# Numerical Simulation of Rockfill Materials Based on Fractal Theory
## 4 Analysis of the Results of the Numerical Test
### Biaxial Compression Test  
To eliminate the influences of different compactness on the macroscopic strength and deformation characteristics of the specimen, the RD (0.89) is taken to prepare the numerically simulated specimens that are made by the four different scale methods, and the specimen size is \(\varphi\) 300 mm \(\times\) 600 mm. In this paper, the linear stiffness model without considering particle breakage is adopted, and the calibration of meso parameters refers to references [29; 30]. The mesounerical parameters are set as follows: the modulus of elasticity is 2.0 GPa; the ratio of the normal contact stiffness of the particle to the tangential contact stiffness is 1.5; the friction coefficient between the particles is 0.48; the friction factor between the particle and the rubber film and that between the particle and the compression plate is 0.001; and the density is 2.45 g/cm\({}^{3}\).  
Figure 3: Relationship between dense degree index and fractal dimension.  
Figure 4 shows the deviatoric stress and volumetric strain with axial strain under different fractal dimensions \(D\). At the initial stage of loading, the deviatoric stress increases rapidly with the axial strain, thereby indicating that the rockfill materials has not yet been completely compacted, and as the axial strain advances, the deviatoric stress increases slowly until the peak deviatoric stress is reached. At that stage, the specimen is wholly shrunk, and the volumetric strain of different specimens is greatly affected by the fractal dimension \(D\). Without considering particle breakage, after reaching the peak deviatoric stress, the deviatoric stress-axial strain curve exhibits softening, and the entire specimen produces a dilatancy effect, a great fractal dimension \(D\) corresponds to a more obvious dilatancy effect. The more fine-grained soil in rockfill materials, the more obvious the shear dilatancy. The numerically simulated deviatoric stress and volumetric strain with axial strain curve fluctuates violently, because particles with a particle size of 5 mm are substituted equivoluminally for particles with a particle size smaller than 5 mm. Without the filling of smaller particles, the sliding and rolling of large particles become relatively easy. Thus, the particle aggregate changes the dilatancy significantly.  
Figure 5 shows that the fractal dimension \(D\) greatly affects the mechanical properties of the numerically simulated specimen. The peak internal friction angle of the specimen and the secant modulus rise with the fractal dimension \(D\), because an increase in the number of fine particles in the specimen increases the friction between particles and intensifies the sliding and rolling of particles, with the peak deviatoric stress increasing. An increase in the content of fine particles improves the filling of particles, thereby forming a more stable and dense structure with minor axial deformation under the initial stress, but the secant modulus of the specimen increases. The Poisson's ratio of the specimen decreases with the fractal dimension \(D\), thereby indicating that a large number of fine particles in the specimen corresponds to evident aggregate interlock behavior and thus weak lateral deformation of the specimens. The peak internal friction angle of the specimen, the secant modulus, and the Poisson ratio show better linear fitting with the fractal dimension \(D\).  
Figure 4: Relation curves of deviatoric stress and volumetric strain with axial strain under different fractal dimensions.