## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Experimental Investigation of TPB-IC Specimens
Therefore, according to the definition of the GMTSED criterion, the tangential strain energy density factor can be calculated from:  
\[C\ =\ \frac{1}{2}r\sigma_{\varphi}\epsilon_{\theta^{\prime}} \tag{11}\]  
Due to the three-dimensional mixed-mode fracture, the plane strain state of \(\sigma_{Z}\) was adopted. According to [PERSON]'s law, \(\epsilon_{\theta^{\prime}}\ =\ \left[\sigma_{\theta^{\prime}}\ -\
u(\sigma_{r}+ \sigma_{Z^{\prime}})\right]/E\), therefore the tangential strain energy density factor \(C\) is:  
\[C\ =\ \frac{1}{4\pi E}\Big{\{}m_{11}K_{i}^{2}+m_{22}K_{II}^{2}+m_{30}K_{III}^{2} +m_{44}(B_{I}\mathcal{K}_{eff})^{2}+m_{12}K_{i}K_{II}+m_{33}K_{I}K_{III}+m_{23 }K_{II}K_{III}+(m_{14}K_{I}+m_{24}K_{II}+m_{34}K_{III})B_{I}\mathcal{K}_{eff} \Big{\}} \tag{12}\]  
where \(\eta\) is the relative critical distance defined as \(\eta\ =\ \sqrt{\frac{2r}{\delta}}\). [PERSON] [45] and [PERSON] [46] used \(\alpha\) to represent the relative critical distance, however, \(\alpha\) was used to represent horizontal deflection angle in the previous section, so \(\eta\) was used here. All the coefficients (\(m_{ij}\)) are shown in Tables 10 and 11. According to the hypothesis of the GMTSED criterion, a mixed-mode fracture occurs when the maximum tangential strain energy density factor C reaches a maximum under the critical load near the crack tip. The angle and condition of crack propagation based on the GMTSED criterion are as follows:  
\[\frac{\partial C}{\partial\theta}\ =\ 0,\frac{\partial C}{\partial\phi}\ =\ 0 \tag{13}\]  
and sufficient conditions:  
\[\frac{\partial^{2}C}{\partial\theta^{2}}\Bigg{\{}0,\frac{\partial^{2}C}{ \partial\theta^{2}}\frac{\partial^{2}C}{\partial\phi^{2}}-\left[\frac{ \partial}{\partial\phi}\left(\frac{\partial C}{\partial\theta}\right)\right] ^{2}\Bigg{\}}0 \tag{14}\]  
The second hypothesis can also be written as:  
\[C(\theta_{0},\phi_{0})\ =\ C_{cr} \tag{15}\]  
Figure 14: Stress state in the \(\theta\)–z plane [46]. where \(\theta_{0}\) and \(\phi_{0}\) are the in-plane and out-of-plane initiation angles, respectively, and are calculated from Equations (13)-(15). When a pure I mode fracture occurs, \(K_{II}~{}=~{}K_{III}~{}=~{}0\), \(\theta_{0}~{}=~{}\phi_{0}~{}=~{}0\), \(K_{I}~{}=~{}K_{IC}\), \(C_{cr}\) is expressed using \(K_{IC}\) as:  
\[C_{cr}~{}=~{}\left\{ \begin{array}{ll}\frac{1}{4\pi E}[1-(2\
u+1)\
u-2\
u B_{I} \eta]K_{IC}^{2}&(plane\;strain)\\ \frac{1}{4\pi E}(1-\
u-\
u B_{I}\eta)K_{IC}^{2}&(plane\;stress)\end{array} \right. \tag{16}\]  
where \(B_{I}\) is the biaxiality ratio for the pure mode I loading. Part of the derivation process can refer to Hua [46]. When different fracture parameters (i.e., \(K_{I}\), \(K_{II}\), \(K_{III}\) and \(T\)) are confirmed, the in-plane and out-of-plane fracture angles can be obtained by Equations (13) and (14), and the mixed-mode fracture toughness ratio can also be predicted by Equations (12) and (16). Figure 15 shows the relationship between mixed-mode I/II and I/III fracture toughness ratios under the plane stress condition. It can be seen that the GMTSED criterion had a high sensitivity in predicting I/II mixed fractures. The ratio of pure mode II fracture toughness change under different positive and negative T-stresses was more than 220%. However, the prediction of a mixed-mode I/III fracture using GMTSED criterion was very poor. As shown in Figure 16, the predicted value of the GMTSED criterion under the plane strain condition with different T-stress had a good sensitivity to I/II and I/III mixed-mode fracture toughness when \(\
u=0.35\), where the value of pure III mode fracture toughness decreased regardless of the positive and negative T-stresses. Therefore, T-stress cannot be ignored in investigations of mixed-mode fractures. In mixed-mode fracture experiments, the mixed-mode fracture toughness is reduced under the influence of positive T-stress, which makes the material more vulnerable to failure.