Abstract
The effect of inclusion size on fatigue behaviour of high strength steels in the very high cycle fatigue (VHCF) regime (>107–109 cycles) is reviewed. Internal fatigue fractures of high strength steels in the VHCF regime initiate mostly at non-metallic inclusions. The critical inclusion size below which it is hard to initiate fatigue cracking of high strength steels in the VHCF regime is found to be about half the critical value characteristic of the high cycle fatigue (HCF) regime (about 105–107 cycles). A stepwise or duplex S–N curve is observed in the VHCF regime. The shape and form of the S–N curves are affected by inclusion size and other factors including surface condition, residual stress, environment and loading modes. Fatigue strength and fatigue life for high strength steels have been found to obey inverse power laws with respect to inclusion size D of the form σ w∝D −n1 and N f∝D −n2 respectively. For fatigue strength, the exponent n 1 has been reported to be ∼0·33 in the literature for the HCF regime and, more recently, to fall in the range 0·17–0·19 for the VHCF regime. For fatigue life, the exponent n 2 is reported to be ∼3 in the HCF regime, and in the range 4·29–8·42 in the VHCF regime. A special area was often observed inside a ‘fish eye’ mark in the vicinity of a non-metallic inclusion acting as the fracture origin for specimens having a long fatigue life. The major mechanisms of formation for this special area are discussed. To estimate the fatigue strength and fatigue life, it is necessary to know the size of the maximum inclusion in a tested specimen, and to be able to infer this value using data from a small volume of steel. The statistics of extreme value (SEV) method and the generalised Pareto distribution (GPD) method are introduced and compared. Finally, unresolved problems and future work required in studying the VHCF of high strength steels are briefly presented.
Introduction
The present review is one of two being published by IMR on the fatigue behaviour of metallic materials in the very high cycle fatigue (VHCF) regime (generally understood to refer to the range from ∼107 to 109 cycles). A complementary review by Zimmerman1 focuses on the damage mechanisms of ductile materials that do not contain non-metallic inclusions; in contrast, in the present review, attention is paid to the influence of inclusions on the fatigue features of high strength steels.
For more than half a century, non-metallic inclusions have been associated with the general problems of fatigue failure in steels.2 – 4 Particularly, in the VHCF regime, the inclusions play a crucial role in fatigue failure of high strength steels. 5 5,6
Conventionally, the fatigue design of structural components is based upon the data in the high cycle fatigue (HCF) regime (about 105–107 cycles). However, the fatigue life of current automobile engines range around 108 cycles; high speed trains range to 109 cycles. It is further noted that at this time interest in fatigue life extends to about 1010 cycles, for example, in turbine engine components.6 There is strong need to investigate the fatigue behaviours of steels in the VHCF regime nowadays.
From a historical perspective, although Kikukawa et al. 7 found for the first time in the mid-1960s that structural steels can fail after 107 cycles, no substantial progress has been made until mid-1980s. Then numerous works had been conducted to explore the effect of inclusions on VHCF behaviours of steels.8 – 17
Existing overviews or monographs on inclusions usually point out the following factors that should be considered in resolving the effects of inclusions on fatigue properties, i.e. the inclusion size, shape, quantity, distribution and composition.2 – 4,18,19
Before mid-1980s, the increased knowledge of the oxide and sulphide inclusions meant that it was possible to gradually decrease the O and S levels of most steels. Now it is possible to make steels with as low as 1–10 ppm of both these elements. It is often a question of cost how far this process should be taken.19 However, every ton of steel made with as low as 1 ppm of O and S still holds 109–1012 inclusions. In other words, about 10–104 inclusions exist in a cubic millimetre, of course most of which are small inclusions. Stress localisation at the interface between the inclusion and matrix is the origin of fatigue failure. This arises from (1) the differential thermal contraction of inclusions and matrix during cooling and (2) the concentration of remote applied stresses due to elastic constant differences between the matrix and the inclusions.4 The difference of the deformability between inclusion and surrounding matrix will stimulate the crack initiation. The qualitative effects of inclusions on the mechanical properties of steel as well as on failures in steel processing and in service are known in outline. 2 4 2,4,19
In the HCF regime, the main conclusions were that two criteria must be fulfilled for an inclusion to be a potential initiation site for a fatigue failure:4 a critical size, depending on the depth of its location below the steel surface (for rotating bending fatigue); a low index of deformability at the actual temperature of the fatigue loading. The most dangerous inclusions are single phase Al2O3, spinels and Ca aluminates, and the least harmful are MnS inclusions.4 The review of Leslie20 summarised the situation before mid-1980s. His major conclusions were: the main effect of inclusions related to fatigue is on crack initiation, but they also have an effect on fatigue crack growth rate. In the HCF range (>105 cycles), nearly all cracks originate from inclusions. In the low cycle fatigue range (<105 cycles), especially at the range of 101–103 cycles, slip band cracks predominate. Fatigue cracks often initiate at cracked or debonded inclusions or in slip bands emanating from inclusions.
In the past two decades, great progress on VHCF performance of steels and related failure mechanisms has been made. 5 5,6 It is now generally accepted that the fatigue properties are significantly affected by the size of inclusions. Studies indicated that the rotating bending fatigue strength of bearing steels decreases with increasing inclusion size21 and the crack initiation depends on the inclusion size and shape, but is almost independent of their composition.22 Figure 1 clearly shows that though the composition of inclusions has certain influence on the fatigue property, the inclusion size plays the most important role indeed in the VHCF regime.21 Recently, Murakami et al. further pointed out that fatigue strength in particular is influenced by the sizes of the largest inclusions. 5 5,23
Harmful index of various inclusions in bearing steel 52100 (y ordinate scale unit is decrease in fatigue strength of 125 MPa at N = 108 cycles)21
The influence of inclusions on fatigue properties is an important topic for both manufacturers and users of steels. So many investigations have been carried out that it is rather difficult to conduct an exact and impartial survey. Here, attention was paid to summarise the effect of inclusion size on fatigue performances of high strength steels in the VHCF regime, especially the quantitative or semiquantitative understanding of those relations.
