## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Numerical Simulation of the TPB-IC Specim  
In this study, the mixed-mode fracture parameters of the TPB-IC specimens were determined accurately based on the ABAQUS software and J-integral method. The finite element analysis defined a series of element groups around the crack tip for generating different closed paths when solving the SIFs and T-stresses. Due to the singularity of the stress-strain field at the crack tip, it was necessary to consider the singular elements at the crack tip, which can be achieved by moving the middle node of the element of the first integral loop around the crack tip to a quarter distance near the crack front [27]. The full field solution of the stress at the crack tip was obtained through finite element analysis. The three-dimensional finite element models of the TPB-IC specimen with various crack height ratios and crack deflection angles were established and analyzed. According to previous fracture studies [36], the parameters of the crack tip of the specimen models were functions of the specimen geometry (B and W), support span (S), crack length (a), load (P) and crack deflection angles (\(\alpha\) and \(\beta\)). The expressions of the fracture parameters were as follows:  
\[K_{I} = \frac{PS}{BW^{3/2}}Y_{I}(a/W,\alpha,\beta) \tag{1}\] \[K_{II} = \frac{PS}{BW^{3/2}}Y_{II}(a/W,\alpha,\beta)\] (2) \[K_{III} = \frac{PS}{BW^{3/2}}Y_{III}(a/W,\alpha,\beta)\] (3) \[T = \frac{PS}{BW^{2}}T^{*}(a/W,\alpha,\beta) \tag{4}\]  
where YI, YII and YIII and T* are the dimensionless SIFs and T-stresses, which are functions of the crack height ratio a/W, horizontal deflection angle \(\alpha\) and vertical deflection angle \(\beta\), respectively. In order to ensure the accuracy of the simulation results, the crack tip region was densified and optimized, and different element sizes of the same models were compared. The results showed that different element sizes had little effect on the results, and the error was less than 1%, as seen in Table 1. The origin of the values in Table 1 will be explained later. Considering comprehensively, 2 mm was determined as the element size for this simulation. The finite element analysis of the specimen models consisted of more than 95,000 solid elements (c3d15 elements at the crack tip, c3d20 elements in the surrounding area near the crack and c3d10 elements far away from the crack), allowing construction of a three-dimensional fracture model with different crack height ratios, and different horizontal and vertical deflection angles. Figure 2 shows a three-dimensional grid pattern of the specimen and a zoomed view of the crack tip region. The material of model is assumed to be linear elastic and isotropic. The material characteristics and geometric dimensions of the TPB-IC specimen for finite element analysis are shown in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Element Sizes** & **T*** & **YI** & **YII** & **YIII** & **Computation Time** \\ \hline
**1 mm** & 3.5864 & 1.2778 & 0.4694 & 0.5165 & 2967 s \\
**2 mm** & 3.5736 & 1.2767 & 0.4698 & 0.5172 & 66 s \\
**3 mm** & 3.5712 & 1.2745 & 0.4687 & 0.5186 & 19 s \\
**4 mm** & 3.5926 & 1.2722 & 0.4691 & 0.5194 & 7 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless T-stresses and SIFs for the finite element model of the TPB-IC specimen of different element sizes when a/W = 0.5, \(\alpha\) = 30\({}^{\circ}\) and \(\beta\) = 30\({}^{\circ}\). The fixed displacement constraints were set at the bottom of the TPB-IC specimen. As shown in Figure 2, the constraint of horizontal and vertical directions on the left side was 45 mm from the center of the bottom, and the constraint of vertical direction on the right side was 45 mm from the bottom center, so as to achieve static constraint of the TPB-IC specimen. The distances between two sections of displacement constraint and the center of the TPB-IC specimen were equal, and the constraint distance (i.e., span S) was fixed at 90 mm. The crack was a natural crack without any constraints.