Strain-controlled fatigue properties of steels and … fatigue properties of steels and some simple...

International Journal of Fatigue 22 (2000) 495–511www.elsevier.com/locate/ijfatigue

Strain-controlled fatigue properties of steels and some simpleapproximations

M.L. Roessle1, A. Fatemi*

Department of Mechanical, Industrial, and Manufacturing Engineering, The University of Toledo, Toledo, Ohio 43606, USA

Received 20 February 1999; received in revised form 5 August 1999; accepted 4 February 2000

Abstract

In this study, first strain-controlled deformation and fatigue data are reported and compared for several steels most commonlyused in the ground vehicle industry. Correlations between monotonic tensile data and constant amplitude strain-controlled fatigueproperties are then investigated, and validity of some of the more commonly used methods of estimating fatigue properties isexamined. A simple method requiring only hardness and modulus of elasticity is proposed for estimation of the strain–life curve.Prediction capability of this method is evaluated for steels with hardness between 150 and 700HB and compared with severalother methods proposed in the literature. The proposed method is shown to provide good approximations of the strain–life curve. 2000 Elsevier Science Ltd. All rights reserved.

Keywords:Strain-controlled fatigue; Fatigue property correlations; Strain-life approximations; Cyclic deformation; Low-cycle fatigue; Fatiguelife prediction

1. Introduction

Mechanical properties determined from monotonictensile tests are often used in the design of components.However, in service, most components experience cyclicloading and the fatigue properties of the materials are,therefore, of utmost importance to design engineers.Over the years, many researchers have attempted todevelop correlations among the monotonic tensile dataand fatigue properties of materials. Such correlations aredesirable, considering the amount of time and effortrequired to obtain the fatigue properties, as compared tothe monotonic tensile properties. If reliable correlationswith reasonable accuracy can be established, durabilityperformance predictions and/or optimization analysescan be performed, while substantially reducing time andcost associated with material fatigue testing.

The traditional approach to predict crack initiation ornucleation life is based on nominal stress, referred to as

* Corresponding author. Tel.:+1-419-530-8213; fax:+1-419-530-8206.

E-mail address:[email protected] (A. Fatemi).1 Formerly Research Assistant, Currently at Tenneco Automotive.

0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0142-1123 (00)00026-8

the S–N approach. The relationship (often referred to asBasquin’s equation) can be represented as:

DS/25sf9(2Nf)b (1)

whereDS/2 is the stress amplitude and 2Nf is the numberof reversals to failure. The two fatigue properties neededin this equation are the fatigue strength coefficient,sf9,and the fatigue strength exponent,b. A second commonand more recent approach is based on local strain,referred to as the strain–life (e–N) method. Thisapproach relates the reversals to failure, 2Nf, to the strainamplitude,De/2. The relationship (often referred to asthe Coffin–Manson relationship) is as follows:

De2

5Dee2

1Dep2

5s9

f

E(2Nf)b1e9f(2Nf)c (2)

whereDe/2, Dee/2, andDep/2 are the total, elastic, andplastic strain amplitudes, respectively. In this equation,the two additional fatigue properties needed are thefatigue ductility coefficient,e9f, and the fatigue ductilityexponent,c. The strain-based approach considers theplastic deformation that often occurs in localized regionswhere cracks nucleate. Also, the strain-based approachmay be regarded as a comprehensive approach describ-ing both elastic and inelastic cyclic behavior of a

496 M.L. Roessle, A. Fatemi / International Journal of Fatigue 22 (2000) 495–511

material. This approach is now commonly used infatigue design, particularly in the ground vehicle indus-try.

This study was undertaken by the American Iron andSteel Institute (AISI) to obtain and compare strain-con-trolled deformation and fatigue properties for 20 steelscommonly used in the ground vehicle industry. Thesedata have not been published previously. The data fromthese steels in addition to 49 other steels published bythe American Society of Metals (ASM) in [1] were thenused to examine correlations among the various mono-tonic and fatigue properties. These include relationshipsamong ultimate tensile strength, fatigue limit, fatiguestrength coefficient, and hardness, as well as estimationof strain–life fatigue properties and approximation ofstrain–life curves.

2. Experimental program

2.1. Materials

The materials used in the experimental investigationconsisted of SAE 1141, SAE 1038, SAE 1541, SAE1050, and SAE 1090 steels. The SAE 1141 steels areresulfurized steels with medium carbon content and havegrain refinement additives. The SAE 1038, SAE 1050,and SAE 1541 steels are plain carbon steels withmedium carbon content. For SAE 1038 and SAE 1050,the maximum allowable amount for Mn is 1.00%,whereas for SAE 1541 the maximum range for Mn isbetween 1.00% and 1.65%. The SAE 1090 steels areplain carbon steels with high carbon content. The overallrange of ultimate tensile strength,Su, was 582 MPa to2360 MPa and the overall range of Brinell hardness,HB,was 163 to 536. Table 1 provides a summary of thematerials used, including material identification, pro-cessing condition, grain structure and size, and hardness.The average grain size is represented by an ASTM grainsize number and is inversely proportional to the averagegrain diameter. Anisotropy was minimal, if any, for allmaterials, as judged by similarities in the transverse andlongitudinal average grain sizes and respective hard-ness values.