On critical inclusion size
It is well known that there is a difference between the effect of large and small inclusions. The former are harmful to the steel, often disastrous, whereas the latter are unavoidable, usually not dangerous, and sometimes can even be used to enhance certain steel properties. The critical border between small and large, the critical inclusion size, is still in most cases neither theoretically known or experimentally established, and it also depends on the property studied.19 The critical inclusion size is not fixed and depends on many factors, including service conditions. Broadly speaking, it is in the range of 5–500 μm. 4 4,24 It decreases with increasing yield stress. In high strength steels, its size will be around the lower bound of the range mentioned above.24
Duckworth and Ineson25 found that the critical inclusion size for rotating bending fatigue depended on the distance below the steel surface and increased from 10 μm just below the surface to 30 μm about 100 μm below the surface by artificially introducing Al2O3 inclusions into bearing steel. For 4340 and 300M steels, the critical inclusion size of 10 μm at the surface was also reported.2 Kawada et al. 26 found a slightly smaller value for the critical inclusion size of 8 μm just below the surface for the rotating bending fatigue. From a study of a series of fatigue failures in different bearing steels, they obtained the relation between critical inclusion size and depth below the steel surface for inclusions which could start a fatigue crack in rotating bending fatigue (in the HCF regime). Figure 2 shows the experimental results of bearing steels.26
Distance from specimen surface versus diameter of inclusions which have initiated fatigue fracture in rotary bending: N, S and V are normal, normal Swedish and vacuum melt bearing steels respectively26
For a fine grained 42CrMoVNb steel, the critical inclusion size for rotating bending fatigue can be 7–8 μm at steel surface,27 and for SAE9254 steel, the critical inclusion size can be about 6 μm.28 In a rolling contact fatigue test, the critical inclusion size as small as 2 μm was found in a bearing steel of GCr15.29 For clarity, the critical inclusion sizes for different steels are listed in Table 1. One can see that the critical inclusion sizes for these high strength steels are about 6–10 μm under rotating bending fatigue testing in the HCF regime (107 cycles).
Critical inclusion size for steels under rotating bending fatigue and contacting fatigue tests
Experimentally, for rotating bending fatigue, the critical inclusion size is generally obtained from extrapolation method, as shown in Fig. 2.26 First, a figure about the relationship between the size (inclusion diameter d, μm) and location (the distance from the inclusion to surface H, μm) of inclusions at which the fatigue crack initiated is drawn out. And then, the lowest line is drawn by passing through two data points, below which no other data point exists. The line is extended to intersect the ordinate at H = 0 μm (i.e. the specimen surface). The intercept is considered as the critical inclusion size of the steel investigated. Apparently, it is assumed implicitly that the lowest critical inclusion size should be located on the specimen surface. However, it is obvious that in this method, a large number of fractured specimens have to be required to assure the accuracy of the deductive results and, since there are generally some inclusions in a specimen that exceed the true value of critical inclusion size, the extrapolated critical inclusion size could be higher in general. Finally, it is noted that this method is valuable to determine the critical inclusion size for rotating bending fatigue because the stress gradient exists in the specimen so that the smaller inclusions as fatigue origins are near the surface and larger ones are close the centre of the specimen. In contrast, for axial loading fatigue, because the stress is rather uniformly distributed along the cross-section of a specimen and the size of inclusion as fatigue origin is rather randomly distributed along the cross-section, this extrapolation method could not be used properly.
Theoretically, Kiessling and Nordberg30 advocated the use of fracture mechanics concepts to estimate the critical defect size for fracture under monotonic loading in different steels. The critical values are 10–100 times too large compared with the critical inclusion size obtained by experiments.
4
4,30 A theory was proposed for determining the critical inclusion size with respect to void formation during hot working,31 and based on the theory, the corresponding critical inclusion sizes for with and without electroslag refined En 52 steel were determined.32 As regards fatigue, based on the dislocation accumulation theory, Tanaka and Mura33 proposed a model concerning fatigue cracks initiated from inclusions. Three cases of fatigue crack initiation at inclusions were classified: (1) from a completely debonded inclusion; (2) from a cracked inclusion impinged by slip bands; (3) from a slip band emanating from an uncracked inclusion. In case (2), the fatigue crack initiation life n
c due to inclusion can be written as33
Recently, the critical inclusion size of high strength steels with given matrix hardness or tensile strength and the surface roughness under VHCF was estimated35 mainly on the basis of Murakami’s theory, i.e. the inclusion effective projected area model.36
–
38 The fatigue strength of high strength steels influenced by inclusion can be written as39
Relationship between critical inclusion size d c and hardness HV of steel matrix35
Figure 4a displays the size of inclusions located at fatigue origin of spring steels and medium carbon Cr–Mo steels. Fatigue tests were conducted by rotating bending (107 cycles, most for medium carbon Cr–Mo steels) and tension–compression ultrasonic fatigue testing (109 cycles, most for spring steels).42 From this figure, the critical inclusion size of about 3 μm can be determined for the spring steel at 109 cycles. Here the critical inclusion size is determined by the minimum inclusion size not by the extrapolation method due to axial loading fatigue mentioned above. Also, from this figure, the critical inclusion size of about 6 μm can be determined for the medium carbon Cr–Mo steels at 107 cycles by extrapolation method. In the VHCF regime (109 cycles), the critical inclusion sizes can be found about 2 and 5 μm for SNCM439 and SUJ2 steels respectively, under rotating bending fatigue in laboratory air.43 It is noted that the critical inclusion size in the HCF regime (∼107 cycles) can be about 6–10 μm as mentioned above; on the other hand, in the VHCF regime (∼109 cycles), the critical inclusion size can be about 3–5 μm, just about a half of the value in HCF.
a Size and position as well as b size distribution of 244 inclusions at fracture origins of high strength steels under rotating bending fatigue and tension–compression ultrasonic fatigue testing42
Figure 4b shows the size distribution of these inclusions that acted as fatigue origins. One can see that the inclusions with sizes of less than 5 μm or greater than 50 μm are rare. Most inclusions found at fracture origins are around 10–30 μm.
Some questions still remain. For example, the model of Tanaka and Mura33 seems rather well for estimating the critical inclusion size with a clear physical picture; however, some material parameters used in their model cannot be well determined. In contrast, the model of Yang et al. 35 seems predicting the critical inclusion size in a reasonable range; however, the convincing mechanism of crack initiation caused by inclusion was not involved.
