General Relation Between Tensile Strength and Fatigue Strength of Metallic Materials - J.C. Pang

Materials Science & Engineering A 564 (2013) 331–341

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Materials Science & Engineering A

0921-50

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General relation between tensile strength and fatigue strengthof metallic materials

J.C. Pang, S.X. Li, Z.G. Wang, Z.F. Zhang n

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China

a r t i c l e i n f o

Article history:

Received 8 May 2012

Received in revised form

7 November 2012

Accepted 27 November 2012Available online 1 December 2012

Keywords:

High-strength materials

Tensile strength

Fatigue strength

Fatigue crack initiation site

Fatigue damage mechanism

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x.doi.org/10.1016/j.msea.2012.11.103

esponding author. Tel.: þ86 24 23971043; fa

ail address: [email protected] (Z.F. Zhang).

a b s t r a c t

With the development of high-strength materials, the existing fatigue strength formulae cannot

satisfactorily describe the relation between fatigue strength sw and tensile strength sb of metallic

materials with a wide range of strength. For a simple but more precise prediction, the tensile and

fatigue properties of SAE 4340 steel with the tensile strengths ranging from 1290 MPa to 2130 MPa

obtained in virtue of different tempering temperatures were studied in this paper. Based on the

experimental results of SAE 4340 steel and numerous other data available (conventional and newly

developed materials), through introducing a sensitive factor of defects P, a new universal fatigue

strength formula sw¼sb(C�Pdsb) was established for the first time. Combining the variation tendency

of fatigue crack initiation sites and the competition of defects, the fatigue damage mechanisms

associated with different tensile strengths and cracking sites are explained well. The decrease in the

fatigue strength at high-strength level can be explained by fracture mechanics and attributed to the

transition of fatigue cracking sites from surface to the inner inclusions, resulting in the maximum

fatigue strength smaxw at an appropriate tensile strength level. Therefore, the universal fatigue strength

formula cannot only explain why many metallic materials with excessively high strength do not display

high fatigue strength, but also provide a new clue for designing the materials or eliminating the

processing defects of the materials.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Fatigue is referred to the degradation of mechanical propertiesleading to failure of a material or a component under cyclicloading. The fatigue strength of materials is often defined as themaximum stress amplitude without failure after a given numberof cycles (e.g., 107 or 109). It is estimated that �90% of servicefailures of metallic components resulted from fatigue. However, itis very time and money consuming to perform fatigue tests.Therefore, many attempts have been made to determine thefatigue strength in a cost-effective way relating fatigue strengthto other mechanical properties, such as yield strength [1], tensilestrength [2–4], hardness [5–7] and so on; accordingly, the rela-tions between fatigue strength and other mechanical propertieshave been of more interest. Engineers and scientists have pro-posed many formulae to describe the relations between fatiguestrength and other mechanical properties [1–7]. In 1870s, Wohler,one of the pioneers in the fatigue field, found that the ratio offatigue strength sw to tensile strength sb for ferrous metals

ll rights reserved.

x: þ86 24 23891320.

followed a simple proportional relation as below [8],

sw ¼ 0:4�0:5ð Þsb ð1Þ

Based on the numerous data of fatigue strength and tensilestrength available for steels, copper and aluminum alloys [2–4] inthe past century, a more general form can be summarized asfollows,

sw ¼msb ð2Þ

However, it is found that the fatigue strength either maintainsconstant or decreases with further increasing the tensile strength[3,4]; in other words, the linear relation in Eq. (2) is no longer heldat high-strength level. The critical tensile strength sbc, abovewhich fatigue strength does not increase correspondingly, themaximum fatigue strengths smax

w and the coefficient m in Eq. (2)for steels, Cu and Al alloys are summarized from Refs. [2–4]. It isapparent that the linear equation cannot adequately be applied toestimate the fatigue strength of those high-strength materials.

On the other hand, in 1980s, another important finding byMurakami [5] is that there is a quantitative relationship amongfatigue strength sw, hardness Hv and inclusion size

ffiffiffiffiffiffiffiffiffiffiareap

in high-strength steels. Soon after, many related tests have been done byultrasonic fatigue testing machines [6,7,9–20] and some modifiedfatigue strength formulae were proposed by Wang et al. [9], Liu

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Fig. 1. Configuration of specimens tested for: (a) tensile strength; (b) VHC fatigue

strength.

