International Journal of Fatigue 60 (2014) 48–56

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International Journal of Fatigue

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Influence of microstructure and surface defect on very high cycle fatigueproperties of clean spring steel

0142-1123/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijfatigue.2013.06.017

⇑ Corresponding author. Tel.: +86 010 68918730; fax: +86 010 68918528.E-mail address: lw_cy@sina.com (W. Li).

Wei Li a,⇑, Tatsuo Sakai b, Masami Wakita c, Singo Mimura c

a School of Mechanical Engineering, Beijing Institute of Technology, Chinab Research Center of Advanced Materials Technology, Ritsumeikan University, Japanc Department of Technology development, Chuo Spring Company Limited, Japan

a r t i c l e i n f o

Article history:Received 13 March 2012Received in revised form 25 October 2012Accepted 14 June 2013Available online 2 July 2013

Keywords:Clean spring steelVery high cycle fatigueLoading conditionSurface grinding defectInhomogeneous microstructure

a b s t r a c t

Very high cycle fatigue (VHCF) properties of a newly developed clean spring steel were experimentallyexamined under rotating bending and axial loading. As a result, this steel represents the duplex S–Nproperty only for surface-induced failure under rotating bending, whereas it represents the single S–Nproperty for surface-induced failure and interior inhomogeneous microstructure-induced failure underaxial loading. Surface small grinding defect-induced failure is the predominant failure mode of this steelin VHCF regime. The surface morphology of the interior inhomogeneous microstructure with distinctplastic deformation is much rougher than that of the ambient matrix, which means the stress concentra-tion resulted from the strain inconsistency between the microstructural inhomogeneity as soft phase andthe ambient matrix as hard phase plays a key role in causing interior crack initiation. Considering theeffect of surface compressive residual stress, the threshold stress intensity factor for surface smalldefect-induced crack propagation of this steel is evaluated to be 2.04 MPam1/2, which means that theshort crack effect plays a key role in causing the surface small defect-induced failure of this steel inthe VHCF regime. From the viewpoint of defect distribution, surface and interior failure probabilitiesare equivalent under a fixed characteristic value of defect density. If the interior defect size is less thanor even equal to the surface defect size, surface defect-induced failure will become the predominant fail-ure mode in VHCF regime, especially under rotating bending.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Based on two failure modes consisting of surface-induced fail-ure and interior-induced failure, some high strength steels withtensile strength exceeding 1200 MPa usually represent a character-istic S–N property called the duplex S–N property or the stepwiseS–N property in the very high cycle fatigue (VHCF) regime [1–3].This characteristic S–N property is mainly embodied in thetransition of failure mode from the surface-induced failurecorresponding to high stress amplitude and short fatigue life tothe interior-induced failure corresponding to low stress amplitudeand long fatigue life. Generally, the interior non-metal inclusion-induced failure is the predominant failure mode of high strengthsteel in the VHCF regime.

Recently, with higher cleanness of steel upgraded steadily, stud-ies show that there may be a critical inclusion size [4,5], belowwhich fatigue crack causing failure of high strength steel in theVHCF regime does not initiate from the interior inclusion butmay initiate from some interior inhomogeneous microstructures

[6–14]. These interior inhomogeneous microstructures are mainlygenerated during the process of heat treatment, closely relating tothe quenching and the tempering temperature. For some steelswith tempered martensite structure [6,7], the inhomogeneousmicrostructure causing interior crack initiation is mostly the bai-nite derived from the imperfect transformation of the austeniteduring quenching. For some duplex-phase steels such as thebainite/martensite steels [8–11], ferrite/pearlite steels [12,13]and ferrite/austenite steels [14], the inhomogeneous microstruc-tures causing interior crack initiation may be the coarse bainite[8–11], the occasionally generated ferrite grain [8] and or the fer-rite grain as matrix [12–14], corresponding to the soft phase inthe microstructure of steel. Furthermore, it should be noted thatthe ‘‘fine granular area’’ (FGA) [15] sometimes can be observedin the vicinity of these homogeneous microstructures [8–10,14]but sometimes cannot be observed even in the life regime beyond108 cycles [6,7,11,12]. Apparently, the interior crack initiation andpropagation mechanisms induced by the inhomogeneous micro-structure are different from that induced by the inclusion, whichwill have to be discussed further.

