# Evaluation of the Methods for Estimating the Fully Reversed Unnotched Fatigue Limits of Steels

Authors: Jing Li; Yuan-ying Qiu; Hai-dong Wang; Zhao-xi Wang
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

Experimental determination of the fatigue limit is expensive considering the time and effort involved. To overcome this drawback, several empirical relations based on monotonic tensile properties and/or hardness have been developed for estimating the fatigue limit. In this study, ten estimation methods are evaluated and compared using the experimental data of 171 steels from literature. The scatter band criterion, standard deviation criterion, and statistical analysis are all used to evaluate and compare the predictability of the considered empirical relations. It is found that the fatigue limit can be well correlated with the Brinell hardness. In the absence of Brinell hardness, the ultimate tensile strength-based direct estimation method may be an alternative to estimate the fatigue limit of steels. It is not recommended to estimate the fatigue limit by using the indirect method on the basis of Basquin’s fatigue properties estimated by the monotonic tensile properties even though the fatigue limit of steel is often determined by the experimental Basquin’s curve at 106 cycles.


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## Body

## 1. Introduction

Fatigue failures occur due to the application of fluctuating stresses that are much lower than the stress required to cause failure during a single application of stress. It has been estimated that fatigue contributes to approximately 90% of all service failures due to mechanical causes [1]. Fatigue is a problem that can affect any part or component subjected to cyclic loading. Automobiles on roads, aircraft wings and fuselages, ships at sea, nuclear reactors, jet engines, and land-based turbines are all subject to fatigue failures. However, fatigue phenomena are poorly understood until recently since they occur unexpectedly at relatively low stress levels [2]. As the century progressed and the use of metals expanded with the increasing use of machines, more and more failures of components subjected to repeated loads are recorded. Today, structural fatigue has assumed an even greater importance as a result of the ever-increasing use of high-strength materials and the desire for higher performance from these materials.The fatigue limit has historically been a prime consideration in long-life fatigue design. For a given material, the fatigue limit has an enormous range, depending on surface finish, specimen or component size, type of loading, temperature, corrosive and other aggressive environments, mean stresses, residual stresses, and stress concentrations [1]. Most long-life S-N fatigue data available in the literature consist of fully reversed uniaxial fatigue strengths or fatigue limits of small, highly polished, unnotched specimens based on 106 to 5 × 108 cycles to failure in laboratory air environment. Therefore, determination of the fatigue limit from tests will be not only time-consuming but also expensive. In contrast, mechanics properties determined from monotonic tensile tests or hardness tests are more convenient than the fatigue tests. Over the years, many researchers have attempted to develop correlations among the monotonic tensile data or hardness and fatigue limit of materials [3–5]. Such correlations are desirable, considering the amount of time and effort required to obtain the fatigue limit, as compared to the monotonic tensile properties or hardness. If reliable correlations with reasonable accuracy can be established, durability performance predictions and/or optimization analyses can be performed, while substantially reducing time and cost associated with material fatigue testing.In the present study, ten methods utilized for estimating the fully reversed unnotched fatigue limits of steels from monotonic tensile properties or Brinell hardness are analyzed. The study is based on 171 steels which cover quite a range in variables that might affect fatigue behavior such as cyclic hardening/softening characteristics, tensile strengths covering the range from 331 to 2585 MPa, and Brinell hardness from 80 to 660.


## 2. The Estimation Methods

### 2.1. The Direct Methods

A common procedure to partially compare materials for their fatigue resistance is to plot the unnotched, fully reversed fatigue limit,σL, obtained under similar ideal laboratory conditions described above, versus the ultimate tensile strength, σu. A traditional idea is that the fatigue limit can be increased by raising the strength of a material, either by the chemical composition of the alloy, or by a heat treatment which increases the hardness. An approximation between the fatigue limit and ultimate tensile strength that is often used for steel is given as follows [1]:(1)σL=0.5σu.A plot of the fatigue limit vs. ultimate tensile strength is provided in Figure1. The tested data shown in this figure are taken from AISI Bar Steel Fatigue Database [6] (Table 1), American Society for Metals (ASM) Reference Handbook [7] (Table 2), and SAE J1099 Technical Report [8] (Table 3). It can be seen that there is a good deal of scatter in the data and that predictions based on equation (1) have poor agreement with the data and are nonconservative for most of steels.Figure 1
Fatigue limit vs. ultimate tensile strength.Table 1
Monotonic deformation properties and fatigue limit of the selected steels from the AISI Bar Steel Fatigue Database [6].