J.C. Pang et al. / Materials Science & Engineering A 564 (2013) 331–341332

et al. [10] and McGreevy et al. [21]; nevertheless, no reportindicated that those relations suit for other high-strength materi-als. In addition, the fatigue strength is found to have linearrelation with hardness or the sum of tensile and yield strengthsonly in lower strength range [2–5]. In a word, there is also nosuitable formula to satisfactorily describe the general relationbetween tensile and fatigue strengths of both high- and low-strength materials.

During recent decades, many new high-strength materials,such as bulk metallic glasses [22], ultrafine or nano-grainedmaterials [23–26] and ultra-high strength steels [27] have beensuccessfully developed; however, their fatigue strengths arefound to be not as high as we expected, even become relativelylower in comparison with their higher tensile strength [27–32].Therefore, this gives rise to two open questions for scientists andengineers: (1) Why do the materials with excessively high tensilestrength not possess high fatigue strength? (2) Is there a moreuniversal equation to describe the general relation betweenfatigue strength and tensile strength in a wide strength range?Therefore, in this study, SAE 4340 steel with a very wide tensilestrength range, one of the excellent quenched and tempered low-alloy steels [2–5,33–36], was employed to study and establish asimple but more precise relation between fatigue strength andtensile strength of materials and provide a better clue for thedesign of high-performance structural materials.

2. Experimental material and procedures

In the current study, SAE 4340 steel bars were received with adiameter of 14 mm under annealing condition and its compositionis given in Table 1. To gain different strength levels, five optimizedheat-treatment procedures as shown in Table 2 were employedand the corresponding specimens are defined as A–E, respectively.The configurations of the tensile and fatigue specimens are shownin Fig. 1. All fatigue samples were polished in the longitudinaldirection using an emery paper having a mesh of 2000#.

The tensile tests were conducted at a strain rate of 2�10�4 s�1;very high-cycle (VHC) fatigue tests were conducted at a frequency of20 kHz up to 109 cycles using a ultrasonic fatigue testing system(Shimadzu USF-2000). To avoid the sample heating, the middlesection of each ultrasonic fatigue specimen was cooled by com-pressed air. All fatigue tests were performed with the sinusoidalwave shape under applied stress ratio of R¼�1. The fatiguestrength was determined by the staircase method in which at leastsix pairs of specimens were tested according to ISO12107:2003.The microstructures of the specimens with different strength levelswere examined by electron backscattered diffraction (EBSD) with

Table 1Chemical composition of SAE 4340 steel (Wt%).

C Mn Si P S Cr Ni Mo Cu As

0.42 0.66 0.25 0.009 0.014 0.74 1.41 0.17 0.11 0.034

Table 2Heat-treatment procedures of SAE 4340 steel.

No. Quenching Tempering

A preheating to 850 1C

for 10 min and quenching in oil

180 1C tempering for 120 min

B 250 1C tempering for 120 min

C 350 1C tempering for 120 min

D 420 1C tempering for 30 min

E 500 1C tempering for 30 min

scanning electron microscope (LEO SUPRA35). The fatigue fracto-graphies were observed by using a Quanta 600 scanning electronmicroscope (SEM).

3. Results

3.1. Microstructures

With increasing the tempering temperature, the body-centered tetragonal (BCT) martensite, which is a supersaturatedsolution of carbon in a-Fe, transforms to different microstructuresas shown in Fig. 2. Referring to the textbook [36,37] and XRDprofiles, the microstructure features of sample A–E are illustratedas follows: the sample A tempered at 180 1C contains manyneedle- or plate-shaped tempered martensites (see Fig. 2(a))and some retained austenites. The sample B tempered at 250 1Cconsists of microstructure (Fig. 2(b)) similar to that of the sampleA, but the size of retained austenite decreases because of itsdecomposition; however, the sample C tempered at 350 1C onlydisplays needle- or plate-shaped tempered martensite. The sam-ples D and E, respectively tempered at 420 1C and 500 1C, exhibittempered troostite with plate-like appearance (Fig. 2(d) and (e)),and the lath width of troostite in sample E increases in compar-ison with sample D because a phase has obviously recovered aftertempering above 400 1C.

3.2. Tensile properties

After different tempering procedures, the specimens A to Eexhibit different tensile properties and their tensile stress–straincurves are shown in Fig. 3(a). It can be seen that the specimens A toE display different yield strength, work-hardening ability, ultimatetensile strength and elongation. Fig. 3(b) and (c) show the relationsamong the strength, elongation to fracture and reduction in areaversus tempering temperature: it can be seen that as temperingtemperature increases, tensile strength successively decreases,however, elongation to fracture and reduction in area successivelyincrease, which are in agreement with other steels [34,35,38].On the other hand, the yield strength first increases slightlyand then decreases, which agrees with 300 M steel [39] and someother ultrahigh-strength steels tempered below 200 1C [34,35].Fig. 3(d) demonstrates the relationships of yield strength andtensile strength versus elongation to failure of the SAE 4340 steel.As tensile and yield strengths increase, elongation to failurecontinuously decreases, which is consistent with the trade-offsbetween strength and elongation of steels [34,35]. By comparison,

Fig. 2. EBSD microstructures of SAE 4340 steel processed at different tempering temperature: (a) 180 1C; (b) 250 1C; (c) 350 1C; (d) 420 1C and (e) 500 1C.

