## 3 Results and Analyses
### Microscale Behaviors
#### 3.2.2 Three-Dimensional Polar Distributions of the Branch Vector Orientations  
Figure 9 illustrates the three-dimensional polar distribution of branch vector orientations for various types of contact forces derived from numerical simulation results. The branch vector is used to identify and characterize contact forces between colliding particles [68], even when multiple contact points exist between them. In DEM simulations, statistical methods can systematically analyze the magnitude, direction, and type of contact forces between particles. Some researchers [69] have to adjust the contact force origins in gravel to accurately capture the contact forces from aggregates rather than pebbles, thereby enhancing the credibility of numerical results by repositioning the source of the contact forces to the center of the aggregates. However, in this study, due to the diverse characteristics of contact forces generated by binary-medium materials--including variations in contact type and stiffness--the contact forces from filling particles (pebbles) are treated as those originating from gravel clumps, with the sources of these contact forces being the centers of the filling pebbles rather than the gravel clumps, as illustrated in the simplified schematic diagram (Figure 9a). Figure 9b illustrates the three-dimensional polar coordinate distribution of branch vector orientations under identical consolidation conditions (\(\sigma_{3}=1000\) kPa) for various gravel contents, with color intensity indicating the proportion. After isotropic consolidation, the branch vectors exhibit isotropy and are uniformly distributed throughout the material. As the loading process progresses, the distribution pattern of the branch vectors gradually transitions from an isotropic state to a concentration along the vertical direction, consistent with the predefined test loading path, as observed by [PERSON] et al. [57]. When the contact force chain skeleton is predominantly formed between sand particles (S-S), the predominance of sand particles (GC = 10%, 20%, and 30%) results in an increased number of contact  
Figure 8: Evolution process of contact normal force with axial strain of specimens with varying gravel contents (the yellow line in the figure represents the average contact force, \(f_{0}^{m}\). forces within the specimen, leading to a top-to-bottom force chain skeleton. At this stage, the distribution frequency in the horizontal direction decreases significantly, while that in the vertical direction increases markedly. As the gravel content increases (GC = 45%, 55%), when the contact force chain skeleton is mainly composed of interactions between gravel clumps and sand particles (G-S), although the distribution remains predominantly vertical, the frequency of horizontal distribution slightly increases. When the contact force chain skeleton is primarily constituted by the contact forces between gravel clumps (G-G) (GC = 70%, 80%), the disparity in the distribution frequency of the vectors in the vertical and horizontal directions is notably reduced. The primary reasons for this phenomenon are twofold. First, in S-RMs with similar void ratios, as the gravel content increases, the sand particle content significantly decreases, leading to a substantial reduction in the number of contact forces generated by particle collisions. This results in decreased overall homogeneity and increased singularity of the mixture. Second, the distribution of the branch vectors of the contact forces between gravel and sand, as well as between gravel clumps, undergoes a slight shift towards the horizontal direction. As the gravel content increases, this shift is attributed to changes in the main body of the contact force chain skeleton, thereby causing variations in the distribution of the contact force branch vectors. Figure 9c presents the distribution of the contact force branch vectors under different consolidation pressures (\(\sigma_{3}=300,\ 500,\ \text{and}\ 1000\ \text{kPa}\)) for the duplicate gravel content.