2.2. Specimen

Identical round specimens were used for all tests. Thespecimen configuration and dimensions are shown inFig. 1. The secondary radius in the gage section causesthe minimum diameter to occur in the middle of the testsection and compensates for the slight stress concen-tration that occurs at the shoulders. The stress concen-tration factor due to this secondary radius in the gagesection is very small,Kt=1.017. The specimens weremachined from the longitudinal direction (i.e. rolling

direction) of the raw material. A commercial round-specimen polishing machine was used to polish thespecimen gage section with polishing marks coincidingwith the specimen’s longitudinal direction. The polishingprocedure produced a mirror-like surface finish withinthe test section for all the materials tested.

2.3. Test equipment and procedures

Servo-controlled hydraulic axial load frames in con-junction with digital controllers were used to conductthe tests. Hydraulically operated grips using universaltapered collets were employed to secure the specimenends in series with the load cell. Epoxy was used to pro-tect the specimen surface from the knife-edges of theextensometer. Significant effort was put forth to alignthe load train (load cell, grips, specimen, and actuator).Misalignment can result from both tilt and offsetbetween the central lines of the load train components.ASTM Standard E1012 [2] was followed to verify speci-men alignment. All tests were conducted at room tem-perature.

The monotonic tension tests were performed using testmethods specified by the ASTM Standard E8 [3]. Afterthe tension tests were concluded, the broken specimenswere reassembled and the final gage length, the finaldiameter, and the neck radius were measured.

All constant amplitude fatigue tests were performedaccording to the ASTM Standard E606 [4]. For eachmaterial tested, 18 specimens at 6 different strain ampli-tudes ranging between 0.15% to 1.5% (with 3 duplicatetests for each strain amplitude) were utilized. All testswere conducted under completely reversed strain con-ditions, R=emin/emax=21. Failure of the specimens wasdefined as the point at which the maximum loaddecreased by approximately 50% because of a crack orcracks being present, as recommended by the ASTMStandard E606 [4]. Strain control was used in all tests,except for long life and run-out tests due to limitationsof the extensometer at high frequencies. For these tests(greater than 105 cycles), strain control was used initiallyto determine the stabilized load. Then load control wasused for the remainder of the test. For the strain-con-trolled tests the applied frequencies ranged between 0.1Hz to 3.5 Hz, with approximately a constant strain rate.For the load-controlled tests the frequency was increasedup to 35 Hz or higher in order to shorten the overall testduration. All tests were conducted using a triangularwaveform.

3. Experimental results

3.1. Monotonic and cyclic deformation behaviors andproperties

The properties determined from the monotonic tensiletests were modulus of elasticity,E, yield strength at

497M.L. Roessle, A. Fatemi / International Journal of Fatigue 22 (2000) 495–511

Table 1Summary of materials

Materiala Material ID Processing condition Grain type ASTM grain size Brinell hardness

SAE 1141 (AlFG) A1 Normalized at 1650°F ferrite/pearlite 10 to 11 223SAE 1141 (AlFG) A2 Reheat, Q and T martensite NA 277SAE 1141 (NbFG) A3 Normalized at 1650°F ferrite/pearlite 7 to 9 199SAE 1141 (NbFG) A4 Reheat, Q and T martensite NA 241SAE 1141 (VFG) A5 Normalized at 1650°F ferrite/pearlite 6 to 8 217SAE 1141 (VFG) A6 Reheat, Q and T martensite NA 252SAE 1141 (VFG) A7 Normalized at 1750°F ferrite/pearlite 6 to 8 229SAE 1038 B1 Normalized at 1650°F ferrite/pearlite 8 to 9 163SAE 1038 B2 Cold size/form ferrite/pearlite 8 to 10 185SAE 1038 B3 Reheat, Q and T ferrite/spher. pearlite 10 to 11 195SAE 1541 C1 Normalized at 1650°F ferrite/pearlite 10 180SAE 1541 C2 Cold size/form ferrite/pearlite 10 195SAE 1050(M) D1 Normalized at 1650°F ferrite/pearlite 9 to 9. 205SAE 1050(M) D2 Hot forge, cold extrude ferrite 7.2 (ferrite) 220SAE 1050(M) D3 Induction through- martensite NA 536

hardenedSAE 1090 E1 Normalized at 1650°F pearlite NA 259SAE 1090(M) E2 Hot form, accelerated cool pearlite/martensite NA 357SAE 1090 E3 Hot form, Q and T martensite NA 309SAE 1090 E4 Hot form, austemper martensite/bainite NA 279SAE 1090(M) E5 Hot form, accelerated cool pearlite/martensite NA 272

a AlFG: aluminum fine grain; NbFG: niobium fine grain; VFG: vanadium fine grain; M: modified chemical composition.