Fig. 3. Tensile properties of SAE 4340 steel: (a) tensile engineering stress–strain curves; (b) the relation between strength and tempering temperature; (c) relations of

elongation and reduction in area vs. the tempering temperature and (d) relation between strength and elongation to fracture.

J.C. Pang et al. / Materials Science & Engineering A 564 (2013) 331–341 333

the elongation to failure of specimen C appears smaller because ofthe tempering brittleness at 350 1C [33].

3.3. Fatigue properties

Six groups of VHC fatigue tests (the highest strength level wasrepeated once more) were conducted to obtain the fatiguestrengths by the staircase method. Fig. 4 shows the crack origin

of VHC fatigue. There are four types of crack origin modes:(a) surface scratch; (b) surface inclusion; (c) subsurface inclusion;(d) inner inclusion, which are in accordance with the crackinitiation modes of high-strength steels [5,6]. Besides, specialgranular bright and rough area in comparison with other area,which is named as Granular-bright-facet (GBF) by Shiozawa et al.[40], appears around the inclusion, as indicated by the blue cyclein Fig. 4(d), but is not shown in Fig. 4(c).

Fig. 4. Crack origin of VHC fatigue: (a) surface scratch; (b) surface inclusion; (c) subsurface inclusion and (d) inner inclusion.

Fig. 5. Relation between the VHC fatigue crack sites ratio and tensile strength.


Broadly speaking, according to the fatigue cracking site, thefour crack initiation modes can be classified into two types: i.e.,surface (surface scratch, surface/subsurface inclusion) and inner(inner inclusion). Fig. 5 displays the statistic ratio of surface andinner cracking sites for VHC fatigue test: the ratio of innercracking sites (RICS, the ratio of the number of samples originatedfrom the inner inclusion site to the total number of failuresamples) gradually increases with increasing tensile strength;however, when tensile strength is lower than �1200 MPa, RICStends to zero, when the tensile strength is higher than�2000 MPa, RICS tends to 100%. The ratio of surface crackingsites (RSCS, the ratio of the number of samples originated fromsurface site to the total number of failure samples) graduallydecreases as tensile strength increases; when tensile strength ishigher than �2000 MPa, RSCS tends to zero. This indicates thatwhen tensile strength is lower than �1200 MPa, nearly all thefatigue cracks originated from surface defects; when tensilestrength is higher than �2000 MPa, nearly all the fatigue cracksinitiated from inner inclusion; when tensile strength is in therange of 1200–2000 MPa, there is a gradual transition for fatiguecrack initiation from surface defects to inner inclusions.

Fig. 6(a) shows the relationship between the fatigue strengthand tensile strength of the SAE 4340 steels heat-treated under

different conditions. It is found that the VHC fatigue strengthsincrease first and then decrease with increasing tensile strength,which is consistent with that of the Cu–Be alloy after differentageing technologies [41]. The highest fatigue strength can be ashigh as 693 MPa, which occurs in the specimen B with a tensilestrength of 1830 MPa that is not the highest tensile strength,implying that a higher tensile strength of materials cannot alwayslead to a higher fatigue strength, which is also found in otherhigh-strength materials [27–32,41]. In contrast, as the tensilestrength increases to a certain extent, a decrease in fatiguestrength can be observed, for example, the specimen A with thehighest tensile strength of 2124 MPa has a fatigue strength of655 MPa, which is lower than the fatigue strength (693 MPa) ofthe specimen B with a tensile strength of 1830 MPa (see Fig. 6a).In order to confirm such results, a repeated VHC fatigue testwas conducted and a fatigue strength of 657 MPa was obtained,which implies that the fatigue testing result for the steel withthe highest tensile strength but with lower fatigue strength isreliable.