Materials
E (GPa)
σ
y (MPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
1541
197
471.2
783.2
1207
55.1
0.8
180
260.0
1541
196
461
905.5
1247
41.7
0.54
195
311.7
1050M
211
449.6
821
1128
49.6
0.7
238
254.5
1050M
202
460
828.5
1065
34.1
0.416
220
360.2
1050M
220
2100
2360
2115
14.7
0.16
536
796.1
1090
219
728.8
1090
966
14
0.15
259
307.1
1090M
217
1065
1388
1427
25
0.29
357
404.1
1090
213
677.6
1147
1151
21.7
0.245
309
346.8
1090M
212
751.3
1252.3
1169
13.6
0.146
279
338.3
1090M
214
759.5
1124
840
37.6
0.47
272
376.0
1141AL
216
457.4
770.6
1207
57
0.85
223
286.0
1141AL
227
814.2
924.8
1405
59
0.88
277
432.0
1141Nb
220
417.8
695.2
1000
53
0.76
199
276.6
1141Nb
217
602.1
801.8
1228
54
0.77
241
341.8
1141V
214
449.9
725.2
1087
49
0.68
217
287.0
1141V
215
609.7
796.8
1243
58
0.88
252
332.2
1141V
220
493.5
789.4
1117
47
0.64
229
295.8
1038
201
330.7
582.2
898
54
0.77
163
222.6
1038
219
359.4
651.6
1051
53
0.76
185
240.8
1038
218
559.8
743
1292
69
1.16
245
317.6
10V45
213
465.2
764.6
1131
48
0.7
212
270.4
10V45
216
605.6
908.6
1197
42
0.5
269
374.3
1020
207
294.9
502.2
849
63.5
1.01
135
194.3
10B21
212
1062.3
1104.8
1573
70.5
1.22
322
536.9
1022
200
457.3
604.1
1587
68.7
1.16
195
230.6
1045
209
509.6
746.6
1151
62
1
222
278.5
1151V
206
452.3
760.8
1319
50.6
0.7
205
266.9
15V24
209
645.7
877.6
1363
61.1
0.9
243
379.2
4130AL
213
1284.5
1482.8
2055
44.3
0.6
442
527.6
4140
207
928.9
1042.9
1519
61.3
1
325
437.4
5140
218
956.7
1039.2
1445
52.7
0.8
305
496.8
5150
210
471.8
867.1
1382
55.8
0.8
245
294.5
51B60
200
1830
1970
1968
21.6
0.23
450
771
9254V
206
1870
2050
2118
35.1
0.4
536
710.2
1070
207
520
659
1360
36.2
0.5
280
339.3
C-70
201
765
964
837
16.8
0.197
241
340.2
15B35
219
866.4
939.9
1747
64.5
1.04
286
467.8
4620
208
687.5
997.8
1530
58.2
0.872
289
368.5
4320
202
920
994
1336
63
0.99
188
547.1
4620
211
892
964
1651.9
61.9
0.966
289
453.5
5120
214
780
1008
1287
58
0.87
252
471.8
9310
197
804
902
1418
70.8
1.23
260
464.1
8822
204
918
1131
1362
56
0.88
252
442.1
8620
203
1357
1764
801.4
13.9
0.15
457
805.7
4320
201
1344.5
1705.3
2197.8
55.5
0.811
445
774.5
4140
197
1297
1390
1521.5
48.3
0.66
377
602.5
4140
200
1305
1400.5
1771
48.2
0.657
380
607.5
4140
204
1330
1537
1705.2
42.1
0.546
410
626.3
4140
205
1314
1445.8
1754.5
52.1
0.736
390
697.3
4140
199
1255.1
1355.1
1332.7
38.8
0.492
370
679
4140
201
1341
1474.2
1786.2
52.5
0.745
396
730.4
8620
207
1401.7
1677.2
1851.7
14.2
0.153
439
731.1
41B17M
213
826.9
872.3
1304
68
1.137
277
465.5
86B20
205
989
1034.1
869
63.7
1.013
294
498.6
4140
213
1669.2
1924.3
1859
15.7
0.163
477
587.7
41B17M
216
863.1
904
1245.4
64.7
1.04
277
480.5
4140
206
1290.7
1380.4
1384.5
23.6
0.269
394
554.81
86B20
206
1198
1502
968.2
59.7
0.909
401
494.8
5160H
209
1604.2
1761.2
1843.8
30.2
0.359
456
736.9
9254
206
1841
2020
3644
51
0.71
508
664.6
15V41
193
665.5
1070.7
766.5
32.6
0.395
303
431.9
4140
201
1363
1514
2071
48
0.65
405
552.4
1552
196
420
964
1526
45
0.59
276
387.2
8620
208
1566
1785
2206.4
39
0.49
461
614.6
4140
203
1157.5
1247.3
1311.3
47.5
0.646
