effects of inclusions on very high cycle fatigue properties of high strength steels

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Effects of inclusions on very high cycle fatigueproperties of high strength steels

S. X. Li*

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 sw!D2n1 and Nf!D2n2 respectively. For fatigue strength, the exponent n1 has

been reported to be y0?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 n2 is reported to be y3 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.

Keywords: Very high cycle fatigue, Inclusion size, High strength steels, Fatigue strength, Fatigue life, Critical inclusion size, Maximum inclusion size, S–Ncurve, Hydrogen content, Review

IntroductionThe present review is one of two being published byIMR on the fatigue behaviour of metallic materials inthe very high cycle fatigue (VHCF) regime (general-ly understood to refer to the range from y107 to109 cycles). A complementary review by Zimmerman1

focuses on the damage mechanisms of ductile materialsthat do not contain non-metallic inclusions; in contrast,in the present review, attention is paid to the influence ofinclusions on the fatigue features of high strength steels.

For more than half a century, non-metallic inclusionshave been associated with the general problems of fatiguefailure in steels.2–4 Particularly, in the VHCF regime, theinclusions play a crucial role in fatigue failure of highstrength steels.5,6

Conventionally, the fatigue design of structuralcomponents is based upon the data in the high cycle

fatigue (HCF) regime (about 105–107 cycles). However,the fatigue life of current automobile engines rangearound 108 cycles; high speed trains range to 109 cycles.It is further noted that at this time interest in fatigue lifeextends to about 1010 cycles, for example, in turbineengine components.6 There is strong need to investigatethe fatigue behaviours of steels in the VHCF regimenowadays.

From a historical perspective, although Kikukawaet al.7 found for the first time in the mid-1960s thatstructural steels can fail after 107 cycles, no substantialprogress has been made until mid-1980s. Then numerousworks had been conducted to explore the effect ofinclusions on VHCF behaviours of steels.8–17

Existing overviews or monographs on inclusionsusually point out the following factors that should beconsidered in resolving the effects of inclusions onfatigue properties, i.e. the inclusion size, shape, quantity,distribution and composition.2–4,18,19

Before mid-1980s, the increased knowledge of the oxideand sulphide inclusions meant that it was possible togradually decrease the O and S levels of most steels. Now

Shenyang National Laboratory for Materials Science, Institute of MetalResearch, Chinese Academy of Sciences, Shenyang 110016, China

*Corresponding author, email [email protected]

92

� 2012 Institute of Materials, Minerals and Mining and ASM InternationalPublished by Maney for the Institute and ASM InternationalDOI 10.1179/1743280411Y.0000000008 International Materials Reviews 2012 VOL 57 NO 2

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it is possible to make steels with as low as 1–10 ppm ofboth these elements. It is often a question of cost how farthis process should be taken.19 However, every ton ofsteel made with as low as 1 ppm of O and S still holds109–1012 inclusions. In other words, about 10–104

inclusions exist in a cubic millimetre, of course most ofwhich are small inclusions. Stress localisation at theinterface between the inclusion and matrix is the origin offatigue failure. This arises from (1) the differentialthermal contraction of inclusions and matrix duringcooling and (2) the concentration of remote appliedstresses due to elastic constant differences between thematrix and the inclusions.4 The difference of thedeformability between inclusion and surrounding matrixwill stimulate the crack initiation. The qualitative effectsof inclusions on the mechanical properties of steel as wellas on failures in steel processing and in service are knownin outline.2,4,19

In the HCF regime, the main conclusions were thattwo criteria must be fulfilled for an inclusion to be apotential initiation site for a fatigue failure:4 a criticalsize, depending on the depth of its location below thesteel surface (for rotating bending fatigue); a low indexof deformability at the actual temperature of the fatigueloading. The most dangerous inclusions are single phaseAl2O3, spinels and Ca aluminates, and the least harmfulare MnS inclusions.4 The review of Leslie20 summarisedthe situation before mid-1980s. His major conclusionswere: the main effect of inclusions related to fatigue is oncrack initiation, but they also have an effect on fatiguecrack growth rate. In the HCF range (.105 cycles),nearly all cracks originate from inclusions. In the lowcycle fatigue range (,105 cycles), especially at the rangeof 101–103 cycles, slip band cracks predominate. Fatiguecracks often initiate at cracked or debonded inclusionsor in slip bands emanating from inclusions.

In the past two decades, great progress on VHCFperformance of steels and related failure mechanismshas been made.5,6 It is now generally accepted that thefatigue properties are significantly affected by the size ofinclusions. Studies indicated that the rotating bendingfatigue strength of bearing steels decreases with increas-ing inclusion size21 and the crack initiation depends onthe inclusion size and shape, but is almost independent

of their composition.22 Figure 1 clearly shows thatthough the composition of inclusions has certaininfluence on the fatigue property, the inclusion sizeplays the most important role indeed in the VHCFregime.21 Recently, Murakami et al. further pointed outthat fatigue strength in particular is influenced by thesizes of the largest inclusions.5,23

The influence of inclusions on fatigue properties is animportant topic for both manufacturers and users ofsteels. So many investigations have been carried out thatit is rather difficult to conduct an exact and impartialsurvey. Here, attention was paid to summarise the effectof inclusion size on fatigue performances of high strengthsteels in the VHCF regime, especially the quantitative orsemiquantitative understanding of those relations.

On critical inclusion sizeIt is well known that there is a difference between theeffect of large and small inclusions. The former areharmful to the steel, often disastrous, whereas the latterare unavoidable, usually not dangerous, and sometimescan even be used to enhance certain steel properties. Thecritical border between small and large, the criticalinclusion size, is still in most cases neither theoreticallyknown or experimentally established, and it alsodepends on the property studied.19 The critical inclusionsize is not fixed and depends on many factors, includingservice conditions. Broadly speaking, it is in the range of5–500 mm.4,24 It decreases with increasing yield stress. Inhigh strength steels, its size will be around the lowerbound of the range mentioned above.24

Duckworth and Ineson25 found that the criticalinclusion size for rotating bending fatigue depended onthe distance below the steel surface and increased from10 mm just below the surface to 30 mm about 100 mmbelow the surface by artificially introducing Al2O3

inclusions into bearing steel. For 4340 and 300M steels,the critical inclusion size of 10 mm at the surface wasalso reported.2 Kawada et al.26 found a slightly smallervalue for the critical inclusion size of 8 mm just below thesurface for the rotating bending fatigue. From a study ofa series of fatigue failures in different bearing steels, theyobtained the relation between critical inclusion size anddepth below the steel surface for inclusions which couldstart a fatigue crack in rotating bending fatigue (in theHCF regime). Figure 2 shows the experimental results ofbearing steels.26

For a fine grained 42CrMoVNb steel, the criticalinclusion size for rotating bending fatigue can be 7–8 mmat steel surface,27 and for SAE9254 steel, the criticalinclusion size can be about 6 mm.28 In a rolling contactfatigue test, the critical inclusion size as small as 2 mmwas found in a bearing steel of GCr15.29 For clarity, thecritical inclusion sizes for different steels are listed inTable 1. One can see that the critical inclusion sizes forthese high strength steels are about 6–10 mm underrotating bending fatigue testing in the HCF regime (107

cycles).Experimentally, for rotating bending fatigue, the

critical inclusion size is generally obtained from extra-polation method, as shown in Fig. 2.26 First, a figureabout the relationship between the size (inclusiondiameter d, mm) and location (the distance from theinclusion to surface H, mm) of inclusions at which thefatigue crack initiated is drawn out. And then, the lowest

1 Harmful index of various inclusions in bearing steel

52100 (y ordinate scale unit is decrease in fatigue

strength of 125 MPa at N5108 cycles)21

Li Effects of inclusions on VHCF properties of high strength steels

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Ltd line is drawn by passing through two data points, below

which no other data point exists. The line is extended tointersect the ordinate at H50 mm (i.e. the specimensurface). The intercept is considered as the criticalinclusion size of the steel investigated. Apparently, it isassumed implicitly that the lowest critical inclusion sizeshould be located on the specimen surface. However, itis obvious that in this method, a large number offractured specimens have to be required to assure theaccuracy of the deductive results and, since there aregenerally some inclusions in a specimen that exceed thetrue value of critical inclusion size, the extrapolatedcritical inclusion size could be higher in general. Finally,it is noted that this method is valuable to determine thecritical inclusion size for rotating bending fatiguebecause the stress gradient exists in the specimen sothat the smaller inclusions as fatigue origins are near thesurface and larger ones are close the centre of thespecimen. In contrast, for axial loading fatigue, becausethe stress is rather uniformly distributed along the cross-section of a specimen and the size of inclusion as fatigueorigin is rather randomly distributed along the cross-section, this extrapolation method could not be usedproperly.