Characteristics of S–N curve
For a long time, S–N (stress amplitude versus the number of cycles to failure) curve in fatigue of materials has been considered as a single curve in which the number of cycles to failure increases with decreasing stress amplitude, eventually approaching the endurance limit below which no apparent fatigue failure occurred. Low strength steels have a clear knee point in the S–N curve by which the fatigue limit can be defined. This phenomenon is due to strain ageing, i.e. forming of the so called Cottrell atmospheres in which dislocations are locked by interstitial solute atoms such as carbon and nitrogen.44 – 47 However, it has been reported that the S–N curve for high strength steels consists of two parts, one that corresponds to short fatigue life at a high stress level due to fracture from a surface origin, and another that corresponds to long fatigue life at a low stress level due to fracture from internal inclusions and other inhomogeneities.16,36,48 – 58 The terminology of stepwise 9 10 9,10,15 or duplex 17 55 17,55,59 S–N curve was used in characterisation of those curves in the VHCF regime. Figure 5 schematically shows the S–N curve in high strength steels with duplex or stepwise curve type.15 It is noted that in reality, the experimental data are with a certain degree of scatter in the fatigue life diagram so that the transition of S–N curve will of course not be sharp. Many possible factors, which may cause fatigue failure in the very high cycle regime, are discussed in detail.48,49,52,53,60 – 63 These factors include environment, 15 43 15,43,62 residual stress,64 hardened layer, 15 64 15,64,65 surface roughness,66 loading types17 (of rotating bending and cyclic axial loading) and inclusion size. 67 67,68
Schematic of typical S–N curve for high strength steels15
Fatigue life behaviour of high strength steels containing non-metallic inclusions can also be described in a multistage or a step-wise S–N curve, as shown schematically in Fig. 6. 60 60,61 In the conventional low cycle fatigue range I, fatigue failures are initiated in most cases at the surface. The conventional HCF fatigue limit (range II) can extend into the VHCF regime, where it is terminated with the onset of the VHCF range III in which fatigue failures occur at stresses below the conventional HCF fatigue limit, originating in most cases from internal defects. Finally, there are reasons to believe that, at sufficiently low stress levels, there will be a true ultimate fatigue limit (range IV). 15 60 15,60,61 It is noted that in the VHCF regime, the conventional HCF fatigue limit eliminates 49 50 49,50,69 and the fatigue strength decreases with increasing number of cycles, which will certainly alter the design codes of component.69
Mughrabi61 considered a cylindrical specimen of gauge length l and diameter d containing inclusions of diameter d
i in a volume concentration n
i, and defined a critical volume density of inclusions
Terent’ev 65 65,70 has put forward a hypothesis that associates the presence of a marked deflection and a horizontal plateau on S–N curves with the presence of a hardened surface layer formed in the process of cyclic deformation at large test bases due to the dominant yielding of the surface layer. This harder barrier layer can also be created in the stage of fabrication of specimens or in hardening surface treatment (for example, nitriding). The barrier effect of the hardened surface layer is preserved in the presence of extrusions, intrusions and non-propagating microcracks in the surface layer.
Based on the VHCF testing results of four high strength spring steels with the same strength class (∼1700 MPa) but containing different inclusion sizes, the S–N curves of high strength steels could be divided into three categories:68 (1) the S–N curve displays a continuous decline and the internal cracks initiated from the large oxide inclusions for commercial 50CrV4 steel in which the average inclusion size on the fatigue origin is about 29 μm; (2) stepwise S–N curves were observed for clean 54SiCrV6 and clean 50CrV4 steels in which the average inclusion cluster sizes are about 11 and 7 μm respectively; (3) for clean 54SiCr6 steel in which the inclusion size is smaller than 1 μm which is definitely smaller than the critical inclusion size, the fatigue cracks did not initiate from inclusions or inclusion clusters but from the region enriched with carbon. In this case, the S–N curve shows that the fatigue failure hardly occurs from about 106–109 cycles; in other words, the fatigue property can be substantially improved in the VHCF regime. The S–N curve can be nominated as fatigue limit type.
It seems that the characteristics of S–N curves of high strength steels can relate to inclusion size as schematically shown in Fig. 7a .42 When the inclusion size is greater than ∼20 μm, the S–N curve displays a continuous decline type; 71 71,72 when the inclusion(or inclusion cluster) size is smaller than ∼20 μm and greater than the critical inclusion size, the stepwise type can be observed; 55 73 55,73,74 when the inclusion size is below the critical inclusion size, a fatigue limit type may be achieved. 68 68,75 These experimental results are qualitatively in accordance with the prediction of the variation trend of S–N curve caused by increasing the inclusion (inhomogeneity) size as shown in Fig. 7b . 15 76 15,76,77
Schematic of influence of inclusion size on S–N curve of high strength steels in very high cycle fatigue regime
For a high carbon chromium bearing steel SUJ2, Sakai et al. 17 17,59 and Shiozawa et al. 55 found duplex S–N curves. A typical diagram is shown in Fig. 8. Recently, Lu et al. conducted an interesting work,78 in which the high carbon chromium bearing steel GCr15 with a similar chemical composition but larger inclusion size was tested under rotating bending and cyclic axial loading; the S–N curves are shown in Fig. 9a , in which the S–N curve of SUJ2 shown in Fig. 8 is also indicated by dashed line for comparison. One can see that there is a clear step on the S–N curve of SUJ2 under rotating bending, while it seems that the step became much shorter for the GCr15 under rotating bending. On the other hand, under cyclic axial loading, the S–N curve of GCr15 declined continuously. The size and the position of inclusions at fatigue origins for the two steels are shown in Fig. 9b . It is evident that the inclusion size is much smaller (around 10 μm) for SUJ2 under rotating bending and a step can be clearly seen on the S–N curve. On the other hand, the inclusion size of GCr15 is much larger, around 20 μm for rotating bending and around 35 μm for axial loading. Since the highly stressed volume in the rotating bending specimen is much smaller than that in the axial loading specimen, the smaller inclusions can be found in the rotating bending testing. This may be one of the reasons why the stepwise or duplex S–N curve can be found more easily in rotating bending testing than axial loading testing. Up to now, it is reported that few S–N curves, for example, 51 56 68 71 73 51,56,68,71,73,79 show the stepwise or duplex characteristic under the axial loading test. Besides the influence by inclusion size, the other underlying reasons should be revealed further.