Fig. 1. Specimen configuration and dimensions (courtesy of Dr T.Topper).

0.2% offset,YS, ultimate tensile strength,Su, percentelongation, %EL, percent reduction in area, %RA, truefracture strength,sf, true fracture ductility,ef, strengthcoefficient,K, and strain hardening exponent,n. The truefracture strength was corrected for necking according tothe Bridgman correction factor [5]. The true fractureductility was calculated from the relationship based onconstant volume assumption:

ef5lnSA0

AfD5lnS 100

100−%RAD (3)

whereAf is the cross-sectional area at fracture andA0 isthe original cross-sectional area. Values ofK andn wereobtained according to the ASTM Standard E646 [6] froma least squares fit of true stress versus true plastic strain

data in log–log scale. A summary of the monotonic ten-sile properties is provided in Table 2.

Midlife hysteresis loop data from the fatigue testswere used to determine the steady-state or stable cyclicdeformation properties. These consisted of the cyclicstrength coefficient, K9, cyclic strain hardeningexponent,n9, and cyclic yield strength,YS9. Values ofK9 andn9 were obtained from a least squares fit of truestress amplitude,Ds/2, versus true plastic strain ampli-tude,Dep/2, data in log–log scale. These properties areoften used to characterize the cyclic deformation curveby a Ramberg–Osgood type equation:

De2

5Dee2

1Dep2

5Ds2E

1SDs2K 9D1/n9

(4)

The cyclic yield strength is analogous to the monotonicyield strength at 0.2% offset.

The cyclic stress–strain curve can be vastly differentfrom the monotonic stress–strain curve. A superimposedplot of the monotonic and cyclic stress–strain curves forSAE 1090 steels (material ID: E) is provided in Fig. 2.As shown in this figure, these high strength steels allcyclic soften. The differences between the cyclic curvesare, however, less than the corresponding monotoniccurves. The deformation curves of the 1090 steel seriesare shown since several of these steels exhibit the bestoverall fatigue resistance, as discussed in the next sec-tion. A summary of the cyclic deformation properties ofall materials tested is included in Table 2.


Tab

le2

Sum

mar

yof

mon

oton

icte

nsile

and

stra

in-c

ontr

olle

dde

form

atio

nan

dfa

tigue

prop

ertie

s

YS/

YS9

K/K

9s

f/sf9

Mat

eria

lID

E(G

Pa)

S u(M

Pa)

%E

L%

RA

n/n9

e f/e

f9b

cS

f(M

Pa)

N t(c

ycle

s)(M

pa/M

Pa)

(MP

a/M

Pa)

(MP

a/M

Pa)

A1

216

457/

424

771

40%

57%

1394

/151

50.

216/

0.20

512

07/1

168

0.85

/0.2

5720.

097

20.

464

286

1876

4A

222

781

4/59

192

532

%59

%12

05/1

277

0.07

4/0.

124

1405

/112

70.

88/0

.30920.

066

20.

514

433

5034

A3

220

418/

405

695

35%

53%

1287

/144

80.

217/

0.20

599

9/11

170.

76/0

.2642

0.09

62

0.46

227

624

439

A4

217

602/

481

802

32%

54%

1199

/125

40.

126/

0.15

412

28/1

080

0.77

/0.3

6120.

079

20.

508

342

1084

5A

521

445

0/44

772

532

%49

%13

21/1

467

0.20

7/0.

191

1087

/125

50.

68/0

.43020.

102

20.

529

287

1160

6A

621

561

0/48

779

734

%58

%12

44/1

270

0.14

1/0.

154

1243

/116

20.

88/0

.53420.

086

20.

555

332

9045

A7

220

493/

481

789

30%

47%

1379

/144

10.

187/

0.17

711

17/1

326

0.64

/0.6

0220.

103

20.

581

296

7608

B1

201

331/

342

582

44%

54%

1106

/134

00.

259/

0.22

089

8/10

430.

77/0

.30920.

107

20.

481

222

2724

8B

221

935

9/35

865

238

%53

%11

86/1

420

0.21

9/0.

222

1051

/100

40.

76/0

.20220.

098

20.

440

241

3278

2B

321

941

0/36

464

945

%67

%11

83/1

330

0.22

1/0.

208

1197

/100

91.

10/0

.22520.

097

20.

460

248

3148

5C

120

547

5/42

478

339

%55

%15

76/1

416

0.23

5/0.

194

/162

20.

80/0

.5152

0.13

52

0.54

822

812

496

C2

205

475/

469

906

29%

42%

1924

/950

0.20

4/0.

114

/104

40.

54/0

.5132

0.08

32

0.55

731

583

78D

121

146

5/42

782

143

%50

%18

19/1

673

0.27

4/0.