3.4. Fatigue strength formula

For better understanding on the varying trend of fatiguestrength for the materials at different tensile strength levels, itis necessary to find a general fatigue strength formula to describethe intrinsic relation between fatigue strength and tensilestrength. Fig. 6(b) demonstrates that the fatigue ratio R (the ratioof fatigue strength sw to tensile strength sb) gradually declineswith increasing tensile strength. From the data in Fig. 6(b), it isinteresting to find that the fatigue ratio R declines approximatelyin a linear relation with increasing tensile strength. To confirmthis tendency, a linear fitting is performed and the fittingequation can be expressed as below,

R¼ sw=sb ¼ 0:70�1:85� 10�4sb ð3Þ

As shown in Fig. 6(c), the scope of fatigue ratio is within the 5%error band. The fatigue strength sw in Eq. (3) can be written in

Fig. 6. (a) Relation between tensile and fatigue strengths of the 4340 steel; (b) relation between tensile strength and fatigue ratio; (c) fitting relation between tensile

strength and fatigue ratio and (d) fitting curves of tensile and fatigue strengths.


a quadratic equation of the tensile strength sb as below,

sw ¼ ð0:70�1:85� 10�4sbÞ � sb ð4Þ

Eq. (4) is drawn in Fig. 6(d), it can be seen that the fatiguestrengths are basically within the 5% error band. This illustratesthat Eq. (4) can well describe the general relation between fatiguestrength and tensile strength of the SAE 4340 steel in a widestrength range.

4. Discussion

4.1. Verification of the general fatigue strength formula

In order to confirm the fatigue strength formula above, it isnecessary to find out more fatigue strength data of metallicmaterials available. First, the rotating bending fatigue strengthdata of some high-strength steels with different strength levels inliterature [38] are used to verify the proposed equation. It is notedthat the fitting relations between fatigue ratio and tensilestrength for SAE 4140, 4340, 2340 and 4063 steels displayed inFig. 7 are also linear and the scope of fatigue ratio is within the 5%error band, which entirely agrees with the general relationproposed above. The fitting equations can be written as below,respectively

R¼ 0:87�2:65� 10�4sb For SAE 4140ð Þ ð5aÞ

R¼ 0:76�1:78� 10�4sb For SAE 4340ð Þ ð5bÞ

R¼ 0:74�1:89� 10�4sb For SAE 2340ð Þ ð5cÞ

R¼ 0:92�2:37� 10�4sb For SAE 4063ð Þ ð5dÞ

therefore, their fatigue strength sw can be expressed in quadraticequations of the tensile strength sb as below, respectively

sw ¼ ð0:87�2:65� 10�4sbÞ � sb For SAE 4140ð Þ ð6aÞ

sw ¼ ð0:76�1:78� 10�4sbÞ � sb For SAE 4340ð Þ ð6bÞ

sw ¼ ð0:74�1:89� 10�4sbÞ � sb For SAE 2340ð Þ ð6cÞ

sw ¼ ð0:92�2:37� 10�4sbÞ � sb For SAE 4063ð Þ ð6dÞ

Eqs. (6a)–(d) are drawn in Fig. 8, it can be seen that thefatigue strengths are basically within the 5% error band. Thisindicates that the quadratic equations can well describe thegeneral relation between fatigue strength and tensile strength ofthose steels.

Second, the fatigue strengths of steels with the minimumsample diameter of 3 mm at a frequency of 20 kHz up to 109

cycles are collected from Refs. [9–20]. Those data are also fittedaccording to the previous method and displayed in Fig. 9 with thefollowing equations,

R¼ 0:67�1:52� 10�4sb, ð7aÞ

sw ¼ 0:67sb�1:52� 10�4s2b ð7bÞ

The most data of fatigue ratios and fatigue strengths are alsowithin the 15% error band, which expresses that the fatiguestrength formula above is also applicable to other steels with avery wide range of tensile strength.

From the analysis above, it is wondering whether the relationabove suits for other steels, even for those non-ferrous metals or

Fig. 7. Fitting relation between tensile strength and rotating bending fatigue ratio of high strength steels: (a) SAE 4140; (b) SAE 4340; (c) SAE 2340 and (d) SAE 4063. (Data

are cited from Ref. [38]).

Fig. 8. Fitting relation between tensile strength and rotating bending fatigue strength of high strength steels: (a) SAE 4140; (b) SAE 4340; (c) SAE 2340 and (d) SAE 4063.

(Data are cited from Ref. [38]).


