# Tensile Properties and Tensile Failure Criteria of Layered Rocks
## 3 Numerical Study of Layered Rocks
### Direct Tension Test on Layered Rock Samples by RFPA\({}^{2D}\)
#### 3.3.3 Failure Mechanism  
Figure 16 shows the failure mechanisms of the layered rock specimens with different dip angles under direct tension, and these correspond to the three main failure modes of layered rock specimens. The layered rocks in Figure 16 is mainly composed of two materials, including rock material and joint material (bedding plane). Figure 16a illustrates that when the dip angle of the bedding plane is equal to 0\({}^{\circ}\), with the increasing axial tensile load, the bedding plane will be fractured by tension while the axial tensile stress is greater than the tensile strength of the bedding plane (T\({}_{0}\)). In this configuration, the layered rock specimen will be fractured along the bedding plane, and the tensile strength of the layered rock specimen can be expressed as \(\sigma_{\rm t}=~{}\sigma_{\rm 1}=~{}T_{0}\). Figure 15: The failure patterns of sandstone of Lyons formation tested in direct tension [39]. Figure 16d shows that, when the dip angle of the bedding plane is equal to \(90^{\circ}\), with the increasing axial tensile load, the bedding plane first reaches its tensile strength, and then the bedding plane is cracked. However, in contrast to Figure 16a, the rock material is subjected to the axial tensile load at the cross section of the layered rock specimen. At the cross section, the bedding plane can be simplified as a single point when the layer thickness is small; therefore, it has little influence on the bearing capacity of the specimen while the bedding plane is first cracked. Then, as the axial tensile load increases to the tensile strength of the rock material (T\({}_{90}\)), the layered rock specimen will be fractured along the cross section, resulting in tensile failure, and the failure plane is generally perpendicular to the loading direction. In this configuration, the tensile strength of the layered rock mass can be expressed as \(\sigma_{\rm t}=~{}\sigma_{1}=~{}T_{90}\). Figure 16b shows that when the dip angle of the bedding plane is greater than \(0^{\circ}\) and less than a specified angle, the layered rock mass is also fractured along the bedding plane. It can be seen in Figure 16b that the normal stress at the bedding plane is \(\sigma_{\rm n}=~{}\sigma_{1}\cos^{2}\theta\). In order to make the bedding plane fracture by tension, the normal stress at the bedding plane should be equal to the tensile strength of the bedding plane, which is expressed as \(\sigma_{\rm n}=~{}T_{0}\). We can then obtain the tensile strength of the specimen as \(\sigma_{\rm t}=~{}\sigma_{1}=~{}T_{0}/\cos^{2}\theta\). This demonstrates that the tensile strength of layered rock mass increases with the increasing dip angle. With the increasing dip angle, the failure patterns of the layered rock specimens change. This will result in a progressive saw-toothed failure, as shown in Figure 16c. It is assumed that the area of the cross section is A, and the area of each small matrix exhibited the saw-toothed shape is A\({}_{\rm i}\), so \(\sigma_{\rm 1}=~{}F/A\) and \(\sigma_{\rm i}=~{}F_{\rm i}/A_{\rm i}\). In order to make each small matrix exhibit the saw-toothed shaped cracks, the tensile stress, \(\sigma_{\rm i}\), needs to satisfy \(\sigma_{\rm i}=T_{90}\). It can then be surmised that \(F_{\rm i}=~{}T_{90}A_{\rm i}\). Therefore, \(F=\Sigma F_{\rm i}=T_{90}\Sigma A_{\rm i}\), \(\sigma_{\rm t}=\sigma_{\rm 1}=F/A=T_{90}\Sigma A_{\rm i}/A\). It can be seen that, when the dip angle of the bedding plane is less than \(90^{\circ}\), \(\Sigma A_{\rm i}<A\), that is to say, \(\sigma_{\rm t}=~{}\sigma_{1}<~{}T_{90}\). We can see that as the layered rock mass generates the progressive saw-toothed failure, the tensile strength of the layered rock mass at this specific angle is less than the tensile strength of the layered rock specimens at a dip angle of \(90^{\circ}\). From the point of view of geometry and elastic mechanics, this can explain the variation of tensile strength of the layered rock specimen with different dip angles of the bedding plane.