409
601.9
4140
207
1167.1
1239.5
1537
47.2
0.64
417
575.2
4140
206
1622.2
1818.3
2016.7
42.6
0.555
477
637.83
4140
208
1261
1332.7
1517.6
44
0.581
375
574.2
8822
212
883.5
945.7
1170.2
67.2
1.115
297
511.2
8822
210
990
1334.7
1542.3
46.9
0.633
301
549.3
8822
194
1135
1215
2690
59.4
0.903
338
608.5
8822
208
1528
1723
3387
50
0.672
449
535.3
8620
212
693.5
991
1411.1
54
0.776
326
380.4
8620
210
795.8
1145.2
1585.7
50.8
0.711
352
430.2
8620
214
989.5
1310.8
1711.2
52
0.736
382
528.2
4320
194
620.5
905
1966
59.2
0.896
261
391.8
4320
194
745
1010
1984
57
0.846
288
466.3
4320
194
867
1238
2336
55.2
0.804
343
676.6
9310
196
666
883
1596
61.4
0.899
255
450
9310
196
850
1085
2155
57.5
0.857
306
531.5
9310
195
990
1201
2172
56.5
0.832
334
533.2
20MnCr5
194
852
1053
1991
56.5
0.832
299
489
20MnCr5
194
1071
1337
2351
52
0.74
366
606.8
1538MV
193
650.3
973.4
1038.7
28.3
0.333
285
439.1
1538MV
194
628
943
1069.8
36.4
0.452
277
382.3Table 2
Monotonic deformation properties and fatigue limit of the selected steels from the ASM Handbook [7].
Materials
E (GPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
A538A
185
1515
1896
67
1.1
405
644.5
A538B
185
1860
2137
56
0.82
460
762.1
A538C
180
2000
2241
55
0.81
480
811.3
AM-350
180
1905
2179
20
0.23
496
612.4
H-11
205
2585
3172
33
0.4
660
1037.2
RQC-100
205
940
1069
43
0.56
290
449.1
RQC-100
205
930
1331
67
1.02
290
449.1
10B62
195
1640
1779
38
0.89
430
673.4
1005–1009
205
360
717
73
1.3
90
157.2
1005–1009
205
470
745
66
1.09
125
218.8
1005–1009
200
415
641
64
1.02
125
187.2
1005–1009
200
345
848
80
1.6
90
111.1
1015
205
415
724
68
1.14
80
167.2
1020
205
440
710
62
0.96
108
156.9
1040
200
620
1048
60
0.93
225
202.0
1045
200
725
1227
65
1.04
225
308.7
1045
200
1450
1862
51
0.72
410
645.0
1045
205
1345
1862
59
0.89
390
541.7
1045
205
1585
2103
55
0.81
450
650.1
1045
205
1825
2275
51
0.71
500
712.7
1045
205
2240
2723
41
0.52
595
841.4
1144
195
930
1158
33
0.51
265
313.3
1144
200
1035
1517
25
0.29
305
429.5
1541F
205
950
1276
49
0.68
290
423.3
1541F
205
890
1276
60
0.93
260
455.1
4130
220
895
1420
67
1.12
258
382.4
4130
200
1425
1820
55
0.79
365
523.3
4140
200
1075
1524
60
0.69
310
571.7
4142
200
1060
1117
29
0.35
310
339.8
4142
205
1415
1827
48
0.66
380
571.7
4142
200
1550
1896
47
0.63
400
513.5
4142
200
1760
1999
42
0.54
450
626.5
4142
200
2035
2068
20
0.22
475
629.9
4142
200
1930
2103
37
0.46
450
570.4
4142
205
1930
2172
35
0.43
475
670.0
4142
205
2240
2654
27
0.31
560
729.9
4340
195
825
1089
43
0.57
243
302.4
4340
200
1470
1558
38
0.48
409
534.1
4340
195
1240
1655
57
0.64
350
549.4
5150
195
1670
1931
42
0.87
430
688.9
52100
205
2015
2193
11
0.12
518
700.4
9262
205
925
1041
14
0.16
260
371.2
9262
195
1000
1220
33
0.41
280
423.0
9262
200
1565
1855
32
0.38
410
811.3
950C
205
565
931
64
1.03
159
205.1
950C
205
565
1000
69
1.19
150
196.6
950X
205
440
752
65
1.06
150
210.5
950X
205
530
1000
72
1.24
156
235.5
980X
195
695
1220
68
1.15
225
330.5Table 3
Monotonic deformation properties and fatigue limit of the selected steels from SAE J1099 [8].