220

/989

0.68

/0.4

3320.

126

20.

512

159

6254

8D

220

346

0/52

382

918

%34

%13

13/1

292

0.16

3/0.

146

/109

40.

42/0

.3092

0.07

52

0.50

236

965

13D

320

320

00/1

796

2360

17%

15%

2837

/353

80.

048/

0.10

9/3

492

0.16

/1.8

7020.

109

21.

040

717

77E

120

373

5/54

510

909%

14%

1765

/161

10.

158/

0.17

4/1

310

0.15

/0.2

5020.

091

20.

496

350

4199

E2

203

950/

730

1388

14%

25%

1980

/166

30.

080/

0.13

3/1

945

0.29

/2.5

8020.

106

20.

777

417

2104

E3

217

650/

627

1147

67%

22%

1895

/187

30.

165/

0.17

6/1

878

0.24

/0.7

0020.

120

20.

600

328

4739

E4

203

760/

645

1251

7%14

%27

57/1

835

0.26

4/0.

168

/192

80.

15/0

.7342

0.12

02

0.64

233

720

90E

520

376

5/61

511

2418

%38

%15

35/1

653

0.10

0/0.

159

/154

70.

47/1

.5702

0.09

32

0.68

340

142

24


Fig. 2. Monotonic and cyclic stress–strain curves for SAE 1141 steels.

3.2. Strain-controlled fatigue behavior and properties

The fatigue strength coefficient,s9f, and exponent,b,

were obtained from a least squares fit of true stressamplitude,Ds/2, versus reversals to failure, 2Nf, data inlog–log scale. The fatigue ductility coefficient,e9f, andexponent,c, were obtained from a similar plot of thetrue plastic strain amplitude,Dep/2, versus reversals tofailure, 2Nf. A typical strain–life curve is shown in Fig.3. It should be noted that significant variations in theseproperties for a given material can exist, depending onthe curve-fitting techniques employed. These variationsmainly result from the method of determining elastic andplastic strains, sensitivity of the fittings and obtainedproperties to the value of modulus of elasticity used, andselection of dependent versus independent variables inperforming least squares fits. Elastic and plastic strainscan either be directly measured from the hysteresisloops, or calculated from Dee/2=Ds/2E andDep/2=De/22Ds/2E. The elastic modulus used for thecalculations can either be measured from monotonic ten-sion tests, or from the hysteresis loops in fatigue tests.In fitting the experimental data to obtain material cyclicproperties, fatigue life should be treated as the dependentvariable (i.e.y value in anx–y plot of spreadsheet data).The variability in properties based on the different afore-mentioned curve-fitting techniques is discussed in [7].

A superimposed plot of strain–life curves for allmaterials is provided in Fig. 4. Due to the large numberof curves in this plot, references for the end points areprovided on the left and right sides of the strain–lifecurves. For the low cycle fatigue region the hierarchicalorder of the end point references indicates whichmaterials are able to endure more plastic deformation.

Conversely, for the high cycle fatigue region the hier-archical order of the end point references indicates whichmaterials are able to endure more elastic deformation.Therefore, materials that occur near the top of both thesereference lists are considered “tough” materials becausethey excel in both life regions. For example, the SAE1090(M) steels (E2 and E5 steels) withpearlite/martensite microstructure, have ultimate tensilestrengths of 1388 MPa and 1124 MPa, respectively, andductility as measured by percent reduction in area of25% and 38%, respectively. These steels are, therefore,tough steels and show superior low cycle as well as highcycle fatigue resistances.

A variation of the strain–life plot can be used to evalu-ate the notched fatigue resistance of materials. The mostwidely used method for estimating notch stress andstrain is Neuber’s rule. The general form of this relation-ship for cyclic loading is given by:

!SDs2 DSDe

2 DE5KtSDS2 D (5)

whereDs/2 andDe/2 are the notch root stress and strainamplitudes,DS/2 is the nominal stress amplitude, andKt

is the elastic stress concentration factor. Neuber’s para-meter, which is represented by the left side of Eq. (5),is plotted in Fig. 5 for all materials. The hierarchicalorder of the end point references in Fig. 5 indicates thenotched fatigue resistance of the materials for the differ-ent life regions. Materials that occur near the top of bothreference lists indicate high notched fatigue resistanceover the entire life range.

Representative macroscopic fatigue crack growth andfinal fracture surface features can be seen from Fig. 6


Fig. 3. Example of true strain amplitude vs. reversals to failure (SAE 1141 VFG steel).

Fig. 4. Superimposed plot of true strain amplitude vs. reversals to failure for all materials.