Theoretically, Kiessling and Nordberg30 advocatedthe use of fracture mechanics concepts to estimate thecritical defect size for fracture under monotonic loadingin different steels. The critical values are 10–100 timestoo large compared with the critical inclusion sizeobtained by experiments.4,30 A theory was proposedfor determining the critical inclusion size with respect tovoid formation during hot working,31 and based on the

theory, the corresponding critical inclusion sizes forwith and without electroslag refined En 52 steel weredetermined.32 As regards fatigue, based on the disloca-tion accumulation theory, Tanaka and Mura33 proposeda model concerning fatigue cracks initiated frominclusions. Three cases of fatigue crack initiation atinclusions were classified: (1) from a completelydebonded inclusion; (2) from a cracked inclusionimpinged by slip bands; (3) from a slip band emanatingfrom an uncracked inclusion. In case (2), the fatiguecrack initiation life nc due to inclusion can be writtenas33

nc~CWI= Dt{2kð Þ2l2ah i

(1a)

C~4m m’zmð Þh2=m’ (1b)

where nc (cycles) is the lifetime until fatigue crackinitiation, Dt (MPa) is the cyclic shear stress amplitude,a (mm) is the size of the inclusion, m and m9 (MPa) are theshear moduli of the matrix and inclusion respectively,WI is the specific fracture energy for a unit interfacialarea, k is the friction stress of dislocation, and l and h arethe length and width of a slip band zone respectively.Assuming that at the stage of crack initiation, the lengthand width of slip band zone are identical, equation (1)can be further simplified.33 However, the parameters ofWI and k cannot be well determined for the bearingsteel, so Chen at al.29 modified the equation above as

nc~KH=Dtma (2)

where KH is the material composition parameter andKH5f(m,m9,k,WI). They obtained the values of KH andm by experiments and summarised the relation amongthe critical inclusion size ac for crack initiation underrolling contact fatigue of bearing steel, the lifetime untilcrack initiation nc and the orthogonal shear stressamplitude Dt29

ac~6:33|1035=Dt8:8nc (3)

When nc536106 and Dt51900 MPa, the critical inclu-sion size of 2?9 mm can be estimated and it is close to theexperimental result of 2 mm. Afterwards, Melander34

adopted finite element method combined with fracturemechanics to estimate critical inclusion size of SAE52100 steel under rolling contact fatigue loadingconditions. He found that at a Hertzian pressure of3000 MPa, the calculated critical inclusion size can bey15 mm that is rather high for comparison with theresults of Chen et al.29

Recently, the critical inclusion size of high strengthsteels with given matrix hardness or tensile strength andthe surface roughness under VHCF was estimated35

Table 1 Critical inclusion size for steels under rotating bending fatigue and contacting fatigue tests

Steel Fatigue type Failure location

Criticalinclusionsize/mm Reference

Bearing steels Rotating bending Sample surface 10 254340, 300M Rotating bending Sample surface 10 2Bearing steels Rotating bending Just below sample surface 8 2642CrMoVNb Rotating bending Just below sample surface 7–8 27SAE9254 Rotating bending Just below sample surface 6 28GCr15 Rolling Contact fatigue Below sample surface 2 29

2 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


94 International Materials Reviews 2012 VOL 57 NO 2

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mainly on the basis of Murakami’s theory, i.e. theinclusion effective projected area model.36–38 The fatiguestrength of high strength steels influenced by inclusioncan be written as39

sw~C HVz120ð Þ= areað Þ1=2h i1=6

(4)

where HV is the Vickers hardness (kgf mm22), (area)1/2

(mm) is the square root of the inclusion projected areaperpendicular to the applied stress axis (in this paper, itis also named as inclusion size for short) and C is aconstant depending on the location of inclusion, and forsurface, subsurface and interior inclusions equal to 1?43,1?41 and 1?56 respectively. On the other hand, there is asimple empirical expression for predicting the fatiguestrength swl of low and medium carbon steels by Vickershardness5,40,41

swl~1:6HV (5)

where swl is in MPa and HV is in kgf mm22. Murakami5

suggested that the swl in equation (5) could express theupper limit of fatigue strength of high strength steels.Because of the existence of inclusions in steels, thefatigue strength influenced by inclusion sw (equa-tion (4)), is always lower than the upper limit swl

(equation (5)), until the size of inclusion decreases to a

critical value. It is obvious that when sw5swl, the criticalinclusion size can be estimated as35

dc~Cloc 1z120=HVð Þ6 (6)

where dc is the critical inclusion size (mm), Cloc is thelocation constant and for surface, subsurface andinterior inclusions equal to 0?813, 0?528 and 0?969respectively.35 Figure 3 shows the calculated results.Since equation (4) can be used to predict the fatiguestrength in the VHCF regime, and in this regime, mostfatigue crack initiated from the interior inclusions ofsteels, the critical inclusion size can be estimated as 5–3 mm when the hardness is 400–600 kgf mm22. Theresults show that the critical inclusion size decreases withincreasing Vickers hardness (or tensile strength) ofsteels.

Figure 4a displays the size of inclusions located atfatigue origin of spring steels and medium carbon Cr–Mo steels. Fatigue tests were conducted by rotatingbending (107 cycles, most for medium carbon Cr–Mosteels) and tension–compression ultrasonic fatigue test-ing (109 cycles, most for spring steels).42 From thisfigure, the critical inclusion size of about 3 mm can bedetermined for the spring steel at 109 cycles. Here thecritical inclusion size is determined by the minimuminclusion size not by the extrapolation method due toaxial loading fatigue mentioned above. Also, from thisfigure, the critical inclusion size of about 6 mm can bedetermined 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 foundabout 2 and 5 mm for SNCM439 and SUJ2 steelsrespectively, under rotating bending fatigue in labora-tory air.43 It is noted that the critical inclusion size in theHCF regime (y107 cycles) can be about 6–10 mm asmentioned above; on the other hand, in the VHCFregime (y109 cycles), the critical inclusion size can beabout 3–5 mm, just about a half of the value in HCF.

Figure 4b shows the size distribution of these inclu-sions that acted as fatigue origins. One can see that theinclusions with sizes of less than 5 mm or greater than50 mm are rare. Most inclusions found at fractureorigins are around 10–30 mm.

Some questions still remain. For example, the modelof Tanaka and Mura33 seems rather well for estimating

3 Relationship between critical inclusion size dc and

hardness HV of steel matrix35

4 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



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the critical inclusion size with a clear physical picture;however, some material parameters used in their modelcannot be well determined. In contrast, the model ofYang et al.35 seems predicting the critical inclusion sizein a reasonable range; however, the convincing mechan-ism of crack initiation caused by inclusion was notinvolved.

Characteristics of S–N curveFor a long time, S–N (stress amplitude versus the numberof cycles to failure) curve in fatigue of materials has beenconsidered as a single curve in which the number of cyclesto failure increases with decreasing stress amplitude,eventually approaching the endurance limit below whichno apparent fatigue failure occurred. Low strength steelshave a clear knee point in the S–N curve by which thefatigue limit can be defined. This phenomenon is due tostrain ageing, i.e. forming of the so called Cottrellatmospheres in which dislocations are locked by inter-stitial solute atoms such as carbon and nitrogen.44–47

However, it has been reported that the S–N curve for highstrength steels consists of two parts, one that correspondsto short fatigue life at a high stress level due to fracturefrom a surface origin, and another that corresponds tolong fatigue life at a low stress level due to fracture frominternal inclusions and other inhomogeneities.16,36,48–58

The terminology of stepwise9,10,15 or duplex17,55,59 S–Ncurve was used in characterisation of those curves in theVHCF regime. Figure 5 schematically shows the S–Ncurve in high strength steels with duplex or stepwise curvetype.15 It is noted that in reality, the experimental data arewith a certain degree of scatter in the fatigue life diagramso that the transition of S–N curve will of course not besharp. Many possible factors, which may cause fatiguefailure in the very high cycle regime, are discussed indetail.48,49,52,53,60–63 These factors include environment,15,43,62

residual stress,64 hardened layer,15,64,65 surface roughness,66

loading types17 (of rotating bending and cyclic axial loading)and inclusion size.67,68

Fatigue life behaviour of high strength steels contain-ing non-metallic inclusions can also be described in amultistage or a step-wise S–N curve, as shown schema-tically in Fig. 6.60,61 In the conventional low cyclefatigue range I, fatigue failures are initiated in mostcases 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 inwhich fatigue failures occur at stresses below theconventional HCF fatigue limit, originating in mostcases from internal defects. Finally, there are reasons tobelieve that, at sufficiently low stress levels, there will bea true ultimate fatigue limit (range IV).15,60,61 It is notedthat in the VHCF regime, the conventional HCF fatiguelimit eliminates49,50,69 and the fatigue strength decreaseswith increasing number of cycles, which will certainlyalter the design codes of component.69

Mughrabi61 considered a cylindrical specimen ofgauge length l and diameter d containing inclusions ofdiameter di in a volume concentration ni, and defined acritical volume density of inclusions

ni,crit~1= dipdlð Þ (7)

Here the meaning of dipdl is the volume of outer surfacelayer of the specimen at gauge section with thickness ofdi. If volume densities ni is below this critical value, itwill be hard to find an inclusion in this volume.Mughrabi61 pointed out that when this condition ismet, fatigue cracks could not initiate at inclusions lyingat the surface. Hence, it is assumed that crack initiationat the surface can only occur by some surface rough-ening process such as extrusions and intrusions by slipbands and should be delayed compared to crackinitiation at inclusions lying at the surface. On the otherhand, when the critical volume density of inclusions isexceeded, crack initiation at inclusions at the surface andsubsequent crack propagation must be faster than anycompeting crack initiation at internal inclusions andsubsequent slow crack propagation (in vacuum62).Under these conditions, surface failures, compared tointernal failures, should of course be dominant.