Fig. 9. (a) Fitting relations (a) between tensile strength and fatigue ratio and (b) relation between tensile strength and VHC fatigue strength of steels with the smallest

diameter of 3 mm. Data are from Ref. [9–20].


not. Thus, many fatigue data of engineering materials, such aswrought steels, aluminum and copper alloys [3,4] were furthercollected. Fig. 10(a), (c) and (e) shows the fitting curves of thefatigue ratios of wrought steels, copper alloys and aluminumalloys available, and the fitting equations can be expressed asfollows, respectively,

R¼ 0:61�1:24� 10�4sb For wrought steelsð Þ, ð8aÞ

R¼ 0:54�3:72� 10�4sb For wrought Cu alloysð Þ ð8bÞ

R¼ 0:53�5:66� 10�4sb For aluminum alloysð Þ ð8cÞ

It can be seen that most data of fatigue ratios are also withinthe 20% error band and the relation between the fatigue ratio andtensile strength of metallic materials still looks like linear. Basedon the results, the fatigue strength sw for the three kinds ofmaterials can be separately fitted by the following quadraticequations:

sw ¼ ð0:61�1:24� 10�4sbÞ � sb For wrought steelsð Þ ð9aÞ

sw ¼ ð0:54�3:72� 10�4sbÞ � sb For wrought Cu alloysð Þ ð9bÞ

sw ¼ ð0:53�5:66� 10�4sbÞ � sb For aluminum alloysð Þ ð9cÞ

In Fig. 10(b), (d) and (f), most fatigue strength sw data arewithin the 20% error band, which tells us that the fatigue strengthformula above is also appropriate for the conventional engineer-ing materials.

Recently, many new high-strength materials have been suc-cessfully developed as mentioned above. We curiously knowwhether the quadratic relation applies to them or not. However,there is almost no fatigue data of the identical materials withvarying tensile strength in a very wide range. Luckily, we havecarried out some fatigue tests on ultrafine grained low carbonsteel [42], and collected some fatigue data of coarse grained[43,44] and ultrafine grained [42,44,45] low-carbon steels andanalyzed them as shown in Fig. 11. The fitting relations betweentensile strength and fatigue ratio as well as fatigue strength canbe written as below:

R¼ 0:60�2:13� 10�4� sb For low carbon steelð Þ, ð10aÞ

sw ¼ ð0:60�2:13� 10�4sbÞ � sb For low carbon steelð Þ ð10bÞ

The data of fatigue ratios and fatigue strengths are also withinthe 10% error band. This indicates that the fatigue strengthformula also fits the low-carbon steel.

After analyzing the fatigue data of four specific materials (SAE4340, 4140, 2340, 4063) and three types of materials (steels, copperalloys, aluminum alloys) the two general conclusions can be drawn:(1) the relation between fatigue ratios and tensile strengths is linear;(2) the relation between fatigue strengths and tensile strengths isquadratic. Therefore, Eqs. (3), (7a), (10a) (5a)–(d) and (8a)–(c) can beexpressed in the general form as below,

R¼ C�P � sb ð11Þ

where C and P are two constants; moreover, each material has itsown constants as listed in Table 3. Furthermore, Eqs. (4), (7b), (10b)(9a)–(c) and (6a)–(d), can be unified in a more universal formula asbelow,

sw ¼ C�P � sbð Þ � sb ð12Þ

4.2. Fatigue damage mechanism of materials

To reveal the reasons caused the results above, it is necessaryto further study the fatigue damage mechanism of the high-strength steel. Since it is inevitable that the real materials containdifferent kinds of defects [2,5], they can be roughly categorizedinto two types: one is the intrinsic defect (ID), such as vacancies,interstitial atoms, dislocations, stacking faults (SF), grain bound-aries (GB) and so on, as illustrated in Fig. 12. The IDs are related tothe materials itself and can not be completely avoided by thetechnology innovation of processing or by controlling the forma-tion of microstructure. The other is the processing defect (PD),including nonmetallic inclusions, cavities, segregation andscratches etc, as shown in Fig. 12. The PDs are associated withthe technology of production or processing and can be avoided inpractice to some extent.

First, for the low-strength ductile materials, fatigue cracksnormally do not nucleate at PDs (inclusion or scratch), but at slipbands (SB) or GBs [46] on the sample surface (see Fig. 12). Thematerials with ID and without PD can be regarded as ideal onesbecause ID mainly controls the microstructure and determinesthe main mechanical properties. Therefore, for the ideal materials,the fatigue strength sw might only depend on its IDs and thecorresponding ideal fatigue strength sI

w should show a propor-tional correlation with its tensile strength sb i.e.,

sIw ¼ C � sb ð13Þ

where, sIw: ideal fatigue strength; C: a constant, which agrees

with the proportional relation in intrinsic flaw regime [21], where‘intrinsic flaw’ is ID mentioned above.

