Influence of small defects and nonmetallic inclusions on the high and very high cycle fatigue strength of an ultrahigh-strength steel

The high and very high cycle fatigue (VHCF) properties of ultrahigh-strength Ck45M steel processed by thermomechanical rolling integrated direct quenching were investigated. S – N tests with smooth and small drilled holes containing specimens as well as near-threshold fatigue crack growth measurements were performed up to 2 (cid:1) 10 10 cycles using ultrasonic-fatigue testing technique. The fatigue strength of smooth specimens is mainly determined by the size of nonmetallic inclusions. For surface defects larger than 80 μ m, the fatigue limit can be correlated with a constant threshold-stress intensity factor. The ﬃﬃﬃﬃﬃﬃﬃﬃﬃ area p -parameter model adequately predicts the fatigue limit for internal defects and for surface defects with sizes between 30 and 80 μ m. VHCF failures from smaller surface defects occur at stress amplitudes below the predicted fatigue limit. The long-crack threshold in ambient air is close to the effective threshold stress intensity factor. In optically dark areas at interior inclusions, cracks grow at mean propagation rates of 10 (cid:3) 15 m/cycles.


| INTRODUCTION
Very high cycle fatigue (VHCF) failure is an important issue for high-strength steels. Due to their low defect tolerance, fatigue cracks can even initiate at small inherent defects such as nonmetallic inclusions. Theoretical maximum fatigue strength-related with the matrix material in the absence of detrimental defects-can be only achieved in the case of super-clean materials and by avoiding component production-related flaws (e.g., insufficient surface finish, scratches, and punch marks). For steels, the upper boundary of the fatigue limit depends on the material hardness (e.g., the Vickers hardness HV) and can be estimated by 1.6ÁHV 1 -which could be demonstrated even for high-strength steels with tensile strengths up to 1.7 GPa. 2 In this case, however, the critical defect size is only a few microns, and larger flaws would already decrease the fatigue strength. 3,4 While it may be possible to produce small defect-free fatigue test specimens, the probability of the presence of detrimental, fatigue-strength decreasing defects increases for large components or production series. This important issue should be considered when laboratory fatigue test results are used for component designs. Bearing in mind that fatigue failure potentially originates from the largest flaw, defect screening of the deployed material in combination with extreme-value rating is useful to predict the maximum defect in relevant, cyclically stressed volumes. 1,5 Furthermore, fatigue tests with materials containing an intentionally increased number and size of defects may serve to trigger relevant fracture mechanisms, such as inclusion initiated fracture under cyclic torsional loading. 6,7 To systematically characterize the defect sensitivity of a material, defects of different sizes should be utilized to determine the transition size between small and large defects. This is important from a fracture mechanics point of view since the thresholdstress intensity factor for small cracks and defects is size dependent, while it is a constant value for long cracks and large defects. 4,8 Several investigations on the VHCF properties of ultrahigh-strength steels-that is, steels with tensile strengths exceeding 2 GPa-have been performed (e.g., Refs. [9][10][11][12][13], the vast majority with bearing steels. These studies mainly focus on fatigue fracture emanating from interior inclusions, and several models have been proposed to explain failure at very high number of load cycles. [14][15][16][17][18][19] A relatively seldom considered, but important, aspect with respect to VHCF is that typical low-and medium-alloy ultrahigh-strength steels exhibit rather poor corrosion resistance. This may lead to an elimination of the classical fatigue limit and can cause surface failures above 10 7 cycles 20 (in the present work, the fatigue limit is generally considered as the maximum stress level below which the lifetime is infinite). Close inspection of fatigue data obtained from tests with ultrahigh-strength steels indeed reveals that surface fracture even occurs in the VHCF regime. 9,15,21,22 Consequently, VHCF failure of steels with tensile strengths in the range of 2 GPa and higher should not be considered as an issue exclusively associated with interior fracture.
In the present work, ultrasonic fatigue tests were performed with thermomechanically processed and subsequently direct quenched Ck45M steel exhibiting a tensile strength of approximately 2.4 GPa. Fatigue fracture from nonmetallic inclusions with sizes between 10 μm and more than 100 μm enabled to study the influence of defect size on fatigue limit and lifetime. Furthermore, small drilled holes were introduced into some specimens to systematically investigate the defect tolerance. Nearthreshold fatigue-crack growth rate (FCGR) measurements were performed, and the threshold-stress intensity factor range considering crack arrest during 10 10 cycles was determined. Fracture mechanisms for both surface and interior failure are discussed considering environmental influences. In addition, the experimental results were evaluated applying fracture mechanics-based prediction models.

