## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Numerical Simulation of the TPB-IC Specim
All finite element analysis results for the SIF and T-stress on different crack height ratios, horizontal deflection angles and vertical deflection angles were obtained. In the finite element model, the standardized parameter of the crack tip along the thickness direction was defined as z/B. This means that z/B = 0 refers to the middle position of the crack, and z/B = \(\pm\)1 refers to the free surface of the TPB-IC specimen. The mode I SIF at the middle section of the thickness, i.e., z/B = 0, was defined as Klim, and the T-stress in the middle section was defined as Tm. The effective SIF was defined as:  
\[K_{eff}\ =\ \sqrt{K_{I}^{2}+K_{II}^{2}+K_{III}^{2}} \tag{5}\]  
By using the finite element analysis results of the crack tip along the thickness divided by Klim and Tm respectively, the normalized SIFs and T-stress can be obtained. Figure 3 shows the normalized results of four groups of crack deflection angles (\(\alpha\) = 0\({}^{\circ}\), \(\beta\) = 0\({}^{\circ}\), \(\alpha\) = 0\({}^{\circ}\), \(\beta\) = 30\({}^{\circ}\), \(\alpha\) = 30\({}^{\circ}\), \(\beta\) = 0\({}^{\circ}\) and \(\alpha\) = 30\({}^{\circ}\), \(\beta\) = 30\({}^{\circ}\)) under the conditions of crack height ratio a/W = 0.5 and span S = 90 mm. According to Figure 3, the three-dimensional effect on the SIFs and T-stress can be observed at different positions of the crack tip. When \(\alpha\) = 0\({}^{\circ}\), \(\beta\) = 0\({}^{\circ}\) (Figure 3a), Klim and Klim were always zero, so the pure mode I fracture can be well realized. In the middle part of the crack, i.e., \(-\)0.6 < z/B < 0.6, there was almost no change in Klim. There was a deviation (about 20%) near the free surface (z = \(\pm\) 1). The investigations of SIFs in a three-dimensional specimen proved the SIF of the free surface decreased relative to the middle interface [36; 38]. For other angles (see Figure \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Numerical Value** \\ \hline
**S** & 90 mm \\
**B** & 20 mm \\
**W** & 40 mm \\
**a** & From 4 to 28 mm \\
**E** & 5 GPa \\
**P** & 100 N \\
**v** & 0.35 \\ \(\alpha\) & From 0\({}^{\circ}\) to 60\({}^{\circ}\) \\ \(\beta\) & From 0\({}^{\circ}\) to 60\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Material characteristics and geometric dimensions given for the finite element model of the TPB-IC specimen. Figure 2: Finite element model and zoomed view of crack tip region for the TPB-IC specimen. 3b-d), all SIFs, including the effective SIF K\({}_{\text{eff}}\), had a sudden change near the free surface, but the portion near the middle of the crack tended to be stable. When \(\alpha=0^{\circ}\) and \(\beta=30^{\circ}\), K\({}_{\text{III}}\) tended to be 0 in the middle position, and the specimen was subjected to a mixed-mode I/II fracture. When \(\alpha=30^{\circ}\), \(\beta=0^{\circ}\), K\({}_{\text{II}}\) tended to be 0 in the middle position, so the specimen was mainly subjected to a mixed-mode I/III fracture. When \(\alpha=30^{\circ}\), \(\beta=30^{\circ}\), the specimen was subjected to a mixed-mode I/II/III fracture. It is worth noting that the negative value here only represents the direction. In order to visualize the size relationship of the SIFs, the absolute values of the SIFs are used directly in later sections. In the four groups of crack deflection angles, the crack tip normalized T-stresses are shown in Figure 3e when \(-1\leq\text{z}/\text{B}\leq 1\). The variation of the normalized T-stress was similar to the SIFs along the thickness direction. There was almost no change in the middle position, but the deviation of normalized T-stress at the free surface was more obvious than the SIFs for different horizontal and vertical deflection angles. For example, when \(\alpha=30^{\circ}\), \(\beta=30^{\circ}\), there was a difference of more than 25 times between the T-stress of \(\text{z}/\text{B}=\pm 1\) at the free surface and the middle of the model.