Materials
E (GPa)
σ
y (MPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
1008
207
234
331
587.5
77.5
1.49
86
92.7
1008
203
255
365
649.3
77.9
1.51
90
100.3
1010
203
200
331
597.1
80.4
1.63
83
116.9
1015
207
228
414
695.5
68
1.14
80
146.3
1020
203
262
441
713.5
61.8
0.96
109
143.9
1020
186
255
393
644.5
64
1.02
108
129.5
1025
207
306
547
889.4
62.6
0.98
158
197.8
1045
207
517
752
1082.9
44
0.58
225
291.1
1045
207
1689
1827
2758.8
51
0.71
500
690.3
1045
207
1862
2241
3159.8
41
0.53
595
728.5
1045
207
424
718
1062.6
48
0.65
192
227.9
1045
207
1729
1956
2705.1
38.3
0.48
500
756.9
1045
207
1275
1344
2137.0
59
0.89
390
512.6
10B21
197
999
1048
1756.4
67.6
1.13
318
482.7
10B21
203
806
834
NA
NA
NA
255
369.6
10B22
203
806
834
NA
NA
NA
255
450.7
15B27
203
772
847
1431.4
69
1.17
250
410.2
15B27
203
854
916
1525.1
66.5
1.09
264
451.2
4130
200
1358
1427
2207.6
54.7
0.79
366
549.4
4130
221
778
896
1499.0
67.3
1.12
259
412.6
4140
207
848
938
NA
NA
NA
293
473.1
4140
200
1895
2033
2439.6
20
0.22
475
663.5
4142
200
1447
1551
2280.0
47
0.63
400
528.2
4142
200
1860
1929
2642.7
37
0.46
450
597.0
4142
207
1378
1413
2091.2
48
0.65
380
547.9
4142
207
1584
1757
2494.9
42
0.54
450
643.1
4142
207
1722
1929
2604.2
35
0.43
475
667.2
4340
200
1371
1468
2027.3
38.1
0.48
409
540.3
4340
190
834
1048
NA
NA
NA
275
429.8
4340
193
634
827
1185.9
43.4
0.57
243
301.9
4340
207
1102
1171
1826.8
56
0.82
327
502.2
5160
203
1488
1584
2212.8
39.7
0.51
430
634.2
0030
207
303
496
724.2
46
0.62
137
196.5
0050A
209
415
787
936.5
19
0.21
192
211.9
0050A
209
402
583
734.6
26
0.30
174
200.7
0050A
211
542
702
NA
NA
NA
206
258.0
8630
207
985
1144
1475.8
29
0.34
305
334.6To improve the predictions, a modified correlation is given by Roessle and Fatemi [4]:(2)σL=0.38σu.Evaluation of equation (2) by using the tested data is also shown in Figure 1. Relatively good predictions can be obtained.It should be mentioned that traditionally most fatigue limit data are obtained from rotating bending fatigue tests, where here the values are from axial fatigue tests. In axial fatigue tests, lower strengths are often obtained due to reasons such as higher probability of defects associated with larger stressed volume and bending resulting from specimen misalignment. Therefore, the smaller coefficient obtained by Roessle and Fatemi [4] can correlate the tested data better. In the present paper, equation (2) is referred to as the σu-based direct method.The fatigue limit can also be estimated from the Brinell hardness, HB. An approximation has been suggested to be used [3], which is given as(3)σL=1.72HB.A plot of the fatigue limit vs. Brinell hardness is provided by using the aforementioned experimental data (Figure2). Similar to equation (1), there is a poor agreement between the data and the predictions obtained from equation (3). A modification provided by Roessle and Fatemi [4] is given as(4)σL=1.43HB.Figure 2
Fatigue limit vs. Brinell hardness.Evaluations show that equation (4) can provide better predictions (Figure 2). Equation (4) is hereafter referred to as the HB-based direct method.