In addition, surface small grinding defects such as scratchesand cavities or surface small inclusion can also induce crack ini-

(a) Under rotating bending

(b) Under axial loading

Fig. 1. Shapes and dimensions of specimens (units: mm).

W. Li et al. / International Journal of Fatigue 60 (2014) 48–56 49

tiation of high strength steel in the VHCF regime. Unfortunately,this failure mode does not arouse popular attention in previousstudies, even though it was observed during the experiment.This is probably attributed to the very low probability of thisfailure mode in the VHCF regime due to the presence of largerinclusions or the special surface treatment of specimen. How-ever, with the development of clean steels with the inclusioncontrol, this failure mode also becomes a research subject wor-thy of attention. Akiniwa et al. [16] found that slip due to sheardeformation played an important role in this crack initiationmechanism for their investigated steel, and the size of straightslip region was very small and was closed to the prior grain size.Shiozawa et al. [17] founded that the minimum stress intensityfactor for surface small defect-induced crack propagation fortheir investigated steel is only 2.5 MPam1/2. Mayer et al. [18]pointed out that the plausible small crack effect should beresponsible for this crack propagation mechanism controlled bythe small stress intensity factor based on the experimental resultof their investigated steel, which was in agreement with theresult reported by Tanaka et al. [19]. Furthermore, Marineset al. [20] pointed out that the crack propagation following ini-tiation induced by surface small grinding defect was a small por-tion of total life if over 107 cycles. However, the detailed crackinitiation mechanism induced by surface small grinding defectsin the VHCF regime is not yet completely understood so farand still needs to be further verified by means of more test re-sults of other high strength steels under different loadingcondition.

In the present study, fatigue tests under rotating bending andunder axial loading were carried out to clarify fatigue propertiesof a newly developed clean spring steel in the VHCF regime, theemphasis was put on the S–N property, the crack initiation andpropagation mechanisms, and the statistical evaluation of inclu-sion size, fatigue limit and competing failure mode.

2. Experimental procedure

2.1. Material and specimen

Material used in this study is a newly developed clean spring steel(JIS-SUP7-T450), whose chemical composition (mass percentage) is:0.6 C, 2.0 Si, 0.9 Mn, 0.014 P, 0.019 S. Firstly, specimens were ma-chined into the shape of hourglass with a certain amount of finishingmargin, and then quenched in vacuum (1173 K � 20 min + oil cool-ing) and tempered (723 K � 60 min + air cooling). The surfaces ofspecimens under rotating bending and axial loading were all grindedin a direction perpendicular to the axis of specimen by the grade600–2000 abrasive paper to the final shapes, as shown in Fig. 1.The minimum diameter d for these two kinds of specimens is all3 mm, and the round-notched radius c of specimens is 7 mm forrotating bending and 15 mm for axial loading, respectively. The cor-responding elastic stress concentration factors Kt are evaluated to be1.06 and 1.04, respectively. The Vickers hardness (HV) distributionalong the cross-section of specimen is almost uniform, about 485kgf/mm2. The tensile strength is evaluated to be 1586 MPa.

Based on the observation by means of scanning electronicmicroscope (SEM), the microstructure of heat-treated material isthe tempered sorbite structure, in which a number of fine granularcarbides are dispersedly distributed in the matrix of ferrite, asshown in Fig. 2a. Furthermore, the prior austenite grain size ismeasured to be about 9.4 lm, as shown in Fig. 2b.

