A. Microhardness and residual stress

The Vickers microhardness of carburized layer and matrix region was measured by using an instrumented nanoindenter G200 (Agilent Technologies, Santa Clara, CA) with the method of continuous stiffness. The values of microhardness as a function of depth from the surface, ζ, are shown in Fig. 3. It can be seen from the figure that the largest value of microhardness almost occurs near the surface. With the increasing of ζ, the value of microhardness tends to decrease and then reaches an approximately constant value with about 500 kgf/mm² at about ζ = 600 μm. This further verifies that the depth of carburized layer is about 600 μm, in accordance with the result shown in Fig. 2(b).

FIG. 3.

Microhardness of carburized layer and matrix region.

In addition, the residual stress on the round-notched surface of specimen was measured along the axis of specimen by using x-ray diffraction. Based on the sin² ψ method with Cr K_α radiation, the tube voltage of 30 kV and the tube current of 9 mA, the value of residual stress can be determined by averaging the values of 4 measurement points on the surface. The measured maximum compressive residual stress occurs on the surface and is about 150 MPa.

B. S–N characteristics

The experimentally obtained S–N diagram of the carburized gear steel under axial loading is shown in Fig. 4. Based on the preliminary SEM observation of crack initiation sites on all fracture surfaces, the interior crack-induced failure is the unique failure mode of the carburized gear steel in the life regime beyond 10⁵ cycles. In consideration of the data scatter, these test data can be divided into two parts. One part corresponds to the relatively short fatigue life (N _f < 3 × 10⁶ cycles), i.e., the common HCF regime. The other part corresponds to the relatively long fatigue life (N _f > 3 × 10⁷ cycles), i.e., the VHCF regime. The separation and transition between two parts of test data are very distinct. Therefore, the duplex S–N curves are used to represent the S–N characteristics of the carburized gear steel in the life regime beyond 10⁵ cycles under axial loading. For the test data in the HCF regime, the common S–N curve with a horizontal part is plotted and indicated by a dashed line in Fig. 4. For the test data in the VHCF regime, the continuously descending S–N curve is plotted and indicated by a solid line in Fig. 4. The horizontal part of dashed line approximately corresponds to the transition stress from the HCF regime to the VHCF regime. This transition stress is about 625 MPa. Furthermore, based on the fitted S–N curve, the fatigue strength corresponding to fatigue life of 10⁹ cycles is evaluated to be 500 MPa.

FIG. 4.

Duplex S–N curves of carburized gear steel under axial loading.

In addition, it is worth to note that the duplex S–N characteristics herein presented by the carburized gear steel are different from the common duplex S–N characteristics ^{Reference Nishijima and Kanazawa6,Reference Sakai, Sato and Oguma7} associated with surface crack-induced failure and interior crack-induced failure. Only corresponding to one interior crack-induced failure, the reason why the duplex S–N characteristics still can present may be attributed to the different interior failure mechanisms in the HCF regime and in the VHCF regime. The detailed analysis will be discussed later.

C. Crack initiation and growth mechanisms

Both in the HCF regime and in the VHCF regime, the interior cracks are all induced from nonmetallic inclusions by means of the SEM observation and the energy dispersive x-ray spectrometer analysis. The main chemical composition of the inclusion is the aluminum oxide (Al₂O₃). It should be noted that these inclusions are all confined to the interior matrix region. This means that the inclusions within the carburized layer hardly become crack nucleus. To a great extent, it is attributed to the effect of carburized layer. Compared with the carburized layer with higher hardness, the matrix is softer. Therefore, the strain inconsistency between the inclusion and the ambient matrix under cyclic loading is more intense, so the crack is more easily initiated from the interface between them.

