# Experimental and Numerical Investigations on the Slate Shearing Mechanical Behavior
## 4 Numerical Simulations
### Failure Behavior of Slate
#### 4.2.3 The Features of the Force Chain  
In order to investigate the micro-mechanism of layered slate under direct shear, the distribution of the particle force chain (FC) in the specimen was monitored and recorded. FC evolution under varying \(\beta\) and \(\sigma_{\rm n}\) values is shown in Figure 15. Using the examples of \(\beta\) = 0\({}^{\circ}\) and \(\sigma_{\rm n}\) = 5 MPa, the impact of \(\sigma_{\rm n}\) and \(\beta\) on the FC evolution is analyzed. In the FC diagram, blue and red lines represent compression and tensile forces, respectively. The thickness of the FC line corresponds to the magnitude of the contact force: A thicker FC indicates a greater contact force.  
At the same \(\beta\), the FC density increases with increasing \(\sigma_{\rm n}\). In the initial loading stage, the contact force distribution of the samples is relatively uniform. Near \(\tau_{\rm P}\), FC becomes more concentrated, and the damaged area expands. With the progression of displacement, FC is more concentrated in the shear slip direction (the density of FC in this direction is more considerable). As cracks initiate, propagate, and penetrate, internal FC gradually breaks and diminishes. Macroscopic fractures initiate with the increased contact force in the loading direction. Corresponding to the shear stress-displacement curve, as \(\sigma_{\rm n}\) increases from low to high, FC continues to break at shear displacements of 0.40 mm, 0.60 mm, 0.72 mm, and 0.85 mm, respectively. This phenomenon aligns with the crack evolution discussed above, which shows the fastest increase in the number of cracks.  
Figure 14: Cracks evolution of slate under different \(\beta\) and \(\sigma_{\rm n}\).  
For samples with different \(\beta\) values at 5 MPa, a noticeable distribution of the FC along the bedding direction is noticed. In the initial loading stage, the contact force distribution in horizontal bedding is more uniform, with a higher FC density and more prominent contact force. As stress increases, FC gradually diffuses toward both ends. At a load of \(0.8\tau_{\text{P}}\), the contact force becomes concentrated at the upper left and lower right ends of the model. At this point, tensile force is more pronounced with \(\beta\) = 45\({}^{\circ}\) and 60\({}^{\circ}\) along the bedding plane, particularly evident at \(\tau_{\text{P}}\). FC concentrates further in the direction of shear slip as  
Figure 15: Force chain evolution of slate under different \(\beta\) and \(\sigma_{\text{n}}\).  
displacement increases, where the contact force is larger, and the FC density is higher. After the \(\tau_{\rm p}\), except for \(\beta=90^{\circ}\), FC at other \(\beta\) values gradually weakens and disappears with crack penetration.  
The locally enlarged image of FC for a sample with \(\beta=45^{\circ}\) and \(60^{\circ}\) at \(\tau_{\rm p}\) is shown in Figure 16. The tensile FC of the sample at \(45^{\circ}\) is mainly distributed along the chains between the particles and primarily found in the matrix layer. However, the tensile FC is predominantly distributed in the bedding plane for the specimen with \(\beta=60^{\circ}\), forming a chain distribution along the bedding direction. The distribution characteristics of the tensile force provide a better explanation for \(\tau_{\rm p}=45^{\circ}>\tau_{\rm p}=60^{\circ}\): when the tensile force along the non-layered surface distribution is more prominent, failure is more complex, which is attributed to the bedding plane (with lower tensile strength) of the rock being weaker than the matrix.