2.2. Fatigue testing method

A cantilever-type fatigue testing machine (RB4-3510-V2) atfrequency of 52.5 Hz and a multi-type fatigue testing machine(PMF4-10) at frequency of 80 Hz were used to carry out the VHCFtests of SUP7-T450 spring steel under rotating bending and axialloading, respectively. Fatigue tests were all performed in an open

(a) Tempered sorbite (b) Prior austenite grain (9.4 m)

(a) (b)

Fig. 2. Microstructure observation of SUP7-T450 spring steel.

0 20 40 60 80-400

-300

-200

-100

0

100

Test data under rotating bending Test data under axial loading Fitting line under rotating bending Fitting line under axial loading

Res

idua

l str

ess

σ r , M

Pa

Depth from the surface δ, μm

Fig. 3. Residual stress distribution of specimen.

0 1 2 3 4 5 6 7 8 9 10

-

Cum

ulat

ive

prob

abili

ty,

F %

1

510203040506070

80

90

99

---------

-

95

Maximum inclusion size

Fig. 4. Gumbel distribution of inclusion size.

50 W. Li et al. / International Journal of Fatigue 60 (2014) 48–56

environment and at room temperature with the constant stress ra-tio R of �1. After the experiment, fracture surfaces of all the failedspecimens were carefully observed by SEM, paying particularattention to the crack initiation sites and crack initiation and prop-agation mechanisms.

3. Experimental results and discussions

3.1. Surface roughness and residual stress

According to JIS-B0601-2001 [21], the values of average rough-ness Ra, maximum height of the profile Ry and 10-points averageroughness Rz,JIS for the round-notched surface of specimen underrotating bending and axial loading were measured, as shown in Ta-ble 1. The surface grinding condition of specimen under axial load-ing is somewhat better than that under rotating bending, but bothof them all meet the experimental requirement.

Furthermore, the distribution of residual stress rr on the round-notched surfaces of these two kinds of specimens was measured byusing X-ray diffraction as a function of depth d from the surface.The approximate relationships between rr and d for rotating bend-ing and axial loading are indicated by a solid line and a dotted linein Fig. 3, respectively. It can be found that the maximum compres-sive rr is measured on the surface for both kinds of specimens witha value of 330 MPa.

3.2. Inspection of inclusion size

The maximum inclusion size in a certain volume V (units: mm3)of this steel was evaluated by using the statistics of extreme values(SEV) method [22]. 57 standard inspection planes S0 with samesize, about 0.02 mm2, were chosen to observe inclusion size.Firstly, it is confirmed that the measured inclusion sizes from1.76 lm to 7.14 lm can be well characterized by the Gumbel dis-tribution, as shown in Fig. 4. Let XV denote the characteristic valueof maximum inclusion size in units of lm, the return period T andthe cumulative probability P(XV) of XV, are given by:

T ¼ V=V0 ð1Þ

PðXmÞ ¼ 1� 1=T ð2Þ

Table 1Measurement of surface roughness of specimens (lm).

Load type Ra Ry Rz,JIS

Rotating bending 0.26 1.23 0.95Axial loading 0.16 0.93 0.74

where V0 is the volume of standard inspection plane with a certainthickness. This thickness value is defined as the average inclusionsize, about 3.74 � 10�3 mm. As a result, the value of V0 is evaluatedto be 2.65 � 10�2 mm3. Therefore, the value of XV can be evaluatedby the following equation:

XV ¼ 3:14� 1:06 ln � ln 1� 2:65100V

� �� �ð3Þ

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108

5

10

15

20

25

30

Axial loading

: 22.01 mm3

13 μm10 μm

Rotating bending

: 2.53 mm3

Inclusion

Max

imum

incl

usio

n si

ze, μ

m

Volume of steel V, mm3

Fig. 5. Predicted maximum inclusion size.

