing stress of particles. The critical stress of the sample with relatively large initial grains and poor grain size distribution is relatively small, so \(B_{r}\) is relatively large. The critical stress of the sample with relatively small initial grains and sound grain size distribution is relatively large, so \(B_{r}\) is relatively small. Figure 9 also indicates that \(B_{r}\) increases with the rise of \(\sigma_{v}\), but the amplification is decreasing. The \(B_{r}\)-\(\sigma_{v}\) curve presents an inferior fovea shape. Therefore, the following explanation could be provided: with the rise of \(\sigma_{v}\), the particle number caused by breakage is increasing constantly and the contact point between particles is also increasing accordingly. Although the total stress imposed on the whole particle is still increasing, the average stress of the contact points between the particles actually declined. Therefore, the amplification of \(B_{r}\) becomes smaller and smaller with the rise of \(\sigma_{v}\). However, at the fractal stage, the amplification of \(B_{r}\) of the samples becomes fairly consistent with the growth of \(\sigma_{v}\), suggesting that the particle breakage is slightly affected by the initial distribution. Figure 10 shows the effect of volumetric strain on relative breakage rate. Different from the \(B_{r}\)-\(\sigma_{v}\) relationship, when the grain breakage develops to a certain degree, especially at the fractal distribution stage when the grain breakage terminates and the stabilized gradation is achieved, \(B_{r}\) increases linearly with the rise of \(\varepsilon_{v}\) appropriately and is slightly affected by the initial distribution and initial particle size. It can be seen in Figures 9 and 10 that, although the breakage rate of the same breakage state  
Figure 8: Influence of relative breakage on fractal dimension. Figure 9: Influence of stress level on relative breakage. is different when it is described using Einav index and Hardin index and the Einav breakage rate is larger than Hardin breakage rate, the rule of change of the rate of breakage described by the two with the vertical stress and volumetric strain is consistent.