A propagating crack-shaped-like fisheye can be observed on the fracture surface, as shown in Figs. 5(a), 5(c), and 5(e). Approximately, the inclusion is located at the center of fisheye. In Fig. 5(a), the somewhat coarse area at the right bottom of the fracture surface is the remaining cross section that existed at the time of fracture and named as “momentary fracture zone (MFZ)”. The region between the fisheye and the MFZ is named as “final crack growth zone (FCGZ)”. It should be noted that the shapes of fisheye and FCGZ can change with the position change of inclusion on the fracture surface. The reason is that the crack growth toward the carburized layer is inevitably restricted due to the higher strength of carburized layer, and the degree of restriction tends to increase with the increase of crack extension. As a result, the rate of crack growth toward the center matrix region is much faster than that toward the carburized layer. Therefore, when the inclusion is closer to the carburized layer, the shape of fisheye is more or less asymmetric, and the shape of FCGZ becomes irregular, as shown in Fig. 5(a). Conversely, when the inclusion is closer to the center of fracture surface, the shapes of fisheye and FCGZ get more and more circular due to the comparatively consistent crack growth rate, as shown in Fig. 5(c).

FIG. 5.

Observation of fracture surfaces: (a) fisheye, FCGZ, and MFZ (σ_a = 700 MPa, N _f = 1,605,400); (b) inclusion without FGA (σ_a = 700 MPa, N _f = 1,605,400); (c) circular FCGZ outside fisheye (σ_a = 750 MPa, N _f = 501,600); (d) inclusion without FGA (σ_a = 750 MPa, N _f = 501,600); (e) fisheye (σ_a = 600 MPa, N _f = 242,416,200); (f) FGA around inclusion (σ_a = 600 MPa, N _f = 242,416,200).

For the interior failure in the HCF regime, the FGA cannot be observed around the inclusion, as shown in Figs. 5(b) and 5(d). Conversely, for the interior failure in the VHCF regime, the FGA can be observable, as shown in Fig. 5(f). From the viewpoints of dislocation movement and irreversible slip, the FGA formation process proposed by coauthor Sakai can be divided into three stages ^{Reference Sakai, Takeda, Shiozawa, Ochi, Nakajima and Nakamura10} : (A) a fine granular layer caused by the intensive polygonization is gradually induced around the subsurface inclusion during a long sequence of the cyclic loading(Formation of fine granular layer), (B) the number of microdebondings is increased gradually and some of them would coalesce to one another (Nucleation and coalescence of microdebondings), and (C) microdebondings are entirely spread over the fine granular layer, and the penny shape crack is finally formed around the subsurface inclusion of the material (completed formation of the FGA). The relevant schematic diagram is shown in Fig. 6, where the longitudinal sections passing through the inclusion are indicated in the left hand side, and the cross sections passing through the same inclusion are indicated in the right hand side. Obviously, the formation of FGA is the largest difference between the failure mechanisms in the VHCF regime and in the HCF regime. Just because of such difference, the duplex S–N characteristics in Fig. 4 are presented.

FIG. 6.

Schematic diagram of the FGA formation process.

Overall, the fatigue process of the carburized gear steel in the HCF regime can be described as inclusion → fisheye → FCGZ → MFZ, while that in the VHCF regime can be described as inclusion → FGA → fisheye → FCGZ → MFZ.

D. Evaluation of SIF

Based on the observation of fracture surfaces, the shapes of inclusion, FGA, and fisheye can be approximately considered to be circular in a two-dimensional plane. The parameters r _inc, r _FGA, and r _fisheye were defined to denote the radii of inclusion, FGA, and fisheye, respectively. Figure 7 shows the relationships between r _inc, r _FGA, and r _fisheye and N _f. It can be seen that the values of r _inc, r _FGA, and r _fisheye are all independent of fatigue life. The values of r _inc for the existence of FGA are only in the range of the 10.84–13.22 μm, while the values of r _inc for the nonexistence of FGA are in the range of 10.13–51.10 μm. The values for the latter are comparatively larger, partially similar to the values of r _FGA with about 25.56–36.19 μm. Furthermore, the values of r _fisheye are in the range of 269.61–727.59 μm.

FIG. 7.

Relationships between r _inc, r _FGA, and r _fisheye and N _f.