S–N curve for high carbon chromium bearing steel SUJ255
It seems reasonable that if the inclusion size is smaller than the critical value, there will be no substantial effect of inclusion on the VHCF failure of steels, and the S–N curve with fatigue limit type may be achieved. However, if the inclusion size is larger, the volume concentration of inclusion is usually higher; therefore, if the inclusion size is greater than a certain value (say, 20 μm), the volume density of inclusion is beyond a critical value, the possibility to cause fatigue failure at surface inclusion increases61 and the transition of surface to interior failure can readily occur; therefore, a continuous decline type can be achieved. When the inclusion size is small and the volume density is low, the stepwise or duplex S–N curve could be observed as mentioned above.
Recently, the effect of residual stress on the transition of surface failure to internal failure has been studied in detail by Shiozawa et al. 79 Figure 10 shows the S–N curves of SNCM439 steel obtained from an axial loading test, in which the low and high residual compressive stresses existed in emery polished and shot peened specimens respectively. The inclusion size is in the range of 5–30 μm and the average is 14·7 μm. The clear stepwise S–N curve can be seen for the low residual stress specimens but not for the high residual stress specimens.
S–N curves of emery polished and shot peened specimens of SNCM439 steel, obtained form an axial loading test79
Investigation results showed that the S–N curve corresponding to internal fracture, which appears in very high cycles of lifetime, was not much affected by the surface and environment conditions but mainly dependent on inclusion size.78 However, the current understanding of the relationship between inclusion size and S–N characteristics was only qualitative in nature as mentioned above. Quantitative analysis on the relationship between inclusion size and S–N characteristics is much needed.
An attempt was made to evaluate how the S–N curve correlates with the inclusion size as follows. The Basquin equation can be used to describe the S–N curve in the HCF regime. Recently, Liu et al.
80 assumed that this equation is also valid to predict the S–N curve in the VHCF regime
If the inclusion size and hardness of a steel are known, the S–N curve can be predicted by equations (8) and (9). For example, the inclusion size ranging from 10 to 28 μm on the fatigue fracture surface of high strength steel 60Si2CrV, combining with the Vickers hardness of 560, the predicted upper (inclusion size with 10 μm) and lower (inclusion size with 28 μm) bounds of S–N curves are shown in Fig. 11.80 It is generally in accordance with experimental results. From these relations, one can see that both fatigue strength coefficient and Basquin exponent relate with inclusion size and steel strength (hardness); therefore, the wide span of inclusion size will certainly cause a strong dispersion of fatigue data in the S–N diagram. This will be one of the factors that account for the statistical nature81 of the fatigue data. Recently, Sakai et al. reported a convincing probabilistic stress life characteristic for bearing steel SUJ2 in the VHCF testing, in which surface initiated fracture and interior initiated fracture (fish eye fracture) are clearly indicated.82
Predicted S–N curves and experimental results of 60Si2CrV80
Although significant progress has been made in understanding the characteristics of S–N curve of high strength steels in the VHCF regime, the convincing mechanisms, particularly the quantitative descriptions of the transition from surface fracture to internal inclusion fracture, are still lacking.
Fatigue strength and fatigue life
In the early experimental studies, it was found that there exists a correlation between inclusion size and fatigue strength provided that the inclusions are with identical chemistry and similar shape.
2
4
25
2,4,25,83 The relation is expressed by2
If the area of a crack at inclusion is denoted by ‘area’, then the maximum value K
Imax of the stress intensity factor along its crack front is given approximately by87
When K
Imax = ΔK
th, the fatigue strength can be determined. Combining equations (15) and (16), and noticing the difference of the unit of the (area)1/2 between those equations, the fatigue strength σ
w can be expressed as equation (4),39 i.e. 
Recently, an expression based on the understanding of the effect of hydrogen during forming granular bright facet (GBF)55 (noting that GBF and ODA are different terminologies for the same special area around an inclusion that acted as the fatigue origin, see the section of mechanisms of failure from internal inclusions below) was developed to predict the fatigue strength of high strength steels in the very high cycle regime (at 109 cycles), it reads as88
The relations between fatigue life and inclusion size in high strength steels were proposed
74
78
74,78,89 based on the assumptions below: (1) for VHCF life or low applied stress amplitude, the fatigue crack initiates from an interior inclusion around which a GBF area exists, and the diameter of the GBF area is several times larger than that of the inclusion; (2) the growing crack in GBF area is always assumed as a micropenny crack and according to Paris et al.’s and Marines-Garcia et al.’s opinions
90
90,91 that the microcrack still obeys Paris equation, i.e. da/dN = C(ΔK)m, where, da/dN is the crack growth rate, ΔK is the stress intensity factor range, and C and m are constants; (3) total fatigue life N
f is approximated by the life of crack growing in whole GBF area. Mayer et al.
89 and Lu et al.
78 investigated the fatigue behaviour of bearing steels and found
Paris constants for some high strength steels
From equations (14), (20) and (21), the dependence of fatigue life on inclusion size D can be summarised as
Index of inclusion size correlating with fatigue life of some high strength steels
Based on the experimental results of the ODA size against the number of cycles, obtained from references
13
97
13,97,98 for quenched and tempered SUJ2, SCM435 and SNCM439 steels, Chapetti et al.
99 obtained the relationship between fatigue life and inclusion size
In the early days, the fatigue life is limited to 107 cycles, and the effect of inclusion on fatigue damage is comparatively not very strong as in the VHCF regime; therefore, a smaller index of 3 could be obtained. In contrast, in the VHCF regime, the inclusion plays a very important role and a greater index such as 4·29–8·42 can be found. The results are summarised in Table 3
According to Murakami’s point of view,5 a hard inclusion could be considered as a small crack in high strength steels, and fatigue strength is not determined by the critical stress for crack initiation but the threshold stress for crack growth. Since the threshold value of stress intensity factor range for small crack is weakly related with inclusion size by a power of 1/3; (for example, see equation (16)), 38 38,100 the fatigue strength in turn depends on inclusion size rather weakly by a power about 1/6. On the other hand, fatigue life of high strength steel is mostly determined by crack propagation rate that is strongly related to inclusion size by Paris law in which the index m could be much greater.
At present, the commercial high strength steels (ultimate tensile strength >∼1500 MPa) usually contain numerous small inclusions and some large inclusions. From an intensive work on fatigue testing, the inclusion sizes on fatigue fracture origin are found to be around 20 μm in average (for example, see Fig. 4b ). If the average inclusion size on fracture surface could be reduced from 20 to 10 μm that is still greater than the critical inclusion size, ∼10% of fatigue strength increase and ∼100 times of fatigue life longer could be expected (where n 1 = 1/6 and n 2 = 6·5 were used in the estimation). Fatigue life is very sensitive to the inclusion size; hence, the high quality steels require very strictly control of the inclusion size so that reliability of the fatigue property could be substantially improved.
