## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Numerical Simulation of the TPB-IC Specim

Table 4: Comparison of experimental and calculated values of mode I fracture toughness. \begin{table}
\begin{tabular}{c c c c} \hline \hline _a/W_ & \(\mathbf{Y_{\rm If}}\) & \(\mathbf{Y_{\rm I}}\) & **Relative Error** \\ \hline
0.1 & 0.8299 & 0.8090 & 2.52\% \\
0.3 & 1.5228 & 1.4887 & 2.24\% \\
0.5 & 2.6649 & 2.6316 & 1.25\% \\
0.7 & 5.6534 & 5.9142 & 4.61\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between formula values and the numerical results of the dimensionless mode I. SIF. In order to further research the relationship between the dimensionless SIF and crack deflection angles \(\alpha\) and \(\beta\), models with span S = 90 mm and crack height ratio \(\alpha/\text{W}=0.5\) were selected. All the simulation results are shown in Figure 5. The effect of the crack deflection angles on one of the fracture parameters is shown by two graphs in each row of Figure 5. For example, according to Figure 5a,b, by increasing \(\alpha\) and \(\beta,Y_{\text{I}}\) decreased but the decrease amplitudes were not the same. The decrease rate of \(Y_{\text{I}}\) increased with the increase of \(\alpha\), but it tended to be mild with the increase of \(\beta\). Each column graph (Figure 5a,c,e) in Figure 5 shows the effect of one crack deflection angle on the three fracture parameters. By increasing horizontal deflection angle \(\alpha\), the value of \(Y_{\text{I}}\) and \(Y_{\text{II}}\) became less, but the magnitude value of \(Y_{\text{III}}\) increased first and then decreased. The maximum value of \(Y_{\text{III}}\) was obtained when \(\alpha=45^{\circ}\). In general, the independent change of the crack deflection angle did not affect the effect of the other angles on the dimensionless parameters, but did affect the value size shown in Figure 5.  
Therefore, the analysis of the simulation results indicated that when the span was fixed and the crack height ratio a/W and \(\beta\) increased, a pure mode II fracture of the TPB-IC specimen could be obtained. If the span was changed, a/W and \(\beta\) of the pure mode II fracture were correspondingly changed. For example, [PERSON] [32] used short specimens with short spans to achieve a pure mode II fracture. Table 5 also shows the corresponding span S, angle value of \(\beta\) and crack height ratio a/W of the pure mode II fracture, or close to pure mode II fracture. According to the previous result, if the specimen suffers a pure mode III fracture, \(K_{\text{II}}\) needs to be kept at zero and the value of \(\alpha\) changes only under the condition of \(\beta=0^{\circ}\). However, no matter what the values of the crack height ratio a/W and \(\alpha\) were, \(K_{\text{I}}\) was not zero, as seen in Figure 5a. Thus, pure mode III fractures and mixed-mode II/III fractures cannot be realized using this specimen, which is similar to the investigations of [PERSON] [41] and [PERSON] [42]. Figure 4: The relationship between dimensionless SIFs and crack height ratio a/W for typical crack inclined angles. (**a**) The normalized SIFs in \(\alpha=0^{\circ}\) and \(\beta=0^{\circ}\). (**b**) The normalized SIFs in \(\alpha=0^{\circ}\) and \(\beta=30^{\circ}\). (**c**) The normalized SIFs in \(\alpha=30^{\circ}\) and \(\beta=0^{\circ}\). (**d**) The normalized SIFs in \(\alpha=30^{\circ}\) and \(\beta=30^{\circ}\). Figure 6 shows the variation of the dimensionless T-stress (i.e., T*) for different crack height ratios a/W. According to Figure 6, no matter what mixed-mode fracture is considered, T* increased with an increasing crack length. Furthermore, when the vertical deflection angle \(\beta\) was small, T* had a negative value for the two curves of \(\beta=0^{\circ}\) in the case of low crack height ratio. Figure 7 shows the variations between the crack deflection angles and T*, but the variations are relatively complex. As shown in Figure 7a, when \(\beta\) was large, by increasing \(\alpha\), T* decreased. When \(\beta\) was small, the change of T* cannot be clearly determined due to its relatively small value and small change.