Based on the theory of fracture mechanics, the SIF range, ΔK _int, for an interior crack with a radius, a, can be given by ^{Reference Marines-Garcia, Paris, Tada, Bathias and Lados26}

where ∆σ is the applied stress range. Accordingly, for the inclusion, FGA, and fisheye, the relevant SIF ranges, ∆K _inc, ∆K _FGA, and ∆K _fisheye, can be expressed as

At the stress ratio of −1, Eq. (2) is rewritten as

where K _max denotes the maximum SIF. In view of the fact that the tension has a predominant effect on the crack growth, Fig. 8 only shows the relationships between K _max–inc, K _max–FGA, and K _{max–fisheye} and N _f. The values of K _max–inc are in the range of 2.33–5.90 MPa m^1/2 and tend to decrease with the increase of N _f, as indicated by a dashed line in Fig. 8. Among them, the values of K _max–inc for the existence of FGA are only in the range of 2.33–2.67 MPa m^1/2, obviously less than these for the nonexistence of FGA in the range of 3.19–5.90 MPa m^1/2. Compared with the threshold value controlling macroscopic crack growth for some high-strength steels, about 4–5 MPa m^1/2, ^{Reference Shiozawa, Murai, Shimatani and Yoshimoto4,Reference Li, Sun, Zhang, Deng and Sakai5,Reference Sakai, Sato and Oguma7} the values of K _max–inc for the existence of FGA in the VHCF regime are obviously smaller. Under this circumstance, it can be concluded that the microcrack growth should play a certain role in the interior crack growth process under the VHCF.

FIG. 8.

Relationships between K _max–inc, K _max–FGA, and K _{max–fisheye} and N _f.

The values of K _max–FGA approximately keep a constant regardless of N _f, as indicated by a dotted line in Fig. 8. The mean value of K _max–FGA is about 3.94 MPa m^1/2 and is very similar to the mean value of K _max–inc for the nonexistence of FGA with about 4.21 MPa m^1/2. Combined with other experimental results, ^{Reference Li, Sun, Zhang, Deng and Sakai5,Reference Sakai, Sato and Oguma7,Reference Shiozawa, Morii, Nishino and Lu9} it can be assumed that K _max–FGA or K _max–inc for the nonexistence of FGA is the threshold value for controlling stable growth of interior macroscopic crack. Furthermore, just like K _max–FGA, the values of K _{max–fisheye} also approximately keep a constant regardless of N _f, as indicated by a solid line in Fig. 8. The mean value of K _{max–fisheye} is about 18.33 MPa m^1/2, similar to the fracture toughness of steel. Therefore, K _{max–fisheye} can be assumed to be the threshold value controlling the unstable growth of interior macroscopic crack.

Based on above-mentioned analysis, therefore, the crack initiation and growth behaviors in the HCF regime can be characterized as (i) crack initiation around the inclusion, (ii) stable macrocrack growth within the fisheye, (iii) unstable macrocrack growth outside the fisheye and within the FCGZ, and (iv) final fracture within the MFZ. On the other hand, the crack initiation and growth behaviors in the VHCF regime can be characterized as (i) crack initiation around the inclusion, (ii) microcrack growth within the FGA, (iii) stable macrocrack growth outside the FGA and within the fisheye, (iv) unstable macrocrack growth outside the fisheye and within the FCGZ, and (v) final fracture within the MFZ. Among them, fatigue lives consumed in the stages of unstable crack growth and final fracture can be negligible. Figure 9 shows the schematic diagram of interior inclusion-induced failure progress within the fisheye.

FIG. 9.

Schematic diagram of interior inclusion-induced failure progress.

E. Evaluation of crack initiation life

In view of the interior failure mechanism associated with the inclusion-induced crack, the crack initiation life, N _i, can be evaluated by using the dislocation dipole accumulation model proposed by Tanaka and Mura, ^{Reference Tanaka and Mura27} and given by

where μ and μ_inc are the shear moduli of homogeneous matrix and inclusion, respectively, W _c is the specific fracture energy for a unit interfacial area, l and h are the semimajor length and semiminor length of elliptic slip band zone, respectively, Δτ is the resolved shear stress range, and τ_f is the critically resolved shear stress for plastic deformation. For this model, it is based on the assumption that all the dislocations in the slip band zone contribute to the crack formation. However, the study shows that generally not all dislocations contribute to the crack formation, at least not immediately at crack initiation. ^{Reference Venkataraman, Chung and Mura28} The number of dislocations that contribute to the crack formation is closely related to the crack length, c. Under this circumstance, the value of W _c can be expressed as ^{Reference Chan29}