### 2.2. The Indirect Methods

In view of the asymptotic character of the S-N curve, the fatigue limit can be defined as the constant stress amplitude at which the fatigue life becomes infinite or fatigue failures do not occur. However, in some cases where fatigue testing must be terminated at a specific large number of cycles, this nonfailure stress amplitude is often referred to as the endurance limit. Even the endurance limit needs not be a fatigue limit, but from the engineering point of view, the term of a fatigue limit is commonly used to refer to either case. Here, the fully unnotched fatigue limitσL at 106 cycles for steels is estimated from the monotonic tensile properties.Many equations have been proposed to describe the S-N curve. The simplest one is Basquin’s equation, which successfully describes the fatigue performance of steels for the high cycle regime [1]:(5)σa=σf′2Nfb,where σa is the fatigue stress amplitude, σf′ is the fatigue strength coefficient, b is the fatigue strength exponent, and 2Nf is the number of reversals to failure.If Basquin’s equation is used to model the S-N curve, the fatigue limit can be determined by the following equation at 106 cycles:(6)σL=σf′2×106b.In the absence of tested data, the fatigue strength coefficient,σf′, and the fatigue strength exponent, b, can be estimated using the monotonic tensile properties or Brinell hardness [4,9–12]. Then, the fatigue limit, σL, can also be estimated indirectly by substituting the predicted σf′ and b into equation (6). This method will be hereafter referred to as indirect method.Based on the indirect method, the fatigue strength coefficient and the fatigue strength exponent can be predicted by six existing estimation methods [4,9–12] firstly, and then the fatigue limit is calculated using equation (6). The six existing estimation methods considered in this paper are listed in Table 4. In this table, E is the modulus of elasticity, σu is the ultimate tensile strength, HB is the Brinell hardness, σf is the true fracture strength, and εf is the true fracture ductility.Table 4
Approximation methods for Basquin’s fatigue properties.
Approximation methods
σ
f
′ (MPa)
b
Four-point correlation method (FPCM) [9]
1.12
σ
u
σ
f
/
σ
u
0.893
−
0.0792
−
0.179
lg
σ
f
/
σ
u
Universal slopes method (USM) [9]
1.9018
σ
u
–0.12
Modified universal slopes method (MUSM) [10]
E
×
0.623
σ
u
/
E
0.832
–0.09
Modified four-point correlation method (MFPCM) [11]
σ
u
1
+
ε
f
1
6
lg
0.16
σ
u
/
E
0.81
−
lg
σ
f
/
E
Medians method (MM) [12]
1.5
σ
u
–0.09
Hardness method (HM) [4]
4.25
HB
+
225
–0.09


## 2.1. The Direct Methods

A common procedure to partially compare materials for their fatigue resistance is to plot the unnotched, fully reversed fatigue limit,σL, obtained under similar ideal laboratory conditions described above, versus the ultimate tensile strength, σu. A traditional idea is that the fatigue limit can be increased by raising the strength of a material, either by the chemical composition of the alloy, or by a heat treatment which increases the hardness. An approximation between the fatigue limit and ultimate tensile strength that is often used for steel is given as follows [1]:(1)σL=0.5σu.A plot of the fatigue limit vs. ultimate tensile strength is provided in Figure1. The tested data shown in this figure are taken from AISI Bar Steel Fatigue Database [6] (Table 1), American Society for Metals (ASM) Reference Handbook [7] (Table 2), and SAE J1099 Technical Report [8] (Table 3). It can be seen that there is a good deal of scatter in the data and that predictions based on equation (1) have poor agreement with the data and are nonconservative for most of steels.Figure 1
Fatigue limit vs. ultimate tensile strength.Table 1
Monotonic deformation properties and fatigue limit of the selected steels from the AISI Bar Steel Fatigue Database [6].
