## 3 Finite Element Analysis  
### Procedure  
A series of elasto-plastic FEA was performed using the commercial FEA code MARC to elucidate the FCG characteristics observed in the mixed mode I/III loading tests and predict the FCG direction, which is \(\theta\) and \(\varphi\) in Figure 5. Therefore, a stationary crack was considered. The three-dimensional (3D) finite element mesh of the round bar specimen and the boundary conditions applied are shown in Figure 16. The plasticity was considered highly localized near the notch under the loading, while other specimen regions remained elastic. Therefore, the area close to the symmetry section for the axial direction was enough for the mesh. The mesh density increased at the crack tip where, for the radial direction, the size of an 8-node brick element was 10 \(\upmu\)m, which was considered well included in the plastic zone ahead. The total numbers of elements and nodes were 14,001 and 15,522, respectively. This mesh and the loading procedure confirmed that the nominal stress value in the ligament ahead of the crack was accurate.  
Figure 15: Fracture surface of RP3 observed using a SEM (arrows indicate the direction of the crack growth). (**a**) Macroscopic appearance of the fracture surface, (**b**) 3 mm from the notch tip (\(\delta=30^{\circ}\)), and (**c**) an enlarged view indicated by a frame in (**b**).  
At the center of the upper surface of the mesh, a pilot node was connected to the remaining nodes of that surface through a multipoint constraint; force (\(P\)) and torque (\(T_{\rm a}\)) were applied to the node; their ranges \(\Delta P\) and \(\Delta T_{\rm a}\) were equal to 55 kN (\(R_{\rm I}\) = 0.05) and 525 Nm (\(R_{\rm III}\) = \(-\)1), respectively. If \(\Delta P\) and \(\Delta T_{\rm a}\) were equal to 55 kN and 525 Nm, respectively, \(\Delta K_{\rm III}/\Delta K_{\rm I}\) became 1.5. The fixed boundary conditions applied on the lower surface of the mesh were of the same as the symmetry condition for the axial direction. The \(a\) value was set at 4.6 mm for the cases of \(\delta\) = 30\({}^{\circ}\), 90\({}^{\circ}\), and 180\({}^{\circ}\) and at 7 mm for \(\delta\) = 120\({}^{\circ}\), representing the actual RP3 test.  
In this study, the model combining nonlinear kinematic hardening rule with the isotropic hardening rule developed by [PERSON] and [PERSON] (2010) (C & L model) was employed.  
\[{}^{ti+\Delta ti}\sigma_{y}={}^{0}\sigma_{y}+Q\Big{[}1-\exp\Big{(}-B^{ti+ \Delta ti}\vec{e}^{p}\Big{)}\Big{]} \tag{14}\]  
and  
\[\mathrm{d}\alpha=\frac{2}{3}h\mathrm{d}e^{p}-\zeta\alpha\mathrm{d}\vec{e}^{p} \tag{15}\]  
where \({}^{ti+\Delta ti}\sigma_{y}\) is the updated yield stress at time \(t_{i}+\Delta t_{i}\), \({}^{0}\sigma_{y}\) is the initial yield stress, \(Q\), \(B\), \(h\), and \(\zeta\) are material constants, \({}^{ti+\Delta ti}\vec{e}^{p}\) is the accumulated effective plastic strain at \(t_{i}+\Delta t_{i}\), \(\alpha\) is the shift of the yield surface center, \(e^{p}\) is the plastic strain, and d implies increment. The material constants for RP and RF are summarized in Table 4 along with Young's modulus and Poisson ratio. The FEA was performed on these two rail steels to clarify the material effect on the FCG rate.