Terent’ev65,70 has put forward a hypothesis thatassociates the presence of a marked deflection and ahorizontal plateau on S–N curves with the presence of ahardened surface layer formed in the process of cyclicdeformation at large test bases due to the dominantyielding of the surface layer. This harder barrier layer canalso be created in the stage of fabrication of specimens orin hardening surface treatment (for example, nitriding).The barrier effect of the hardened surface layer ispreserved in the presence of extrusions, intrusions andnon-propagating microcracks in the surface layer.

Based on the VHCF testing results of four highstrength spring steels with the same strength class(y1700 MPa) but containing different inclusion sizes,

6 Schematic fatigue life diagram of high strength steels

containing inclusions60,61

5 Schematic of typical S–N curve for high strength steels15



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the S–N curves of high strength steels could be dividedinto three categories:68 (1) the S–N curve displays acontinuous decline and the internal cracks initiated fromthe large oxide inclusions for commercial 50CrV4 steelin which the average inclusion size on the fatigue originis about 29 mm; (2) stepwise S–N curves were observedfor clean 54SiCrV6 and clean 50CrV4 steels in which theaverage inclusion cluster sizes are about 11 and 7 mmrespectively; (3) for clean 54SiCr6 steel in which theinclusion size is smaller than 1 mm which is definitelysmaller than the critical inclusion size, the fatigue cracksdid not initiate from inclusions or inclusion clusters butfrom the region enriched with carbon. In this case, theS–N curve shows that the fatigue failure hardly occursfrom about 106–109 cycles; in other words, the fatigueproperty can be substantially improved in the VHCFregime. The S–N curve can be nominated as fatigue limittype.

It seems that the characteristics of S–N curves of highstrength steels can relate to inclusion size as schemati-cally shown in Fig. 7a.42 When the inclusion size isgreater than y20 mm, the S–N curve displays acontinuous decline type;71,72 when the inclusion(orinclusion cluster) size is smaller than y20 mm andgreater than the critical inclusion size, the stepwise typecan be observed;55,73,74 when the inclusion size is belowthe critical inclusion size, a fatigue limit type may beachieved.68,75 These experimental results are qualita-tively in accordance with the prediction of the variationtrend of S–N curve caused by increasing the inclusion(inhomogeneity) size as shown in Fig. 7b.15,76,77

For a high carbon chromium bearing steel SUJ2,Sakai et al.17,59 and Shiozawa et al.55 found duplex S–Ncurves. A typical diagram is shown in Fig. 8. Recently,Lu et al. conducted an interesting work,78 in which thehigh carbon chromium bearing steel GCr15 with asimilar chemical composition but larger inclusion sizewas tested under rotating bending and cyclic axialloading; the S–N curves are shown in Fig. 9a, in whichthe S–N curve of SUJ2 shown in Fig. 8 is also indicatedby dashed line for comparison. One can see that there isa clear step on the S–N curve of SUJ2 under rotatingbending, while it seems that the step became muchshorter for the GCr15 under rotating bending. On the

a types of S–N curve relating to inclusion size;42 b variation trend of S–N curve caused by increasing inclusion (inhomo-geneity) size15,76,77

7 Schematic of influence of inclusion size on S–N curve of high strength steels in very high cycle fatigue regime

8 S–N curve for high carbon chromium bearing steel

SUJ255



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Ltd other hand, under cyclic axial loading, the S–N curve of

GCr15 declined continuously. The size and the positionof inclusions at fatigue origins for the two steels areshown in Fig. 9b. It is evident that the inclusion size ismuch smaller (around 10 mm) for SUJ2 under rotatingbending and a step can be clearly seen on the S–N curve.On the other hand, the inclusion size of GCr15 is muchlarger, around 20 mm for rotating bending and around35 mm for axial loading. Since the highly stressedvolume in the rotating bending specimen is muchsmaller than that in the axial loading specimen, thesmaller inclusions can be found in the rotating bendingtesting. This may be one of the reasons why the stepwiseor duplex S–N curve can be found more easily inrotating bending testing than axial loading testing.Up to now, it is reported that few S–N curves, forexample,51,56,68,71,73,79 show the stepwise or duplexcharacteristic under the axial loading test. Besides theinfluence by inclusion size, the other underlying reasonsshould be revealed further.

It seems reasonable that if the inclusion size is smallerthan the critical value, there will be no substantial effectof inclusion on the VHCF failure of steels, and the S–Ncurve with fatigue limit type may be achieved. However,if the inclusion size is larger, the volume concentrationof inclusion is usually higher; therefore, if the inclusionsize is greater than a certain value (say, 20 mm), thevolume density of inclusion is beyond a critical value,the possibility to cause fatigue failure at surfaceinclusion increases61 and the transition of surface tointerior failure can readily occur; therefore, a continuousdecline type can be achieved. When the inclusion size issmall and the volume density is low, the stepwise orduplex S–N curve could be observed as mentionedabove.

Recently, the effect of residual stress on the transitionof surface failure to internal failure has been studied indetail by Shiozawa et al.79 Figure 10 shows the S–Ncurves of SNCM439 steel obtained from an axialloading test, in which the low and high residualcompressive stresses existed in emery polished and shotpeened specimens respectively. The inclusion size is inthe range of 5–30 mm and the average is 14?7 mm. Theclear stepwise S–N curve can be seen for the low residual

stress specimens but not for the high residual stressspecimens.

Investigation results showed that the S–N curvecorresponding to internal fracture, which appears invery high cycles of lifetime, was not much affected by thesurface and environment conditions but mainly depen-dent on inclusion size.78 However, the current under-standing of the relationship between inclusion size andS–N characteristics was only qualitative in nature asmentioned above. Quantitative analysis on the relation-ship between inclusion size and S–N characteristics ismuch needed.

An attempt was made to evaluate how the S–N curvecorrelates with the inclusion size as follows. The Basquinequation can be used to describe the S–N curve in theHCF regime. Recently, Liu et al.80 assumed that thisequation is also valid to predict the S–N curve in theVHCF regime

sa~s’f 2Nfð Þb (8)

where sa is the stress amplitude, Nf is the number ofcycles to failure, sf9 is the fatigue strength coefficient andb is the Basquin exponent. In the VHCF regime, sf9 and

9 a S–N curves for high carbon chromium bearing steel GCr15 under rotating bending and cyclic axial loading condi-

tions, and S–N curve of SUJ2 as shown in Fig. 8 is inserted as dashed line for comparison, and b size and depth of

inclusion as fatigue origin for two steels78

10 S–N curves of emery polished and shot peened speci-

mens of SNCM439 steel, obtained form an axial load-

ing test79



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b correlate with inclusion size and hardness of thesteel.80 Based on the knowledge of fatigue strengths atboth high cycle (y106) and very high cycle (109), theyfound80

s’f~1:12 HVz120ð Þ9=8= areað Þ1=2h i1=8

(9a)

b~1=3lg 1:35 HVz120ð Þ1=16areað Þ1=2

h i1=48� �

(9b)

where HV is the Vickers hardness (kgf mm22) and(area)1/2 (mm) is the inclusion size.

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 to28 mm on the fatigue fracture surface of high strengthsteel 60Si2CrV, combining with the Vickers hardness of560, the predicted upper (inclusion size with 10 mm) andlower (inclusion size with 28 mm) bounds of S–N curvesare shown in Fig. 11.80 It is generally in accordance withexperimental results. From these relations, one can seethat both fatigue strength coefficient and Basquinexponent relate with inclusion size and steel strength(hardness); therefore, the wide span of inclusion size willcertainly cause a strong dispersion of fatigue data in theS–N diagram. This will be one of the factors thataccount for the statistical nature81 of the fatigue data.Recently, Sakai et al. reported a convincing probabilisticstress life characteristic for bearing steel SUJ2 in theVHCF testing, in which surface initiated fracture andinterior initiated fracture (fish eye fracture) are clearlyindicated.82

Although significant progress has been made inunderstanding the characteristics of S–N curve of highstrength steels in the VHCF regime, the convincingmechanisms, particularly the quantitative descriptionsof the transition from surface fracture to internalinclusion fracture, are still lacking.