Materials
E (GPa)
σ
y (MPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
1541
197
471.2
783.2
1207
55.1
0.8
180
260.0
1541
196
461
905.5
1247
41.7
0.54
195
311.7
1050M
211
449.6
821
1128
49.6
0.7
238
254.5
1050M
202
460
828.5
1065
34.1
0.416
220
360.2
1050M
220
2100
2360
2115
14.7
0.16
536
796.1
1090
219
728.8
1090
966
14
0.15
259
307.1
1090M
217
1065
1388
1427
25
0.29
357
404.1
1090
213
677.6
1147
1151
21.7
0.245
309
346.8
1090M
212
751.3
1252.3
1169
13.6
0.146
279
338.3
1090M
214
759.5
1124
840
37.6
0.47
272
376.0
1141AL
216
457.4
770.6
1207
57
0.85
223
286.0
1141AL
227
814.2
924.8
1405
59
0.88
277
432.0
1141Nb
220
417.8
695.2
1000
53
0.76
199
276.6
1141Nb
217
602.1
801.8
1228
54
0.77
241
341.8
1141V
214
449.9
725.2
1087
49
0.68
217
287.0
1141V
215
609.7
796.8
1243
58
0.88
252
332.2
1141V
220
493.5
789.4
1117
47
0.64
229
295.8
1038
201
330.7
582.2
898
54
0.77
163
222.6
1038
219
359.4
651.6
1051
53
0.76
185
240.8
1038
218
559.8
743
1292
69
1.16
245
317.6
10V45
213
465.2
764.6
1131
48
0.7
212
270.4
10V45
216
605.6
908.6
1197
42
0.5
269
374.3
1020
207
294.9
502.2
849
63.5
1.01
135
194.3
10B21
212
1062.3
1104.8
1573
70.5
1.22
322
536.9
1022
200
457.3
604.1
1587
68.7
1.16
195
230.6
1045
209
509.6
746.6
1151
62
1
222
278.5
1151V
206
452.3
760.8
1319
50.6
0.7
205
266.9
15V24
209
645.7
877.6
1363
61.1
0.9
243
379.2
4130AL
213
1284.5
1482.8
2055
44.3
0.6
442
527.6
4140
207
928.9
1042.9
1519
61.3
1
325
437.4
5140
218
956.7
1039.2
1445
52.7
0.8
305
496.8
5150
210
471.8
867.1
1382
55.8
0.8
245
294.5
51B60
200
1830
1970
1968
21.6
0.23
450
771
9254V
206
1870
2050
2118
35.1
0.4
536
710.2
1070
207
520
659
1360
36.2
0.5
280
339.3
C-70
201
765
964
837
16.8
0.197
241
340.2
15B35
219
866.4
939.9
1747
64.5
1.04
286
467.8
4620
208
687.5
997.8
1530
58.2
0.872
289
368.5
4320
202
920
994
1336
63
0.99
188
547.1
4620
211
892
964
1651.9
61.9
0.966
289
453.5
5120
214
780
1008
1287
58
0.87
252
471.8
9310
197
804
902
1418
70.8
1.23
260
464.1
8822
204
918
1131
1362
56
0.88
252
442.1
8620
203
1357
1764
801.4
13.9
0.15
457
805.7
4320
201
1344.5
1705.3
2197.8
55.5
0.811
445
774.5
4140
197
1297
1390
1521.5
48.3
0.66
377
602.5
4140
200
1305
1400.5
1771
48.2
0.657
380
607.5
4140
204
1330
1537
1705.2
42.1
0.546
410
626.3
4140
205
1314
1445.8
1754.5
52.1
0.736
390
697.3
4140
199
1255.1
1355.1
1332.7
38.8
0.492
370
679
4140
201
1341
1474.2
1786.2
52.5
0.745
396
730.4
8620
207
1401.7
1677.2
1851.7
14.2
0.153
439
731.1
41B17M
213
826.9
872.3
1304
68
1.137
277
465.5
86B20
205
989
1034.1
869
63.7
1.013
294
498.6
4140
213
1669.2
1924.3
1859
15.7
0.163
477
587.7
41B17M
216
863.1
904
1245.4
64.7
1.04
277
480.5
4140
206
1290.7
1380.4
1384.5
23.6
0.269
394
554.81
86B20
206
1198
1502
968.2
59.7
0.909
401
494.8
5160H
209
1604.2
1761.2
1843.8
30.2
0.359
456
736.9
9254
206
1841
2020
3644
51
0.71
508
664.6
15V41
193
665.5
1070.7
766.5
32.6
0.395
303
431.9
4140
201
1363
1514
2071
48
0.65
405
552.4
1552
196
420
964
1526
45
0.59
276
387.2
8620
208
1566
1785
2206.4
39
0.49
461
614.6
4140
203
1157.5
1247.3
1311.3
47.5
0.646
409
601.9
4140
207
1167.1
1239.5
1537
47.2
0.64
417
575.2
4140
206
1622.2
1818.3
2016.7
42.6
0.555
477
637.83
4140
208
1261
1332.7
1517.6
44
0.581
375
574.2
8822
212
883.5
945.7
1170.2
67.2
1.115
297
511.2
8822
210
990
1334.7
1542.3
46.9
0.633
301
549.3
8822
194
1135
1215
2690
59.4
0.903
338
608.5
8822
208
1528
1723
3387
50
0.672
449
535.3
8620
212
693.5
991
1411.1
54
0.776
326
380.4
8620
210
795.8
1145.2
1585.7
50.8
0.711
352
430.2
8620
214
989.5
1310.8
1711.2
52
0.736
382
528.2
4320
194
620.5
905
1966
59.2
0.896
261
391.8
4320
194
745
1010
1984
57
0.846
288
466.3
4320
194
867
1238
2336
55.2
0.804
343
676.6
9310
196
666
883
1596
61.4
0.899
255
450
9310
196
850
1085
2155
57.5
0.857
306
531.5
9310
195
990
1201
2172
56.5
0.832
334
533.2
20MnCr5
194
852
1053
1991
56.5
0.832
299
489
20MnCr5
194
1071
1337
2351
52
0.74
366
606.8
1538MV
193
650.3
973.4
1038.7
28.3
0.333
285
439.1
1538MV
194
628
943
1069.8
36.4
0.452
277
382.3Table 2
Monotonic deformation properties and fatigue limit of the selected steels from the ASM Handbook [7].