| MATERIAL AND EXPERIMENTAL PROCEDURE
Fatigue tests were performed with thermomechanically processed (TMP) and subsequently direct quenched (DQ) Ck45M steel (TMP-DQ). The chemical composition determined by spark optical emission spectroscopy is 0.440C-0.667Mn-0.294Si-0.009P-0.036S-0.218Cr-0.221Cu-0.0002B-0.034Mo (weight %). A 200 Â 80 Â 60 mm block machined from the casting was soaked at 1150 C for 1 h and thermomechanically rolled in two stages, as shown in Figure 1. Stage 1 comprised hot rolling in three passes above the no-recrystallization temperature (T nr ) to a thickness of $31.5 mm with $0.2 strain/pass. Stage 2 comprised controlled rolling in the T nr regime (below $950 C) comprising three passes ($0.2 strain/pass) to a thickness of 15.5 mm with the finish rolling temperature at $830 C (above the ferrite start temperature, A r3 ). Immediately after hot rolling, the sample was DQ in water to room temperature. Tensile tests were performed according to ASTM E8M standard.
Specimen shapes used for S-N and near-threshold FCGR tests are shown in Figure 2A,B, respectively. All specimens were machined with their longitudinal axis in rolling direction. The surface in the gauge section of machined specimens was ground with emery paper up to F I G U R E 1 Schematic diagram of the laboratory rolling schedule #2000 and electropolished to remove any residual stresses.
Small drilled holes were introduced into the surface of some specimens. Hole diameters, d, were 50, 100, 200, and 300 μm, and the depths of holes, h, were the same as the diameter; that is, h = d. In addition, 2-and 3-hole defects were introduced with hole diameters of 50 μm oriented perpendicular to the loading direction; see Figure 2C.
Fully reversed tension-compression fatigue tests were performed using ultrasonic fatigue testing equipment with a cycling frequency of about 19 kHz. The vibration of one specimen's end is measured with an induction coil and used to control cyclic loading in a closed-loop circuit; that is, ultrasonicfatigue experiments are displacement controlled. The specimen is cycled in resonance, and the resonance frequency is controlled in a second closed-loop circuit. To prevent self-heating of specimens during testing, pulsed loading was applied in addition to compressed-air cooling. A detailed description of the ultrasonic testing technique can be found in Mayer 23 . Tests were performed in ambient air at approximately 23 C and 50% relative humidity (laboratory environment with air conditioning and controlled humidification).
In the FCGR specimens, a single-edge notch with a length of 1 mm was introduced by electrical discharge machining, acting as a well-defined crack initiation site. Measurements were started at crack growth rates of about 10 À8 m/cycle, and the stress intensity factor, ΔK, was decreased in steps of 3-7% (with the smaller steps in the near-threshold regime) after a crack extension of at least 50 μm was detected. This was repeated until the threshold, ΔK th,lc , was reached-which was determined as the stress intensity factor range where no crack extension could be observed within 10 10 cycles. Fatigue crack growth was observed with a digital camera and a zoom lens allowing magnifications up to 680-fold (magnifications between 420-and 680-fold were used, with higher magnification at lower FCGRs) and a resolution of approximately 1 μm. Extremely slow crack growth with rates as low as 2 Â 10 À14 m/cycles was observed, which could be only measured due to the high testing frequency (in the near-threshold regime, continuous loading at approximately 19 kHz was applied). It is noted that accumulating 10 10 cycles nevertheless takes more than 6 days, resulting in a measurement time of several weeks for the determined FCGR curve.

| RESULTS
3.1 | Microstructural and static mechanical property evaluation Figure 3A presents a typical field-emission scanning electron micrograph (FE-SEM) of the investigated TMP-DQ processed steel after etching. The microstructure consists mainly of fine packets and blocks of martensite. Straining of austenite prior to quenching resulted in refinement and randomization of packets and blocks of martensite. Untransformed, so-called retained austenite (RA) present as films between martensitic laths is difficult to discern from FE-SEM micrographs. Thin film examinations in transmission electron microscope (TEM) including bright and dark field imaging clearly revealed that the material contains highly dislocated lath martensite structure separated by fine RA films ( Figure 3B). The dark field image ( Figure 3C) confirms the presence of film-like RA. The volume fractions of martensite and RA are estimated 96% and 4%, respectively, by X-ray diffraction analysis corroborated well by TEM results. Furthermore, energydispersive X-ray spectroscopy (EDX) of the micrograph shown in Figure 3A revealed presence of aluminum/calcium oxide (Al 2 O 3 -CaO) and manganese sulfide (MnS) inclusions with circular and elongated morphology, respectively.
The experimentally determined high yield strength (0.2% proof stress of $1613 MPa) in combination with ultrahigh tensile strength ($2439 MPa) and reasonable total elongation ($3%) corroborate the observed microstructure as shown in Figure 3A-C. The nanolath martensitic matrix imparts the high strength, whereas the finely divided, film-like interlath RA provides the work hardening capacity. Furthermore, the high hardness (717 HV10) correlates well with the determined tensile strength.