By substituting Eq. (5) into Eq. (4), we have

Furthermore, the applied stress range can be related to the shear stress range by using the Taylor factor M, leading to

where σ_w is the critical stress below which the crack cannot initiate. In the HCF regime, it corresponds to the transition stress from the HCF regime to the VHCF regime. Whereas in the VHCF regime, it corresponds to the fatigue strength at life of 10⁹ cycles. Furthermore, the exponent of N _i can be generalized to a variable α (0 < α < 1), which is based on the better agreement with the experimental data. ^{Reference Chan29} Also, studies show that the value of α is not a constant but depends on the stacking fault energy and on the degree of slip irreversibility. ^{Reference Venkataraman, Chung and Mura28,Reference Cheng and Laird30} Under this circumstance, Eq. (7) can be rewritten as

when α = 1/2, Eq. (8) has the same power law relation as Eq. (7). Furthermore, the shear modulus is related to the Young's modulus and given by

and

where E and ν denote the Young's modulus and Poisson's ratio of the homogeneous matrix, respectively, and E _inc and ν_inc denote the Young's modulus and Poisson's ratio of the inclusion, respectively. For the matrix of steel, the values of E and ν are 204,000 MPa and 0.3, respectively. For the inclusion of Al₂O₃, the values of E _inc and ν_inc are about 400,000 MPa and 0.25, respectively. ^{Reference García, Mantič and Graciani31} For the interior failure without the FGA formation, the value of l in Eq. (8) is generally defined as the half of grain size. ^{Reference Tanaka and Mura27,Reference Chan29} However, for the interior failure with the FGA formation, herein it is assumed that the elliptic slip band zone spreads over the whole FGA. That is, the value of l is approximately equal to the radius of FGA. In view of the fact that the value of h is generally not available, for these two failure modes, h is all regarded as a fitting parameter just like α. Therefore, based on the S–N data and the sizes of FGA and inclusion, the corresponding values of α and h can be determined, and the crack initiation lives at different values of c can be obtained.

Corresponding to different values of c, the crack initiation life curves evaluated in the HCF regime and in the VHCF regime are indicated by dashed and solid lines in Fig. 10, respectively. Both in the HCF regime and in the VHCF regime, the crack initiation lives at a given stress amplitude all tend to increase with the increase of crack length. At a given crack length, the needed crack initiation lives are shorter at higher stress amplitude, whereas at lower stress amplitude those can occupy the vast majority of the total fatigue life. For example, corresponding to c = 1 μm in the HCF regime, the evaluated crack initiation life at stress amplitude of 650 MPa is almost equivalent to one experimental fatigue life, whereas that at stress amplitude beyond 750 MPa is much less than the total fatigue life. Approximately, the crack initiation life curve at c = 5 μm is in agreement with the test data in the HCF regime. On the other hand, in the VHCF regime, the crack initiation lives corresponding to c = 20 μm are approximately in good agreement with the test data. It should be noted that the crack length of 20 μm is very similar to the difference between the radii of inclusion and FGA.

FIG. 10.

Evaluated crack initiation lives at different crack sizes.

