## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Numerical Simulation of the TPB-IC Specim
Even though the deviation between the free surface and the middle position was large, according to the research of [PERSON] and [PERSON] [47], the singularity of the stress field near the free surface or corner is different from the traditional definition of the stress-strain field at the crack tip. Thus, the simulation results near the free surface of the specimen can be ignored. In the study of the effects of thickness and Poisson's ratio by [PERSON] and [PERSON] [30], the SIFs and T-stress in the middle position of the crack were taken as the simulation result, and a good research achievement was seen. Our study also used that method. The SIFs and T-stresses in the middle of the crack (z/B = 0) were taken as the finite element simulation result of the TPB-IC specimen. It was necessary to verify the correctness of the simulation results. The existing SIF formulas in the relevant literature were used to compare with the numerical results in this study. The usual formula for the dimensionless SIF of mode I fracture of three-point bending specimen is [7]:  
\[Y_{lf}\ =\ 2.9\Big{(}\frac{a}{W}\Big{)}^{1/2}-4.6\Big{(}\frac{a}{W}\Big{)}^{ \frac{3}{2}}+21.8\Big{(}\frac{a}{W}\Big{)}^{\frac{5}{2}}-37.6\Big{(}\frac{a}{W }\Big{)}^{\frac{7}{2}}+38.7\Big{(}\frac{a}{W}\Big{)}^{9/2} \tag{6}\]  
Taking the results of a/W of 0.1, 0.3, 0.5 and 0.7 for comparison, the data in Table 3 were obtained. From Table 3, it is shown that the relative error of the calculation value and the formula was within 5%, and the relative error was 1.25% which is minimal with a/W = 0.5. At the same time, according to the experimental data of [PERSON] [7], the fracture load and specimen parameters of 25 \({}^{\circ}\)C at room temperature were used in the numerical modeling of this study. The corresponding K\({}_{\rm I}\) value was obtained from the numerical solution seen Table 4. It is shown in Tables 3 and 4 that the numerical results obtained in this study had satisfactory accuracy and effectiveness. The verification of Y\({}_{\rm II}\) and Y\({}_{\rm III}\) can also be obtained by the same method. The fracture parameters K\({}_{\rm I}\), K\({}_{\rm II}\), K\({}_{\rm III}\) and T of the specimen were found by numerical simulation, and can be transformed into dimensionless expressions (Y\({}_{\rm I}\), Y\({}_{\rm II}\), Y\({}_{\rm III}\) and T*) by Equations (1)-(4), respectively. Figure 4 shows the variation of Y\({}_{\rm I}\), Y\({}_{\rm II}\) and Y\({}_{\rm III}\) under different crack height ratios for a/W = 0.1-0.7 and four groups of typical horizontal and vertical deflection angles. It can be seen from Figure 4 that all the ranges of the crack height ratio did not change the mixed-mode fracture. The crack height ratio a/W had the same effect on the variations of the dimensionless SIFs. The value of Y\({}_{\rm I}\), Y\({}_{\rm II}\) and Y\({}_{\rm III}\) increased with the increase of crack height ratio, and the change rate of Y\({}_{\rm I}\) clearly increased with high crack height ratio. When the vertical deflection angle \(\beta\) = 0\({}^{\circ}\), no matter what the values of the crack height ratio a/W and the horizontal deflection angles \(\alpha\) were, Y\({}_{\rm II}\) was almost constant at zero. In this case, the specimen was subjected to a mixed-mode I/III fracture. When the horizontal deflection angle \(\alpha\) = 0\({}^{\circ}\), similarly, Y\({}_{\rm III}\) was almost constant at zero, and the specimen suffered a mixed-mode I/II fracture. This phenomenon is consistent with the investigation of the mixed-mode fracture of [PERSON]'s edge-notched disc and semicircle disk [26; 30; 36; 37; 38]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**W (mm)**} & \multirow{2}{*}{**B (mm)**} & \multirow{2}{*}{**a (mm)**} & \multirow{2}{*}{**P (N)**} & \multicolumn{2}{c}{**K\({}_{\rm IC}\) Experimental Data**} & \multirow{2}{*}{**K\({}_{\rm IC}\) Numerical Results**} \\  & & & & [7] (**Mpm12**) & \\ \hline
10.03 & 5.15 & 4.99 & 95.6 & 0.9771 & 0.9903 \\ \hline \hline \end{tabular}
\end{table}