Fig. 5. Superimposed plot of Neuber’s parameter vs. reversals to failure for all materials.

for the Q and T A2 steel (Fig. 6(a) and (b)) and thenormalized A3 steel (Fig. 6(c) and (d)). Figure 6 (a) and(c) are from low cycle fatigue (LCF) tests, whereas Fig.6(b) and (d) are from high cycle fatigue (HCF) tests.The final fracture surfaces represented by the darkerregions in each figure are rougher than the relativelysmooth fatigue crack growth regions (lighter colorregions). Also, LCF crack growth regions (Fig. 6(a) and(c)) are somewhat rougher than HCF crack growthregions (Fig. 6(b) and (d)), indicating higher crackgrowth rates. Fatigue crack growth regions in HCF testsoccupy a larger portion of the fracture surface, comparedto fracture surfaces from LCF tests, indicating longercracks sustained at lower loads. “Beach marks” areabsent from the fracture surfaces, indicating uniformstraining throughout the cracking phase. A “sunrise” pat-tern can be seen in Fig. 6(b), as the crack grew onslightly different levels or planes, simultaneously. Thecenter of the rays point to the crack nucleation site. Typi-cal microscopic features of the fracture surfaces can beseen from the fractographs shown in Fig. 7 for the Qand T A2 steel (Fig. 7(a)) and the normalized A3 steel(Fig. 7(b)). These SEM fractographs show the fatiguecrack growth region, indicating cracking and de-cohesion of inclusions. Both materials sustained largeplastic deformations, resulting in ductile fracture.

Table 2 includes a summary of the strain-controlledfatigue properties. As may be seen in this table, thefatigue strength exponent,b, ranged from20.066 to20.135, with an average value of20.10, while thefatigue ductility exponent,c, ranged from20.44 to

21.04, with an average value of20.57. The fatiguelimit, Sf, and the transition fatigue life,Nt, are also givenin Table 2. The fatigue limit was calculated from Eq. (1),usingNf=106 cycles. The transition fatigue life indicateswhen a material will experience equal amounts of elasticand plastic strains, as shown in Fig. 3. It can be obtainedby setting equal the elastic strain amplitude,Dee/2=s9

f/E(2Nf)b, and the plastic strain amplitude,Dep/2=e9f(2Nf)c,resulting in the following equation:

Nt512S s9

f

e9fED1/(c−b)

. (6)

4. Correlations among tensile data and fatigueproperties

As mentioned previously, it is often desirable to esti-mate fatigue behavior of a material from easily andquickly obtainable material properties such as hardnessand tensile data, with a reasonable degree of accuracy.Therefore, many correlations among the monotonic ten-sile data and fatigue properties of materials have beenproposed. In this section, the data presented in the pre-vious section are used to evaluate some of the more com-monly used correlations, as well as to develop a simplemethod for estimating material strain–life curve. Inaddition to the 20 steels from this study, data from 49other steels were selected from the American Society for


Fig. 6. Representative macroscopic fractographs at 15×.

Metals (ASM) reference book [1] to evaluate corre-lations among the monotonic and fatigue properties.These steels include plain carbon steels, resulfurized car-bon steels, chromium–molybdenum steels, nickel–chro-mium–molybdenum steels, chromium steels, silicon–manganese steels, high strength low-alloy steels, andalloy steel pressure vessel plate steels. By using a totalof 69 steels, Brinell hardness,HB, ranged from 80 to660, ultimate tensile strength,Su, ranged from 345 MPato 2585 MPa, and percent reduction in area, %RA,ranged from 11% to 80%. Therefore, a broad range ofsteels is used for the correlations.

4.1. Correlations among tensile strength, hardness,fatigue limit, and transition life

A commonly used approximation of the ultimate ten-sile strength,Su, from Brinell hardness,HB, for low andmedium strength carbon and alloy steels is representedby a linear relationship as:

Su>3.45HB (MPa) (7)

A plot of ultimate tensile strength vs. Brinell hardnessis provided in Fig. 8. It may be seen from this figurethat the approximation from Eq. (7) agrees well withexperimental data forHB,350. A least squares fit usinga second-order polynomial results in the following corre-lation with R2=0.96:

Su50.0012(HB)213.3(HB) (MPa) (8)

For steels with hardness less than 500 HB, the fatiguelimit, Sf, has been suggested to be estimated from theBrinell hardness as follows [8]:

Sf>1.72HB (MPa) (9)

A plot of the fatigue limit vs. Brinell hardness is pro-vided in Fig. 9. As can be seen, there is a poor agreementbetween the data and the predictions obtained from Eq.(9). A linear least squares fit through the data withR2=0.91 results in the following relationship:


Fig. 7. Representative SEM fractographs. (a) SAE 1141 Q&T steel(A2) fatigue fracture surfaceea=1.5% at 500X. (b) SAE 1141 nor-malized steel (A3) fatigue fracture surface (ea=0.23%) at 1000X.