W. Li et al. / International Journal of Fatigue 60 (2014) 48–56 51

The detailed solution process is seen in Ref. [23]. The predictedmaximum inclusion size tends to increase with the increase of thevolume of steel, as shown in Fig. 5. Furthermore, the control vol-umes of specimen used in this study under rotating bending andaxial loading, VRB and VAL, are given by: [3]

VRB ¼ 0:25pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � ½c� 0:5ðd=

ffiffiffid3p� d�

2q

ð1�uÞðdþ d=ffiffiffiffiup Þ2 ð4Þ

VAL ¼ 0:25plðdÞ2 ð5Þ

where the parameter u is empirically chosen in the range0:95 P u P 0:9 under rotating bending, here let u ¼ 0:9; theparameter l is defined as the length of variable cross-section ofspecimen under axial loading at which the stress value on thecross-section is 0.9 times that on minimum cross-section at l/2.Therefore, the values of VRB and VAL are calculated to be 2.53 mm3

and 22.01 mm3, respectively. Based on Eq. (3), the evaluated maxi-mum inclusion size corresponding to VRB is about 10 lm, less thanthat corresponding to VAL about 13 lm.

3.3. S–N property

Fig. 6 shows the S–N diagrams of SUP7-T450 spring steel underrotating bending and axial loading. Based on the SEM observationof fracture surfaces, all fatigue failure of this steel under rotatingbending is induced from the surface of specimen. All S–N datacan be divided into two parts, corresponding to two fatigue life re-gions: short life region (about 850 MPa 6 ra 6 1100 MPa with

103 104 105 106 107 108 109 1010

500

600

700

800

900

1000

1100

1200

Surface failure under axial loading Interior failure under axial loading

Surface failure for short life under rotating bending

Surface failure for long life under rotating bending

Str

ess

ampl

itude

σa ,

MPa

Number of cycles to failure Nf

Fig. 6. S–N diagrams of SUP7-T450 spring steel under RB and AL.

N < 106 cycles) and very long life region (about 650 MPa 6 ra

6 950 MPa with N > 107 cycles). Furthermore, it should be notedthat under stress amplitude of 850 MPa 6 ra 6 950 MPa thedispersion degree of fatigue life is considerable, simultaneouslycorresponding to the short fatigue life and the very long fatiguelife. Under this circumstance, the conventional S–N curve is notsuitable to evaluate the scatter of S–N data, especially for the S–Ndata under stress amplitude of 850 MPa 6 ra 6 950 MPa. There-fore, the duplex S–N curves are proposed to represent the VHCFS–N property of this steel under rotating bending. According tonon-linear and linear S–N curve models proposed by the JSMS-SD-6-04 [24], two S–N curves, respectively corresponding to shortlife region and very long life region, are plotted by a solid line and adotted line in Fig. 6.

On the other hand, fatigue failure of this steel under axial load-ing corresponds to two failure modes: surface-induced failure andinterior-induced failure. However, it should be noted that only onedata point for interior-induced failure was obtained, the surface-induced failure also occurred in the very long life region ofN > 107 cycles, just like under rotating bending. Obviously, the du-plex S–N property cannot be observed by means of the currentlyavailable S–N data. Therefore, the single S–N curve is proposed torepresent the VHCF S–N property of this steel under axial loading.Based on the same non-linear S–N curve model proposed by theJSMS-SD-6-04, the S–N curve is plotted by a dashed line in Fig. 6.

Apparently, it can be found that the surface-induced failure isthe predominant failure mode of this steel in the VHCF regime un-der whatever loading condition, which is different from the con-ventional failure character of some high strength steels in theVHCF regime, where the interior-induced failure prevails. Further-more, fatigue strength of this steel corresponding to 108 cycles un-der rotating bending is evaluated to be 854 MPa, larger than thatunder axial loading with about 647 MPa. This is attributed to thedifferent stress distributions along the cross-sections of specimensunder these two loading conditions. Under the same stress ampli-tude, the stress is uniform on the cross-sections of specimens un-der axial loading, whereas the stress is the largest on the surfacelayer and gradually decreases to zero at the center of cross-sectionsof specimens under rotating bending, i.e. the so-called stress gradi-ent effect.

