## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Experimental Investigation of TPB-IC Specimens
This is why the displacement and fracture were the largest when \(\alpha=60^{\circ}\) and \(\beta=60^{\circ}\). The peak fracture load curves at different angles are shown in Figure 12. When the horizontal deflection angle \(\alpha\) increased, the fracture load increased in a small range globally, and the maximum change was less than 1.46 times (for the vertical deflection angle \(\beta=30^{\circ}\)). However, with the increase of vertical deflection angle \(\beta\), the fracture load increased gradually, and the maximum change was more than 3.13 times at \(\alpha=15^{\circ}\). Although certain errors were considered and reduced in the experiments, the deviation of each group of experimental fracture loads still occurred. This may be due to the distribution of internal microcracks and microdefects, or manual machining and grinding the crack tip by using blades. By substituting the critical peak load obtained from the experiment into Equations (1)-(4), where the corresponding geometric factors Y\({}_{\rm I}\), Y\({}_{\rm II}\) and Y\({}_{\rm III}\) are extracted from the above diagram (Figure 5), and referring to Table 6, the fracture toughness of different mixed-mode fractures can be obtained. The experimental results, including the fracture load and corresponding fracture  
Figure 10: The fractured TPB-IC specimen when \(\alpha=30^{\circ}\) and \(\beta=30^{\circ}\). (**a**) The front of the fractured specimens. (**b**) The back of the fractured specimens. (**c**) A close view of fracture surface. was about 0.387, and KII/KIC was about 0.307. When \(\alpha\) = 60, \(\beta\) = 0, KI/KIC was about 0.426, and KIII/KIC was about 0.361. As mentioned above, the mixed-mode fracture toughness is also related to the presence of T-stress in the TPB-IC specimens. Therefore, when analyzing the fracture toughness of TPB-IC specimens, the traditional MTS criterion, MTSN criterion and MTSED criterion cannot be used for accurate predictions. However, it was shown that the generalized maximum tangential strain energy density (GMTSED) criterion can significantly improve the prediction of experimental results by considering the influence of T-stress, in addition to KI, KII and KIII. Under the loading condition of general mixed-mode fractures, the elastic stress field at the crack tip can be expressed as a tensor S:  
\[\text{S}\ =\ \left[ \begin{array}{ccc}\sigma_{r}&\sigma_{r\theta}&\sigma_{rz}\\ \sigma_{\theta r}&\sigma_{\theta}&\sigma_{\theta z}\\ \sigma_{\text{zr}}&\sigma_{\theta z}&\sigma_{z}\end{array} \right] \tag{8}\]  
where \(\sigma_{r}\), \(\sigma_{\theta}\), \(\sigma_{z}\), \(\sigma_{r\theta}\), \(\sigma_{z0}\) and \(\sigma_{rz}\) are the stress components near the crack tip, which are available in literature [54], and r and \(\theta\) are the coordinates of the crack tip, as illustrated in Figure 13.  
The stress state in the initial coordinate plane (\(\theta\)-z) was assumed to be equal to the angle of rotation of element \(\phi\) around the r-axis at the initial stage of crack propagation (Figure 14), so the stress tensor after coordinate transformation \(S^{\prime}\) is expressed as:  
\[S^{\prime}\ =\ \text{QSQ}^{T}\ =\ \left[ \begin{array}{ccc}\sigma_{r}&\sigma_{ r\theta^{\prime}}&\sigma_{rz^{\prime}}\\ \sigma_{\theta^{\prime}r}&\sigma_{\theta^{\prime}}&\sigma_{\theta^{\prime}z^{ \prime}}\\ \sigma_{Zr^{\prime}}&\sigma_{Zr^{\prime}}&\sigma_{z^{\prime}}\end{array} \right] \tag{9}\]  
Figure 13: Mixed-mode I/II/III stress fields near the crack tip [46]. Q is the transformation matrix and is expressed as:  
\[\text{Q}\ =\ \left[ \begin{array}{ccc}1&0&0\\ 0&cos\phi&-sin\phi\\ 0&sin\phi&cos\phi\end{array} \right] \tag{10}\]  
The GMTSED criterion considering the combined effect of tangential stress and tangential strain in three-dimensional fracture parameter was used to predict the fracture test results of sandstone materials.