Fatigue strength and fatigue lifeIn the early experimental studies, it was found that thereexists a correlation between inclusion size and fatiguestrength provided that the inclusions are with identicalchemistry and similar shape.2,4,25,83 The relation isexpressed by2

Kf!D1=3 (10)

where Kf is the strength reduction factor, i.e. the ratio ofthe fatigue strengths in a steel without and withinclusions of diameter D. We rewrite equation (10) as

Kf~swi=sw!D1=3 (11)

where swi and sw are the fatigue strengths without andwith inclusion of diameter D respectively. We have

sw!swi=D1=3 (12)

Early researchers84–86 observed a trend that fatigue life isinversely related to the total inclusion content r, whichcan be written as

Nf!1=r (13)

Supposing that the inclusion content is proportional tothe number of inclusions Ni and the volume of a singleinclusion, i.e. r!NiD

3, then from above expression, wehave

Nf!1= NiD3

� �(14)

At present, the most widely used approach to estimatethe fatigue strength of high strength steels containinginclusion is the so called (area)1/2 parameter modelproposed by Murakami et al.38,48 In addition, theyemphasise the important effect of hydrogen trapping atnon-metallic inclusions and its effect on the size of theoptically dark area (ODA) in the case of internalfracture.

If the area of a crack at inclusion is denoted by ‘area’,then the maximum value KImax of the stress intensityfactor along its crack front is given approximately by87

KImax~0:5sa p areað Þ1=2h i1=2

(15)

Here KIm ax is the applied stress intensity factor given inMPa m1/2, where sa is the applied stress (MPa) and(area)1/2 is the crack size (m). The threshold for crackgrowth can be written5

DKth~3:3|10{3 HVz120½ � areað Þ1=2h i1=3

(16)

where DKth is in MPa m1/2 and HV is in kgf mm22, and(area)1/2 is in mm.

When KImax5DKth, the fatigue strength can bedetermined. Combining equations (15) and (16), andnoticing the difference of the unit of the (area)1/2

between those equations, the fatigue strength sw canbe expressed as equation (4),39 i.e.

sw~C HVz120ð Þ= areað Þ1=2h i1=6

where C51?56 for VHCF of high strength steels.

Recently, an expression based on the understandingof the effect of hydrogen during forming granular brightfacet (GBF)55 (noting that GBF and ODA are differentterminologies for the same special area around aninclusion that acted as the fatigue origin, see the sectionof mechanisms of failure from internal inclusions below)was developed to predict the fatigue strength of highstrength steels in the very high cycle regime (at 109

cycles), it reads as88

11 Predicted S–N curves and experimental results of

60Si2CrV80



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sVHw ~2:7 HVz120ð Þ15=16

.areað Þ1=2

h i3=16

(17)

Supposing that the square root of an inclusion areais proportional to the diameter of the inclusion, i.e.(area)1/2!D, then from equations (12), (4) and (17), thedependence of fatigue strength on inclusion size can besummarised as

sw!1=D1=3 (18a)

sw!1=D1=6 (18b)

sw!1=D3=16 (18c)

Or it is summarised as

sw!1=Dn1 (19)

The values of n1 are 0?333 (1/3), 0?167 (1/6) and 0?188 (3/16) for the works of early researchers, Murakami et al.and Liu et al. respectively. In general, the fatiguestrength dependence on inclusion diameter was moresubstantial (0?333) in early days, possibly because theamount and size of inclusions in those days had not beenwell controlled. The Murakami et al.’s index (0?167) andLiu et al.’s index (0?188) are rather close, and they canbe used for the clean steels in the VHCF regime.

The relations between fatigue life and inclusion size inhigh strength steels were proposed74,78,89 based on theassumptions below: (1) for VHCF life or low appliedstress amplitude, the fatigue crack initiates from aninterior inclusion around which a GBF area exists, andthe diameter of the GBF area is several times larger thanthat of the inclusion; (2) the growing crack in GBF areais always assumed as a micropenny crack and accordingto Paris et al.’s and Marines-Garcia et al.’s opinions90,91

that the microcrack still obeys Paris equation, i.e. da/dN5C(DK)m, where, da/dN is the crack growth rate, DKis the stress intensity factor range, and C and m areconstants; (3) total fatigue life Nf is approximated by thelife of crack growing in whole GBF area. Mayer et al.89

and Lu et al.78 investigated the fatigue behaviour ofbearing steels and found

Nf~ 2= C(m{2)½ �f g pð Þ1=2=2h i{m

areað Þ1=2h i1{m=2

s{ma

(20)

where C and m are constants in Paris’s equation, (area)1/2 isthe inclusion size and sa is the applied stress amplitude.Experimentally, the constants m and C were determinedfor bearing steels SUJ2,74 100Cr689 and GCr15,78 and forlow alloy steel SNCM43979 (Table 2). At the same time,Yang et al.92 studied the spring steels of 60Si2CrV and54SiCrV6 and found

Nf~ 2= C m{2ð Þ½ �f g pð Þ1=2=4h i{m

r0ð Þ1{m=2l{ms{ma (21)

where C, m and sa have the same meaning above, r0 is theradius of inclusion and l is a proportional coefficientrelated to hydrogen content and called ‘hydrogen influencefactor’, in other words, the higher hydrogen content mayimpair the fatigue property of steels, and l is the indicationof deterioration induced by hydrogen and can be expressedas93–95

l~1z0:09 CHð Þ2 (22)

where CH is hydrogen concentration in ppm. The values ofm for 60Si2CrV and 54SiCrV6 are also listed in Table 2.

From equations (14), (20) and (21), the dependence offatigue life on inclusion size D can be summarised as

Nf!D{3N{1i (23a)

Nf!D1{m=2s{ma (23b)

Nf!D1{m=2l{ms{ma (23c)

From the expressions above, one can see that the fatigue lifeof high strength steels strongly relates to inclusion size;however, the applied stress amplitude and hydrogen contentand other factors also display a strong effect on the fatiguelife of high strength steels in VHCF regime. Therefore, in thefatigue testing, the data for fatigue life versus inclusion sizeusually exhibit much scatter. Fortunately, in some works,the relations between fatigue life and inclusion size can befound as mentioned above. We rewrite the relation betweenfatigue life and inclusion size as

Nf!1=Dn2 (24)

For the steels in early days, n2532. At present, if theexponent m of Paris equation in the VHCF regime is knownfrom experiment, the index n2 can be expected as n25m/2–1;therefore, for SUJ2,74 n256?1; for 100Cr6,89 n256?25; forGCr15,78 n256?19; for 60Si2CrV and 54SiCrV6, n255?77and n258?42 respectively.92 Note that for 100Cr6,89 n254?8is also found by direct inspection of the experimental data.For the Mn–Si–Cr series bainite/martensite dual phasesteels, n254?29 and 4?60 are found by direct inspection ofthe experimental data.96 All these data are summarised inTable 3. The experimental data of fatigue life correlatingwith inclusion size for 100Cr689 and 60Si2CrV92 are shownin Fig. 12.

Based on the experimental results of the ODA sizeagainst the number of cycles, obtained fromreferences13,97,98 for quenched and tempered SUJ2,SCM435 and SNCM439 steels, Chapetti et al.99

obtained the relationship between fatigue life andinclusion size

Table 2 Paris constants for some high strength steels

Steel Ultimate tensile strength/MPa Paris constant C Paris constant m Reference

SUJ2 2316 3.44610221 14.2 74100Cr6 2387 4.86610221 14.5 89GCr15 2300 1.48610221 14.37 78SNCM439 1955 5.88610222 15.2 7960Si2CrV 2365 … 13.54 9254SiCrV6 1729 … 18.83 92



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Nf~ 4:47 HVz120ð Þ=Dsth½ �48R{8

i (25)

where Dsth is the threshold stress range for pure fatiguecrack propagation99 and Ri is the radius of the inclusion;one can see that for those steels, n258 can be obtained asindicated in equation (25).

In the early days, the fatigue life is limited to 107

cycles, and the effect of inclusion on fatigue damage iscomparatively not very strong as in the VHCF regime;therefore, a smaller index of 3 could be obtained. Incontrast, in the VHCF regime, the inclusion plays a veryimportant role and a greater index such as 4?29–8?42 canbe found. The results are summarised in Table 3

According to Murakami’s point of view,5 a hardinclusion could be considered as a small crack in highstrength steels, and fatigue strength is not determined bythe critical stress for crack initiation but the thresholdstress for crack growth. Since the threshold value ofstress intensity factor range for small crack is weaklyrelated with inclusion size by a power of 1/3; (forexample, see equation (16)),38,100 the fatigue strength inturn depends on inclusion size rather weakly by a power

about 1/6. On the other hand, fatigue life of highstrength steel is mostly determined by crack propagationrate that is strongly related to inclusion size by Paris lawin which the index m could be much greater.

At present, the commercial high strength steels(ultimate tensile strength .y1500 MPa) usually containnumerous small inclusions and some large inclusions.From an intensive work on fatigue testing, the inclusionsizes on fatigue fracture origin are found to be around20 mm in average (for example, see Fig. 4b). If theaverage inclusion size on fracture surface could bereduced from 20 to 10 mm that is still greater than thecritical inclusion size, y10% of fatigue strength increaseand y100 times of fatigue life longer could be expected(where n151/6 and n256?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 theinclusion size so that reliability of the fatigue propertycould be substantially improved.