F. Evaluation of crack growth life

Based on above-mentioned analysis and assumption, the crack growth life in the VHCF regime mainly consists of two parts: (i) microcrack growth life within the FGA, ${N_{{\rm{p}}\left( {{\rm{inc}} \to {\rm{FGA}}} \right)}}$ , and (ii) macrocrack growth life outside the FGA and within the fisheye, ${N_{{\rm{p}}\left( {{\rm{FGA}} \to {\rm{fisheye}}} \right)}}$ . The crack growth life in the HCF regime only corresponds to the macrocrack growth life outside the inclusion and within the fisheye, ${N_{{\rm{p}}\left( {{\rm{inc}} \to {\rm{fisheye}}} \right)}}$ . Taking account of the difference of crack growth behaviors in these two life regimes, a uniform crack growth equation, i.e., Paris–Hertzberg–McClintock equation, ^{Reference Paris, Marines-Garcia, Hertzberg and Donald32} was adopted to evaluate the crack growth life and given by

where b is the Burger's vector, ΔK _eff is the effective SIF range. For the microcrack growth within the FGA in the VHCF regime, Eq. (11) combined with Eq. (1) can be rewritten as

where ${{\Delta K_{{\rm{inc}}} } \mathord{\left/ {\vphantom {{\Delta K_{{\rm{inc}}} } {\left( {E\sqrt b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E\sqrt b } \right)}} = 1$ corresponds to the threshold location without crack growth, i.e.,

Integrating Eq. (12) from r _inc to r _FGA, the value of ${N_{{\rm{p}}\left( {{\rm{inc}} \to {\rm{FGA}}} \right)}}$ is obtained as

In view of the different growth behaviors between the microcrack and the macrocrack, one must integrate an adjusted growth law from the transition point (micro to macro) with a rate reduced by the factor “1/x ³”. ^{Reference Cheng and Laird30} The schematic diagram of crack growth behavior from microcrack to macrocrack is shown in Fig. 11. At stress ratio of −1, the value of x is generally taken to be 3. ^{Reference Cheng and Laird30} Therefore, based on Eq. (12), the macrocrack growth rate is expressed as

FIG. 11.

Schematic diagram of interior crack growth from microcrack to macrocrack.

Integrating Eq. (15) from r _FGA to r _fisheye, the value of ${N_{{\rm{p}}\left( {{\rm{FGA}} \to {\rm{fisheye}}} \right)}}$ is obtained as

Based on Eqs. (14) and (16), the total crack growth life in the VHCF regime, ${N_{\rm{p}}}$ , is given by

Similar to the crack growth process from r _FGA to r _fisheye in the VHCF regime, the crack growth life in the HCF regime can be obtained as

The evaluated crack growth lives are shown in Fig. 12, where those in the HCF regime are in the range of 1.2 × 10⁵–2.5 × 10⁵ cycles, and those in the VHCF regime are in the range of 4.9 × 10⁵–6.1 × 10⁵ cycles. They all tend to increase with the decrease of stress amplitude. Furthermore, if the fatigue life obtained from experiment is denoted by N _exp, the relationship between the stress amplitude and the ratio of N _p to N _exp can be shown in Fig. 13. More directly, it can be seen from Fig. 13 that the proportion of crack growth life in the total fatigue life tends to decrease with the decrease of stress amplitude. Especially in the VHCF regime, the crack growth life only accounts for a very tiny fraction of the total fatigue life. Conversely, at higher stress amplitude in the HCF regime, the crack growth life can occupy the vast majority of the total fatigue life.

FIG. 12.

Evaluated crack growth lives in the HCF/VHCF regimes.

FIG. 13.

Relationship between σ_a and N _p/N _exp.

G. Prediction of total fatigue life

Based on Eqs. (8) and (17), the predicted fatigue life for the interior inclusion-induced failure with the FGA in the VHCF regime, N _pre-VHCF, can be obtained as

Moreover, based on Eqs. (8) and (18), the predicted fatigue life for the interior inclusion-induced failure without the FGA in the HCF regime, N _pre-HCF, can be obtained as

Figure 14 shows the comparison between the predicted and experimental lives in the HCF regime and in the VHCF regime. It can be seen that the agreement is fairly good within the factor-of-three boundaries. Overall, the results are satisfactory, especially in the VHCF regime. This means that from the viewpoint of crack initiation and growth, the above-established life prediction method for the interior inclusion-induced failure in the HCF regime and in the VHCF regime can be acceptable. Moreover, the characterization of interior crack initiation and growth behaviors in the HCF regime and in the VHCF regime is reasonable.

FIG. 14.

Comparison between the predicted and experimental results.