## 3 Results and Analyses
### Microscale Behaviors
#### 3.2.4 Analysis of Fabric Tensor in Contact Force Chain Skeletons
On the other hand, the second-order tensor, \(\mathbf{F}_{ij}\), can also be expressed in integral form through probability density, \(f(\mathbf{n})\), as presented in Equation (6):  
\[\mathbf{F}_{ij}=\int_{n}f(\mathbf{n})n_{i}n_{j}d\mathbf{n} \tag{6}\]  
In three dimensions, the probability density function of the entity distribution can be expressed as Equation (7), which is derived from a truncated second-order Fourier expansion [72]:  
\[f(\mathbf{n})=\frac{1}{4\pi}\big{(}1+\mathbf{G}_{ij}n_{i}n_{j}\big{)} \tag{7}\]  
Figure 10:0 Contact geometry: contact normal, \(n_{i}^{c}\), and contact orientation, \(\mathbf{\alpha}_{i}^{c}\). where \(\mathbf{G}_{ij}\) is deviatoric and symmetric, referred to as the deviatoric fabric tensor. The tensor, \(\mathbf{G}_{ij}\), can be calculated based on the second-order structure tensor, \(\mathbf{F}_{ij}\):  
Figure 10:1  
Consequently, the structure tensor, \(\mathbf{F}_{ij}\), represents the average orientation vector, \(\mathbf{n}^{\mathbf{c}}\), associated with three modalities of intermolecular contact force chains. The dimensionless fabric anisotropy tensor, \(\mathbf{G}_{ij}\), widely utilized in the literature [73], is directly proportional to a partial component of the fabric tensor, \(\mathbf{F}_{ij}\). Notably, \(\mathbf{G}_{ij}\) scales with a specific volume, as suggested by [PERSON] [74]. The norm \(a^{n}\) (scalar), which quantifies fabric anisotropy, and the direction tensor \(\mathbf{n}_{\mathbf{G}}\) of fabric tensor \(\mathbf{G}_{ij}\) are defined as follows:  
Figure 10:2  
Figure 10:3  
where the direction tensor \(\mathbf{n}_{\mathbf{G}}\) satisfies \(\mathbf{n}_{\mathbf{G}}:\mathbf{n}_{\mathbf{G}}\)= 1 and \(\text{tr}\left(\mathbf{n}_{\mathbf{G}}\right)\) = 0. Figure 13 illustrates the evolution of the contact normal anisotropy coefficient (\(a^{n}\)) for S-RM specimens during shear loading under varying gravel contents and consolidation pressures. The anisotropy coefficient, \(a^{n}\), of all S-RMs exhibits a pronounced initial upward trend induced by loading, particularly evident at a gravel content of 20%. This behavior remains consistent across different consolidation states. Upon reaching a specific axial strain (\(\epsilon>16\%\)), the characteristic values of contact normal anisotropy, \(a^{n}\), stabilize significantly, with substantial differences observed among these stable values. This indicates that the specimens have entered a critical state, although the characteristics of this critical state vary considerably depending on the gravel content. Upon a meticulous examination of Figure 13, it is evident that the contact normal anisotropy characteristics of the specimens tend to exhibit uniformity as the gravel content in the S-RM varies at 10%, 20%, and 30%. This observation is further supported by the skeleton distribution of the contact force chain depicted in Figure 10a, reaffirming that the primary structure of the soil-rock sample's skeleton predominantly comprises contact forces between sand particles (\(f_{n}^{ss}\)) at these gravel contents. Furthermore, there is a discernible trend toward consistent evolution of contact normal anisotropy characteristics when gravel-sand contact forces (\(f_{n}^{ss}\)) become predominant (GC = 45%, 55%) or when gravel-gravel contact forces (\(f_{n}^{ss}\)) take precedence (GC = 70%, 80%). These findings highlight the importance of understanding how variations in gravel content affect both the structural composition and mechanical behavior of S-RMs. Simultaneously, it has been determined that the sand particles, serving as the primary constituent of the contact force chain skeleton, exhibit the highest and most stable anisotropic characteristic values. Additionally, the contact forces generated between gravel clumps and sand particles form the structural framework of the contact force chain skeleton. Conversely, specimens with gravels as their main skeletal component demonstrate lower and more stable anisotropic characteristic values.