# Numerical Simulation of Rockfill Materials Based on Fractal Theory
## 1 Introduction  
Rockfill materials have been used as a main filling material for earth-rockfill dams, especially with the construction of high earth-rock dams in recent years. Thus, their functional characteristics need to be acquired accurately. In practical engineering, the particle size of rockfill materials is 400-600 mm and even reaches more than 1000 mm. Given the limitations of laboratory test instruments, rockfill materials that exceed the permissible maximum particle size needs to be scaled. To determine the mechanical properties of rockfill materials on site, analog simulation is adopted in laboratory tests to fabricate test specimens that are consistent with the prototype rockfill materials in terms of the internal structure, thereby determining the engineering features of the prototype rockfill materials. Thus far, many scholars at home and abroad have conducted extensive research on factors that influence the size effect of rockfill materials from the aspects of scale method [1; 2; 3; 4], specimen size [5; 6; 7], particle shapes [8; 9; 10], and scale ratio [11; 12; 13]. Their research methods are still dominated by conventional triaxial tests [2; 3; 4; 6; 7; 8], and laboratory tests are limited by test conditions and costs, as well as difficulties in analyzing changes in the internal mesoscopic mechanisms of rockfill materials. Numerical simulation can be used to compensate for the shortcomings of laboratory tests, allowing for the real-time monitoring of the law of evolution of rockfill materials in the internal crack, energy and mesostructured [14; 15]. Since its introduction [16], fractal theory has made some valuable achievements in the aspects of rock fracture [17], soil particle morphology [18; 19], and rockfill materials breakage [20; 21; 22]. Fractal geometry theory was drawn into the gradation of rock-fill materials. It is found that the truncation error is one of the main factors affecting the density scale effect of rockfill materials [23]. [PERSON] et al. [20] studied the correspondence between the relative density (RD) of rockfill materials and the fractal dimension based on fractal theory and explained the four common scale methods. [PERSON] et al. [24] found that, with particles being crushed increasingly, the initially distributed particles show similar fractal distributions. [PERSON] and [PERSON] [25] selected two methods for calculating the fractal dimension and explored the relations among the fractal dimension and the size, shape, and properties of particles. [PERSON] et al. [26] found that the fractal dimension increases with the confining pressure in the test specimen, and the fractal dimension depends on the particle size distribution. [PERSON] et al. [27] observed crushed particles of a rock fracture zone under an optical microscope and found that the fractal dimension that was obtained on site was apparently higher than those measured in laboratory tests but close to the theoretical fractal dimension. The above findings are mainly concentrated in the aspects of fractal dimension and particle breakage, scale methods, and mechanical properties, and the research techniques are mostly based on [PERSON]'s fractal model for mass distribution [19]. At present, most scholars have studied the mechanical properties of rockfill materials using the quality fractal model proposed by [PERSON], but the fractal model is not reported with the number of particles between different particle groups as the statistical number in the newspaper. In this paper, with the use of the particle flow code in two dimensions and on the basis of fractal theory, a fractal model is established with the number of particles of different particle fractions as the statistics to explore the fractal characteristics of the particle size distribution. Using the fractal model, the original grading curve of rockfill materials is scaled in four conventional scales.