# Tensile Properties and Tensile Failure Criteria of Layered Rocks
## 3 Numerical Study of Layered Rocks
### Brazilian Disc Test of Layered Rock Specimens by RFPA\({}^{\text{2D}}\)
#### 3.2.2 Results and Analysis  
Figure 6 shows the tensile strength of the layered rock specimens under the numerical Brazilian test, which is calculated by Equation (1). The tensile strength of the layered rock specimens decreases with the increasing dip angle; the maximum tensile strength is 11.49 MPa at a dip angle of 0\({}^{\circ}\), which corresponds to the direct tension strength at a dip angle of 90\({}^{\circ}\). Figure 7 shows the horizontal stress at the center of the specimen vs. the vertical loading displacement. The pre-peak behavior of the specimens with different dip angles is quite similar, showing a linear elastic behavior. Figures 6 and 7 indicate that the tensile strength and loading curves of the layered rock specimens under the Brazilian test are very sensitive to the dip angle.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Materials** & **Parameters** & **Elastic Modulus** & **Tensile Strength** \\ \hline \multirow{2}{*}{Rock} & Homogeneity index & 4 & 4 \\  & Mean value & 50,000 MPa & 45 MPa \\ \hline \multirow{2}{*}{Joint} & Homogeneity index & 3 & 3 \\  & Mean value & 1000 MPa & 4 MPa \\ \hline \hline \end{tabular}
\end{table}
Table 1: Elemental parameters of the rock and joint material.  
Figure 5: Numerical layered rock specimens under the Brazilian disc test.  
Figure 8 presents the failure modes of the layered rock specimens under the Brazilian test. It can be clearly seen that the failure mode of the specimen at 0\({}^{\circ}\) is similar to the isotropic rocks. The cracks generate at the center of the specimen, then propagate in a direction towards to the loading ends, and finally penetrate the whole specimen. After that, there are some small shear cracks near the loading ends. The main failure mechanism in this case is rock material tensile splitting. The tensile strength of the specimen at 0\({}^{\circ}\) is 11.49 MPa, which is similar to the tensile strength in the direct tension test (11.42 MPa) and the laboratory test (12.50 MPa). The failure modes of the specimens at 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) are quite complicated, and they show a mixed failure mode combining tensile splitting and shear failure. With the increasing dip angle, the shear cracks along the bedding planes increase. Shear cracks generate at the contacts of the rock material and the joint material, but they do not propagate any farther along the bedding planes. After that, the tensile cracks generate in the rock material, then propagate to shear cracks. Finally, the shear cracks and tensile cracks penetrate the specimen, which results in the specimen failure. The failure modes of the specimens at 60\({}^{\circ}\) and 75\({}^{\circ}\) are relatively simple, showing a shear failure along the bedding planes. Shear cracks generate at the contacts of the rock material and the  
Figure 6: Tensile strength of layered rock specimens under the Brazilian test.  
Figure 7: Horizontal stress (tensile stress) at the center of specimen vs. vertical displacement.  
joint material, and the cracks propagate along the bedding plane until they penetrate the whole specimen, which finally result in the specimen shear failure. The main crack does not cross the center of the specimen, and no tensile cracks are generated in the center of the disc. The failure mode of the specimen at 90\({}^{\circ}\) is similar to that of the specimen at 0\({}^{\circ}\); the cracks generate almost in the center of the disc and propagate towards the loading ends until the cracks penetrate the whole specimen. The tensile strength of the specimen at 90\({}^{\circ}\) is 1.06 MPa, which is much lower than the tensile strength of the specimen at 0\({}^{\circ}\). The main failure mechanism in this case is joint material tensile splitting. In conclusion, there are three main failure modes of the layered rock specimens under the Brazilian test, including tensile splitting failure, mixed tensile splitting and shear failure, and shear failure.