Fig. 10. Fitting relations of the rotating bending fatigue ratio and fatigue strength vs. tensile strength for different engineering materials, (a) and (b) for wrought steels at

106 cycles; (c) and (d) for copper alloys at 108 cycles; (e) and (f) for aluminum alloys at 5�108 cycles. (Data are collected from Ref. [3,4]).


For the grain-refinement strengthening, the smaller grain sizeis, the larger will be the total boundary surface area per unitvolume. Therefore, this can increase the pileup of dislocation andmake the motion of dislocation difficult, resulting in the increaseof tensile strength, as explained by the well-known Hall–Petchrelation. Therefore, the relation between tensile strength andgrain size D�1/2 is linear, which can be expressed as

sbp1ffiffiffiffiDp ð14Þ

For fatigue strength, small sized grains can lead to the reducedflaw sizes and increase difficulties for the imposed stress con-centration at the flaw to exceed the critical stress of the material,thus suppressing early crack nucleation and propagation. Fatigue

strength also follows the Hall–Petch relation [24,25] as blew,

swp1ffiffiffiffiDp ð15Þ

Therefore, the relation between tensile strength and fatiguestrength can be regarded as linear, which can be verified bythe experimental results of some ultrafine-grained materials inRefs. [25,26].

Second, for the high-strength materials, fatigue cracks do notfrequently initiate along SB or GBs and other IDs, but at PDs(inclusion or scratch in Fig. 12). At this time, the fatigue strengthof materials can be considered as PD-controlled, and the effects ofIDs on fatigue strength could be minified to some extent. If thematerial can be considered as an isotropic solid with a crack,hence the fracture mechanics can be applied. By which, Murakami

Fig. 11. (a) Relation between tensile strength and fatigue ratio and (b) relation between tensile strength and fatigue strength: CG [43,44], UFG [42,44,45].

Table 3Values of the constants C, P, M and E and corresponding sbc and smax

w .

Material C P/10�4

MPa�1

M E/GPa sbc/MPa smaxw /MPa Error

band (%)

Our work 0.70 1.85 38.5 208 1891 662 75

SAE 4140 0.87 2.65 54.9 207 1642 714 75

SAE 4340 0.76 1.78 37.0 208 2135 811 75

SAE 2340 0.74 1.89 39.1 207 1958 724 75

SAE 4063 0.92 2.37 49.1 207 1941 893 75

Steel alloya 0.67 1.52 31.5 207 2204 738 715

Steel alloyb 0.61 1.24 25.7 207 2460 750 720

LC Steelc 0.60 2.13 45.3 208 1408 423 710

Cu alloy 0.54 3.72 44.3 119 726 196 720

Al alloy 0.53 5.66 40.6 71.7 468 124 720

a VHC fatigue under tension and compression.b Rotating bending fatigue.c Low carbon steel.

Fig. 12. Schematic illustration of fatigue mechanisms and strength formula with

tensile strength increasing.


et al. [5] found a well-known fatigue strength formula related tothe parameters such as hardness Hv and the inclusion size


as detail below. If the area of a crack at inclusion is denoted by‘area’, then the maximum value, K Imax of the stress intensity factoralong its crack front is given approximately by [47]

K Imax ¼ 0:5sa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiareap

qð16Þ

here, K Imax is the applied stress intensity factor given in MPa m1/2,where sa is the applied stress (MPa) and


is crack size (m).The threshold for crack growth can be written [5]

DKth ¼ 3:3� 10�3 HVþ120ð Þffiffiffiffiffiffiffiffiffiffiareap� �1=3

ð17Þ

where DKth is in MPa m1/2 and HVis in kgf mm�2, andffiffiffiffiffiffiffiffiffiffiareap

is inmm. When K Imax¼DKth, the fatigue strength can be determined.Combining Eqs. (16) and (17), and noticing the difference of theunit of the


between those equations, the fatigue strengthsw can be expressed as below [48],

sw ¼ C HVþ120ð Þ=ffiffiffiffiffiffiffiffiffiffiareap� �1=6

ð18Þ

where, C: material constant relative to crack initiation site.Eq. (18) is widely used to predict the fatigue strength of high-strength steel with inclusion. While, the fatigue strength mea-sured by staircase method is a statistical value; If the specimenswere prepared with the same batch of steel bars but different heattreatments, the average inclusion sizes contained in those speci-mens should be close to each other. Therefore, the inclusion sizeffiffiffiffiffiffiffiffiffiffi

areap

in Eq. (18) can be considered as a constant, thus, Eq. (18)becomes

sw ¼ C 00 Hvþ120ð Þ ð19Þ

where, C00 ¼C0ffiffiffiffiffiffiffi

areapð Þ

1=6. In practice, the hardness Hv and tensile

strength sb show linear relation [2,39]; so that Eq. (19) isapproximately consistent with Eq. (13). It implies that the rela-tion between tensile strength and fatigue strength of one mate-rial, no matter whether in low-strength level or in high-strengthlevel, is still linear in theory.

