## 5 Results and Analysis  
### Fractal Behavior of Particle Size Distribution during the Crushing Process  
Figure 4 shows the particle size distribution curve of each sample at each stress level. In Figure 4, a small number of fine particles were produced due to the crushing in sample T\({}_{1}\) and T\({}_{3}\) with relatively large initial particle size at the minimum stress level of 0.8 MPa. However, for T\({}_{2}\) sample with relatively small initial particle size, this kind of situation occurred when the stress level is 1.6 MPa. For the initially evenly distributed T\({}_{1}\) sample and T\({}_{2}\) sample, with the rise of the stress level, the grain breakage degree intensifies and the samples tend to be unevenly distributed. In Figure 4, at the highest stress level of 204.8 MPa, although the fine grains produced by crushing increased substantially, most of the initial large particles are retained and did not disappear due to the crushing. This may be explained by the fact that, during the crushing process, the big particles are cushioned by smaller particles around them, which enhances the anti-crushing ability of big particles. Therefore, smaller particles are more likely to be broken. [PERSON] et al. [24] observed the buffering effect through test and numerical simulation, and found that, when the large particles were cushioned, their neighboring small particles were crushed continuously. The fractal behavior of particle size distribution can be preliminarily investigated with the particle size distribution curves in Figure 4.  
Is represented by the double logarithmic coordinate axis, which is shown in Figure 5. The shape of the particle size distribution curve was observed. It was found that, at the low stress level, the curve shape is similar to that on the semi-log axis, which is of the concave upward shape. Figure 4: Evolution of particle size distributions (semi-logarithmic axes): (**a**) T1 sample, (**b**) T2 sample and (**c**) T3 sample. However, with the rise of the stress level, the curve turns straight from bending, and the fractal features are gradually apparent. At high stress level(\(\text{\textasciitif{25.6-204.8} MPa}\) for T\({}_{1}\) sample and 51.2-204.8 MPa for T\({}_{2}\) sample and T\({}_{3}\) sample), the particle size distribution curve is appropriately a straight line, and the fractal features are apparent. During the grain crushing process, the transformation of different initial distribution of the soil particles to the fractal distribution is a gradually changing process. The fractal characteristics of the particle size distribution under each stress level can be deeply learned through carrying out linear regression analysis based on the measured particle size distribution data and Equation (1). The results are shown in Figure 6. The fractal dimension \(D\) obtained and corresponding fitted relation coefficient \(R\) are listed in Table 2. The value of \(D\) rises with the growth of grain breakage. The significance level of self-similarity of particle size distribution data also greatly improves. Figure 5: Evolution of particle size distributions (double logarithmic axes): (**a**) T1 sample, (**b**) T2 sample and (**c**) T3 sample. Figure 6: \(Cont.\)  
The self-similarity of particle size means that, although the grain size is different, the grains all maintain the same crushing probability. [PERSON] [8] provided the following explanation: the stress distribution of all particles can be maximized with the fractal distribution of the particles. The probability of breakage is not controlled by the particle's self-strength, but is controlled by the size of its adjacent particles. Once the self-similar fractal distribution is achieved, the particle has the same crushing probability in all directions, so the self-similarity is maintained. However, [PERSON] et al. [25] held that the crushing probability is not only controlled by the size of adjacent particles, but also related to the number of contact points between particles.