## 6 Conclusions  
In this paper, the uniaxial side confined compression test under high stress condition was carried out to study the evolutionary rule and fractal behavior of the particle size distribution of the rare earth tailings during the crushing process at different initial state. The following six points are concluded:  
1. With the development of particle breakage, the quartz sand-gravel with different initial distribution and particle size all tends towards self-similar fractal distribution and this process is related to the breakage amount of the particles.
2. The particle breakage growth can be described using the increased fractal dimension. For the particles of the rare earth tailings at different initial status, the measured data of the particle size distribution with the fractal dimension greater than 2.2 all show fairly strict self-similarity, so that this value can be taken as the lower limit value of the fractal dimension when the particle size distribution tends towards fractal distribution.
3. The significance level of the self-similarity of the measured data of the particle size distribution are very sensitive to the prediction results of the particle size distribution model. The particle size distribution predicted by the fairly strict self-similar fractal parameters are quite consistent with the actually measured data.
4. In the same breakage state, the breakage rate defined by Hardin index is smaller than that defined by the Einav index. However, the two kinds of breakage rate are consistent in terms of the variation rule with the vertical stress and volumetric stress.
5. The ratio of the volumetric strain to the breakage rate remains constant, and is slightly affected by the evenness of the initial distribution and particle size when the particle size distribution tends towards fractal distribution. This feature can be utilized to identify the fractal distribution. It also means that, if the particle size distribution is fractal in the experiment, there is no need to terminate the test to carry out particle size analysis, and the particle breakage rate can be estimated only through measuring the volumetric strain.
6. Under the condition of high stress, the particle crushing process of rare earth tailings tends to be of fractal distribution, which provides a possibility for the model prediction of particle size distribution. However, when the particle size distribution of the particle breakage is described as fractal distribution, the degree of particle breakage should be examined to confirm whether it can facilitate the transition to fractal distribution and fairly strict self-similar fractal parameters shall be adopted when carrying out model prediction to avoid relatively large error of the predicted results. Further research needs to be done to learn the fractal behavior of other types of soil samples in the crushing process, and to clarify the effect of different soil sample properties on the fractal behavior of particle size distribution.  
In fact, it is difficult to perform tests using higher stress levels than 204.8 MPa due to the limitation of the instrument. Our future work will focus on the high stress levels.  
Conceptualization, W.Z.; Data curation, F.Y.; Formal analysis, F.Y.; Funding acquisition, W.Z.; Project administration, W.Z.; Supervision, A.C.; Writing--original draft, and F.Y.; Writing--review and editing, A.C.  
The authors gratefully acknowledge the financial support from the Project supported by the National Natural Science foundation of China (Nos. 51504102, 51764014, and 51404111), China Postdoctoral Science Foundation (No. 2017M622099), National key technologies Research & Development program (No. 2017 YFC0804601), and Program of Qingjiang Excellent Young Talents, Jiangxi University of Science and Technology.  
The authors declare no conflicts of interest.