| S-N tests with smooth specimens
Fatigue test results with smooth S-N specimens are plotted in Figure 4. Tests were performed at stress amplitudes between σ a = 570 and 700 MPa. There is a huge scatter F I G U R E 4 S-N diagram for smooth specimens. Runout specimens are marked with arrows of fatigue lifetimes at each stress amplitude, up to five orders of magnitude. No fatigue limit could be determined, and failure even occurred after 2.56 Â 10 10 cycles. Two specimens were stopped after more than 1 Â 10 10 cycles without failure (marked with arrows in Figure 4).
FE-SEM images of typical fracture surfaces are shown in Figure 5. Failure mainly occurred from nonmetallic inclusions located at the surface or in the interior ( Figure 5A,B) that were identified as Al 2 O 3 -CaO with EDX. For some specimens, the cause of failure could not be identified ( Figure 5C,D, marked as "unknown" in Figure 4). EDX analyses revealed no differences in chemical composition at the crack initiation location and the matrix. The high-magnification image in Figure 5C shows both the specimen surface (top) and the fracture surface (bottom), which was achieved by tilting the specimen in the SEM. A structure similar to bainite is visible at the surface which might be the source of failure (see, e.g., the investigations on bearing steels by Murakami 1 ). Further, some specimens failed from surface pits-as will be shown in Section 4.3-that probably originated from inclusions during electropolishing of specimens. From a fracture mechanics point of view, small pits can be treated similarly to surface inclusions, 1 and no further differentiation between pits and surface inclusions will be made in the following.
One specimen was polished with 1-μm diamond slurry before testing at σ a = 625 MPa in order to achieve a mirror-like surface. After N = 1.20 Â 10 10 cycles, the surface in the gauge section was examined with a light microscope. Some locations with crack-like features were found around nonmetallic inclusions. The specimen was afterward tested at σ a = 650 MPa for another N = 1.00 Â 10 10 cycles, and the same locations were reinvestigated. A slight increase in crack length could be observed which proofs the existence of nonpropagating cracks. Examples of nonpropagating cracks at an inclusion are shown in Figure 6A.

| S-N tests with specimens containing drilled holes
Fatigue tests with specimens containing drilled holes were performed up to 10 9 cycles, and the results are plotted in Figure 7. The S-N curves exhibit clear kneepoints between 10 5 and 10 6 cycles. Only one specimen failed due to crack initiation at a drilled hole in the VHCF regime (d = 100 μm, σ a = 575 MPa, N f = 7.47 Â 10 8 cycles). Another specimen with a 1-hole defect (d = 50 μm) failed after N f = 6.06 Â 10 8 cycles at σ a = 625 MPa, but fracture originated from another location at the surface rather than from the drilled hole. Therefore, this specimen is marked as runout in Figure 7.
The runout specimens were observed for nonpropagating cracks after fatigue testing. However, due to a slight etching of the specimens' surface that occurred during electropolishing, it was in some cases difficult to decide whether a crack was present or not. Therefore, some runout specimens were polished with 1-μm diamond slurry and fatigue loaded for another 10 7 cycles. Clear evidence of a nonpropagating crack could be only provided at the edge of a 1-hole defect with d = 100 μm; see Figure 6B.

| Near-threshold fatigue crack growth tests
The FCGR test results are shown in Figure 8. The stress intensity factor ranges were calculated according to following equation 24 : F I G U R E 7 S-N diagram for specimens containing drilled holes. Runout specimens are marked with arrows 0Á(a/W) 4 that was deduced from finite element calculations for single-edge notched ultrasonic-fatigue specimens as shown in Figure 2B. 25 In Equation 1, Δσ is the remote stress range (calculated from the measured strain range multiplied with the Young's modulus), a is the crack length (including notch depth), and W is the specimen width (W = 15 mm). Crack growth rates between 3 Â 10 À8 and 2 Â 10 À14 m/cycles were observed, which means that propagation rates significantly below one Burgers vector per load cycle could be measured. Data points marked with arrows indicate measurements where no crack propagation could be observed within 10 10 cycles. The mean value of the determined long-crack threshold-stress intensity factor range is ΔK th,lc = 2.4 MPa ffiffiffiffi ffi m p . Note that the whole stress range (tension and compression part) was used to calculate the stress intensity factor range at R = À1; that is, the threshold-stress intensity factor amplitude as well as the maximum stress intensity factor at the threshold is 1.2 MPa ffiffiffiffi ffi m p : In order to verify crack growth at extremely low propagation rates and the low threshold stress intensity factor range, three specimens were tested (marked with different symbols in Figure 8). The curves measured with different specimens are in good accordance with each other; hence, it can be assumed that the determined results are valid.

| Observation of nonpropagating cracks
Nonpropagating cracks with lengths of approximately 10 μm could be observed at both nonmetallic surface inclusions and small drilled holes (as shown in Figure 6). This is a surprising observation since self-arrested cracks are rarely observed in ultrahigh-strength steels. 26,27 The presence of nonpropagating cracks after cyclic loading at subcritical loads (i.e., at or below the fatigue limit stress amplitude) is a clear indication that fracture mechanics approaches might be used to predict the fatigue strength in the presence of small defects. This is due to the fact that the fatigue limit is determined by the critical condition for crack propagation rather than for crack initiation, which means that the fatigue limit can be correlated with the threshold-stress intensity.
