# Tensile Properties and Tensile Failure Criteria of Layered Rocks
## 3 Numerical Study of Layered Rocks
### Direct Tension Test on Layered Rock Samples by RFPA\({}^{2D}\)
#### 3.3.2 Results and Analysis  
The complete stress-strain relations of the layered rock specimens with dip angles, \(\theta\), of \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\), \(85^{\circ}\), and \(90^{\circ}\) are shown in Figure 10. All stress-strain curves exhibit strong linear elasticity before the rock samples reach their peak tensile strength. The relationship between the stress and strain shows that the tensile strength increases as the dip angle increases from \(0^{\circ}\) to \(90^{\circ}\). This means that the tensile strength is anisotropic due to the existence of weak joints. The critical strain, at the point where the specimens reach the peak tensile load, also increases when the dip angle increases from \(0^{\circ}\) to \(85^{\circ}\). However, the critical strain of the specimen with a dip angle of \(90^{\circ}\) decreases compared with the specimen at \(85^{\circ}\). We can consider that the critical strain may be influenced by the failure modes of the layered rock specimens. Figure 10: Complete stress–strain curves for the layered rock specimens. Figure 9: Layered rock specimens with different dip angles subjected to direct tension. The relationships between tensile strength and the tensile modulus with different dip angles are shown in Figure 11. The tensile strength of the specimen with a dip angle of 0\({}^{\circ}\) is very close to the tensile strength of the specimen with a dip angle of 15\({}^{\circ}\). We can also see that, by varying the dip angle, the tensile modulus of the specimens follows the same trend as the tensile strength. As the dip angle increases from 0\({}^{\circ}\) to 90\({}^{\circ}\), the tensile modulus increases gradually. It is obvious that the strength and tensile modulus of the specimens with the lower dip angles, from 0\({}^{\circ}\) to 15\({}^{\circ}\), are closer to those of the joint material, whereas the strength and tensile modulus of the specimens with higher dip angles, from 75\({}^{\circ}\) to 90\({}^{\circ}\), are closer to the tensile modulus of the rock material. We can conclude that the deformation behavior and tensile strength of layered rocks with lower dip angles depend on the existence of joints. However, the influence of joints on the layered rocks with higher dip angles is slight. Figure 12 shows the simulated tensile strength under direct tension, which has a good agreement with experimental results. Figure 11: The tensile strength and tensile modulus of specimens with different dip angles. Figure 12: Tensile strength of the layered rock mass with different dip angles [40]. Figure 13 shows typical failure patterns of specimens with different dip angles. The fracture process of the specimen with a dip angle of 85\({}^{\circ}\) is displayed in Figure 14. The colors in Figures 13 and 14 represent the value of minimal principal stress. We can observe that the fracture of specimens with a lower dip angle, from 0\({}^{\circ}\) to 60\({}^{\circ}\), is caused by the propagation of joints, and the fractures occur along joints from one side to the other. More than two fractures are observed in the specimens with a dip angle from 0\({}^{\circ}\) to 45\({}^{\circ}\), whereas only one fracture can be seen along the joint from the upper-right to the lower-left corner in the specimen with a dip angle of 60\({}^{\circ}\). However, for the specimens with dip angles of 75\({}^{\circ}\), 80\({}^{\circ}\), and 85\({}^{\circ}\), fractures propagate partly along and partly through the joints. The failure plane exhibits a saw-toothed shape, and the same phenomenon was also observed in [PERSON]'s tests [39] and [PERSON]'s experiments [40]. For the specimens with dip angles of 75\({}^{\circ}\) and 80\({}^{\circ}\), the cracks initiate along the joints from the left or right boundary, then propagate along the joints. Before they reach the other end, the new cracks initiate and propagate in the rock material perpendicular to  
Figure 14: The failure process of the specimen with a layer dip angle of 85\({}^{\circ}\).