Another question is: how large is the difference offatigue strength between 107 and 109 cycles? Bathiaset al.63 found that the difference of fatigue strength,

12 Fatigue life correlating with inclusion size for a 100Cr689 and b 54SiCrV6, where s is stress amplitude (MPa), Nf is

fatigue life (cycles) and r0 is radius of inclusion (m)92

Table 3 Index of inclusion size correlating with fatigue life of some high strength steels

SteelUltimate tensilestrength/MPa

Index ofinclusion size n2

Method todetermine the index

Fatiguetesting/cycle Reference

SUJ2 2316 6.1 m514.2 109 74n25m/2–1

100Cr6 2387 6.25 m514.5 109 89n25m/2–1

4.8 By direct experimentalGCr15 2300 6.19 m514.37 56108 78

n25m/2–160Si2CrV 2365 5.77 m513.54 109 92

n25m/2–154SiCrV6 1729 8.42 m518.83 109 92

n25m/2–1SNCM439 1955 6.6 m515.2 109 79

n25m/2–1Mn–Si–Cr 1451 4.29 By direct experimental 109 96B/M dual phase 1581 4.60SUJ2 2316 8 By analysing the

experimental data109 99

SCM435 1950SNCM439 1955



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between 106 and 109 cycles, decreases by 50–200 MPafor some alloys and steels. Recently, Hong et al.101

collected 58 S–N curves for low strength and highstrength steels.51,63,64,71,89,100,102–122 The difference offatigue strength between 107 and 109 cycles for steelswith different tensile strengths is shown in Fig. 13.101 Ina previous work, the fatigue strength of high strengthsteels at 106 cycles was estimated as80

sw~2 HVz120ð Þ= areað Þ1=2h i1=6

(26)

At 109 cycles, the fatigue strength sVHw can be given by

equation (17), then the difference Dsw~sw{sVHw can be

predicted by the values of hardness and inclusion size ofthe steels. Here the inclusion size of 20 mm was used inthe calculation for all steels and it was assumed thatfatigue strength at 106 cycles has no big difference withthat at 107 cycles. The calculated results represented bysolid line is inserted in Fig. 13. One can see that for highstrength steels (sb.y1500 MPa), the experimental dataare around the predicted solid line though they are muchdispersed, and the upper bound is about 200 MPa. Itshows that the prediction is roughly in accordance withthe experimental results for the steels with a tensilestrength over 1500 MPa. However, the predictionis not good for the medium and low strength steels

(sb,y1200 MPa), and all the data are below thepredicted curve. The major reason may be that in thisstrength range, the fatigue strength is relatively insensi-tive to inclusions; thus, other failure mechanisms willalso operate. Zimmerman addresses the VHCF beha-viour and failure mechanisms of defect insensitive ordefect free materials in a complementary review.1

From this figure, one can see that the difference of thefatigue strength between 107 and 109 cycles is less than25 MPa for about one-third of the steels at differentstrength levels. For low strength steels or high strengthsteels with inclusion size less than the critical value, thefatigue limit type of S–N curve may exist, so that thedifference of fatigue strength between 107 and 109 cyclesis small. However, for high strength steels, if theinclusion size is not less than the critical value, someof which still show the slight difference of fatiguestrength between 107 and 109 cycles, the underlyingreasons need further investigation.

Mechanism of failure from internalinclusionsInternal fatigue fracture origins of high strength steels inVHCF are mostly at non-metallic inclusions. In thevicinity of a non-metallic inclusion at the fracture origin,a dark area was often observed inside a fish eye mark forspecimens with a long fatigue life. Murakami et al. madeobservations by optical microscopy and named this areaas ‘optically dark area’ (ODA).48,52,53 Sakai et al. madeobservations by scanning electron microscopy andtransmission electron microscopy and the ‘fine granulararea’ (FGA) around the inclusion at the fracture originwas found.59,123–126 By using scanning electron micro-scopy, a rough area beside the inclusion at the fractureorigin was found by Shiozawa et al. and it was named asthe ‘granular bright facet’ (GBF) zone.55,127 This featureis also found and named as ‘rough surface area’ (RSA)by Ochi et al.103 Now it is clear that the differentterminologies employed by different authors refer to thesame microscopic feature. The typical fracture surfaceand its characteristic zones are shown in Fig. 14.

The size of the ODA (GBF, FGA and RSA) increaseswith increasing fatigue life, and the ODA (GBF, FGAand RSA) cannot be found on fracture surfaces ofspecimens which failed at a small number of cycles.Recent studies have confirmed that the fatigue crackinitiation period can account for a very large fraction offatigue life57,60–62,73,99 in the VHCF regime; therefore,

14 a typical fractography of high strength steels under VHCF50 and b schematic of characteristic zones on fracture

surface

13 Difference of fatigue strengths between 107 and 109

cycles Dsw for steels with different tensile strengths

sb:101 solid line represents value predicted by present

author



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investigation of the forming mechanisms of ODA (GBF,FGA and RSA) is essential.

It is quite common for heat treated high streng-th steels to contain hydrogen around non-metallicinclusions.128,129 Recently, Takai et al.129,130 andMurakami et al.131 directly verified the presence ofhydrogen trapped at the interface of inclusions usingsecondary ion mass spectrometry. Hydrogen trappedaround a non-metallic inclusion was also observed bytritium microautoradiography as shown in Fig. 15.132

Murakami et al. proposed a hydrogen embrittlementmechanism for the ODA formation. By the super-position of applied cyclic loading, residual stresses andexistence of hydrogen, the initiation and growth of crackfrom internal non-metallic inclusions take place due tohydrogen assisted fatigue. The process is characterisedby a formation of ODA.48,52,53 After slow fatigue crackgrowth in the dark area beside the inclusion, the size ofthe crack exceeds a critical value given by the thresholdfor pure fatigue propagation, and the crack growswithout the assistance of hydrogen afterwards. In theODA formation process, the hydrogen enhanced loca-lised plasticity133,134 will cause a highly localised plasticfailure. In this process, since a very complicated stressdistribution exists near the inclusion and at themicrocrack tip, the hydrogen assisted crack growthmay promote the activation of different slip planes andhence promotes the formation of rougher fracturesurface. The rough area around inclusion can reflectthe light in an optical microscope and hence, a dark areacan be observed. Figure 16 is the schematic illustrationof the mechanism for VHCF fatigue failure coupled withhydrogen.48,52,53,57

Based on the mechanism of crack growth assisted byhydrogen and the model of local stress intensity factor atthe crack tip due to the influence of hydrogen,135 acriterion for the formation of GBF (ODA, FGA andRSA) is proposed.100 Suppose a penny shape microcrackgrows from the internal inclusion, the crack size is(area)1/2, and the total stress intensity factor at the cracktip is KT, which is the superimposition of intensityfactors exerted by applied stress KImax and the hydrogeninfluence kH. It is known that KImax increases with cracksize (for example, see equation (15)). kH can be writtenas100

kH~AmC0 areainð Þ1=2h i3

sa=sy

� �areað Þ1=2

h i{5=2

(27)

where A is a dimensionless constant, m is the shearmodulus, C0 is the initial mass fraction of hydrogen,(areain)1/2 is the inclusion size, sa is the applied stressamplitude, sy is the yield strength and (area)1/2 is themicrocrack size. kH decreases drastically with increasingcrack size due to the hydrogen which is mostly trappedby an inclusion nearby100 (see equation (27), the indexon the microcrack size is {5=2). When the total stressintensity factor KT is greater than the threshold of crackgrowth (KGBF)th, the crack will propagate. Figure 17schematically shows the criterion. The thick solid linerepresents the crack growth threshold that increases withincreasing crack size for microcrack (for example, seeequation (16), not a constant as for macrocrack). In case1, if the inclusion size is large and the applied stress ishigh, the initiated crack is also large (inclusion itself is

15 Hydrogen trapped around non-metallic inclusion

(Al2O3) observed by tritium autoradiography132

16 ODA formation mechanism for very high cycle fatigue failure from internal inclusion with initial fatigue crack growth

assisted by hydrogen (schematic was firstly drawn by Murakami et al.48 and modified by Chapetti et al.57)

17 Schematic of criterion for GBF crack growth100



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part of microcrack), and the applied stress intensityfactor KImax is great enough (represented by straightdashed line), the crack will continue to propagatewithout hydrogen assistance until the specimen isbroken. This is the case of no GBF (ODA, FGA andRSA) formation. In case 2, the applied stress intensityfactor KImax is lower than the threshold and thehydrogen assistant intensity factor kH is necessary topromote crack growth; therefore, a GBF (ODA, FGAand RSA) can form in this process; however, when thecrack size increases, the hydrogen assistant intensityfactor kH decreases drastically as indicated by the thinsolid line. At this time, with the growing crack, thehydrogen assistance can be neglected, but if the appliedstress intensity factor happens to be large enough, thespecimen will break but with a GBF (ODA, FGA andRSA) left on the fracture surface. In case 3, similar tocase 2, the GBF (ODA, FGA and RSA) can form due tohydrogen assistance, and with further growth, thehydrogen assistance can be neglected, but the appliedstress intensity factor can be still lower than thethreshold. In this case, a GBF (ODA, FGA and RSA)forms inside a specimen without fracture. From thismodel, the fatigue strength of high strength steelsdepending on the inclusion size in the VHCF regimewas deduced as equation (17).88 Figure 18 shows theexperimental result of fatigue strength at 109 cycles for17 high strength steels with different inclusion sizes. Theprediction by equation (17) is basically within 15%error.

