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.
Fremy 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. Bold 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),
KI (
ΔKI), and
KII (
ΔKII), were greater than or equal to 0.5, a long coplanar crack growth was observed. Wong 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 (
δ) appeared. The effective values of
ΔKI and
ΔKII (
ΔKIeff and
ΔKIIeff, respectively) and the
δ effect on the crack growth were considered and crack growth models were proposed in the
ΔKIeff and
ΔKIIeff forms. They reported that
ΔKIeff is a control parameter determining the crack growth direction and that the
δ increase encourages crack branching; they also proposed a branch criterion in terms of
ΔKIeff and
δ. Akama and Susuki [
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 Wong et al., they defined crack growth models using
ΔKIeff and
ΔKIIeff 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 Doquet and Pommier [
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
ΔKII/
ΔKI 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. Doquet et al. [
12] investigated through FEA the influence of static and 90° out-of-phase cyclic mode II loading superimposed to the cyclic mode I on the plasticity- and asperity-induced closure for an aluminum alloy. The applied mode II could either increase or decrease
ΔKIeff depending on the stress ratio and loading path; in some cases, it had opposite effects on the two closure mechanisms. In the case of 90° out-of-phase mixed mode loading, mode II drastically reduced or even suppressed the plasticity-induced closure. To clarify the incipient direction of the crack path observed by Bold et al. [
8], Dahlin and Olsson [
13] investigated several criteria while using the FEA approach; they concluded that a maximum tangential stress (MTS) range criterion based on an elasto–plastic stress field can appropriately predict the incipient crack growth direction in ductile metals. Yu et al. [
14] studied the FCG behavior in a thin-walled tubular specimen of aluminum alloy under non-proportional mixed mode I and II loading, showing that the FCG significantly differed from that under mode I or proportional mixed mode loading. Although they obtained a long and stable shear mode growth, they could not apply the commonly accepted MTS criterion in most of the non-proportional loading cases.
As indicated above, previous studies on coplanar crack growth under non-proportional mixed mode I/II loading cycles apparently focused on rail steel. Only Akama and Susuki [
10] considered also the wheel steel case, but their elucidation of the phenomena associated with crack growth under such loading cycles was not sufficient. In particular, the reason why the growth direction of the crack changes depending on the loading conditions and the type of steel was not clarified. Besides, the crack growth models by Wong et al. [
9] have been pointed out as arbitrary. Therefore, in this study, crack growth data from non-proportional mode I/II loading cycles were re-constructed by using a reliable equivalent SIF range and FEA was performed to elucidate the crack growth behavior under these loading cycles. The reliable criteria for predicting the crack path direction of non-proportional mixed mode loading were used, and the results were compared with corresponding experimental results.
This paper, part 1 of two companion papers, presents FCG under non-proportional mixed mode I/II loading and is organized as follows. The introduction starts in this section with a brief overview of past papers for the crack growth under non-proportional mixed mode I/II loading cycles.
Section 2 describes the detailed method of the experiments conducted and presents the experimental results. In
Section 3, the FEA model for predicting the crack path direction is presented and the results are indicated.
Section 4 gives detailed considerations and discussions by comparing the experimental and FEA results. Finally, the important results obtained in this study are summarized in
Section 5.
4. Discussion
The coplanar growth rates were plotted against ΔKI, ΔKII, and ΔKIeff for RP; the effective values were obtained considering that the crack closure and locking ratios were changed with respect to δ. ΔKII gave satisfactory correlations compared to ΔKI, suggesting that the crack growth rates were more influenced by mode II loading than by mode I loading. Moreover, ΔKIeff 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 ΔKv, which considered the interaction of the two loading cycles; although ΔKv was invented for proportional loading, it gave satisfactory correlations.
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°, 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
ΔKII/
ΔKI = 2.5 and
ΔKII/
ΔKI = 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 Fujii 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 δ increase could encourage crack branching. This can be elucidated by FEA; when δ increased, Δσmax/Δτmax increased and the Δσmax plane was oriented toward the branch direction, whereas the Δτmax 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 μm. 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 (
rp) 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
where
τys is the yield stress of the material under shear. Based on all the conditions analyzed using RP, which were
ΔKII = 16.6 MPa√m and
τys = 293 MPa, the
rp was calculated as 128 μm. Therefore, the points at which the stresses were evaluated were still far inside the plastic zone.
The stress evaluation was performed in the 50th loading cycle, although the steady-state stress distribution was not achieved because the distortion of elements increased, deteriorating the FEA accuracy. MARC provides a function to subdivide the mesh when the element distortion is large; one of the criteria is the maximum change in an interior angle from the initial angle for triangle elements and the recommended angle is 40°. For the ARP2 case, this criterion was violated at the 100th cycle and, hence, the stresses were evaluated at the 50th cycle with a margin.
However, remeshing was not performed in this study; the crack growth was expected to occur during the loading periods under the present testing conditions. In the ARP2 case, for example, ΔKI and ΔKII were 11.1 MPa√m and 16.6 MPa√m, respectively. Based on the crack growth law expressed in Equations (10) and (11), the crack growth length in 50 loading cycles was 3.4 μm, and the stress redistribution near the crack tip occurred according to this crack extension.
ΔCTOD and ΔCTSD were obtained for both RP and RF via FEA since they can influence the crack growth rate. In fact, a vector crack tip displacement (CTD) criterion has been proposed by Li [
24] to correlate mixed mode FCG rates. This criterion was based on the concept that the vector CTD range (ΔCTD) is the driving force for FCG. Here, the vector ΔCTD corresponded to the vector summation of ΔCTOD and ΔCTSD. The FEA results indicated that ΔCTOD and ΔCTSD of RF were smaller than those of RP, probably because the former was a high-strength steel (
Table 2); besides, RF exhibited much smaller residual CTOD and its opening distance under no loading compared to RP. Since there were irregularities on the crack faces, there is a possibility that the mode II loading might have been attenuated, especially in RF. This implies that the crack growth rate in RF was lower than that in RP, explaining the experimental results.
5. Conclusions
Fatigue tests were conducted on rail and wheel steel specimens to obtain FCG rates under mixed mode I/II loading cycles, which can simulate rolling contact conditions, by using an in-plane biaxial fatigue machine. Simplified cycles were applied to the analysis of RCF cracks, including the effect of fluid trapped inside the cracks. To elucidate the experimental results, a FEA was also performed. The main findings can be summarized as follows.
(1) When the stress intensity range ratio between mode II and I increased, the crack tended to branch. This happened because, when this ratio increased, the direction of the maximum tangential stress range changed from coplanar growth to branch one.
(2) As the degree of overlap between the mode I and II stress intensities increased, the crack easily branched; in particular, the MTS range became superior to the maximum shear stress range and its plane oriented toward the branch direction.
(3) Coplanar crack growth rates were lower in head hardened rail steel than in normal rail steel. This was due to the smaller CTDs in head hardened rail steel, which can influence the growth rate, compared to normal rail steel.
(4) The branch crack growth rate varied considerably depending on the stress intensity range ratio even within the same rail steel and, unlike the coplanar growth rate, could not be correlated by a single line because the plastic zone size increased with the shear part of the loading as a result of the T-stress and this larger plastic zone gave higher growth rates.
(5) Clear striation patterns were not found near the crack tip region. Therefore, the main driving force for the crack growth under the investigated non-proportional mixed mode I/II loading was thought to be the mode II loading.