## 2 Determining \(G\) for I-II Mixed Mode Crack
### Discussion of Numerical Results
#### 2.2.2 Experiment Verification  
To research the law of crack propagation under combined loading, the uniaxial oblique crack experiment was carried out. For an oblique crack under a tensile loading shown in Figure 10, the stress intensity factors can be calculated by Formula (5):  
\[\left. \begin{array}{l}K_{I}=\sigma\sqrt{\pi a}\sin^{2}\beta\\ K_{II}=\sigma\sqrt{\pi a}\sin\beta\cos\beta\end{array} \right\} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \(\sigma=-10.4\times 10^{5}\) MPa, \(\tau=2.5\times 10^{5}\) MPa & & & & \\ \hline \(\theta\) & \(\mathbf{k_{I}(0)}\) from ABAQUS & \(\mathbf{k_{II}(0)}\) from ABAQUS & \(\mathbf{G(0)}\) for Calculation & \(\mathbf{G}\) for Simulation \\ (deg) & (\(\mathbf{MPa\sqrt{mm}}\)) & \(\mathbf{MPa\sqrt{mm}}\) & (\(\mathbf{m}\)) & (\(\mathbf{m}\)) \\ \hline \(30^{\circ}\) & \(1.31\times 10^{5}\) & \(9.90\times 10^{5}\) & 4.84 & 4.84 \\ \(60^{\circ}\) & \(-5.01\times 10^{5}\) & \(6.26\times 10^{5}\) & 3.12 & 3.17 \\ \(90^{\circ}\) & \(-7.72\times 10^{5}\) & \(1.30\times 10^{5}\) & 2.98 & 2.98 \\ \(120^{\circ}\) & \(-6.88\times 10^{5}\) & \(-2.42\times 10^{5}\) & 2.58 & 2.58 \\ \(150^{\circ}\) & \(-4.02\times 10^{5}\) & \(-3.22\times 10^{5}\) & 1.29 & 1.29 \\ \(180^{\circ}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & 0.00 & 0.00 \\ \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \(\sigma=0\) MPa, \(\tau=10^{5}\) MPa & & & & \\ \hline \(\theta\) & \(\mathbf{k_{I}(0)}\) from ABAQUS & \(\mathbf{k_{II}(0)}\) from ABAQUS & \(\mathbf{G}\) for Calculation & \(\mathbf{G}\) for Simulation \\ (deg) & \(\mathbf{MPa\sqrt{mm}}\) & \(\mathbf{MPa\sqrt{mm}}\) & (\(\mathbf{m}\)) & (\(\mathbf{m}\)) \\ \hline \(30^{\circ}\) & \(2.91\times 10^{6}\) & \(1.18\times 10^{6}\) & 47.82 & 47.82 \\ \(60^{\circ}\) & \(1.85\times 10^{6}\) & \(1.46\times 10^{6}\) & 26.84 & 26.84 \\ \(90^{\circ}\) & \(8.14\times 10^{5}\) & \(1.18\times 10^{6}\) & 10.02 & 10.01 \\ \(120^{\circ}\) & \(1.50\times 10^{5}\) & \(5.97\times 10^{5}\) & 1.84 & 1.84 \\ \(150^{\circ}\) & \(-8.86\times 10^{4}\) & \(1.02\times 10^{5}\) & 0.09 & 0.09 \\ \(180^{\circ}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & 0.00 & 0.00 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of G for two different methods. Figure 10: Uniaxial oblique crack. Reference [22] verified the validity of specimen for mixed mode fracture experiment. The composition of the 6061-T6 aluminum alloy which is used to manufacture test specimens is shown in Table 2. Three groups were divided by inclination angle (30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\)), and each group has three samples, as shown in Figure 11a. Samples are processed by middle-speed WEDM, and the accuracy is 0.2 mm. Figure 11b shows the Universal Hydraulic Testing Machine, type LD26.105. The samples with oblique crack were of the exerted displacement boundary condition on the top end at a speed of 0.2 mm/s, and fixed on the bottom end until the crack started propagation. As shown in Figure 12a, it was revealed that when the inclination angle was 90\({}^{\circ}\)(Mode I crack), the crack extended in a self-similar manner that propagated along the direction of the initial crack. While the phenomenon was unsuitable to describe 30\({}^{\circ}\) and 60\({}^{\circ}\) (mixed I-II crack) extension situations, crack propagation under these two angles deviated from the initial direction, which illustrated that the assumption for the mixed-mode crack extension path of Formula (1) was wrong. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Chemical Composition** & **Cu** & **Si** & **Fe** & **Mn** & **Mg** & **Zn** & **Cr** & **Ti** & **other** & **AL** \\ \hline Ratio & 25\% & 60\% & 70\% & 15\% & 85\% & 25\% & 16\% & 15\% & margin \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Composition of 6061-T6 aluminum alloy. Figure 11: Samples and Universal Hydraulic Testing Machine. (**a**) Three groups samples; (**b**) Universal Hydraulic Testing Machine. Figure 12: Crack propagation path.