The effects of hydrogen on the fatigue behaviours ofhigh strength steels in the VHCF regime have beenstudied recently, for example.93–95,136–138 The fatiguestrength depending on inclusion size, hardness andhydrogen concentration has been studied, it reads,94,95

s�w~1:56 HVz120ð Þ=l= areað Þ1=2h i1=6

(28)

where l is the hydrogen influence factor as mentionedabove (equation (22)). It can be obtained approximatelyby the ratio of fatigue strength of a high strength steelwith very low hydrogen content over that with higherhydrogen content, i.e.

l~sw=s�w (29)

where sw is the fatigue strength of steel withouthydrogen charging and with very low hydrogen content,and s�w is the fatigue strength of the same steel with

hydrogen charging and with higher hydrogen content.Both sw and s�w are determined at 109 cycles. Figure 19

shows the fitting curve and the expression of hydrogeninfluence factor 1/l. The fatigue strength decreasesdrastically with increasing hydrogen content. Whenthe hydrogen content increases to 1–2 ppm, the fati-gue strength decreases by 10–30% correspondingly.Figure 20 shows the comparison of the fatigue strengthsbetween calculated values by equation (28) and experi-mental results for several steels with different hydrogencontents;95 it seems that those data coincide rather well.

Now there are three equations to estimate the fatiguestrength based on the inclusion size under stressamplitude control at stress ratio of R521. Murakamiequation (equation (4)) was developed on the bases ofthe inclusion effective projected area model and fracturemechanics. No detailed hydrogen assisted mechanismwas involved. It can be used to estimate the fatiguestrength of high strength steels with low hydrogencontent (for different R ratios, Murakami5 and Ba-thias and Paris6 provided the modified equations).Equation (17) was developed by the model of GBFformation which is influenced by hydrogen near theinclusion. Both equations (4) and (17) are quite similar.It is noted that equation (17) is also used to estimate thefatigue strength of high strength steels with lowhydrogen content. The reason is that the hydrogenconcentration at crack tip near the inclusion cannot bedetermined accurately, so that the parameter in thisequation has to be determined experimentally byanalysing the fatigue strengths of many high strengthsteels with low hydrogen content.88 For high hydrogencontent, an alternative method was used to evaluate thefatigue strength, in which the hydrogen content in thetested sample was measured experimentally with certainaccuracy. Equation (28) can be used to estimate thefatigue strength of high strength steels with highhydrogen content.

All these models to predict the fatigue strength39,88,95

and fatigue life78,89,92 are essentially microcrack growthapproach, in which the specific microcrack initiation andearly growth process depending on the microstructural

18 Relationship between sw/(HVz120)15/16 and inclusion

size (areain)1/2 for 17 high strength steels with differ-

ent inclusion sizes88

19 Fitting curve of hydrogen influence factor for total

hydrogen content42



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barriers49,62 has not been well involved. However, thebasic characteristic of the microcrack initiation andearly growth has been intensively studied recently. Fromthe schematic of a typical fractography of high strengthsteels in the very high cycle fatigue regime (Fig. 14b),one can see that the fatigue failure process can bedivided into several stages: at first, the nucleation of amicrocrack at an inclusion with a size equivalent to theinclusion, and the microcrack nucleation life is Ni1; next,the early slow growth of the microcrack beside theinclusion in the region of GBF (ODA/FGA/RSA), andthe fatigue life in this region is Ni2; at last, the crackpropagation in the fish eye region which results in thefinal fracture, and the crack propagation life is Np. Thetotal fatigue life Nf should be as Nf5Ni1zNi2zNp.

For high strength steels in the HCF regime, if aninclusion is without GBF (ODA, FGA and RSA) areaon the fracture surface, crack initiation at the inclusiongenerally occurs very early in the fatigue process and thecrack propagation life usually takes up a large percen-tage of the total life.139 On the other hand, in the VHCFregime, the situation is quite different. Murakami et al.53

found that the number of cycles required for the crack togrow from the ODA border to the fish eye border in theCr–Mo steel SCM435 ranges from 105 to 106 cycles,which is a small fraction of the total fatigue life in theVHCF regime. Wang et al.16 analysed the crackpropagation life Np for Cr–Si and Cr–V spring steelsand found that Np is less than 1% for the total fatiguelife over 108 cycles. Chapetti et al.57 reached the similarstatements by systematic analysis of the data of bearingsteel SUJ2.55 Kazymyrovych et al.140 measured thestriation spacing on the fracture surface of AISI H11tool steel and found that the fatigue life consumedwithin and beyond fish eye is only a very small fraction(about 104–105 cycles) of the total fatigue life (about108–109 cycles). Paris et al.90,113 have shown that in theVHCF regime, the fatigue crack growth constitutes onlyabout 1% of the total fatigue life. All these results showthat the crack propagation life Np in the fish eye regionand over can be neglected without dispute in the VHCFregime for high strength steels.

Now it is necessary to reveal the features of themicrocrack nucleation and early slow growth process. Itis indeed that one difficulty will always be where todraw the borderline between crack initiation andpropagation.61,62 Since the inclusion at the initiationsite and the related GBF (ODA, FGA and RSA) areaare usually very small (less than 100 mm141 in general),and the microcrack growth in this region is very slow,many researchers define that the microcrack nucleationand slow growth in the GBF (ODA, FGA and RSA)area are the constituents of crack initiation stage, forexample, Refs. 61 and 62. From the review mentioned inthe above paragraph, the fraction of crack propagationlife Np in the fish eye region and over can be less than1%; therefore, the fraction of fatigue life in the crackinitiation stage, Ni5Ni1zNi2, can be greater than 99%;in other words, it is indeed that the fatigue life iscontrolled by the crack initiation stage for high strengthsteels in the VHCF regime.

As regards the nucleation life Ni1, if a premicrocrackdoes exist in an inclusion or at the interface between theinclusion and matrix before fatigue (sometimes it can beinduced by rolling or other processing routes), thenucleation life could be considered as null at the firstapproximation. It is further noted that for an inclusionwith GBF (ODA, FGA and RSA) area on the fracturesurface, crack nucleation at the inclusion generallyoccurs at about 5–10% of the total fatigue life.142

Those results indicate that the nucleation life Ni1 forhigh strength steels in the VHCF regime could be arather small fraction of the total fatigue life. Therefore,the microcrack slow growth in the GBF (ODA, FGAand RSA) area will play a key role in controlling thefatigue behaviour of the high strength steels. Kuroshimaand Harada143 and Tanaka and Akiniwa74 assumed thatin the GBF (ODA, FGA and RSA) area, the Parisequation can also be used and the Paris constants can bedetermined by experiment. This method is effective toinvestigate the fatigue behaviours of high strength steelsin the VHCF regime with the merit of simplicity. On theother hand, it is worth to develop more subtle models toevaluate the crack initiation stage. Wang et al.73

estimated the microcrack initiation life at the inclusionbased on a modified Tanaka and Murra’s model.33 Theyfound that the crack initiation life occupies the over-whelming part of the total fatigue life.

The microcrack initiation process is of very impor-tance for the type I materials including pure metals anddefect free materials1 as emphasised by Mughrabi60,61,144

and Lukas and Kunz.145 And it is also very importantfor VHCF of high strength steels as mentioned above;however, the model involving the specific mechanismand microstructual features to evaluate the crackinitiation process is much needed.

Shiozawa et al.127 proposed another mechanism thatmultiple microcracks are initiated by decohesion ofspherical carbide from the matrix around an inclusion.Figure 21 schematically shows that discrete microcracksalong boundaries between the carbide particles and thematrix and their coalescence generate the characteristicmorphology inside the rough surface area. They name itas GBF area. Recently, they also made detailedobservation of the features of GBF on the fracturesurface as shown in Fig. 22.79 It clearly shows that aGBF area is very rough compared with the other area

20 Comparison of calculated fatigue strength Ds�w,cal withexperimental results Ds�w,exp, for high strength steelswith different hydrogen contents: both fatiguestrengths are normalised by hardness95



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inside the fisheye (Fig. 22a). Figure 22b shows the birdview over the GBF area and the matrix of the specimen,and Fig. 22c shows the distribution of carbon near aninclusion (Al2O3) on the fracture surface. It is believedthat detection of carbon enrichment in the GBF arearesults from the remaining carbide particles, which arecreated on the fracture surface around an inclusionduring fatigue fracture process.

Sakai126 proposed a new mechanism that consists ofthree steps (Fig. 23): (1) a fine granular layer caused bythe intensive polygonisation is gradually induced aroundthe interior inclusion during long sequence of the cyclicloading; (2) nucleation and coalescence of microdebond-ings; (3) microdebondings are entirely spread over the

fine granular layer and the penny shape crack is finallyformed.

Recently Huang et al. proposed that the ODA (GBF,FGA and RSA) could be formed by irreversiblepersistent slip bands in a bearing steel.146 Clear evidenceto verify this mechanism is still needed.

