## 2 Discrete Element Modeling and Energy Consumption Calculation  
### Discrete Element Modeling of Soil-Rock Mixtures  
The Particle Flow Code 3D (PFC3D) software was employed for the numerical simulations conducted in this study. The S-RM specimens used had a fixed coarse particle content of 50%. The mixing degree index (\(I\)) values, ranging from 0.0 to 1.0, were determined using the formulation proposed by [PERSON] et al. [17]. The brief calculation principle is as follows:  
The curves in Figure 1 represent the variation of coarse and fine aggregate content \(V\!f\) with the normalized depth \(H/HB\) (\(HB\) is the height of the mixture) for different particle mixes, where the Arbitrary Line (black) is the arbitrary mix curve, and the Separation Line (blue) and the Uniform mixing Line (red) are the complete segregation curve and the uniform distribution curve, respectively. When the Arbitrary Line and the Uniform mixing Line do not coincide exactly, the total area \(A\)s (shaded in Figure 1) enclosed by the two curves represents the degree of coarse aggregate segregation in the graded aggregates. The area bounded by the Separation Line, the Uniform mixing Line, and Lines AB and CD is the area of complete segregation (as shown in areas \(A\)i and \(A\)ii). The mixing degree index (\(I\)) in this paper was calculated using Equation (1):  
\[I=1-\frac{A_{s}}{A_{\mathrm{i}}{+}A_{\mathrm{ii}}} \tag{1}\]  
Figure 2 illustrates the \(I\) value versus the depth of the curves for the six groups of specimens. In the figure, \(H\) represents the height from the bottom of the specimen, while \(HW\) denotes the total height. The specimens had dimensions of 800 mm \(\times\) 800 mm \(\times\) 600 mm and were enclosed by five rigid walls. The sample preparation process is as follows:  
(1) Divide the specimen height into 10 equal layers. Generate specific numbers of coarse and fine particles in each layer based on the initial void ratio target of 0.7 and relative coarse/fine particle proportions in Figure 2.  
(2) Reset the interparticle contact forces and velocities every 5 cycles until 20,000 cycles to eliminate excessive contacts. (3) Allow the particles to settle under gravity to produce the initial specimen. The final I values match the targets. The top of the specimen was equipped with an 800 mm \(\times\) 800 mm \(\times\) 2 mm loading plate, which contains a cluster composed of 6241 sub-spheres with a radius of 1 mm. The resulting model is depicted in Figure 2. To simulate the vibrations induced by field vibratory rollers, an excitation force was applied to the plate. This force followed a cosine  
Figure 1: Schematic representation of the principles of calculating the indicators. Figure 2: Variation curve of coarse particles with normalized depth under different \(I\) value. wave pattern, and matched the magnitude and frequency of the field vibration compaction, as described by Equation (2):  
\[F=-F_{0}(\cos(2\pi f\Delta t)+1) \tag{2}\]  
where \(F\) presents the excitation force; \(F_{0}\) = 500 kN; \(f\) is the loading frequency; and \(\Delta t\) is the loading time. Given the focus of this study on particle motions and energy dissipation, important parameters included dashpot coefficients, sliding friction, and rolling friction. The contact stiffness and other relevant parameters were directly derived from triaxial model verifications and incorporated into the analysis [18]. Linear contact models were employed for particle interactions. Research indicates that both linear and nonlinear Hertz models can effectively represent the behaviors of granular materials [19]. The normal stiffness of a particle, \(k_{\rm n}\), is expressed in Equation (3):  
\[k_{\rm n}=\pi E_{c}r/(r_{1}+r_{2}) \tag{3}\]  
where \(E_{c}\) is the particle effective modulus; \(r_{1}\) and \(r_{2}\) are the contacting particle radii; and \(r\) is the smaller radius.