As demonstrated by Nisitani 28 , the root radius of a notch or a defect, ρ, influences the fatigue limit if the value of ρ is above a critical value. This critical notch root radius, ρ 0 , depends on the material. If ρ ≤ ρ 0 , the fatigue limit is determined by the threshold condition for crack propagation-which means that nonpropagating cracks are observable (see Figure 9). A notch with a small crack at its root may be regarded equivalent to a crack 1 (i.e., at the crack tip, ρ is virtually zero-independent of the root radius of the initial notch). Thus, fracture mechanics principles are applicable. If ρ > ρ 0 , the threshold condition for crack initiation determines the fatigue limit. In this case, notch-fatigue concepts may serve to predict the fatigue strength rather than a fracture mechanics F I G U R E 8 Fatigue crack growth rate data at R = À1. Different symbols mark various test specimens F I G U R E 9 Schematic on the existence of nonpropagating cracks between the crack initiation and the propagation limit (according to Nisitani 28 ) [Colour figure can be viewed at wileyonlinelibrary.com] approach. For example, the relative stress gradient approach by Siebel and Stieler 29 has been successfully applied for martensitic stainless steels. 4 However, the applicability of this approach to small defects needs to be further investigated.
Values of ρ 0 typically vary between 100 and 500 μmwith a tendency of decreasing values of ρ 0 with increasing tensile strength. 28 However, a systematic investigation of martensitic stainless steels with tensile strengths up to 1 GPa 4,30 has shown that ρ 0 can be as low as 25-100 μm. Therefore, it might be assumed that the critical defect size for steels with tensile strengths above 2 GPa is even smaller and that small drilled holes (and maybe even nonmetallic inclusions) rather behave like blunt notches. 26 The observation of a nonpropagating crack at a 100-μm hole as shown in Figure 6B, however, confutes this assumption (although it is conceivable that local microstructural inhomogeneities may act similarly to small, sharp notches) and suggests that fracture mechanics principles can be applied-as will be discussed in the following sections.

| Fracture mechanics evaluation
The stress intensity factor ranges, ΔK, for nonmetallic inclusions and drilled holes were calculated by the widely used equation proposed by Murakami 1 : with f = 0.65 for surface defects and f = 0.5 for interior defects. Equation 2 gives the maximum value of ΔK along the front of the defect in MPa ffiffiffiffi ffi m p if the cyclic stress range, Δσ, is in MPa and ffiffiffiffiffiffiffiffiffi area p is in m. The size parameter ffiffiffiffiffiffiffiffiffi area p introduced by Murakami and Endo 31 is the square root of the projection area of the defect perpendicular to the loading direction, and values for ffiffiffiffiffiffiffiffiffi area p were determined from FE-SEM fractrographs as shown in Figure 5. Sizes of ffiffiffiffiffiffiffiffiffi area p = 13-116 μm were determined for nonmetallic inclusion and ffiffiffiffiffiffiffiffiffi area p = 10-56 μm for surface pits. Runout specimens were retested at a higher stress amplitude to enable determination of the crackinitiating defect's size.
In Figure 10, the calculated values of ΔK are plotted versus the number of cycles to failure. Failure from surface inclusions (or pits) and interior inclusions is represented by open and solid symbols, respectively ( Figure 10A). Further, interior inclusions with sizes of ffiffiffiffiffiffiffiffiffi area p < 60 μm (gray symbols) and ffiffiffiffiffiffiffiffiffi area p > 60 μm (black symbols) are separately plotted. With this classificationand in contrast to the S-N curve shown in Figure 4clear fatigue-lifetime curves could be plotted. The separation of small and large interior inclusions indicates that (for a given stress intensity factor range) fatigue lifetimes increase when the inclusion sizes become larger. This might be a hint that the threshold-stress intensity factor range, ΔK th , is size dependent (in the nonmetallic inclusions size range).
