## 1 Introduction  
When railway wheels repeatedly pass over the rails, rolling contact fatigue (RCF) cracks occur on both the wheel tread and the railhead. The accumulation of large plastic shear strains generates these cracks, which initially grow in the plastic flow direction of the resulting microstructure [1]; when their growth reaches a depth of several millimeters, they are unaffected by the plastic deformation layer and are then subjected to a mixed mode I/II/III loading, growing as coplanar cracks at shallow angles to the surface [2; 3; 4; 5]. The mode II/III loading cycle is almost proportional, while the mode I/II and I/III cycles are non-proportional. [PERSON] et al. [6] conducted experiments in non-proportional mixed mode I/II/III loading conditions to investigate the load path effect on fatigue crack growth (FCG) in 316L stainless steel. They applied the'star' load path to a cruciform specimen containing an inclined crack by using a six-actuator servo-hydraulic testing machine; the addition of mode III loading steps to a mode I/II loading sequence increased the FCG rate even when the crack path was not significantly modified. However, the tested material was not rail or wheel steel and the load history did not simulate the one experiencing RCF cracks. Moreover, only a coplanar crack growth below 2 mm was obtained. Many studies focused on the coplanar crack growth of rail steel under non-proportional mixed mode I/II loading cycles. Figure 1 shows how mode I and II loadings generated in the rail and wheel when the fluid penetrates into the crack. Details will be described in the next section. [PERSON] et al. [7]performed fatigue tests, sequentially applying cyclic mode I and II loadings on rail steel biaxial cruciform specimens. When they removed the mode I loading before applying the fully reversed mode II and the ratio between the nominal range values of the mode I and II stress intensity factors (SIFs), \(K_{\text{I}}\) (\(\Delta K_{\text{I}}\)), and \(K_{\text{II}}\) (\(\Delta K_{\text{II}}\)), were greater than or equal to 0.5, a long coplanar crack growth was observed. [PERSON] et al. [8,9] carried out sequential mixed mode I/II loading experiments on cruciform specimens; when they applied mode II loading before removing mode I, a degree of overlap (\(\delta\)) appeared. The effective values of \(\Delta K_{\text{I}}\) and \(\Delta K_{\text{II}}\) (\(\Delta K_{\text{Ieff}}\) and \(\Delta K_{\text{Ieff}}\), respectively) and the \(\delta\) effect on the crack growth were considered and crack growth models were proposed in the \(\Delta K_{\text{Ieff}}\) and \(\Delta K_{\text{Ieff}}\) forms. They reported that \(\Delta K_{\text{Ieff}}\) is a control parameter determining the crack growth direction and that the \(\delta\) increase encourages crack branching; they also proposed a branch criterion in terms of \(\Delta K_{\text{Ieff}}\) and \(\delta\). [PERSON] and [PERSON] [10] conducted non-proportional mixed mode I/II FCG tests on rail and wheel steels in practical use in Japan. Based on the study of [PERSON] et al., they defined crack growth models using \(\Delta K_{\text{Ieff}}\) and \(\Delta K_{\text{Ieff}}\) and found that the cracks branched easily in wheel steel compared to rail steel. They also examined the fracture surfaces, but no clear striation pattern was observed. Sequential mode I/II FCG tests on rail steel were also performed by [PERSON] and [PERSON] [11]. They used tubular specimens and gold microgrids with a 4 mm pitch, which were laid ahead of the crack tips and along the crack surfaces to observe the residual shifts; hundreds of microns of constant-speed coplanar growth were obtained when \(\Delta K_{\text{II}}/\Delta K_{\text{I}}\) ranged from 1 to 4. They also investigated the role of plasticity and crack face friction via finite element analysis (FEA) and concluded that the maximum growth rate criterion rationalized the crack path observed in the non-proportional loadings. [PERSON] et al.