## 3 Results
### Failure Process Analysis
The Mohr strength theory can be expressed as [54]:  
\[\sigma_{1}-\beta\sigma_{3}=\sigma_{t} \tag{3}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum principal stress and the minimum principal stress respectively, and \(\beta\) is the ratio of tensile (\(\sigma_{t}\)) to compressive strength (\(\sigma_{c}\)). In fact, the maximum principal stress, \(\sigma_{1}\), can be considered as the tensile stress, \(\sigma_{x}\), on the surface of the Brazilian disc, while the minimum principal stress, \(\sigma_{3}\), is the compressive stress, \(\sigma_{y}\), correspondingly. Moreover, the tensile and compressive stress on the disc surface can be derived by the VIC-3D software using the plane stress solution:  
\[\sigma_{x}=\frac{E}{1-\
u^{2}}\Big{(}\varepsilon_{x}+\
u\varepsilon_{y}\Big{)} \tag{4}\]  
\[\sigma_{y}=\frac{E}{1-\
u^{2}}\Big{(}\varepsilon_{y}+\
u\varepsilon_{x}\Big{)} \tag{5}\]  
where \(\varepsilon_{x}\) and \(\varepsilon_{y}\) are tensile and compressive strain, respectively. Consequently, the equivalent stress, \(\sigma_{M}\), can be derived as:  
\[\sigma_{M}=\sigma_{x}-\beta\sigma_{y} \tag{6}\]  
Figure 6: **(a)** Seven virtual extensometers assigned equally along the loading diameter; **(b)** the lateral strains history during the loading process. It is well known that the crack initiation point in the Brazilian test is generally located near the vertical diameter of the disc. Therefore, this paper mainly focuses on the equivalent stress distribution on this path. For the purpose of a better comparison, the final results are normalized by the maximum equivalent stress of each rock specimen, and plotted in Figure 7. It was found that the possible crack initiation points of all specimens are all located in the central part of the disc. In other words, these specimens can be regarded as exhibiting a valid failure mode [12]. Additionally, the current results show a good agreement with a previous numerical study performed by [PERSON] and [PERSON] [54]. In other words, the maximum equivalent stress (individually directed by arrows) based on Mohr theory generally tends to occur far away from the disc center for an increasing ratio of tensile to compressive strength in rocks. It can be seen from the insert figure in Figure 7 that the vertical distance between the maximum equivalent stress point and the center of the disc varies significantly with the ratio of tensile to compressive strength increasing, and the Basalt specimen exhibits the largest distance while the Marble specimen shows the lowest one. However, it can be found that the result is aberrant for the red sandstone specimen, despite the fact that its tensile to compressive strength ratio is slightly higher than that of the granite specimen. This observation could be explained by the fact that, for relatively hard materials (e.g., the granite specimen), a smooth transition from the loaded rim to the unloaded part of the specimen is not allowed, and the fracture tends to occur further away from the disc center [12]. Therefore, it can be recognized that the maximum equivalent stress of a disc specimen is co-controlled by the material stiffness and the tensile-compression ratio when the loading condition is fixed. For the purpose of simple description, the results of the Type II tests are analyzed continuously. For this test specification, two regions of dominant horizontal extension strain were located at the top and bottom of the disc prior to failure, and the minimum tensile strain was located at the disc center, as shown in Table 3. In order to quantitatively compare the distribution of tensile strain, the strain profiles across the vertical diameter are shown in Figure 8. It can be seen that the strain concentration at the loading points on both ends of the granite and basalt specimen is the most severe, which may indicate a localized premature failure. The fracturing process of specimen G2-1 is investigated as an example.