Each of these mechanisms would be conceivable insome cases; however, none of these mechanisms can bebroadly accepted without challenge.126 However, thehydrogen assisted fatigue cracking for high strengthsteels in the VHCF regime could be encounteredfrequently in practice, and the related mechanism isworth further studying.

Estimation of maximum inclusion sizeUp to now, it seems that once we have reliableinformation on inclusion size as well as other relatedparameters, such as hardness and hydrogen content, thefatigue strength and fatigue life could be estimat-ed reasonably by corresponding equations mentionedabove for some high strength steels. Now it is wellknown that the inclusion size at the fatigue origin ismuch greater than the inclusion size measured bystandard metallographic inspection. Generally, theinclusion at the fatigue origin is the maximum one inthe highly stressed test volume of a specimen or acomponent. The question is how to predict the size ofthe maximum inclusion in a tested specimen or in a verylarge volume of steel using data from a small volume ofsteel. Recently, much progress has been made.147 Thestatistics of extreme value (SEV) method is based onmeasuring the maximum size of inclusions in randomlychosen areas or volumes and it has been intensivelyinvestigated by Murakami and co-workers,14,36–38,148–150

for discrimination between superclean steels and theestimation of the maximum size of inclusions in a largevolume of steel. The second method is the generalisedPareto distribution (GPD) method, developed byAtkinson et al.,151–155 where the size of all inclusionslarger than a chosen threshold value is measured. BothSEV and GPD methods allow data on inclusion sizes insmall volume of steel to be used to predict the maximuminclusion size in a large volume of steel.

The basic concept of extreme value theory is thatwhen a fixed number of data points following a basicdistribution are collected, the maximum and minimumof each of these sets also follow Gumbel distribution156

a step I; b step II; c step III; d fracture surface21 Schematic of GBF formation by coalescence of decohesion of carbide with matrix127

a surface roughness along cutting plane around inclu-sion; b bird view of GBF area and matrix observed byscanning probe microscope (SPM); c carbon distribu-tion around inclusion detected by electron probemicroanalyser (EPMA)79

22 Characteristic of GBF area



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as follows

G(z)~ exp { exp {(z{l)=a½ �f g (30)

where G(z) is the probability that the largest inclusion isno greater than size z, and a and l are the scale andlocation parameters respectively. For simplicity, thereduced variable y is used, i.e.

y~ z{lð Þ=a (31)

Then from equation (30), the distribution function G(z)is rewritten as

H(y)~ exp { exp ({y)½ � (32)

The key point is to determine the parameters a and l byexperiment. In the practical measurement,14,36–38,148–150

a standard inspection area S0 (mm2) is defined. The areaof the maximum inclusion in S0 is measured. Then, theinclusion size (square root of the area (areamax)1/2) iscalculated. This is repeated for N areas of S0 as shownschematically in Fig. 24.

The values of (areamax,i)1/2are ranged, starting from

the smallest, and ranked with i51, 2, …, N. Then(areamax,1)

1/2((areamax,2)

1/2(…((areamax,N)1/2. The cumu-

lative probability of the ith inclusion that is no greater than

inclusion size zi can be calculated simply by

H yið Þ~i= Nz1ð Þ (33)

From equations (32) and (33), we have

yi~{ ln { ln i=(Nz1)½ �f g (34)

Now if yi is plotted versus inclusion size zi, a straight line canbe found and the parameters a and l can be obtained by theslope and intercept (equation (31) that is a linear equation) orby the other methods such as maximum likelihoodmethod.147

For the prediction of inclusion size in a large volumeof steel V, the return period T is defined as

T~V=V0 (35)

where V0 is the standard inspection volume and isdefined

V0~S0h (36)

where h is the mean value of the measured inclusion size.The characteristic size zV of the maximum inclusion involume V can be considered as the size that is expectedto be exceeded exactly once in volume V. It is noted thatT represents the number of times to measure the

23 Illustration of fatigue crack initiation process around inclusion126

a sample; b S0; c zi5(areamax,i)1/2

24 Sketch of measurement of maximum inclusion size in SEV method, modified from Ref. 147



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inclusion sizes in a volume V, each time in a randomlyselected standard volume V0. Therefore

1=T~1{G zVð Þ (37)

The right side of equation (37) indicates the probabilityto exceed the inclusion size zV; the left side means that inthe volume V, there is only one time to reach zV in Ttimes of measurement. Then substituting equation (30)into equation (37), and combining with equation (31),we have the maximum inclusion size zV in volume V

zV~ayzl (38)

where y521n {2ln [(T21)/T]}. Now a, l as well as yare all known so that zV can be determined byequation (38).

Atkinson et al.151–155 predict the maximum inclusionsize in a certain volume of steel based on the GPD

F (x)~1{ 1zj x{uð Þ=s’½ �{1=j(39)

where s9.0 is a scale parameter and j (2‘,j,‘) is ashape parameter. If x is the size of inclusions, F(x) is theconditional probability that an inclusion is no greaterthan x, given that it is at least u. The key point is also todetermine the parameters s9 and j by experiment. Thedata needed for the GPD method are the number andsize of inclusions larger than a certain size u, asillustrated in Fig. 25, which is different from that forthe SEV method.

When the GPD is used to model the sizes of inclusionswhich exceed a threshold u, the characteristic size of themaximum inclusion xV in a volume V can be found asfollows.151 The expected number of inclusions in volumeV exceeding a size x is equal to the product of theexpected number in V exceeding u and the probability,given that the inclusion is larger than u and the size ofinclusions is larger than x. This last probability is givenimmediately by equation (39). Thus, if NV(u) denotes theexpected number of exceedances of u in unit volume, thesize xV which is expected to be exceeded exactly once involume V satisfies

NV(u)V 1{F (xV)½ �~1 (40)

From equations (39) and (40), xV can be solved as151

xV~u{ s’=jð Þ 1{ NV(u)V½ �jn o

(41)

Here, the determination of the critical threshold u that isused in equation (40) to estimate the maximum inclusionsize in volume V is a major issue. One way151 is to plotthe mean excess versus the threshold u. The mean excessfor a particular threshold is found by taking thedifference between the size of each inclusion and thethreshold and averaging all these differences. This isrepeated for a whole series of potential thresholds andthe results are summarised on the mean excess plot. Anillustrative example is shown in Fig. 26.147 The plot isinspected to find where it becomes reasonably linear,rather than displaying the steep curve that tends to occurat the left hand end. The point beyond which it becomesreasonably linear is termed the critical threshold u. Forgraphical estimation of s9 and j, the line is drawn: theintercept on the vertical axis gives s9/(12j), and theslope is j/(12j). These two equations can then be solvedfor rough estimates of the s9 and j parameters. Anotherway is to apply the maximum likelihood method todetermine these parameters.151 At last, NV(u) can bedetermined approximately according to the number ofintercepted inclusions per unit area on the polishedsurface and the average of the inclusion size.151,153,157

Now the maximum inclusion size in volume V can beestimated by equation (41).

For the SEV method, the size of maximuminclusion increases infinitely with increasing steel

a sample; b S0; c xi5(areai)1/2 (xi.u)

25 Sketch of measurement of inclusion size which is greater than threshold u in GPD method, modified from Ref. 147

26 Illustrative mean excess plot for GPD method147



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volume. However, for GPD method, one of the maincharacteristics is that there is an upper limit for theestimated inclusion size147 (Fig. 27157). This feature is abig help for the steel makers to evaluate the quality oflarge amount of steels. However, for the fatigue testingin the laboratory, the highly stressed volume in thegauge section of specimen is usually very limited, forexample, for an axial hourglass shaped specimen withdiameter of 3 mm and the gauge length 5 mm, the testvolume is about 35?3 mm3 and its weight is about0?28 g. Supposing that 15 samples are used to determinethe fatigue strength by staircase method, then the weightof the total tested volume is about 4 g, i.e. 461023 kg.Only if larger specimen and more specimens are used,the test volume can be larger. However, if rotatingbending specimen is used, the highly stressed volume canbe smaller due to stress gradient. Therefore, for theconventional specimens, the weight of tested volume canbe around 1023–1021 kg, and in this range, thepredictions of the maximum inclusion size betweenSEV and GPD are very close as shown in Fig. 27. Forclean steels, the inclusion content is very low, and ineach inspection area, very few inclusions can be found;therefore, the SEV method may be preferential forestimation of the maximum inclusion size due to itssimplicity and time saving.

If the maximum inclusion size is estimated, the lowerbound of fatigue strength can be evaluated38 by usingthe equation (4) or (17).

Now it is confirmed that one adopts the estimatedmaximum inclusion size to predict the lower bound offatigue strength rather well,14,38,42 for example, seeFig. 28.14 The question is how the fatigue strength canbe predicted more precisely, based on inclusion sizeestimated by the present statistical methods. Experi-mentally, the fatigue strength is determined by severalspecimens, for example, at least six pairs of effectivedata are needed to determine the fatigue strength bystaircase method. It generally involves different inclu-sion sizes at fatigue origins of the broken specimens. Inprevious work, it was shown that if the average inclusion

size at fatigue origins was used to estimate the fatiguestrength, a better precision will be achieved.88 Now thequestion is how to use the information of maximuminclusion size and distribution that is obtained by SEVor GPD method to estimate the fatigue strength moreprecisely, not the lower bound of fatigue strength byusing the maximum inclusion size.