However, a question is raised as below. For our fatigue test, thespecimens A to E were prepared with the same batch of steel barsbut different heat treatments, the inclusion size contained inthose specimens should be the same in the statistical meaning.Therefore, the influence of the inclusion size on fatigue strengthof those specimens could be neglected. According to Eqs. (13) or(19), the fatigue strength should be proportional to the tensilestrength; however, it is inconsistent with the experiment resultsas mentioned above. The reason may be as follows. By analyzingof the relation between DKth and tensile strength sb of 300 Msteel in Ref. [49], as shown Fig. 13, it is found that DKth linearlydecreases with tensile strength sb increasing. The linear relationcan be written in the form like:

DKthpC1�C2 � sb ð20Þ

Fig. 13. Relation between threshold stress intensity range DKth at R¼0.05 and

tensile strength of 300 M steel tempered by different temperatures, adapted from

Ref. [49].


where, C1 and C2 are material constants. The decrease of DKth canbe also attributed to the enhancement of defect sensitivity withtensile strength increasing.

Now, the resistance DKth for fatigue crack growth near aninclusion in steels with a wide strength range should be modified.Here, as an assumption, the DKth in Eq. (17) is modified as

DKth ¼ C HVþ120ð Þffiffiffiffiffiffiffiffiffiffiareap� �1=3

C01�C02sb

� �ð21Þ

Combining Eqs. (16) and (21), the fatigue strength formula canbe written as:

sw ¼C00 Hvþ120ð Þ C01�C02 � sb

� �ffiffiffiffiffiffiffiffiffiffiareap� �1=6

ð22Þ

Considering the inclusion size as a constant in average in allfatigue tests, and the linear relation between tensile strength sb

and hardness Hv as mentioned above, Eq. (22) can be furthersimplified as Eq. (12), i.e., sw¼(C�P�sb)�sb. This indicates thatto a first approximation, the quadratic relation can be explainedby fracture mechanics. As for the fatigue damage mechanisms, itmay be illustrated in outline as follows, thus, Eq. (12) can berewritten as

sw ¼ C�P � sbð Þ � sb ¼ Csb�Ps2b ¼ s

Iwþs

Pw ð23Þ

From the analysis above, it is known that the first term on theright side of Eq. (23), i.e., Eq. (13) as mentioned above, is the idealfatigue strength, and it is valid no matter whether in low-strengthlevel or in high-strength level, and the parameter C is named asintrinsic factor due to the influence of the intrinsic defects. On theother hand, the second term in Eq. (23) can be expressed by,

sPw ¼�P � s2

b ð24Þ

This indicates that the reduction of fatigue strength due to thecompetitive effects of ID and PD, and the parameter P is named asthe fatigue sensitivity coefficient to processing defects.

In fact, a real material contains both IDs and PDs. At low-strength level, sP

w is so small that the influence of PD on fatiguestrength can be completely negligible and fatigue cracks appear atSBs or GBs (see Fig. 12); and Eq. (23) becomes equivalent toEq. (13), i.e., sw � sI

w ¼ C � sb, which can be called as ID regime inFig. 12. With increasing the tensile strength, sP

w becomes gradu-ally large (violet line in Fig. 12) and can not be negligible.However, the influence of ID on the fatigue strength is stilldominative, fatigue cracks will emerge from SB, GB, scratchesand so on, the quadratic Eqs. (12) or (23) can be approximatelyexpressed as Eq. (2), i.e., sw¼msb, it is similar to Eq. (13) that iscontrolled completely by IDs. However, the coefficient m shouldbe slightly smaller than the constant C in Eq. (13) due to the

influence of PDs. Eq. (2) has been widely applied to mostengineering materials [2–5]. This can be named as main ID regime(see Fig. 12). With further increasing the tensile strength, in thiscase, sP

w becomes so large (violet line in Fig. 12) that the influenceof PD on the fatigue strength is significant; therefore, in thistensile strength level it is defined as main PD regime (see Fig. 12).In this regime, the fatigue cracks often nucleate at scratches,surface or subsurface inclusion (SI), internal inclusion (II), andeven that crack initiation at internal inclusion becomes the mainevent in those high-strength steels [5,6] with tensile strengthincreasing (main PD regime, light green zone in Fig. 12). Thefatigue strength is significantly affected by the size of inclusions[5,7]. If the inclusion size is very different, the fatigue data will bevery scatter and Eq. (18) can suit well.

