## 4 Discussion  
In the case of RP and WT, a coplanar growth was obtained when \(\delta\) was smaller than or equal to \(90^{\circ}\) regardless of the \(\Delta K_{\rm III}\) / \(\Delta K_{\rm I}\) value and the growth rate was well correlated with \(\Delta K_{\rm s}\), as shown in Figure 9. According to Equation (6), \(\Delta K_{\rm I}\) was equal to only \(4\%\) of \(\Delta K_{\rm III}\), suggesting a strong dependence of the FCG rate on the mode III loading. Conversely, in the RF case, some factory-roof fractures were observed at \(\delta\) = \(60^{\circ}\) and \(90^{\circ}\), and no good correlation with \(\Delta K_{\rm s}\) was obtained. The FEA results shown in Figure 21 indicated that the CTOD variation in one loading cycle was smaller for RF than for RP under the same loading condition. In non-proportional mixed mode loading cycles, fatigue cracks follow the direction of the \(\Delta\sigma_{\rm max}\) or \(\Delta\tau_{\rm max}\) plane depending on if these growth rates on these planes is faster [23]. When the CTOD is small, the actual crack faces have irregularities and, hence, are likely to make contact. When the crack faces make contact, the friction can make the coplanar FCG rate smaller relative to the branch FCG rate. Therefore, factory-roof fractures, which are traces of branch FCG, should have appeared. Figure 21: Variations of half the crack tip opening displacements behind the crack tip at 160 th cycle of (**a**) RP and (**b**) RF. (\(\Delta K_{\rm III}\)/\(\Delta K_{\rm I}\) = 1.5, \(\delta\) = \(90^{\circ}\)). When \(\delta\) increased to 120\({}^{\circ}\), branch cracks also grew even in RP. According to the FEA results shown in Figure 19, increasing \(\delta\) increased \(\Delta\sigma_{\text{max}}\) with respect to \(\Delta\tau_{\text{max}}\) and turned it toward the branch direction. Conversely, \(\Delta\tau_{\text{max}}\) remained on the coplanar direction. The coplanar FCG rates under the loading cycles adopted herein are considered to have a strong dependence on \(K_{\text{III}}\). Therefore, it should also have a relatively strong dependence on \(\Delta\tau_{\text{max}}\), while the branch FCG rates are generally considered to have a relation with \(\Delta\sigma_{\text{max}}\). These facts conclude that cracks tend to branch when \(\delta\) increases. We determined that when \(\delta\) increases, the \(\sigma_{1\,\text{max}}\) plane turns toward the branch direction, as indicated in Table 5. However, it may be unsuitable to use the MPS criterion to predict the FCG direction under the loading conditions adopted herein. This criterion assumes that the crack grows in mode I. When the coplanar FCG planes were observed using SEM, no clear striation, which is evidence of FCG caused by mode I loading, was observed. The SEM observations provided no evidence to support crack face contact in case of RP at \(\Delta K_{\text{III}}/\Delta K_{\text{I}}=1.5\) and \(\delta=90^{\circ}\). This was confirmed by the FEA results, in which the crack was always widely open (see Figure 21a). Rubbing has been suggested as the reason for the FCG termination. Under the testing conditions in the present study, the crack faces were opened during the experiments. If the crack face contacts with its mating face and the generated friction attenuates the mode III loading, the crack is considered to be arrested. If there is no contact, a long shear mode crack becomes possible under this loading condition. Although the branch FCG rates were plotted against \(\Delta K_{\text{I}}\), a good correlation was not obtained because \(\Delta K_{\text{I}}\) does not consider the \(\Delta K_{\text{III}}\) attenuation due to the crack face contact. Therefore, \(\Delta K_{\text{eq}}\), which includes this attenuation, was proposed, providing a FCG law with a fairly good correlation. Even for the RF steel, where some factory-roof fractures arose on the fracture surface, a better correlation was obtained when using \(\Delta K_{\text{eq}}\) (see Figure 12). In these cases, the occurrence of some crack face contacts was also considered.