3.4. Observation of fracture surface

As above mentioned, regardless of short life or long life region,all fatigue failure of specimens under rotating bending is inducedfrom the surface of specimen. In the short life region, fatigue crackinitiation can be divided into two modes: one is induced by thesurface grinding defect such as the slender flaw, as shown inFig. 7a; the other one is induced by the surface inclusion, as shownin Fig. 7b. The surface flaw-induced failure is the main failure modeof this steel in this life region. In the long life region, fatigue crack isall induced from the surface slender flaw, as shown in Fig. (c). Bycontrast, it can be found that the crack initiation and propagationbehaviors under the same stress amplitude of 950 MPa shown inFig. 7a and c, respectively corresponding to short fatigue life andlong fatigue life, is obviously different, where the former initiatesfrom the whole root of flaw and propagates layer by layer, butthe latter only initiates from a certain weak point at the root offlaw and propagates in the form of point-scattering, just like thecrack initiation and propagation behaviors induced by the inclu-sion shown in Fig. 7b. Maybe, this is the reason why fatigue life un-der the same stress amplitude varies so much.

On the other hand, fatigue failure of specimens under axialloading is composed of the surface-induced failure and the inte-rior-induced failure. For the surface-induced failure, all fatiguecracks are induced from the surface grinding notch-shaped defects

(a) Surface flaw-induced failure (b) Surface inclusion-induced failure

( a=950 MPa, Nf =49220) ( a=850 MPa, Nf =373380)

(c) Surface flaw-induced failure

( a=950 MPa, Nf =39371260)

(d) Surface notch-induced failure (e) Surface notch-induced failure

( a=750 MPa, Nf =1243500) ( a=625 MPa, Nf =11407700)

(c)

(d) (e)

(a) (b)

(f) Fish-eye without inclusion (g) Inhomogeneous microstructure

( a=675 MPa, Nf =60012000) ( a=675 MPa, Nf =60012000)

(f) (g)

Fig. 7. SEM observation of fracture surfaces.

52 W. Li et al. / International Journal of Fatigue 60 (2014) 48–56

regardless of fatigue life, as shown in Fig. 7d and e. For the only oneinterior-induced failure, a clear and isolated fish-eye can be

observed on the fracture surface, as shown in Fig. 7f. Simulta-neously, a white rough area almost occurs at the centre of

Fig. 8. EDS analysis of interior crack initiation region.

600 700 800 900 1000 11000

10

20

30

40

Average value=15.01μm

Average value=14.46 μm

Under rotating bending Under axial loading

√ (ar

easu

r-d),

μm

Stress amplitude σa, MPa

Fig. 10. Relationship betweenffiffiffiffiffiffiffiffiffiffiareap

sur�d and ra.

W. Li et al. / International Journal of Fatigue 60 (2014) 48–56 53

fish-eye, the ultra-high magnification photograph of which isshown in Fig. 7g. It can be found that a number of very small con-vex particles and concave holes are densely distributed at this area,but no inclusion or small inclusion cluster can be observed bymeans of the energy dispersive X-ray spectrometer (EDS) analysisshown in Fig. 8. Unfortunately, now we are not sure that what thechemical composition of this white rough area is. However, it canbe concluded that this rough area should correspond to the inho-mogeneous microstructure generated during the heat-treatmentof steel. In view of the rougher surface morphology of inhomoge-neous microstructure with distinct plastic deformation in compar-ison with ambient matrix, the stress concentration generated bythe strain inconsistency between the microstructural inhomogene-ity as soft phase and the ambient matrix as hard phase plays a keyrole in causing interior crack initiation.