When the tensile strength reaches very high level, nearly allfatigue cracks originate from PDs (inner inclusion, segregation), sP

w

becomes very large, which can be called as PD regime shown inFig. 12. As tensile strength increasing, the critical inclusion size [50]of steels becomes smaller and smaller, more and more inclusionscan become fatigue crack initiation sites and the fatigue sensitivityto defects is much more serious [5]. Therefore, fatigue strengthdecreases as tensile strength further increases.

4.3. Prediction of the maximum fatigue strength

If introducing Young’s modulus E of materials into Eq. (24),one can get

sPw ¼�M � s2

b=E¼�M � sb � eb ð25Þ

where M¼P� E, eb can be defined as the maximum tensile elasticstrain. From Eq. (12), the values of the maximum fatigue strengthsmax

w and critical tensile strength sbc can be obtained as follows:

smaxw ¼

C2

4P¼

EC2

4Mð26Þ

sbc ¼C

2P¼

EC

2Mð27Þ

From Eqs. (26) and (27), it can be concluded that smaxw and sbc

are dependent on C, and P or E, C and M. According to their values,the corresponding smax

w and sbc can be calculated, as listed inTable 3. Compared with the error bands in Table 3, the errorbands of one type of materials such as steels, copper or aluminumalloys are within 20%, which is the maximum. The error bands ofmany steels under the same experimental condition are within15%; and within 10% for low-carbon steel, with 5% for a specificmaterials (SAE 4140, 4340, 2340 and 4063), which is minimum.This indicates that the less are the influencing factors of fatigueexperiments, the more accurate the formula predicts. Besides, it iswell known that the newly developed high-strength materialsoften possess very poor fatigue properties [27–32,41], which canbe explained that their tensile strengths are already greater thantheir sbc, as illustrated in Fig. 12.

For the quadratic equation, the fatigue strength and themaximum fatigue strength can be designed or improved byadjusting the constants P (M) and C. For example, with surfaceroughness of specimens increasing, the fatigue strength continu-ously decreases [4]; with the size of inclusion increasing, fatiguestrength continuously decreases [5,7], which will affect theconstant P (M). With refinement of grain size, the fatigue strengthalso increases [24], which should make a contribution to constantC. Therefore, the universal fatigue formula may not only correctlypredict the fatigue strength, but also provide a possible clue toimprove the fatigue strength by choosing and designing micro-structure or eliminating the processing defects of the materials inindustry.


Notice that the universal fatigue strength formula above isproposed on the basis of smooth specimens during tension-compression fatigue tests under the stress ratio of R¼�1.Actually, there are some other factors affecting the fatiguestrength, including surface condition, specimen size, loadingmode and so on. Small changes of those factors can also affectthe fatigue properties, making analytical prediction of fatigue lifeor strength difficult. Therefore, it is necessary to quantify theeffects of those factors on the fatigue strength in the future.

5. Summary and conclusions

After a systemic study on the fatigue strength sw of SAE 4340steel tempered at different tempering temperatures, a universalfatigue strength formula sw¼sb(C�P�sb) is firstly proposed bytaking into account the constants C and P. Through inspection ofnumerous data available of other materials such as steels, copperand aluminum alloys, it is shown that this universal fatiguestrength formula can well describe the fatigue strength sw ofdifferent materials with increasing their tensile strengths sb.

Combining the variation tendency of fatigue crack sites and thecompetition of defects, the fatigue damage mechanisms with thechange of tensile strength are explained well. The decrease in thefatigue strength at high-strength level can be explained by fracturemechanics and attributed to the transition of fatigue cracking sitesfrom surface to the inner inclusions, resulting in the maximumfatigue strength smax

w at an adequate tensile strength level.In addition, the maximum fatigue strength smax

w and thecritical tensile strength sbc can be calculated and the universalfatigue strength formula can explain why many metallic materialswith excessively high strength do not display high fatiguestrength. The current universal fatigue strength formula mayprovide a new clue or principle to achieve the higher fatiguestrength sw of materials by adjusting the constants P and C in thefuture.

Acknowledgements

The authors would like to thank Mrs. C. L. Dai and Dr. H. F. Zoufor their helps of the SEM and EBSD observations. This work issupported by National Natural Science Foundation of China(NSFC) under Grant Nos. 50625103, 50890173, 50931005 andthe National Basic Research Program of China under grant No.2012CB631006.

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General Relation Between Tensile Strength and Fatigue Strength of Metallic Materials - J.C. Pang