In contrast, the fatigue limits determined with specimens containing drilled holes can be related to a constant stress intensity factor range of around ΔK th ≈ 12 MPa ffiffiffiffi ffi m p F I G U R E 1 0 Stress intensity factor ranges versus number of cycles to failure for (A) nonmetallic inclusions and (B) drilled holes. Runout specimens are marked with arrows as shown in Figure 10B; 1-hole defects with hole diameters of d = 200 and 300 μm exhibit almost the same ΔK-N curves, but fatigue lifetimes are shorter for 2-and 3-hole defects when ΔK is above the threshold. This can be explained by the significantly smaller notch root radius of ρ = d/2 = 25 μm for the latter, which results in higher stress concentration factors. However, it is important to notice that although the stress concentration factor may influence the fatigue lifetimes, the fatigue limit is rather determined by the stress intensity factor. Murakami and Endo 32 proposed a simple model to predict the size-dependent threshold-stress intensity factor range for small cracks and defects, which has been successfully applied to several materials. 1 Beside the size parameter ffiffiffiffiffiffiffiffiffi area p -which is eponymous for the model-only the Vickers hardness, HV, must be known to calculate the threshold-stress intensity factor range 1 : with g = 3.3 Â 10 À3 for surface cracks/defects and g = 2.77 Â 10 À3 for interior cracks/defects. In Equation 3, in μm, and HV in kgf/mm 2 . The applicability of Equation 3 is limited to small defects; that is, the threshold-stress intensity factor should be below the constant long-crack threshold, ΔK th,lc . From tests with specimens containing small drilled holes (Figure 7), a constant threshold of $12 MPa ffiffiffiffi ffi m p can be assumed ( Figure 10B). However, this value is significantly larger than the long-crack threshold of ΔK th,lc = 2.4 MPa ffiffiffiffi ffi m p determined by fatigue crack growth measurements in ambient air ( Figure 8). Deviations from ΔK th,lc for large defects have been reported for martensitic stainless steels 4 and spheroidal cast iron, 33 but they are mainly found at load ratios different from R = À1 and could be explained by the influence of mean stress on the crack tip constraints. 4 The extremely slow crack growth rates measured in the near-threshold regime as shown in Figure 8 rather suggest environmental influences that enable crack growth at extremely low stress intensity factors. Also, the value of 2.4 MPa ffiffiffiffi ffi m p is in the range of the effective thresholdstress intensity factor range of steels, but crack growth at stress intensities close to ΔK th,eff should be only relevant at high load ratios-and not at fully-reversed loading. Therefore, in the following, the long-crack threshold, ΔK th,lc , will be distinguished from the threshold-stress intensity factor range for large defects, ΔK th,ld , which is according to Figure 10B approximately ΔK th,ld = 12 MPa ffiffiffiffi ffi m p . The transition size between small and large defects, ffiffiffiffiffiffiffiffiffi area p trans , can be hence calculated by following equation 30 : Transition sizes of ffiffiffiffiffiffiffiffiffi area p trans,s = 82 μm for surface defects and ffiffiffiffiffiffiffiffiffi area p trans,i = 139 μm for interior defects are calculated for the investigated ultrahigh-strength steel. Similar transitions sizes were determined for martensitic stainless steels as reported in Schönbauer et al. 4,30 Comparison between the transition sizes for surface and interior defects-as well as the stress intensity factor ranges according to Equation 2-reveals that the size of an interior defect (in ffiffiffiffiffiffiffiffiffi area p ) must be by a factor of 1.69 larger to be equally detrimental than a surface defect; see also Murakami. 1 In Figure 11, the stress intensity factor range, ΔK, is plotted versus the defect size, ffiffiffiffiffiffiffiffiffi area p . Solid lines mark the threshold values according to Equation 3 and ΔK th,ld , respectively. Since ΔK th,ld was determined from the experimental results, all failed specimens containing drilled holes with sizes of ffiffiffiffiffiffiffiffiffi area p > ffiffiffiffiffiffiffiffiffi area p trans,s lie per definition above the prediction line ( Figure 11A). But, surface inclusions and pits were all smaller than the transition size of 82 μm, and hence, the applicability of Equation 3 needs to be verified. For specimens that failed below one million cycles (large, solid circles), the ffiffiffiffiffiffiffiffiffi area p -parameter model is accurate within ±10%. If failure occurred after 10 6 cycles, however, the prediction becomes nonconservative. The stress intensity factor for the specimen that failed from a 10-μm pit, for example, was 27% below the predicted threshold value. For interior inclusions, the prediction according to Equation 3 also slightly overestimates the threshold values; see Figure 11B. The specimen that fractured after 2.59 Â 10 10 cycles ( Figure 5B) failed 13% below the calculated fatigue strength.
By substituting the stress intensity factor range and the cyclic stress range in Equation 2 by its threshold, ΔK = ΔK th , and the fatigue limit, Δσ = 2Áσ w , respectively, the fatigue limit stress amplitude, σ w , can be derived. For small cracks or defect, ΔK th is given by Equation 3, and it follows 1 with h = 1.43 for surface cracks/defects and h = 1.56 for interior cracks/defects.
For large defect, ΔK th = ΔK th,ld , and the fatigue limit stress amplitude can be calculated with 30 where i = 434 for surface defects and i = 564 for interior defects.
With this, the experimental results can be plotted in Kitagawa-Takahashi diagrams, as demonstrated in Figure 11C,D, which allows to correlate the fatigue limit with the defect size.
Note that Equations 3, 5, and 6 are only applicable for R = À1. In the presence of mean stresses, a term accounting for the load ratio must be included as described in Schönbauer et al. 1,30 For martensitic stainless steels, the transition size (Equation 4) was found to be independent of load ratio. 4 Although it is expected that the mean load sensitivity increases with tensile strength, systematic investigations with ultrahigh strength steels at different load ratios are necessary to give further insight.

