# Experimental and Numerical Investigations on the Slate Shearing Mechanical Behavior
## 4 Numerical Simulations
### Failure Behavior of Slate
#### 4.2.2 Crack Evolution  
The specimen cracking and damage are macroscopically exhibited as the initiation, propagation, and penetration of microcracks. Quantitative analysis of the damage process can be carried out through the variation in the number of microcracks. The crack evolution with displacement under different conditions is similar. Taking the sample with \(\beta=0^{\circ}\) and \(\sigma_{\mathrm{n}}=5\) MPa as an example, the relationship between shear stress, displacement, and the crack number is presented in Figure 13a. The crack evolution process can be divided into four stages with the progression of displacement:  
Figure 12: Shear stress–displacement curves of numerical simulation for slate with (**a**) different \(\sigma_{\mathrm{n}}\) when \(\beta=0^{\circ}\) and (**b**) different \(\beta\) when \(\sigma_{\mathrm{n}}=10\) MPa. Stage I--elastic stage: The shear stress--displacement curve is approximately linear, and there are almost no cracks during this stage. Stage II--crack stable propagation: As the interaction force between particles exceeds its bond strength, particle bonds break, generating tensile and shear cracks. At this stage, cracks are generated locally without connecting and penetrating. Therefore, the number of cracks increases slowly and remains relatively stable. Stage III--unstable expansion stage: With increasing load, particle bond failure continues to rise, rapidly increasing the number of cracks. The shear stress--displacement curve exhibits prominent nonlinear growth, primarily due to the mutual expansion, intersection, and penetration of cracks. Figure 13: Variation in number of cracks during shearing for slate: (**a**) for the tensile, shear, and total crack (\(\beta\) = 0°, \(\sigma_{\text{n}}\) = 5 MPa); the total crack under different (**b**) \(\sigma_{\text{n}}\) (\(\beta\) = 0°); and (**c**) \(\beta\) (\(\sigma_{\text{n}}\) = 5 MPa). Stage IV--post-peak stage: After reaching the \(\tau_{\rm P}\) of the specimen, the shear stress decreases rapidly. The growth rate of the crack number slows down and gradually stabilizes. Inside the specimen, prominent macroscopic and secondary cracks are forming [37]. The variation in crack number with the displacement of specimens at different \(\sigma_{\rm n}\) with \(\beta\) = 0\({}^{\circ}\) is shown in Figure 13b, where the red points indicate the shear displacement at the peak shear stress. Crack evolution for specimens under different \(\sigma_{\rm n}\) follows a similar trend. For the same \(\beta\), an increase in \(\sigma_{\rm n}\) leads to a higher crack number. Notably, at higher normal stress levels, the increase rate of crack number is lower when it reaches the same displacement in the stable propagation stage. This observation suggests that normal stress effectively inhibits microcrack generation. The crack evolution under varying \(\beta\) values is presented in Figure 13c (\(\sigma_{\rm n}\) = 5 MPa). The number of cracks and the corresponding increase rate at the peak shear stress moment for specimens with vertical bedding are higher than those with horizontal bedding. However, the crack growth rate at the peak shear stress moment for other \(\beta\) values is similar. The growth rate of the number of cracks gradually slows down with increasing displacement and ultimately stabilizes. The total number of cracks initially increases, then decreases, and subsequently increases with the increase in \(\beta\). In conjunction with the crack evolution depicted in Figure 14, when \(\beta\) is within 30-60\({}^{\circ}\), shear cracks initially initiate along the bedding plane of the sample during the early loading stage. As displacement increases, internal stress redistributes, and the slip effect along the bedding plane is constrained by normal stress. Subsequently, cracks start appearing in the matrix and steadily increasing in number; nevertheless, the crack generation rate is slower than that in the bedding plane, culminating in shear failure either along or through the bedding plane.