Materials
E (GPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
A538A
185
1515
1896
67
1.1
405
644.5
A538B
185
1860
2137
56
0.82
460
762.1
A538C
180
2000
2241
55
0.81
480
811.3
AM-350
180
1905
2179
20
0.23
496
612.4
H-11
205
2585
3172
33
0.4
660
1037.2
RQC-100
205
940
1069
43
0.56
290
449.1
RQC-100
205
930
1331
67
1.02
290
449.1
10B62
195
1640
1779
38
0.89
430
673.4
1005–1009
205
360
717
73
1.3
90
157.2
1005–1009
205
470
745
66
1.09
125
218.8
1005–1009
200
415
641
64
1.02
125
187.2
1005–1009
200
345
848
80
1.6
90
111.1
1015
205
415
724
68
1.14
80
167.2
1020
205
440
710
62
0.96
108
156.9
1040
200
620
1048
60
0.93
225
202.0
1045
200
725
1227
65
1.04
225
308.7
1045
200
1450
1862
51
0.72
410
645.0
1045
205
1345
1862
59
0.89
390
541.7
1045
205
1585
2103
55
0.81
450
650.1
1045
205
1825
2275
51
0.71
500
712.7
1045
205
2240
2723
41
0.52
595
841.4
1144
195
930
1158
33
0.51
265
313.3
1144
200
1035
1517
25
0.29
305
429.5
1541F
205
950
1276
49
0.68
290
423.3
1541F
205
890
1276
60
0.93
260
455.1
4130
220
895
1420
67
1.12
258
382.4
4130
200
1425
1820
55
0.79
365
523.3
4140
200
1075
1524
60
0.69
310
571.7
4142
200
1060
1117
29
0.35
310
339.8
4142
205
1415
1827
48
0.66
380
571.7
4142
200
1550
1896
47
0.63
400
513.5
4142
200
1760
1999
42
0.54
450
626.5
4142
200
2035
2068
20
0.22
475
629.9
4142
200
1930
2103
37
0.46
450
570.4
4142
205
1930
2172
35
0.43
475
670.0
4142
205
2240
2654
27
0.31
560
729.9
4340
195
825
1089
43
0.57
243
302.4
4340
200
1470
1558
38
0.48
409
534.1
4340
195
1240
1655
57
0.64
350
549.4
5150
195
1670
1931
42
0.87
430
688.9
52100
205
2015
2193
11
0.12
518
700.4
9262
205
925
1041
14
0.16
260
371.2
9262
195
1000
1220
33
0.41
280
423.0
9262
200
1565
1855
32
0.38
410
811.3
950C
205
565
931
64
1.03
159
205.1
950C
205
565
1000
69
1.19
150
196.6
950X
205
440
752
65
1.06
150
210.5
950X
205
530
1000
72
1.24
156
235.5
980X
195
695
1220
68
1.15
225
330.5Table 3
Monotonic deformation properties and fatigue limit of the selected steels from SAE J1099 [8].