Some problemsRecently, great progress has been made to studythe effects of inclusions on VHCF behaviour ofhigh strength steels. Existing monographs5,6 andreviews38,49,57,77,99,126,158 have highlighted the significantachievements that we have made and some problems thatwe have to solve. Some of these problems are listed below.

Experimental facilities and related limitationsCurrently, there are two major methods to test the VHCFproperties of high strength steels, i.e. ultrasonic fatiguetest with very high frequency of around 20 kHz11,159–162

and modified fatigue testing machines with severalspecimens tested simultaneously at a conventionalfrequency.82,123,126 In ultrasonic fatigue testing, theexperimental data can be obtained in a short period thatis very helpful for improving the steel quality ordeveloping new steels. However, the temperature raise,163

particularly the frequency dependence164,165 of fatigueproperty, should be clarified in further work. It is alsonecessary to use conventional testing machines tocompare with the results tested by high frequency.166

Results depend on frequency when time dependentprocesses like environmental effects (corrosive mediumand high temperature) or strain rate dependence ofmaterial behaviour exist.77 In modified rotating bendingfatigue testing, although several specimens are used in onemachine, the time consumed may still be a barrier forVHCF testing. Recently, other high frequency testsystems with the frequencies of several hundreds to afew thousands of Hertz were also developed for VHCFtesting, mostly for the non-ferrous alloys.167–169 Usingthose test systems to intensively investigate the VHCFbehaviours of high strength steels is desired. It will help us

27 Comparison of characteristic size of maximum inclu-

sion and maximum likelihood confidence intervals

estimated by SEV and GPD methods in 40 sample

areas157

28 Predictions of lower fatigue strength by SEV method

for 10 and 100 specimens and experimental results14



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to obtain more VHCF data and evaluate the frequencyeffect56 of high strength steels.

Fatigue fracture mechanismAs mentioned in previous section, severalmechanisms48,126,127,140,146 have been proposed forfracture initiation at inclusions in the VHCF regime. Itseems that those mechanisms can be sketchily dividedinto three categories: (1) hydrogen assisted fracture; (2)carbide cluster induced fracture; and (3) dislocationconfigurations such as cyclic slip irreversibility,60,61,145

slip bands or polygonisation induced fracture. Thosethree mechanisms may be intimately related with eachother. For example, since the hydrogen enhancedlocalised plasticity will certainly promote the intensivedislocation activity, under a very complicated stressdistribution around the inclusion, multislip will operateand in turn the slip bands or polygonisation could beeasily formed during cyclic loading. Also, the carbideswill be the strengthening phase when they are small anduniformly distributed or they can act as inclusions whenthey are large and cluster together.170 In the later case,the enriched hydrogen around the carbides probablyalso plays a role in the fracture. It is expected that amore comprehensive and convincing model can beproposed soon.

Fatigue origins initiated from other defects ormicrostructuresIn the VHCF regime, Mughrabi61 points out that forductile single phase type I materials that do not containinclusions the fatigue crack initiates at persistent slipbands or cyclic slip irreversibility. Progress in this aspecthas been reviewed by Zimmermann.1 For type IImaterials, e.g. high strength steels containing non-metallic inclusions, fatigue crack initiates at slip bands,grain boundaries, pores, non-metallic inclusions orparticles at surface sites in the HCF regime, but it shiftsto interior sites in the VHCF regime, mostly at non-metallic inclusions. However, even for the high strengthsteels in the VHCF regime, fatigue crack initiation sitecan also be observed in an internal non-defect area ormatrix microstructure that is not associated with pre-existing defects. This type of crack origin is named as‘non-defect crack origins’.104,171 Actually, these originscould be large and soft phase (bainite) in bainite/martensite duplex phase steels48,172 or large and softgrains in non-ferrous alloys,173 or carbon or chromiumenriched regions in some spring steels68 or otherinhomogeneities. How do these inhomogeneities174

compete with inclusions in the VHCF regime? Up tonow, no detailed investigation has been undertaken.

Another problem is that although the inclusion sizewas emphasised in the review, the effect of compositionof inclusion on the VHCF property is still worthinvestigating. In the quality control of the steel makingprocess, the detailed information of chemical composi-tion of inclusion that acted as fatigue origin will helpsteel makers to conduct adequate measures.

StandardisationThe standard procedures on fatigue testing and dataanalysing in the VHCF regime should be accomplishedin the future. Since the highly stressed volumes inspecimens are varied for different fatigue test facilities,the size effect71,175,176 of specimens on fatigue testing

should be evaluated and the standard specimens shouldbe suggested firstly. As regards the assessment ofinclusions in steels, the international standard such asISO 4967:1998(E) is widely used to assess the suitabilityof a steel for a given use, and plays a very important rolein steel industry. However, since it is difficult to achievereproducible results owing to the influence of the testoperator, even with a large number of specimens,177 thesupplement methods such as SEV and GPD arerecommended, and an intensive work is still needed instandardisation of these methods for the sake ofobtaining the results with a certain reproducibility. Inthis process, the determination of inclusion size shouldbe evaluated and standardised further. In Murakami’smodel, the inclusion size is determined as (area)1/2,where the inclusion effective projected area can bemeasured as follows. On the fatigue facture surface, thesingle inclusion with spherical or ellipsoid shape can befrequently found in a scanning electron microscope, thearea of inclusion can be obtained directly by using theimage analysis software, or by measuring the diameterof a spherical inclusion or the lengths of major andminor axes of the ellipsoid inclusion, and hence, the areacan be calculated by formula. For irregularly shapedinclusions, including inclusion clusters, the effective areaof inclusion is estimated by considering a smoothcontour which envelopes the original irregular shape.5

In SEV and GPD methods, the inclusion size on thesample surface was measured in an optical microscopeby the method mentioned above. It is noted that in someworks, the (area)1/2 model was not used, so the inclusionsize may be expressed by diameter that is directlymeasured in a microscope. Different methods lead tomore diversity of the data, so that a standardisedmethod to measure the inclusion size is needed.

Last but not least, the inspection of keycomponents171 in the VHCF regime and the recommen-dations of design code69 based on VHCF are welcomed.

SummaryThe effects of inclusions on VHCF properties of highstrength steels have been reviewed with emphasis on theinclusion size. The critical inclusion size in the VHCFregime is about 3–5 mm that is smaller than that in theHCF regime. In the VHCF regime, the S–N curvecharacteristics are influenced by inclusion size, residualstress, roughness and other factors. The dependence offatigue strength and fatigue life on inclussion size can beexpressed as sw!1=Dn1 and Nf!1=Dn2 respectiviely forsome high strangth steels. In the VHCF regime, theexponents n1 and n2 are found to fall in the rages 0.17–0.19 and 4.29–8.42 correspondingly. Fatigue life is moresensitive to inclusion size than fatigue strength.Hydrogen content and other factors also strongly affectthe fatigue life and fatigue strength for high strengthsteels in the VHCF regime. The statistias of extremevalue (SEV) and the generalised Pareto distribution(GPD) methods are recommended to predict themaxmum inclussion size in a certain volume of steel.However, the prediction of the maximum inclusion sizein a group of specimens, and hence prediction of thefatigue strength of the steel, should be further investi-gated and standardised. Convincing mechanisms for theinitiation of fatigue fracture at inclusions as well as the



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competing mechanisms between inclusions and otherinhomogeneities are much needed.

As Thornton has pointed out:3 ‘Due to the numerousfactors involved, any general correlation between non-metallic inclusions and mechanical properties is highlyimprobable. While it is possible to obtain a plausiblerelationship in a particular situation, the factorsinvolved must be standardized to such an extent thatapplication of the results is extremely limited’. In otherwords, more caution is required in applying therelationships reviewed above. However, those quantita-tive or semiquantitative relationships will certainly helpto explore the VHCF mechanisms and improvethe quality of high strength steels in future. Thoserelationships will be examined and modified in thecoming works, and a more thorough understanding ofthe effects of inclusions on VHCF properties of highstrength steels is in prospect.

Acknowledgements

This work was financially supported by the NationalKey Basic Research and Development Program ofChina (no. G2004CB619100). The author thanksProfessor Y. Q. Weng and Professor W. J. Hui inCentral Iron and Steel Research Institute, Beijing,China, for their longstanding cooperation. Thanks goto Professor Z. F. Zhang and Professor Z. G. Wang fortheir helpful advice, and to my colleagues ProfessorQ. Y. Wang in Sichuan University, China andProfessor Z. G. Yang and Professor G. Y. Li for theirsupport in the work in the past many years. Thanks alsogo 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 fortheir 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 andProfessor L. Z. Sun and Professor T. Zhai in the USAand Professor Z. M. Sun in Japan provided me someoutmoded but important references, so I must expressmy sincere thanks for their kindness.

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effects of inclusions on very high cycle fatigue properties of high strength steels

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