Another question is: how large is the difference of fatigue strength between 107 and 109 cycles? Bathias et al.
63 found that the difference of fatigue strength, between 106 and 109 cycles, decreases by 50–200 MPa for some alloys and steels. Recently, Hong et al.
101 collected 58 S–N curves for low strength and high strength steels.51,63,64,71,89,100,102
–
122 The difference of fatigue strength between 107 and 109 cycles for steels with different tensile strengths is shown in Fig. 13.101 In a previous work, the fatigue strength of high strength steels at 106 cycles was estimated as80
Difference of fatigue strengths between 107 and 109 cycles Δ σ w for steels with different tensile strengths σb:101 solid line represents value predicted by present author
From this figure, one can see that the difference of the fatigue strength between 107 and 109 cycles is less than 25 MPa for about one-third of the steels at different strength levels. For low strength steels or high strength steels with inclusion size less than the critical value, the fatigue limit type of S–N curve may exist, so that the difference of fatigue strength between 107 and 109 cycles is small. However, for high strength steels, if the inclusion size is not less than the critical value, some of which still show the slight difference of fatigue strength between 107 and 109 cycles, the underlying reasons need further investigation.
Mechanism of failure from internal inclusions
Internal fatigue fracture origins of high strength steels in VHCF are mostly at non-metallic inclusions. In the vicinity of a non-metallic inclusion at the fracture origin, a dark area was often observed inside a fish eye mark for specimens with a long fatigue life. Murakami et al. made observations by optical microscopy and named this area as ‘optically dark area’ (ODA). 48 52 48,52,53 Sakai et al. made observations by scanning electron microscopy and transmission electron microscopy and the ‘fine granular area’ (FGA) around the inclusion at the fracture origin was found.59,123 – 126 By using scanning electron microscopy, a rough area beside the inclusion at the fracture origin was found by Shiozawa et al. and it was named as the ‘granular bright facet’ (GBF) zone. 55 55,127 This feature is also found and named as ‘rough surface area’ (RSA) by Ochi et al. 103 Now it is clear that the different terminologies employed by different authors refer to the same microscopic feature. The typical fracture surface and its characteristic zones are shown in Fig. 14.
a typical fractography of high strength steels under VHCF50 and b schematic of characteristic zones on fracture surface
The size of the ODA (GBF, FGA and RSA) increases with increasing fatigue life, and the ODA (GBF, FGA and RSA) cannot be found on fracture surfaces of specimens which failed at a small number of cycles. Recent studies have confirmed that the fatigue crack initiation period can account for a very large fraction of fatigue life57,60 – 62,73,99 in the VHCF regime; therefore, investigation of the forming mechanisms of ODA (GBF, FGA and RSA) is essential.
It is quite common for heat treated high strength steels to contain hydrogen around non-metallic inclusions. 128 128,129 Recently, Takai et al. 129 129,130 and Murakami et al. 131 directly verified the presence of hydrogen trapped at the interface of inclusions using secondary ion mass spectrometry. Hydrogen trapped around a non-metallic inclusion was also observed by tritium microautoradiography as shown in Fig. 15.132
Hydrogen trapped around non-metallic inclusion (Al2O3) observed by tritium autoradiography132
Murakami et al. proposed a hydrogen embrittlement mechanism for the ODA formation. By the superposition of applied cyclic loading, residual stresses and existence of hydrogen, the initiation and growth of crack from internal non-metallic inclusions take place due to hydrogen assisted fatigue. The process is characterised by a formation of ODA. 48 52 48,52,53 After slow fatigue crack growth in the dark area beside the inclusion, the size of the crack exceeds a critical value given by the threshold for pure fatigue propagation, and the crack grows without the assistance of hydrogen afterwards. In the ODA formation process, the hydrogen enhanced localised plasticity 133 133,134 will cause a highly localised plastic failure. In this process, since a very complicated stress distribution exists near the inclusion and at the microcrack tip, the hydrogen assisted crack growth may promote the activation of different slip planes and hence promotes the formation of rougher fracture surface. The rough area around inclusion can reflect the light in an optical microscope and hence, a dark area can be observed. Figure 16 is the schematic illustration of the mechanism for VHCF fatigue failure coupled with hydrogen. 48 52 53 48,52,53,57
Based on the mechanism of crack growth assisted by hydrogen and the model of local stress intensity factor at the crack tip due to the influence of hydrogen,135 a criterion for the formation of GBF (ODA, FGA and RSA) is proposed.100 Suppose a penny shape microcrack grows from the internal inclusion, the crack size is (area)1/2, and the total stress intensity factor at the crack tip is K
T, which is the superimposition of intensity factors exerted by applied stress K
Imax and the hydrogen influence k
H. It is known that K
Imax increases with crack size (for example, see equation (15)). k
H can be written as100
Schematic of criterion for GBF crack growth100
Relationship between σw/(HV+120)15/16 and inclusion size (area in)1/2 for 17 high strength steels with different inclusion sizes88
The effects of hydrogen on the fatigue behaviours of high strength steels in the VHCF regime have been studied recently, for example.93
–
95,136–138 The fatigue strength depending on inclusion size, hardness and hydrogen concentration has been studied, it reads,
94
94,95
Fitting curve of hydrogen influence factor for total hydrogen content42
Comparison of calculated fatigue strength
Now there are three equations to estimate the fatigue strength based on the inclusion size under stress amplitude control at stress ratio of R = −1. Murakami equation (equation (4)) was developed on the bases of the inclusion effective projected area model and fracture mechanics. No detailed hydrogen assisted mechanism was involved. It can be used to estimate the fatigue strength of high strength steels with low hydrogen content (for different R ratios, Murakami5 and Bathias and Paris6 provided the modified equations). Equation (17) was developed by the model of GBF formation which is influenced by hydrogen near the inclusion. Both equations (4) and (17) are quite similar. It is noted that equation (17) is also used to estimate the fatigue strength of high strength steels with low hydrogen content. The reason is that the hydrogen concentration at crack tip near the inclusion cannot be determined accurately, so that the parameter in this equation has to be determined experimentally by analysing the fatigue strengths of many high strength steels with low hydrogen content.88 For high hydrogen content, an alternative method was used to evaluate the fatigue strength, in which the hydrogen content in the tested sample was measured experimentally with certain accuracy. Equation (28) can be used to estimate the fatigue strength of high strength steels with high hydrogen content.