3.5. Evaluation of stress intensity factor

Based on the fractography, a crack size parameterffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp

isused to denote the sizes of surface grinding defects including sur-face flaw, surface inclusion and surface notch. For the surface slen-der flaw, its size is approximately equal to

ffiffiffiffiffiffi10p

time of its depth g[22]. Fig. 9 shows the relationships between g and Nf under rotat-ing bending and axial loading. The values of g under rotating bend-ing are scattered over the range of 2.85–15.81 lm with an average

103 104 105 106 107 108 109

1

10

100

Average value=7.98 μm

Average value=6.29 μm

Under rotating bending Under axial loading

Dep

th o

f su

rfac

e de

fect

η, μ

m

Number of cycles to failure Nf

Fig. 9. Relationship between g and Nf.

value of 6.29 lm, while that under axial loading are scattered overthe range of 5.04–10.52 lm with an average value of 7.98 lm.They are all independent of fatigue life and are randomly generatedduring the grinding process of specimen. Furthermore, it can befound that the values of g are obviously larger than the above-mea-sured surface roughness. This means that surface roughness mea-surement underestimates the depth of these surface defectswhen the needle cannot exactly follow the contour of the surfacedefect.

Fig. 10 shows the relationships betweenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur � d

pand ra

under rotating bending and axial loading. The values offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur � d

punder rotating bending are scatted in the range of

6.81–27.85 lm with an average value of 15.01 lm, while that un-der axial loading are scatted in the range of 9.61–20.69 lm with anaverage value of 14.46 lm. Obviously, most of them are all largerthan the maximum inclusion sizes evaluated by SEV method underrotating bending and AL. Simultaneously, considering the potentialaccelerative effect of external environment [25], it is not surprisingthat the surface defect-induced failure becomes the predominantfailure mode of this steel in the VHCF regime.

Furthermore, considering the effect of surface compressiveresidual stress shown in Fig. 3, the stress intensity factor rangeDKsur�d at the front of surface defect is given by [16]:

DKsur�d ¼ 0:65ðra � rrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dpq

ð6Þ

where the value of rr can be evaluated by the fitting line in Fig. 3,combining with the depth of surface defect shown in Fig. 9. Therelationship between DKsur�d and Nf =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp

is indicated by a so-lid line in Fig. 11. It can be found that the values of DKsur�d tend todecrease with the increase of Nf=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp

. Compared with thethreshold stress controlling surface long crack propagation of highstrength steel under R = 0.1, about 4–5 MPam1/2 [1], most of valuesof DKsur�d is distinctly smaller. Based on the fitted curve shown inFig. 11, the threshold value DKsur�th controlling surface small de-fect-induced crack propagation for SUP7-T450 spring steel is evalu-ated to be 2.04 MPam1/2, which is somewhat less than thatevaluated by Mayer et al. for their investigated steel under the ef-fect of surface compressive residual stress, 2.5 MPam1/2 [17].According to one of most typical characteristics of small crack, i.e.small crack often exhibit lower threshold value than long crack[26], and combined with the fact that the sizes of surface defectsare only on the order of a few grain diameters, it can be inferred thatthe growth process of micstructurally small fatigue cracks [27]whose crack sizes are small in comparison to characteristicmicrostructural dimensions such as the grain size plays a key role

108 109 1010 1011 1012 1013

0

1

2

3

4

5

6

log(ΔKsur-d

-2.04)=-0.53log[Nf /√(area

sur-d)]+5.08

Nf /√(area

sur-d) , cycles/m

Under rotating bending Under axial loading

ΔKsu

r-d, M

Pam

1/2

Fig. 11. Relationship between DKsur�d and Nf=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp

.

103 104 105 106 107 108 109

800

1000

1200

1400

1600

1800

2000

Fitting line under rotating bending Fitting line under axial loading

Under rotating bending Under axial loading

σ a×(

area

sur-

d)1/12

, MPa

μm1/

6

Number of cycles to failure Nf

Fig. 13. Relationship between ra � (areasur-d)1/12 and Nf.