Sf51.43HB (MPa) (10)

The fatigue limit can also be estimated from the ultimatetensile strength. An approximation that is often used forsteels is as follows:

Sf>0.5Su for Su#1400 MPa

Sf>700 MPa forSu.1400 MPa(11)

The reason for a constant value of fatigue limit for steelswith Su.1400 MPa is thought to be due to the role ofinclusions [9]. At such high strengths, fatigue limitbecomes more dependent on the size and dispersion ofinclusions and other impurities. In Fig. 10, a plot of thefatigue limit vs. ultimate tensile strength is provided. Itcan be seen from this figure that there is a large amountof scatter in the data and that predictions based on Eq.(11) have poor agreement with the data and are noncon-

servative for most of the steels. It should be mentionedthat traditionally most fatigue limit data are obtainedfrom rotating bending fatigue tests, whereas here thevalues are from axial fatigue tests. In axial fatigue tests,lower strengths are often obtained due to reasons suchas higher probability of defects associated with largerstressed volume and bending resulting from specimenmisalignment. A fit through all of the data in Fig. 10and a fit through data havingSu#1400 MPa were foundto be equivalent. Therefore, a correlation withR2=0.86was obtained as follows:

Sf50.38Su (12)

The transition fatigue life,Nt, indicates when amaterial will experience equal amounts of cyclic elasticand plastic strains. Knowing the transition life for amaterial allows plastic and elastic strain dominated liferegimes to be identified and the appropriate life predic-tion approach to be selected accordingly. Landgraf [10]has shown that there is a dependence of transition lifeon Brinell hardness for steels. Since hardness variesinversely with ductility, the transition life decreases asthe hardness increases. The relationship shown byLandgraf may be expressed as follows:

log(2Nt)56.12620.0083HB (13)

A semi-log plot of transition life vs. Brinell hardness isshown in Fig. 11. It may be seen from this figure thatthe correlation obtained from a least squares fit of thedata is in close agreement with predictions from Eq.(13). This correlation withR2=0.89 is given by:

log(2Nt)55.75520.0071HB. (14)

4.2. Approximations of strain–life properties

The fatigue strength coefficient,s9f, is analogous to the

true fracture strength,sf, obtained from a tensile test.Several relations for estimating fatigue strength coef-ficient from true fracture strength have been proposed,with estimates ranging from 0.92sf to 1.15sf. As canbe seen from Table 2, there is no strong correlationbetweens9

f andsf. The fatigue strength coefficient wasfound to be more dependent upon the Brinell hardnessand ultimate tensile strength of the steel. Plots of fatiguestrength coefficient vs. Brinell hardness and ultimate ten-sile strength are shown in Fig. 12 and Fig. 13, respect-ively. These figures show relatively good linear leastsquares fits for the data, represented by:

s9f54.25HB1225 (MPa) (15)

s9f51.04Su1345 (MPa) (16)

The approximation from the literature shown in Fig. 13


Fig. 8. Ultimate tensile strength vs. Brinell hardness.

Fig. 9. Fatigue limit vs. Brinell hardness.

is in very close agreement with the linear fit representedby Eq. (16).

The fatigue ductility coefficient,e9f, is thought to beof the same order as the true fracture ductility,ef,obtained from a tensile test. The proposed relations inthe literature predict the fatigue ductility coefficient to

be from 0.15ef to 1.1 ef. As can be seen from Fig. 14,there is no clear correlation betweene9f andef. Therefore,using the true fracture ductility to approximate thefatigue ductility coefficient can result in significant error.

As shown by Morrow [11] through energy arguments,a relationship can be derived between the cyclic strain


Fig. 10. Fatigue limit vs. ultimate tensile strength.

Fig. 11. Transition life vs. Brinell hardness.

hardening exponent,n9, and the fatigue strength and duc-tility exponents (b andc). The relationships that followwere developed from a variety of metals and illustratedusing data for SAE 4340 steel [11]:

b5−n9

1+5n9c5

−11+5n9

(17)

These equations, however, resulted in poor agreements


Fig. 12. Fatigue strength coefficient vs. Brinell hardness.

Fig. 13. Fatigue strength coefficient vs. ultimate tensile strength.

with the data from the 20 steels in this study. Values ofn9 for the ASM data were not available. The fatiguestrength exponent,b, ranged from20.057 to 20.140

with an average value of20.09, and the fatigue ductilityexponent,c, ranged from20.39 to21.04 with an aver-age value of20.60.


Fig. 14. Fatigue ductility coefficient vs. true fracture ductility.