| VHCF failure from surface defects
In steels, fatigue failure from the surface typically occurs within 10 7 cycles or earlier. 1,34 This was also observed for the investigated steel in the presence of small drilled F I G U R E 1 1 Stress intensity factor range versus defects size for (A) surface defects and (B) interior defects and Kitagawa-Takahashi diagrams for (C) surface defects and (D) interior defects. Open symbols mark runout specimens holes where-with one exception-all specimens failed before 10 6 or survived 10 9 cycles. Surface failure in the VHCF regime is typically attributed to environmental effects. In ambient air, this is mainly an issue for highstrength steels (see, e.g., Nishimura et al. 35 ) since corrosion resistance tendentially decreases with increasing tensile strength.
In the Kitagawa-Takahashi diagram plotted in Figure 12, a short-dashed, red line denotes the stress amplitudes above which nonpropagating cracks could be observed. Symbols with centerlines mark the two specimens with nonpropagating cracks that are shown in Figure 6. The size of the nonmetallic inclusion shown in Figure 6 was estimated as ffiffiffiffiffiffiffiffiffi area p = 10 μm (assuming a semi-circular shape). In addition, a dashed, red line represents the experimentally determined long-crack threshold of ΔK th,lc = 2.4 MPa ffiffiffiffi ffi m p . Above the black, solid prediction line-representing Equations 5 and 6-fatigue failure is expected, while no failure should occur if applied stress amplitudes are below this line. The horizontal part of the prediction line-which declines at a crack length of 1.3 μm-marks the upper bound of the fatigue strength which can be estimated (for steel) by the simple equation σ w = 1.6ÁHV. 1 A possible explanation for the observed failure in the hatched area below the predicted fatigue limit in Figure 12 (small solid circles) will be given in the following: At small, sharp defects such as nonmetallic inclusions, cracks can initiate at stress amplitudes even significantly below the fatigue limit. These cracks would arrest in a benign environment due to the stress gradient and the built-up of crack closure mechanisms. [36][37][38] However, in a chemically active environment-and ambient air might be assumed as a weak corrosive atmosphere for ultrahigh-strength steels-the crack growth threshold is decreased resulting in a lowering of the original fatigue limit. 20 In other words, the gap between the crack initiation and the crack propagation limit, which are appreciably different in noncorrosive environments, becomes smaller. This assumption is strongly supported by the extremely low near-threshold FCGRs and thresholdstress intensity factor measured in ambient air (Figure 8). The corrosive influence of ambient air becomes especially important for surface defects with sizes below 30 μm. In the presence of comparably large drilled holes, where the threshold-stress intensity factor is size independent and the notch root is relatively large, the crack-initiation limit is only slightly below the propagation limit. The corrosive atmosphere-which is too weak to decrease the crackinitiation limit, for example, by pitting-therefore, may not cause a significant reduction in fatigue strength.
Furthermore, DQ ultrahigh-strength steels are highly susceptible to hydrogen embrittlement 39 due to high residual stresses exerted by the large fraction of martensite in the final microstructure. 40 Hydrogen can be trapped at different sites such as dislocations, grain boundaries, precipitates, or inclusions. Depending on the activation energy, weak or reversible hydrogen traps must be distinguished from irreversible traps where a significant amount of hydrogen can be captured without considerable impact on cracking. 41 Reversible traps, in contrast, may act as a supplier of diffusible hydrogen to the crack tip. 42 The observation of intergranular crack paths supports the assumption that hydrogen embrittlement affects fatigue crack growth. However, further investigations on the effects of environment and hydrogen on the nearthreshold fatigue crack growth are necessary to give more insight.
The fracture surface of the specimen that failed from a small pit in the VHCF regime, at a stress amplitude 26% below the predicted fatigue limit according to Equation 5, is shown in Figure 13A. Transgranular, cleavagelike facets in the near vicinity of the pit with a nonplanar crack path are visible. Similar-but intergranular-facets are observable adjacent to the crack-initiating inclusion shown in Figure 13B. Such near-threshold facets are expected to be preliminarily environmentally induced 43,44 and can be, for example, observed when fatigue cracks initiate at corrosion pits. 45,46 It is further well-known that dissolved hydrogen benefits intergranular cracking. 47,48 It might be considered that these inter-and transgranular facets can be treated as effective flaws.
F I G U R E 1 2 Kitagawa-Takahashi diagram for surface defects. Specimens with nonpropagating cracks (shown in Figure 6) are highlighted by red symbols with centerlines [Colour figure can be viewed at wileyonlinelibrary.com] The sizes of the resulting effective defects are marked in Figure 13, and these effective defect sizes were used to calculate the fatigue limits according to Equation 5. Plotting the applied stress amplitudes normalized by the predicted fatigue limits (Equations 5 and 6) versus the number of cycles to failure, as shown in Figure 14, reveals that an increase in the effective defect size serves well to explain fatigue failure in the VHCF regime from inclusions and pits that would be nondetrimental under benign environmental conditions.