All these models to predict the fatigue strength 39 88 39,88,95 and fatigue life 78 89 78,89,92 are essentially microcrack growth approach, in which the specific microcrack initiation and early growth process depending on the microstructural barriers 49 49,62 has not been well involved. However, the basic characteristic of the microcrack initiation and early growth has been intensively studied recently. From the schematic of a typical fractography of high strength steels in the very high cycle fatigue regime (Fig. 14b ), one can see that the fatigue failure process can be divided into several stages: at first, the nucleation of a microcrack at an inclusion with a size equivalent to the inclusion, and the microcrack nucleation life is N i1; next, the early slow growth of the microcrack beside the inclusion in the region of GBF (ODA/FGA/RSA), and the fatigue life in this region is N i2; at last, the crack propagation in the fish eye region which results in the final fracture, and the crack propagation life is N p. The total fatigue life N f should be as N f = N i1+N i2+N p.
For high strength steels in the HCF regime, if an inclusion is without GBF (ODA, FGA and RSA) area on the fracture surface, crack initiation at the inclusion generally occurs very early in the fatigue process and the crack propagation life usually takes up a large percentage of the total life.139 On the other hand, in the VHCF regime, the situation is quite different. Murakami et al. 53 found that the number of cycles required for the crack to grow from the ODA border to the fish eye border in the Cr–Mo steel SCM435 ranges from 105 to 106 cycles, which is a small fraction of the total fatigue life in the VHCF regime. Wang et al. 16 analysed the crack propagation life N p for Cr–Si and Cr–V spring steels and found that N p is less than 1% for the total fatigue life over 108 cycles. Chapetti et al. 57 reached the similar statements by systematic analysis of the data of bearing steel SUJ2.55 Kazymyrovych et al. 140 measured the striation spacing on the fracture surface of AISI H11 tool steel and found that the fatigue life consumed within and beyond fish eye is only a very small fraction (about 104–105 cycles) of the total fatigue life (about 108–109 cycles). Paris et al. 90 90,113 have shown that in the VHCF regime, the fatigue crack growth constitutes only about 1% of the total fatigue life. All these results show that the crack propagation life N p in the fish eye region and over can be neglected without dispute in the VHCF regime for high strength steels.
Now it is necessary to reveal the features of the microcrack nucleation and early slow growth process. It is indeed that one difficulty will always be where to draw the borderline between crack initiation and propagation. 61 61,62 Since the inclusion at the initiation site and the related GBF (ODA, FGA and RSA) area are usually very small (less than 100 μm141 in general), and the microcrack growth in this region is very slow, many researchers define that the microcrack nucleation and slow growth in the GBF (ODA, FGA and RSA) area are the constituents of crack initiation stage, for example, Refs. 61 and 62. From the review mentioned in the above paragraph, the fraction of crack propagation life N p in the fish eye region and over can be less than 1%; therefore, the fraction of fatigue life in the crack initiation stage, N i = N i1+N i2, can be greater than 99%; in other words, it is indeed that the fatigue life is controlled by the crack initiation stage for high strength steels in the VHCF regime.
As regards the nucleation life N i1, if a premicrocrack does exist in an inclusion or at the interface between the inclusion and matrix before fatigue (sometimes it can be induced by rolling or other processing routes), the nucleation life could be considered as null at the first approximation. It is further noted that for an inclusion with GBF (ODA, FGA and RSA) area on the fracture surface, crack nucleation at the inclusion generally occurs at about 5–10% of the total fatigue life.142 Those results indicate that the nucleation life N i1 for high strength steels in the VHCF regime could be a rather small fraction of the total fatigue life. Therefore, the microcrack slow growth in the GBF (ODA, FGA and RSA) area will play a key role in controlling the fatigue behaviour of the high strength steels. Kuroshima and Harada143 and Tanaka and Akiniwa74 assumed that in the GBF (ODA, FGA and RSA) area, the Paris equation can also be used and the Paris constants can be determined by experiment. This method is effective to investigate the fatigue behaviours of high strength steels in the VHCF regime with the merit of simplicity. On the other hand, it is worth to develop more subtle models to evaluate the crack initiation stage. Wang et al. 73 estimated the microcrack initiation life at the inclusion based on a modified Tanaka and Murra’s model.33 They found that the crack initiation life occupies the overwhelming part of the total fatigue life.
The microcrack initiation process is of very importance for the type I materials including pure metals and defect free materials1 as emphasised by Mughrabi 60 61 60,61,144 and Lukas and Kunz.145 And it is also very important for VHCF of high strength steels as mentioned above; however, the model involving the specific mechanism and microstructual features to evaluate the crack initiation process is much needed.
Shiozawa et al. 127 proposed another mechanism that multiple microcracks are initiated by decohesion of spherical carbide from the matrix around an inclusion. Figure 21 schematically shows that discrete microcracks along boundaries between the carbide particles and the matrix and their coalescence generate the characteristic morphology inside the rough surface area. They name it as GBF area. Recently, they also made detailed observation of the features of GBF on the fracture surface as shown in Fig. 22.79 It clearly shows that a GBF area is very rough compared with the other area inside the fisheye (Fig. 22a ). Figure 22b shows the bird view over the GBF area and the matrix of the specimen, and Fig. 22c shows the distribution of carbon near an inclusion (Al2O3) on the fracture surface. It is believed that detection of carbon enrichment in the GBF area results from the remaining carbide particles, which are created on the fracture surface around an inclusion during fatigue fracture process.
Schematic of GBF formation by coalescence of decohesion of carbide with matrix127
Characteristic of GBF area
Sakai126 proposed a new mechanism that consists of three steps (Fig. 23): (1) a fine granular layer caused by the intensive polygonisation is gradually induced around the interior inclusion during long sequence of the cyclic loading; (2) nucleation and coalescence of microdebondings; (3) microdebondings are entirely spread over the fine granular layer and the penny shape crack is finally formed.
Illustration of fatigue crack initiation process around inclusion126
Recently Huang et al. proposed that the ODA (GBF, FGA and RSA) could be formed by irreversible persistent slip bands in a bearing steel.146 Clear evidence to verify this mechanism is still needed.