4.3. Approximations of the strain–life curve fromtensile properties

Several researchers have developed empiricalrelations to predict strain–life fatigue behavior usingmonotonic tensile properties. Some of these relationsprovide good approximations for a variety of materials.In a study conducted by Park and Song [12], six suchmethods were evaluated and compared. These consistedof the Universal Slopes and the Four-Point Correlationmethods by Manson [13], the Modified Universal Slopesmethod by Muralidharan and Manson [14], the UniformMaterial method by Ba¨umel and Seeger [15], the Modi-fied Four-Point Correlation method by Ong [16], and themethod proposed by Mitchell et al. [8]. A total of 138materials were used in the study including unalloyedsteels, low-alloy steels, high-alloy steels, aluminumalloys, and titanium alloys, with low-alloy steels provid-ing the most data. Amongst the correlations compared,those proposed by Muralidharan and Manson [14], Ba¨u-mel and Seeger [15], and Ong [16] yielded good predic-tions according to Park and Song. The Modified Univer-sal Slopes method was concluded to provide the bestcorrelation. This method is given by:

De2

50.623SSu

ED0.832

(2Nf)−0.09 (18)

10.0196(ef)0.155SSu

ED−0.53

(2Nf)−0.56

Another study that compared correlations for estimat-ing fatigue properties from monotonic tensile data is byOng [17]. In that study, the ASM data for 49 steels wereused and the correlations compared were the modifiedand original versions of the Four-Point Correlationmethod, the Original Universal Slopes method, and theMitchell et al. method. The best predictions were shownto result from the Modified Four-Point Correlationmethod.

4.4. Proposed approximation of strain–life curve fromhardness and modulus of elasticity

A new method for estimation of the strain–life curveis proposed. This method evolved from a careful exam-ination of the previous methods with the goal of derivinga strain–life approximation equation that required theleast amount and the most common material properties.These properties are hardness and modulus of elasticity.

To derive this relationship, each of the four strain–lifefatigue constants was first estimated based on the dataobtained from the 69 steels used in this study. Thefatigue strength coefficient,s9

f, showed a relativelystrong correlation with hardness as indicated by Fig. 12and represented by Eq. (15). The fatigue ductility coef-ficient,e9f, is found from Eq. (6) in terms of the transitionfatigue life, Nt, as:

e9f5s9

f(2Nt)b

E(2Nt)c (19)


where the nominator is the transition fatigue strength,St,corresponding toNt, St=s9

f (2Nt)b. A strong correlation(with R2=0.97) was found betweenSt and hardness,HB,represented by the following relation:

St50.004 (HB)211.15 (HB) (20)

Substituting the right side of Eq. (20) for the nominatorof Eq. (19), and using Eq. (14) to relateNt in the denomi-nator of Eq. (19) to hardness,HB, results in:

e9f50.004(HB)2+1.15HB

E[10(5.755−0.0071HB)]−0.56 (21)

A simpler second-order polynomial represents Eq. (21)with a very close agreement for 150,HB,700 as fol-lows:

e9f50.32(HB)2−487(HB)+191000

E(22)

The fatigue strength exponent,b, ranged from20.057to 20.140 with the average value of20.09. This averagevalue is the same as the constant value used by the Modi-fied Universal Slopes method given by Eq. (18). There-fore, the fatigue strength exponent,b, was estimated tohave a constant value of20.09. The fatigue ductilityexponent,c, ranged from20.39 to21.04 with the aver-age value of20.60. However, since this value is closeto that in the Modified Universal Slopes method, thefatigue ductility exponent,c, was also approximated asa constant value of20.56.

After substituting the approximated fatigue constantsinto the strain–life equation (Eq. (2)), the final form ofthe proposed correlation is as follows:

De2

54.25(HB)+225

E(2Nf)−0.09 (23)

10.32(HB)2−487(HB)+191000

E(2Nf)−0.56

This approximation uses only hardness and modulus ofelasticity (in MPa) as inputs for strain–life approxi-mation, both of which are either commonly available, oreasily measurable. Eq. (23) provides reasonably accuratepredictions for steels with a Brinell hardness larger than150. As can be seen from this equation, with an increasein hardness the fatigue strength coefficient increases andthe fatigue ductility coefficient decreases, which agreeswith expectations.

To evaluate prediction capabilities of this method,comparisons were made between this proposed relationand the three methods yielding the best predictions inthe study by Park and Song [12] mentioned previously.These three methods consisted of the Modified UniversalSlopes method, the Uniform Material method, and theModified Four-Point Correlation method. For eachmethod, a log–log plot of predicted vs. experimental

strain amplitudes was made using data from the 69steels. Experimental strain amplitude data were obtainedfrom strain–life curve for each material at fatigue livesof 103, 104, 105 and 106 reversals. All of the threemethods resulted in reasonable predictions, even thoughsome differences existed in different life regimes. As anillustration, the plot for the Modified Universal Slopesmethod is shown in Fig. 15. For comparison, predictionof strain amplitudes based on the proposed method isshown in Fig. 16. Only strain amplitudes between 0.1%to 1% are shown in these figures in order to illustratethe spread of the data, since this range is usuallythe practical range of strain–life data. However, all ofthe calculated data are included for the percentage ofdata values tabulated in Fig. 15 and Fig. 16. It may beseen that the proposed method results in reasonablyaccurate and somewhat better predictions of the strainamplitudes. For example, 80% of the predicted strainamplitudes based on the proposed equation are within afactor of ±1.2 of the experimental strain amplitudes,whereas 70% of the predicted strain amplitudes arewithin this factor for the Modified Universal Slopesmethod.

