# Numerical Simulation of Rockfill Materials Based on Fractal Theory
## 3 Fractal Model  
The particle shape and size distribution of rockfill materials, as well as the clearance between particles and surface behaviors, have certain influences on compactness. These features are random in a certain range and are difficult to describe in the conventional language of mathematics. Therefore, the fractal dimension \(D\) is taken as a macro-meso mechanical index to study rockfill materials, with the following basic definition based on fractal theory:  
\[N\propto\left(\frac{1}{r}\right)^{D} \tag{1}\]  
where \(r\) represents the graphic measure, \(N\) represents the measured value from the corresponding \(r\) graph, and \(D\) represents the fractal dimension of the graph. Given that the numerically simulated specimen makes obtaining the number of particles of different particle fractions easier, the particle aggregate inside the numerically simulated specimen is considered a measure, and the composition of particles is a structural pattern without characteristic length. In this paper, a fractal model that is different from that presented by [PERSON] [21] is established, with the ratio of the upper limit \(d_{i}\) for the particle size of the particle fraction to the maximum particle size \(d_{max}\) of the specimen as the measure, and the ratio of the number of particles of the particle fraction to the total number of particles of the specimen as the measured value. Then, the particle size distribution function reflected by the number of particles is  
\[\frac{N_{i}}{N_{sum}}\propto\left(\frac{d_{i}}{d_{max}}\right)^{-D} \tag{2}\]  
where \(N_{i}\) represents the number of particles with a particle size within the range of [\(d_{i-1}\),\(d_{i}\)], \(N_{sum}\) represents the total number of particles of the specimen, \(d_{i}\) represents the upper limit for the particle size of the particle fraction, and \(d_{max}\) represents the maximum particle size in the specimens. With the introduction of a proportional constant \(C\), Equation (2) can be expressed by  
\[\frac{N_{i}}{N_{sum}}=C\left(\frac{d_{i}}{d_{max}}\right)^{-D} \tag{3}\]  
From Equation (3), the number \(N_{i}\) of particles in the particle fraction is related to the proportional constant \(C\), the upper limit \(d_{i}\) for the particle size of the particle fraction,  
Figure 1: Grading curves. the maximum particle size \(d_{max}\) inside the specimen, and the fractal dimension \(D\) of the specimen. This equation gives the basic mathematical model for the particle size distribution of the numerically simulated specimen. The numerically simulated specimen during particle formation has no particle breakage, so \(N_{i}\propto(d_{i})^{-D}\). With the introduction of another proportional constant \(\lambda\), Equation (3) can be expressed by  
\[N_{i}=\lambda(d_{i})^{-D} \tag{4}\]  
Both sides of Equation (4) are operated logarithmically. Then  
\[\lg(N_{i})=-D\lg(d_{i})+\lg\lambda \tag{5}\]  
Equation (5) presents the method of solving the fractal dimension. The double logarithmic chart of the number of particles in the particle fraction \(N_{i}\) and the upper limit for the particle size of the particle fraction \(d_{i}\) are plotted. The regression curve is fitted by using the least squares method, and the inverse of the slope of the line is also the fractal dimension \(D\). The number of particles in the numerically simulated specimen is shown in Table 1. Linear fitting \(\lg(d_{i})\) and \(\lg(N_{i})\) is shown in Figure 2.  
The fitting results in Table 1 and Figure 2, except for the relatively poor linear correlation (correlation coefficient R\({}^{2}\) = 0.838) that corresponds to the numerically simulated specimen from the equivalent substitution, indicate that the numerically simulated specimens from the other three scale methods show better linear fitness, with a correlation coefficient within the range of 0.935-0.992. These findings indicate that the fractal dimension \(D\) can basically reflect the changing trend of particle size distribution.