Each of these mechanisms would be conceivable in some cases; however, none of these mechanisms can be broadly accepted without challenge.126 However, the hydrogen assisted fatigue cracking for high strength steels in the VHCF regime could be encountered frequently in practice, and the related mechanism is worth further studying.
Estimation of maximum inclusion size
Up to now, it seems that once we have reliable information on inclusion size as well as other related parameters, such as hardness and hydrogen content, the fatigue strength and fatigue life could be estimated reasonably by corresponding equations mentioned above for some high strength steels. Now it is well known that the inclusion size at the fatigue origin is much greater than the inclusion size measured by standard metallographic inspection. Generally, the inclusion at the fatigue origin is the maximum one in the highly stressed test volume of a specimen or a component. The question is how to predict the size of the maximum inclusion in a tested specimen or in a very large volume of steel using data from a small volume of steel. Recently, much progress has been made.147 The statistics of extreme value (SEV) method is based on measuring the maximum size of inclusions in randomly chosen areas or volumes and it has been intensively investigated by Murakami and co-workers,14,36 – 38,148–150 for discrimination between superclean steels and the estimation of the maximum size of inclusions in a large volume of steel. The second method is the generalised Pareto distribution (GPD) method, developed by Atkinson et al.,151 – 155 where the size of all inclusions larger than a chosen threshold value is measured. Both SEV and GPD methods allow data on inclusion sizes in small volume of steel to be used to predict the maximum inclusion size in a large volume of steel.
The basic concept of extreme value theory is that when a fixed number of data points following a basic distribution are collected, the maximum and minimum of each of these sets also follow Gumbel distribution156 as follows
Sketch of measurement of maximum inclusion size in SEV method, modified from Ref. 147
The values of (area
max,i)1/2are ranged, starting from the smallest, and ranked with i = 1, 2, …, N. Then (area
max,1)1/2⩽(area
max,2)1/2⩽…⩽(area
max,N)1/2. The cumulative probability of the ith inclusion that is no greater than inclusion size z
i can be calculated simply by
For the prediction of inclusion size in a large volume of steel V, the return period T is defined as
Atkinson et al.
151
–
155 predict the maximum inclusion size in a certain volume of steel based on the GPD
Sketch of measurement of inclusion size which is greater than threshold u in GPD method, modified from Ref. 147
When the GPD is used to model the sizes of inclusions which exceed a threshold u, the characteristic size of the maximum inclusion x
V in a volume V can be found as follows.151 The expected number of inclusions in volume V exceeding a size x is equal to the product of the expected number in V exceeding u and the probability, given that the inclusion is larger than u and the size of inclusions is larger than x. This last probability is given immediately by equation (39). Thus, if N
V(u) denotes the expected number of exceedances of u in unit volume, the size x
V which is expected to be exceeded exactly once in volume V satisfies
Illustrative mean excess plot for GPD method147
For the SEV method, the size of maximum inclusion increases infinitely with increasing steel volume. However, for GPD method, one of the main characteristics is that there is an upper limit for the estimated inclusion size147 (Fig. 27 157). This feature is a big help for the steel makers to evaluate the quality of large amount of steels. However, for the fatigue testing in the laboratory, the highly stressed volume in the gauge section of specimen is usually very limited, for example, for an axial hourglass shaped specimen with diameter of 3 mm and the gauge length 5 mm, the test volume is about 35·3 mm3 and its weight is about 0·28 g. Supposing that 15 samples are used to determine the fatigue strength by staircase method, then the weight of the total tested volume is about 4 g, i.e. 4×10−3 kg. Only if larger specimen and more specimens are used, the test volume can be larger. However, if rotating bending specimen is used, the highly stressed volume can be smaller due to stress gradient. Therefore, for the conventional specimens, the weight of tested volume can be around 10−3–10−1 kg, and in this range, the predictions of the maximum inclusion size between SEV and GPD are very close as shown in Fig. 27. For clean steels, the inclusion content is very low, and in each inspection area, very few inclusions can be found; therefore, the SEV method may be preferential for estimation of the maximum inclusion size due to its simplicity and time saving.
Comparison of characteristic size of maximum inclusion and maximum likelihood confidence intervals estimated by SEV and GPD methods in 40 sample areas157
If the maximum inclusion size is estimated, the lower bound of fatigue strength can be evaluated38 by using the equation (4) or (17).
Now it is confirmed that one adopts the estimated maximum inclusion size to predict the lower bound of fatigue strength rather well, 14 38 14,38,42 for example, see Fig. 28.14 The question is how the fatigue strength can be predicted more precisely, based on inclusion size estimated by the present statistical methods. Experimentally, the fatigue strength is determined by several specimens, for example, at least six pairs of effective data are needed to determine the fatigue strength by staircase method. It generally involves different inclusion sizes at fatigue origins of the broken specimens. In previous work, it was shown that if the average inclusion size at fatigue origins was used to estimate the fatigue strength, a better precision will be achieved.88 Now the question is how to use the information of maximum inclusion size and distribution that is obtained by SEV or GPD method to estimate the fatigue strength more precisely, not the lower bound of fatigue strength by using the maximum inclusion size.
Predictions of lower fatigue strength by SEV method for 10 and 100 specimens and experimental results14
Some problems
Recently, great progress has been made to study the effects of inclusions on VHCF behaviour of high strength steels. Existing monographs 5 5,6 and reviews 38 49 57 77 99 126 38,49,57,77,99,126,158 have highlighted the significant achievements that we have made and some problems that we have to solve. Some of these problems are listed below.
Experimental facilities and related limitations
Currently, there are two major methods to test the VHCF properties of high strength steels, i.e. ultrasonic fatigue test with very high frequency of around 20 kHz11,159 – 162 and modified fatigue testing machines with several specimens tested simultaneously at a conventional frequency. 82 123 82,123,126 In ultrasonic fatigue testing, the experimental data can be obtained in a short period that is very helpful for improving the steel quality or developing new steels. However, the temperature raise,163 particularly the frequency dependence 164 164,165 of fatigue property, should be clarified in further work. It is also necessary to use conventional testing machines to compare with the results tested by high frequency.166 Results depend on frequency when time dependent processes like environmental effects (corrosive medium and high temperature) or strain rate dependence of material behaviour exist.77 In modified rotating bending fatigue testing, although several specimens are used in one machine, the time consumed may still be a barrier for VHCF testing. Recently, other high frequency test systems with the frequencies of several hundreds to a few thousands of Hertz were also developed for VHCF testing, mostly for the non-ferrous alloys.167 – 169 Using those test systems to intensively investigate the VHCF behaviours of high strength steels is desired. It will help us to obtain more VHCF data and evaluate the frequency effect56 of high strength steels.
