## 5 Shear Fracture Behavior by FCEM  
Prior work has documented fracture behavior of slate such as large strength anisotropy in both laboratory experiments and FCEM, while the tensile fracture patterns and failure processes of slate have been carefully compared experimentally and numerically. However, these studies did not pay attention to the shear fracture that is also fundamental in evaluating the failure properties of slate. Thus, the objective of this section is to examine the mode II fracture toughness (\(K_{IIC}\)) and its shear fracture behavior by conducting a series of asymmetric semi-circular bend (ASCB) numerical tests for slate with appropriate values for S\({}_{1}\) and S\({}_{2}\) (see Figure 2) under the same loading rate as above [25; 26; 49]. According to reference [49], the \(K_{IIC}\) of the TI slate can be evaluated by the following formula;  
\[K_{IIC}=\frac{P_{max}\sqrt{\tau b}}{2 RB}Y^{\prime\prime} \tag{14}\]  
Figure 17: The variation of \(K_{IC}\) and \(K_{IIC}\) under different bedding angles obtained by experimental and numerical tests.  
where \(Y^{\prime\prime}\) is the dimensionless stress intensity factor under the plane stress state that correlates with a/R, S\({}_{1}\)/R, and S\({}_{2}\)/R; here its value is 1.7822.  
We found that for the oriented angle of the bedding of 0\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\), and divider type, the corresponding numerical peak loads for the mode II loading were 852.4, 1018.9, 855.2, 804.4, 496.3, and 1142.4 N, respectively. Figure 17 depicts the variation of fracture toughness \(K_{IC}\) and \(K_{IIC}\) with different bedding angles obtained from the experimental and numerical tests. As indicated in the figure, we found that the values of experimental \(K_{IC}\) are consistent with the numerical \(K_{IC}\). However, the shear fracture toughness changes in a complicated way. As the bedding angle increase from 0\({}^{\circ}\) to 90\({}^{\circ}\), the numerical \(K_{IIC}\) values of the layered slate change sequentially as follows: 0.215, 0.257, 0.216, 0.203, and 0.125 MPa\(\surd\)m, then sharply increase to 0.289 MPa\(\surd\)m for the divider type. Accordingly, the numerical \(K_{IIC}\) exhibits an increasing trend first from 0\({}^{\circ}\) to 30\({}^{\circ}\) then decreases from 30\({}^{\circ}\) to 90\({}^{\circ}\), while ultimately sharply increasing from 90\({}^{\circ}\) to divider type. Therefore, it can be concluded that the present numerical values of fracture toughness \(K_{IC}\) and \(K_{IIC}\) predicted by the FCEM are fairly good.  
The complete shear fracture patterns and internal fracture trajectories of simulated ASCB specimens are shown in Figure 18. In the cases for bedding angle of 0\({}^{\circ}\) under mode II loading, the crack initiates from the notch tip with an angle relative to the vertical notch line, and the fracture propagates in a stable manner before rapidly swaying when the crack encounters the horizontal bedding, and then it travels up to the top boundary of the loading (see Figure 18a). With the increase of inclined bedding angle from 30\({}^{\circ}\) to 60\({}^{\circ}\) covered divider type, the fracture patterns of the simulations are also essentially the same, while the fracture patterns are very complex with greater interaction between weak planes and the fracture, which not only enhance the peak load but also affect the interaction pattern of the fracture plane (see Figure 18b,d,f). However, with a bedding angle of 90\({}^{\circ}\) under mode II loading, the fracture pattern is illustrated as a vertical straight-through crack from the notch tip to the upper loading end (see Figure 18e), which may be mainly affected by the artificial weak planes in the computational ASCB model.