For each method, predicted vs. experimental fatiguelives are also plotted. For each steel, fatigue lives werecalculated at 1.5%, 1.0%, 0.6%, 0.35%, 0.2%, and0.15% strain amplitudes. Newton’s iterative procedurewas used to solve Eq. (2) for life, at each strain ampli-tude. These plots are shown in Fig. 17 and Fig. 18 for theModified Universal Slopes and the proposed methods,respectively. The proposed method results in somewhatbetter and more conservative predictions over the entirefatigue life regime. It should be emphasized, however,that the proposed method only requires hardness andmodulus of elasticity of the material.

5. Summary and conclusions

Material data for twenty steels commonly used in theground vehicle industry were presented and comparisonswere made between monotonic deformation, cyclicdeformation, and strain-controlled fatigue properties ofthese steels. The data from these steels in addition to 49other steels published by the American Society of Metals(ASM) were then used to examine correlations amongthe various monotonic and fatigue properties. Theseincluded relationships among ultimate tensile strength,fatigue limit, fatigue strength coefficient, and hardness,as well as estimation of strain–life fatigue properties andapproximation of strain–life curves. Validity of some ofthe more commonly used methods of estimating fatigueproperties found in the literature was evaluated andimprovements upon some of these correlations are sug-gested. Based on the discussions in the preceding sec-tions, the following conclusions can be drawn:


Fig. 15. Prediction of total strain amplitude by the Modified Universal Slopes method.

Fig. 16. Prediction of total strain amplitude by the proposed method.


Fig. 17. Prediction of fatigue life by the Modified Universal Slopes method.

Fig. 18. Prediction of fatigue life by the proposed method.


1. A strong correlation exists between hardness and ulti-mate tensile strength of steels. A second order poly-nomial is found to provide a better fit to the data, ascompared with a commonly used linear fit.

2. Correlations between the fatigue limit and ultimatetensile strength were weak, with significant scatter. Acommonly used estimate of fatigue limit as half ofthe ultimate tensile strength was found to be noncon-servative for the great majority of steels. Correlationof fatigue limit with hardness was found to be betterthan with the ultimate tensile strength.

3. A relatively strong correlation was found between thetransition fatigue life and hardness. A relationshipproposed in the literature provided good predictionsof the transition fatigue life based on hardness formost of the data.

4. No strong correlation was found between the fatiguestrength coefficient and the true fracture strength. Bet-ter correlations of the fatigue strength coefficient werefound with the ultimate tensile strength and hardness.

5. No correlations were found between the fatigue duc-tility coefficient and the true fracture ductility. Usingthe true fracture ductility to approximate the fatigueductility coefficient can result in significant error.

6. Three relationships estimating the strain–life curvefrom monotonic tensile properties were evaluated.These three estimates had been found to provide goodapproximations of the strain–life curve in anotherstudy and consist of the Modified Universal SlopesMethod, the Uniform Material Method, and the Modi-fied Four-Point Correlation Method. Similar and satis-factory predictions from these three methods werealso obtained for the data in this study.

7. A simple method is proposed for estimation of thestrain–life curve. This method only requires hardnessand modulus of elasticity as inputs, both of whichare either commonly available or easily measurable.Prediction capability of the proposed method wasevaluated for steels with hardness in the rangebetween 150 and 700 HB. The proposed method isshown to provide good approximations of the strain–life curve and result in somewhat better predictionsof the strain amplitudes or fatigue lives over the entirefatigue life regime, compared to several methods pro-posed in the literature.

Acknowledgements

We would like to thank the American Iron and SteelInstitute (AISI) for financial support of this project. Wealso acknowledge Dr T. Topper and his research assist-

ants from the University of Waterloo for their collabor-ative efforts in obtaining the material properties for SAE1541, SAE 1050, and SAE 1090 steels used in this study.

References

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[3] ASTM Standard E8-96a, Standard Test Methods for TensionTesting of Metallic Materials. Annual Book of ASTM Standards,vol. 03.01. American Society for Testing and Materials, WestConshohocken, PA. 1997:56–76.

[4] ASTM Standard E606-92, Standard Practice for Strain-ControlledFatigue Testing. Annual Book of ASTM Standards, vol. 03.01.American Society for Testing and Materials, West Consho-hocken, PA. 1997:523–537.

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[8] Mitchell MR, Socie DF, Caulfield EM. Fundamentals of ModernFatigue Analysis. Fracture Control Program Report No. 26, Uni-versity of Illinois, USA. 1977:385-410.

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[13] Manson SS. Fatigue: a Complex Subject—Some SimpleApproximations. Experimental Mechanics—Journal of theSociety for Experimental Stress Analysis 1965;5(7):193–226.

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[17] Ong JH. An evaluation of existing methods for the prediction ofaxial fatigue life from tensile data. International Journal ofFatigue 1993;15(1):13–9.

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