Fatigue fracture mechanism
As mentioned in previous section, several mechanisms 48 126 127 140 48,126,127,140,146 have been proposed for fracture initiation at inclusions in the VHCF regime. It seems that those mechanisms can be sketchily divided into three categories: (1) hydrogen assisted fracture; (2) carbide cluster induced fracture; and (3) dislocation configurations such as cyclic slip irreversibility, 60 61 60,61,145 slip bands or polygonisation induced fracture. Those three mechanisms may be intimately related with each other. For example, since the hydrogen enhanced localised plasticity will certainly promote the intensive dislocation activity, under a very complicated stress distribution around the inclusion, multislip will operate and in turn the slip bands or polygonisation could be easily formed during cyclic loading. Also, the carbides will be the strengthening phase when they are small and uniformly distributed or they can act as inclusions when they are large and cluster together.170 In the later case, the enriched hydrogen around the carbides probably also plays a role in the fracture. It is expected that a more comprehensive and convincing model can be proposed soon.
Fatigue origins initiated from other defects or microstructures
In the VHCF regime, Mughrabi61 points out that for ductile single phase type I materials that do not contain inclusions the fatigue crack initiates at persistent slip bands or cyclic slip irreversibility. Progress in this aspect has been reviewed by Zimmermann.1 For type II materials, e.g. high strength steels containing non-metallic inclusions, fatigue crack initiates at slip bands, grain boundaries, pores, non-metallic inclusions or particles at surface sites in the HCF regime, but it shifts to interior sites in the VHCF regime, mostly at non-metallic inclusions. However, even for the high strength steels in the VHCF regime, fatigue crack initiation site can also be observed in an internal non-defect area or matrix microstructure that is not associated with pre-existing defects. This type of crack origin is named as ‘non-defect crack origins’. 104 104,171 Actually, these origins could be large and soft phase (bainite) in bainite/martensite duplex phase steels 48 48,172 or large and soft grains in non-ferrous alloys,173 or carbon or chromium enriched regions in some spring steels68 or other inhomogeneities. How do these inhomogeneities174 compete with inclusions in the VHCF regime? Up to now, no detailed investigation has been undertaken.
Another problem is that although the inclusion size was emphasised in the review, the effect of composition of inclusion on the VHCF property is still worth investigating. In the quality control of the steel making process, the detailed information of chemical composition of inclusion that acted as fatigue origin will help steel makers to conduct adequate measures.
Standardisation
The standard procedures on fatigue testing and data analysing in the VHCF regime should be accomplished in the future. Since the highly stressed volumes in specimens are varied for different fatigue test facilities, the size effect 71 175 71,175,176 of specimens on fatigue testing should be evaluated and the standard specimens should be suggested firstly. As regards the assessment of inclusions in steels, the international standard such as ISO 4967:1998(E) is widely used to assess the suitability of a steel for a given use, and plays a very important role in steel industry. However, since it is difficult to achieve reproducible results owing to the influence of the test operator, even with a large number of specimens,177 the supplement methods such as SEV and GPD are recommended, and an intensive work is still needed in standardisation of these methods for the sake of obtaining the results with a certain reproducibility. In this process, the determination of inclusion size should be evaluated and standardised further. In Murakami’s model, the inclusion size is determined as (area)1/2, where the inclusion effective projected area can be measured as follows. On the fatigue facture surface, the single inclusion with spherical or ellipsoid shape can be frequently found in a scanning electron microscope, the area of inclusion can be obtained directly by using the image analysis software, or by measuring the diameter of a spherical inclusion or the lengths of major and minor axes of the ellipsoid inclusion, and hence, the area can be calculated by formula. For irregularly shaped inclusions, including inclusion clusters, the effective area of inclusion is estimated by considering a smooth contour which envelopes the original irregular shape.5 In SEV and GPD methods, the inclusion size on the sample surface was measured in an optical microscope by the method mentioned above. It is noted that in some works, the (area)1/2 model was not used, so the inclusion size may be expressed by diameter that is directly measured in a microscope. Different methods lead to more diversity of the data, so that a standardised method to measure the inclusion size is needed.
Last but not least, the inspection of key components171 in the VHCF regime and the recommendations of design code69 based on VHCF are welcomed.
Summary
The effects of inclusions on VHCF properties of high strength steels have been reviewed with emphasis on the inclusion size. The critical inclusion size in the VHCF regime is about 3–5 μm that is smaller than that in the HCF regime. In the VHCF regime, the S–N curve characteristics are influenced by inclusion size, residual stress, roughness and other factors. The dependence of fatigue strength and fatigue life on inclussion size can be expressed as 
As Thornton has pointed out:3 ‘Due to the numerous factors involved, any general correlation between non-metallic inclusions and mechanical properties is highly improbable. While it is possible to obtain a plausible relationship in a particular situation, the factors involved must be standardized to such an extent that application of the results is extremely limited’. In other words, more caution is required in applying the relationships reviewed above. However, those quantitative or semiquantitative relationships will certainly help to explore the VHCF mechanisms and improve the quality of high strength steels in future. Those relationships will be examined and modified in the coming works, and a more thorough understanding of the effects of inclusions on VHCF properties of high strength steels is in prospect.
Footnotes
Acknowledgements
This work was financially supported by the National Key Basic Research and Development Program of China (no. G2004CB619100). The author thanks Professor Y. Q. Weng and Professor W. J. Hui in Central Iron and Steel Research Institute, Beijing, China, for their longstanding cooperation. Thanks go to Professor Z. F. Zhang and Professor Z. G. Wang for their helpful advice, and to my colleagues Professor Q. Y. Wang in Sichuan University, China andProfessor Z. G. Yang and Professor G. Y. Li for their support in the work in the past many years. Thanks also go to Dr Y. B. Liu, Dr Y. D. Li and Dr J. M. Zhang, and to Master J. F. Zhang and Master S. M. Chen for their contributions to the review. Particularly, Dr Y. B. Liu redrew some of the figures used in this review. Professor Q. Y. Wang in Sichuan University, China and Professor L. Z. Sun and Professor T. Zhai in the USA and Professor Z. M. Sun in Japan provided me some outmoded but important references, so I must express my sincere thanks for their kindness.