## 3 Results and Analyses
### Microscale Behaviors
#### 3.2.4 Analysis of Fabric Tensor in Contact Force Chain Skeletons
When the primary load-bearing force within the skeletal force chain is sustained by contact forces between gravel clumps (GC = 70%, 80%), the average coordination number reaches its equilibrium state last and exhibits minimal stability. This phenomenon remains unaffected by consolidation pressure. The observed behavior can be attributed to an inverse relationship between the gravel clump and sand particle content within a constant-density S-RM, resulting in pronounced specimen heterogeneity. Additionally, irregularities in gravel shapes contribute to unstable internal contact forces. These findings underscore that skeletal structure significantly influences the stability characterization value of the average coordination number (\(CN\)), whereas consolidation pressure exerts a comparatively minor impact. Consequently, these observations unequivocally demonstrate that the average coordination number serves as a reliable indicator of the critical state within the particle material and delineates the primary composition of the force chain skeleton. * The fabric anisotropic coefficient, \(a^{n}\):  
The microscopic examination of fabric and its evolution plays a pivotal role in analyzing the mechanical response of granular materials--this analysis is an indispensable element for establishing constitutive relationships and elucidating fabric anisotropy. The quantification of fabric is typically achieved through the utilization of the average fabric tensor, \(F\), which must be defined and linked to the statistical distribution of unit vectors along the principal axis of elongated particles, contact normal directions, void vectors,scan line directions, or other micro-mechanically oriented entities. In the numerical model, granule materials consist exclusively of rigid particles or clumps, facilitating inter-particle interaction via contact forces within confined contact areas, resulting in three distinct types of contacts (\(f_{m}^{ss}\), \(f_{h}^{gg}\), and \(f_{h}^{gs}\)). At the microscale, the interaction between particles or clumps involves significant characteristic quantities such as contact force and relative displacement. For multiple agglomerates of rigid clumps and particles in contact (refer to Figure 12), the vector connecting the centers of the collision particles is denoted as \(I_{i}^{c}\), and the unit normal vector corresponding to the contact is represented by \(\mathbf{n}_{i}^{c}\). Due to irregular shapes of gravel clumps, the contact normal vector is orthogonal to the contact surface, and the orientation of the contact force \(\mathbf{c}\) in the direction \(\mathbf{a}_{i}^{c}\) is defined as the angle between the contact normal and the XY plane. The distribution of contacts can be characterized using symmetric second- or fourth-order tensors [71]. In this study, second-order tensors are preferred due to their simplicity  
Figure 11: The evolutionary trend of the average coordination number \(CN\) in soil–rock mixture samples. and sufficient precision in capturing the principal characteristics of anisotropy. The structure tensor, \(\mathbf{F}_{ij}\), is defined as follows:  
\[\mathbf{F}_{ij}=\frac{2}{\left(N_{g}+N_{s}\right)}\sum_{c\in\mathbf{C}}n_{i}^{c}n_{j}^{c} \tag{5}\]  
where \(\left(N_{g}+N_{s}\right)\) denotes the total number of sand particles and gravel clumps in the specimen; the contact set, \(\mathbf{C}\), is the aggregate of the contact force, \(\mathbf{c}\). \(n_{i}^{c}\) is the \(c\)th unit contact normal vector. \(\mathbf{F}_{ij}\) is a symmetric tensor with the first trace, \(\text{tr}\left(\mathbf{F}_{ij}\right)=1\).