| Fatigue limit in the presence of interior defects
The normalized stress amplitudes versus number of cycles to failure for interior nonmetallic inclusions are plotted in Figure 15. Compared to Figure 10, where ΔK was plotted versus the number of cycles, scatter in fatigue data could be significantly reduced. This is due to the fact that not only the size of inclusions but also the sizedependent threshold-stress intensity factor range (Equation 3) is considered. Several examples of normalized S-N curves have been published showing a similar reduction in scatter. 9,49-51 However, there is still a tendency of longer fatigue lifetimes for larger inclusions noticeable in Figure 15. This is similar to specimens containing drilled holes where fatigue lifetimes are shorter when the holes are smaller (and/or the notch root radius is reduced), see Figures 10B and 14, and might be explained by differences in stress distribution at small and large inclusions.
In the case of an accurate prediction of σ w , no failure should occur below σ a /σ w = 1. In the VHCF regime, however, the ffiffiffiffiffiffiffiffiffi area p -parameter model delivers slightly nonconservative predictions, as shown in Figure 15. For example, fracture occurred at a stress amplitude 13% below the fatigue limit calculated with Equation 5 (i.e., at σ a /σ w = 0.87), however, after an extremely high number of cycles of more than 2 Â 10 10 . Therefore, the prediction might be considered as acceptably accurate. Nevertheless, assuming that the explanation given in the last section (i.e., that the fatigue limit is reduced by an environmental interaction and dissolved hydrogen, and Equation 5 generally serves well to predict the fatigue limit of the investigated steel in the presence of surface defects) is correct, there must be a reason for the observed reduction in fatigue strength in the case of internal fracture in the VHCF regime. Also, similar (nonconservative) results are reported for other ultrahigh-strength steels, 52,53 although Equation 5 is in good agreement with experimentally determined fatigue strengths in the VHCF regime for high-strength steels (with tensile strength below 2 GPa), 50,54-56 even in the case of failure from internal inclusions.
Fatigue fracture from the interior forms a so-called fish eye morphology, and a fine-grained area is often observed around the crack-initiation location if failure occurs in the VHCF regime. This area is mostly called "optically dark area" (ODA) 14,53 since it appears dark when observed with an optical microscope-but bright with an SEM-or "fine granular area" due to its morphology 15 and can be seen in Figure 5B,C. Hydrogen trapped by nonmetallic inclusions can significantly influence the formation of ODAs as reported by Murakami et al. 14,53 , and the concomitant, localized embrittlement may explain the reduction of the fatigue strength as displayed in Figure 15 (i.e., similar effects as expected for surface inclusions might be expected). But, ODAs can also be observed in the absence of inclusions, as shown in Figure 5C, as long as crack initiation is from the interior. In contrast, ODAs have never been observed at surface inclusions-as long as tests were performed in ambient environment. In high vacuum, however, fine granular appearing fracture surfaces can be produced around small surface defects 57 and even during longcrack propagation in the near-threshold regime. 17,58 Tests with specimens containing small surface defects conducted in high-vacuum and air by Ooka et al. 59 showed that fatigue lifetimes are significantly increased in vacuum. Failure occurred even in the VHCF regime, while the S-N curves in air exhibited knee-points at around 10 5 cycles. Nevertheless, fatigue failure in vacuum occurred at stress amplitudes below the fatigue limit determined in ambient air. Figure 16A shows the fracture surface of the specimen that initially survived 1.10 Â 10 10 cycles at 625 MPa (marked as runout specimen in Figure 15). Afterward, the specimen was retested at 700 MPa, and failure occurred after 1.23 Â 10 7 cycles. In the vicinity of the crack-initiating inclusion, a pronounced ODA is visible (the border of the ODA is marked with arrows in Figure 16A). Such an area is not identifiable at the fracture origin of another specimen that was exclusively tested at a stress amplitude of 700 MPa, although the F I G U R E 1 4 Normalized stress amplitude, σ a /σ w , versus number of cycles to failure for surface defects. Curved arrows mark the increase of normalized stresses amplitude by the use of the effective defects size instead of the inclusion/pit size. Runout specimens are marked with horizontal arrows F I G U R E 1 5 Normalized stress amplitude, σ a /σ w , versus number of cycles to failure for interior inclusions. The runout specimen is marked with an arrow and was retested at a higher stress amplitude (half-solid symbol) inclusion size and the number of cycles to failure were comparable; see Figure 16B. It can be therefore assumed that formation of the ODA visible in Figure 16A took place during cycling at 625 MPa. Since ODA formation consumes the vast majority of the fatigue lifetime, the mean crack growth rate inside the fine-grained area was approximately 10 À15 m/cycle. This is extremely slow and 1-3 orders of magnitude below FCGRs measured indirectly by two-step tests 60 and directly by long-crack measurements in high vacuum. 44,61 Hong et al. 62 report mean crack growth rates in ODAs between 10 À11 and 10 À12 m/cycle for lifetimes of 10 6 -10 7 cycles and between 10 À12 and 10 À13 m/cycle for lifetimes of 10 7 -4Á10 8 cycles. At fatigue lifetimes exceeding 10 10 cycles, hence, mean growth rates of 10 À15 m/cycle are conceivable. Although the border of the ODA is more difficult to distinguish, similar crack growth rates are estimable for the specimen that failed after 2.59 Â 10 10 cycles (fracture surface shown in Figure 5B).
