## 4 Discussion  
The coplanar growth rates were plotted against \(\Delta K_{\rm I}\), \(\Delta K_{\rm II}\), and \(\Delta K_{\rm Jeff}\) for RP; the effective values were obtained considering that the crack closure and locking ratios were changed with respect to \(\delta\). \(\Delta K_{\rm II}\) gave satisfactory correlations compared to \(\Delta K_{\rm I}\), suggesting that the crack growth rates were more influenced by mode II loading than by mode I loading. Moreover, \(\Delta K_{\rm Jeff}\) also provided relatively satisfactory correlations; however, these correlations were not sufficient because the crack growth contributions of mode I and II loading cycles were not mutually independent and, hence, the single mode range alone could not represent the crack growth rates uniquely. Therefore, the crack growth rates were plotted against \(\Delta K_{\rm v}\), which considered the interaction of the two loading cycles; although \(\Delta K_{\rm v}\) was invented for proportional loading, it gave satisfactory correlations. Figure 16: Ranges of (**a**) crack tip opening and (**b**) sliding and (**c**) residual crack tip opening displacement of RP (solid line) and RF (dashed line). (\(\Delta K_{\rm{I}}\)/\(\Delta K_{\rm{I}}\) = 1.5, \(\delta\) = 30\({}^{\circ}\)). The branch crack growth rates could not be represented by a single line, as shown in Figure 11. All the cruciform specimens were tested under biaxial loading conditions and, in these cases, the non-singular stress called the \(T\)-stress acted at the crack tips. If the crack angle is not 45\({}^{\circ}\), as in the branch crack case, the \(T\)-stress affects the plasticity near the crack tip and the plastic zone size increases with the shear part of the loading. Because a larger plastic zone yields higher growth rates [19], a separation of the data for \(\Delta K_{\rm II}/\Delta K_{\rm I}=2.5\) and \(\Delta K_{\rm II}/\Delta K_{\rm I}=2.0\) was observed, i.e., when the rate of shear loading was higher, the growth rates became higher. The fractographic observation by SEM showed no clear striation patterns on the fracture surfaces near the crack tips, while roughness due to friction was observed in the case of RF1 and WT3. Such surface damage usually arises from the interaction between the crack faces under shear mode loading, which was reported in [PERSON] et al. [23]; therefore, the main crack growth mechanism observed in this study was assumed as caused by mode II loading. The experimental results suggested that a \(\delta\) increase could encourage crack branching. This can be elucidated by FEA; when \(\delta\) increased, \(\Delta\alpha_{\rm max}/\Delta\tau_{\rm max}\) increased and the \(\Delta\alpha_{\rm max}\) plane was oriented toward the branch direction, whereas the \(\Delta\tau_{\rm max}\) plane remained on the coplanar plane. The widely accepted criterion to predict the crack growth direction under non-proportional loading states that the crack selectively grows along the fastest growth direction. Based on this criterion, the FEA results can explain the experiment results. The stresses were evaluated at the center of the elements around the crack tip, whose size was 25 \(\upmu\)m. Because the MTS range criterion based on an elasto-plastic stress field is considered, these elements should be included inside the plastic zone ahead of the crack tip. The plastic-zone size under the investigated mixed mode I/II loading was not clear. Here, the cyclic crack tip plastic zone size (\(r_{\rm p}\)) was derived from the analytical solutions based on pure mode I and mode II loading. Under the FEA conditions adopted, the size developed by mode II loading was larger than that resulting from mode I. If the size extension due to the stress redistribution is not taken into account, the size developed by mode II loading can be roughly estimated for both plane stress and strain conditions as  
\[r_{\rm p}=\left(\frac{1}{2\pi}\right)\left(\frac{\Delta K_{\rm II}}{2\tau_{ \rm ys}}\right)^{2} \tag{16}\]  
where \(\tau_{\rm ys}\) is the yield stress of the material under shear.