54 W. Li et al. / International Journal of Fatigue 60 (2014) 48–56

in causing small surface defect-induced failure in the VHCF regime,and spends a significant fraction of fatigue life.

3.6. Evaluation of surface fatigue limit

Fatigue limit rsur-w corresponding to surface defect-inducedfailure under R = �1, can be evaluated by the following equation[28]:

rsur�w ¼ 1:43� ðHVþ 120Þ=ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp

Þ1=6 ð7Þ

It should be noted that herein evaluated fatigue limit is thethreshold stress for non-propagation of crack originating from aninitial defect or crack for the metallic materials containing smalldefects or cracks [22]. The ratio ra/rsur-w between the appliedstress amplitude and the fatigue limit is given by:

ra

rsur�w¼ ra � ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiareasur�dp Þ1=6

1:43� ðHV þ 120Þ ð8Þ

Fig. 12 shows the relationships between ra/rsur-w and Nf underrotating bending and axial loading. It can be found that the valuesof ra/rsur-w under these two loading types all tend to decrease withthe increase of Nf, which are larger than 1 even in the VHCF regime.This also means that the occurrence of surface defect-induced fail-ure in the VHCF regime is reasonable although fatigue limit issometimes underestimated by using the Eq. (7) [29,30].

103 104 105 106 107 108 109

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Under rotating bendingUnder axial loading

σ a / σsu

r-w

Number of cycles to failure Nf

Fig. 12. Relationship between ra/rsur-w and Nf.

Furthermore, based on Eq. (8), the modified S–N curve modescan be represented by the relationship between ra � ðareaÞ1=12

and N, and are given by:

Modified linear S� N curve mode : ðra � ðareaÞ1=12Þg� N

¼ Q ð9Þ

Modified non-linear S� N curve mode

: ððra � ðareaÞ1=12Þ � nÞg� N ¼ Q ð10Þ

where Q, g and n are all material-independent constants. Based onEqs. (9) and (10), modified S–N diagrams for surface-induced failureof SUP7-T450 spring steel under rotating bending and axial loadingare shown in Fig. 13. It can be found that the duplex or step-wise S–N property of this steel under rotating bending, indicated by a solidline in Fig. 13, can be described properly, which means that theabove-mentioned representation of S–N property of this steel underrotating bending based on duplex S–N curves in Fig. 6 is reasonableand acceptable. Under axial loading, this steel still represents thecontinuously descending S–N property, indicated by a dashed linein Fig. 13, just like it in Fig. 6.

3.7. Evaluation of competing failure modes

The occurrence probability of defect on the surface or in theinterior of specimen directly affects the fatigue failure mode ofspecimen, which can be effectively evaluated by using 2D Poissondefect distribution [31,32]. Firstly, the area A of minimum cross-section of specimen used in this study can be classified into twoareas: the interior area Aint and the surface annular area Asur witha width Wsur encircled by Aint, as shown in Fig. 14. The width ofthe surface annular area is approximately equal to the defect sizeoccurring on the surface of specimen. Thus, the occurrence proba-bility P(n) of the number n of random defects can be given by:

PðnÞ ¼ e�kAint ðkAintÞn

n!ð11Þ

where k is the defect density. Based on Eq. (11), the occurrenceprobabilities of no defect and one or more defects in Aint are givenby:

Pðn ¼ 0Þ ¼ e�kAint ð12Þ

Pðn P 1Þ ¼ 1� Pðn ¼ 0Þ ¼ 1� e�kAint ð13Þ

Thus, the surface and interior failure probability, Psur and Pint,are expressed as follows:

Fig. 14. Definition of Aint and Asur.