It is often concluded that ODA formation ends when the stress intensity factor of the crack exceeds a constant value (e.g., the long-crack threshold). 15,[62][63][64] As shown in Figure 11-assuming the same threshold-stress intensity factor for long cracks in vacuum as determined for large surface defects, that is, ΔK th,ld ≈ 12 MPa ffiffiffiffi ffi m p -it might be however expected that the threshold-stress intensity factor is rather size dependent (as assumed by Murakami et al. 53 ) and that the transition size according to Equation 4 is 139 μm. Since the border of ODAs is difficult to distinguish in the investigated material (similarly to other materials such as 17-4PH stainless steel 56 ), it is futile to evaluate the stress intensity factors of ODAs (which might rather deliver expected outcomes than objective results). Assuming that environmental conditions of interior cracks can be accurately simulated, 65,66 it seems to be more appropriate to perform FCGR measurements in high vacuum and to correlate the stress intensities of small defects and inclusions with the determined thresholds.

| CONCLUSIONS
Ultrasonic fatigue tests up to more than 10 10 cycles were performed with ultrahigh-strength Ck45M steel processed by thermomechanical rolling and subsequent direct quenching. The fracture origins of smooth specimens were mainly nonmetallic inclusions located at the surface or in the interior of specimens. Systematic investigation on the defect tolerance was conducted with specimens containing small drilled holes. Following main results were obtained: F I G U R E 1 6 Fractographs: (A) specimen tested at σ a = 700 MPa, N f = 1.23 Â 10 7 cycles (half-solid symbol in Figure 15). The specimen was previously tested at σ a = 625 MPa for 1.10 Â 10 10 cycles (runout specimen in Figure 15) where the ODA (marked with arrows in the high-magnification image) was formed. (B) Specimen tested at σ a = 700 MPa, N f = 5.57 Â 10 6 cycles. No pronounced ODA is visible around the inclusion 1. The fatigue limit in the presence of surface defects larger than approximately 80 μm can be correlated with a constant threshold-stress intensity factor range of ΔK th,ld ≈ 12 MPa ffiffiffiffi ffi m p . For smaller defects, the threshold-stress intensity factor becomes size dependent, and the ffiffiffiffiffiffiffiffiffi area p -parameter model serves well to predict the fatigue strength in the high cycle fatigue regime. Applicability of fracture mechanics for fatigue limit prediction is supported by the observation of nonpropagating cracks in specimens cycled below the fatigue limit. 2. A long-crack threshold-stress intensity factor range of ΔK th,lc = 2.4 MPa ffiffiffiffi ffi m p was determined in ambient air by near-threshold FCGR measurements at R = À1. Propagation rates as low as 2 Â 10 À14 m/cycles were measured. Crack growth at extremely low propagation rates and the low threshold value were associated with environmental effects and dissolved hydrogen. 3. VHCF failure from small surface defects (<30 μm) occasionally occurred at stress amplitudes below the fatigue limit predicted by the ffiffiffiffiffiffiffiffiffi area p -parameter model. Crack growth at stress intensities below the crack length-dependent threshold-stress intensity factor is fostered by the corrosive influence of ambient air and hydrogen embrittlement. 4. Failure from the interior occurred even beyond 2 Â 10 10 cycles, indicating mean crack growth rates of approximately 10 À15 m/cycles within the ODA that is formed around interior inclusions. It is assumed that the extremely low growth rates are associated with crack propagation under high-vacuum condition.

ACKNOWLEDGMENTS
The authors would like to express their gratitude to Mr. Seppo Järvenpää for his valuable assistance during this research. The financial support of the Austrian Science Fund (FWF) under project number P 29985-N36 and the Academy of Finland under the auspices Genome of Steel (Profi3) project #311934 is acknowledged.

DATA AVAILABILITY STATEMENT
The data that support the finding of this study are included within the paper. stress intensity factor range ΔK th threshold stress intensity factor range for small cracks or defects ΔK th,eff effective threshold stress intensity factor range ΔK th,lc threshold stress intensity factor range for long cracks ΔK th,ld threshold stress intensity factor range for large defects Δσ stress range ρ notch root radius ρ 0 critical notch root radius σ a stress amplitude (σ a = Δσ/2) σ w fatigue limit stress amplitude at fullyreversed loading (R = À1) σ w,ld fatigue limit stress amplitude in the presence of large defects ORCID Bernd M. Schönbauer https://orcid.org/0000-0001-8964-2357 Herwig Mayer https://orcid.org/0000-0002-8872-5393