## 3 Finite Element Analysis
### Analytical Results  
The analytical conditions (Table 5) were decided based on the actual experiments described in Section 2. Figures 14 and 15 show some results about the distribution of the tangential (\(\sigma_{\theta}\)) and shear stresses (\(\tau_{\mathrm{r\theta}}\)), respectively, in the first loading cycle for ARP2; the maximum \(\Delta\sigma_{\theta}\) appeared between \(-45^{\circ}\) and \(-67.5^{\circ}\), whereas the maximum \(\Delta\tau_{\mathrm{r\theta}}\) was observed exactly at \(0^{\circ}\).  
Figure 12: Fracture surface near the crack tip; (**a**) RP2, (**b**) WT3, and (**c**) RF1. Arrows indicate the direction of crack growth.  
When using the C&L model, several loading cycles are required to achieve a steady-state stress cycle; therefore, at least 100 loading cycles were initially planned for the simulation. However, as the number of cycles increased, the deformation of the crack tip elements increased accordingly and, hence, an accurate analysis was considered no longer possible after the 50 th cycle. Thus, the evaluations were performed at the 50 th loading cycle and, at this point, a steady-state stress cycle could not be achieved.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(\mathbf{E}\) (MPa) & \(\mathbf{\
u}\) & \(\mathbf{{}^{0}}\mathbf{\sigma_{y}}\) (MPa) & \(\mathbf{Q}\) & \(\mathbf{b}\) & \(\mathbf{h}\) (MPa) & \(\mathbf{\zeta}\) \\ \hline RP & 183,008 & 0.3 & 508 & \(-\)208 & 24.2 & 85,248 & 193 \\ \hline RF & 182,778 & 0.3 & 684 & \(-\)264 & 1.27 & 88,615 & 185 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Material properties used in FEA.  
Figure 13: Finite element mesh and boundary conditions for the cruciform specimen under mixed mode I/II loading.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & \(\Delta\mathbf{K_{\Pi}}\)/\(\Delta\mathbf{K_{\text{I}}}\) & \(\mathbf{\delta}\) (degree) & \(\mathbf{F}\) (N) & \(\mathbf{S}\) (N) & **Material** \\ \hline ARP1 & 1.5 & 90 & 0 \(\leftrightarrow\) 26,667 & \begin{tabular}{c} \(-\)20,000 \(\leftrightarrow\) \\ 20,000 \\ \end{tabular} & RP \\ \hline ARP2 & 1.5 & 120 & 0 \(\leftrightarrow\) 26,667 & \begin{tabular}{c} \(-\)20,000 \(\leftrightarrow\) \\ 20,000 \\ \end{tabular} & RP \\ \hline ARP3 & 1.5 & 30 & 0 \(\leftrightarrow\) 26,667 & \
\begin{tabular}{c} \(-\)20,000 \(\leftrightarrow\) \\ 20,000 \\ \end{tabular} & RP \\ \hline ARF1 & 1.5 & 30 & 0 \(\leftrightarrow\) 26,667 &
\begin{tabular}{c} \(-\)20,000 \(\leftrightarrow\) \\ 20,000 \\ \end{tabular} & RF \\ \hline \hline \end{tabular}
\end{table}
Table 5: FEA conditions.  
Figure 14: Tangential stress distribution near the crack tip for ARP2 (\(\Delta K_{\rm II}\)/\(\Delta K_{\rm I}\) = 1.5, \(\delta\) = 120\({}^{\circ}\)) at the (**a**) 90\({}^{\circ}\) (maximum \(K_{\rm I}\)), (**b**) 150\({}^{\circ}\) (maximum \(K_{\rm II}\)), and (**c**) 330\({}^{\circ}\) (minimum \(K_{\rm II}\)); contour levels in MPa and the deformation magnification is 5\(\times\).  
Figure 15: Shear stress distribution near the crack tip for ARP2 (\(\Delta K_{\rm II}/\Delta K_{\rm I}=1.5\), \(\delta=120^{\circ}\)) at the (**a**) \(90^{\circ}\) (maximum \(K_{\rm I}\)), (**b**) \(150^{\circ}\) (maximum \(K_{\rm II}\)), and (**c**) \(330^{\circ}\) (minimum \(K_{\rm II}\)); contour levels in MPa and the deformation magnification is \(5\times\).