W. Li et al. / International Journal of Fatigue 60 (2014) 48–56 55

Surface failure : Psur ¼ ð1� e�kAsur Þ ð14Þ

Interior failure : Pint ¼ ð1� e�kAint Þe�kAsur ð15Þ

For the specimen tested in this study, A = 7.06 mm2, i.e. Aint +Asur = 7.06 mm2. If the observed maximum inclusion size and pre-dicted maximum inclusion sizes under rotating bending and axialloading by SEV method, about 7 lm, 10 lm and 13 lm, are consid-ered as the maximum defect size causing crack initiation containedin Asur, i.e. the values of Wsur are equal to 7 lm, 10 lm and 13 lm,respectively, the corresponding relationships between Pint and Psur,

and k can be evaluated by using Eqs. (14) and (15), as shown inFig. 15. Firstly, it can be seen from Fig. 15 that surface and interiorfailure probability curves under each Wsur all intersect at a point.At this point, the values of Psur and Pint are equivalent, about 0.5.That is, the occurrence probabilities of these two failures are thesame under a fixed characteristic value of k. For Wsur = 7 lm,10 lm and 13 lm, the characteristic values of k are evaluated tobe 10.5 mm�2, 5.2 mm�2 and 4.5 mm�2, respectively. It is interest-ing to note that the characteristic values of k tend to decrease withthe increase of Wsur. This is because the larger Wsur means largerdefect size, these two failure probabilities can reach a balancedchange under the relatively low defect density.

Furthermore, when the value of k is far greater than the charac-teristic value, the occurrence probability of surface defect-inducedfailure is much greater than that of interior defect-induced failure.

10-3 10-2 10-1 100 101 102 103

0.0

0.2

0.4

0.6

0.8

1.0

1.2

λw=16

=4.5 mm-2

λw=13

=5.2 mm-2

λw=7

=10.5 mm-2

Psur

(Wsur

=7 μm)

Pint

(Wsur

=7 μm)

Psur

(Wsur

=10 μm)

Pint

(Wsur

=10 μm)

Psur

(Wsur

=13 μm)

Pint

(Wsur

=13 μm)

Fail

ure

prob

abil

ity

Defect densityλ, mm-2

Fig. 15. Relationships between Psur and Pint, and k under different Wsur.

Because of the higher defect density, the more likely defect occurson the surface of specimen. When the defect density is high en-ough, i.e. defects simultaneously occur on the surface and in theinterior of specimen, the surface defect-induced failure becomesthe predominant failure mode, especially combined with the effectof external environment. Therefore, it can be concluded that if thesizes of interior defects such as inclusion and inhomogeneousmicrostructure are less than the surface grinding defect size al-ready existed on the surface specimen, interior defect-induced fail-ure is almost impossible to take place even in the VHCF regime. Forclean high strength steel with inclusion control, the surface smallgrinding defect-induced failure will become a noteworthy failuremode in the VHCF regime, especially under rotating bending.

4. Conclusions

Main conclusions obtained in this study are summarized asfollows:

1. SUP7-T450 spring steel represents the duplex S–N propertyonly for surface-induced failure under rotating bending,whereas it represents the single S–N property for surface-induced failure and interior inhomogeneous microstructure-induced failure under axial loading. Surface small grindingdefect-induced failure is the predominant failure mode of thissteel in the VHCF regime.

2. The surface morphology of interior inhomogeneous microstruc-ture as crack initiation site is rougher than that of the ambientmatrix, which means the stress concentration resulted from thestrain inconsistency between the inhomogeneous microstruc-ture as soft phase and the ambient matrix as hard phase playsa key role in causing interior fisheye-shaped crack initiation.

3. Considering the effect of surface compressive residual stress,the threshold stress intensity factor for surface small defect-induced crack propagation, DKsur�th, of this steel is evaluatedto be 2.04 MPam1/2, and the short crack effect plays a key rolein causing the surface small defect-induced failure in the VHCFregime.

4. From the viewpoint of defect distribution, surface and interiorprobabilities are equivalent under a fixed characteristic valueof defect density. If the interior defect size is less than or evenequal to the surface defect size, surface defect-induced failurewill become the predominant failure mode even in the VHCFregime, especially under rotating bending.

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