Materials
E (GPa)
σ
y (MPa)
σ
u (MPa)
σ
f (MPa)
RA (%)
ε
f
HB
σ
L (MPa)
1008
207
234
331
587.5
77.5
1.49
86
92.7
1008
203
255
365
649.3
77.9
1.51
90
100.3
1010
203
200
331
597.1
80.4
1.63
83
116.9
1015
207
228
414
695.5
68
1.14
80
146.3
1020
203
262
441
713.5
61.8
0.96
109
143.9
1020
186
255
393
644.5
64
1.02
108
129.5
1025
207
306
547
889.4
62.6
0.98
158
197.8
1045
207
517
752
1082.9
44
0.58
225
291.1
1045
207
1689
1827
2758.8
51
0.71
500
690.3
1045
207
1862
2241
3159.8
41
0.53
595
728.5
1045
207
424
718
1062.6
48
0.65
192
227.9
1045
207
1729
1956
2705.1
38.3
0.48
500
756.9
1045
207
1275
1344
2137.0
59
0.89
390
512.6
10B21
197
999
1048
1756.4
67.6
1.13
318
482.7
10B21
203
806
834
NA
NA
NA
255
369.6
10B22
203
806
834
NA
NA
NA
255
450.7
15B27
203
772
847
1431.4
69
1.17
250
410.2
15B27
203
854
916
1525.1
66.5
1.09
264
451.2
4130
200
1358
1427
2207.6
54.7
0.79
366
549.4
4130
221
778
896
1499.0
67.3
1.12
259
412.6
4140
207
848
938
NA
NA
NA
293
473.1
4140
200
1895
2033
2439.6
20
0.22
475
663.5
4142
200
1447
1551
2280.0
47
0.63
400
528.2
4142
200
1860
1929
2642.7
37
0.46
450
597.0
4142
207
1378
1413
2091.2
48
0.65
380
547.9
4142
207
1584
1757
2494.9
42
0.54
450
643.1
4142
207
1722
1929
2604.2
35
0.43
475
667.2
4340
200
1371
1468
2027.3
38.1
0.48
409
540.3
4340
190
834
1048
NA
NA
NA
275
429.8
4340
193
634
827
1185.9
43.4
0.57
243
301.9
4340
207
1102
1171
1826.8
56
0.82
327
502.2
5160
203
1488
1584
2212.8
39.7
0.51
430
634.2
0030
207
303
496
724.2
46
0.62
137
196.5
0050A
209
415
787
936.5
19
0.21
192
211.9
0050A
209
402
583
734.6
26
0.30
174
200.7
0050A
211
542
702
NA
NA
NA
206
258.0
8630
207
985
1144
1475.8
29
0.34
305
334.6To improve the predictions, a modified correlation is given by Roessle and Fatemi [4]:(2)σL=0.38σu.Evaluation of equation (2) by using the tested data is also shown in Figure 1. Relatively good predictions can be obtained.It should be mentioned that traditionally most fatigue limit data are obtained from rotating bending fatigue tests, where here the values are from axial fatigue tests. In axial fatigue tests, lower strengths are often obtained due to reasons such as higher probability of defects associated with larger stressed volume and bending resulting from specimen misalignment. Therefore, the smaller coefficient obtained by Roessle and Fatemi [4] can correlate the tested data better. In the present paper, equation (2) is referred to as the σu-based direct method.The fatigue limit can also be estimated from the Brinell hardness, HB. An approximation has been suggested to be used [3], which is given as(3)σL=1.72HB.A plot of the fatigue limit vs. Brinell hardness is provided by using the aforementioned experimental data (Figure2). Similar to equation (1), there is a poor agreement between the data and the predictions obtained from equation (3). A modification provided by Roessle and Fatemi [4] is given as(4)σL=1.43HB.Figure 2
Fatigue limit vs. Brinell hardness.Evaluations show that equation (4) can provide better predictions (Figure 2). Equation (4) is hereafter referred to as the HB-based direct method.


## 2.2. The Indirect Methods

In view of the asymptotic character of the S-N curve, the fatigue limit can be defined as the constant stress amplitude at which the fatigue life becomes infinite or fatigue failures do not occur. However, in some cases where fatigue testing must be terminated at a specific large number of cycles, this nonfailure stress amplitude is often referred to as the endurance limit. Even the endurance limit needs not be a fatigue limit, but from the engineering point of view, the term of a fatigue limit is commonly used to refer to either case. Here, the fully unnotched fatigue limitσL at 106 cycles for steels is estimated from the monotonic tensile properties.Many equations have been proposed to describe the S-N curve. The simplest one is Basquin’s equation, which successfully describes the fatigue performance of steels for the high cycle regime [1]:(5)σa=σf′2Nfb,where σa is the fatigue stress amplitude, σf′ is the fatigue strength coefficient, b is the fatigue strength exponent, and 2Nf is the number of reversals to failure.If Basquin’s equation is used to model the S-N curve, the fatigue limit can be determined by the following equation at 106 cycles:(6)σL=σf′2×106b.In the absence of tested data, the fatigue strength coefficient,σf′, and the fatigue strength exponent, b, can be estimated using the monotonic tensile properties or Brinell hardness [4,9–12]. Then, the fatigue limit, σL, can also be estimated indirectly by substituting the predicted σf′ and b into equation (6). This method will be hereafter referred to as indirect method.Based on the indirect method, the fatigue strength coefficient and the fatigue strength exponent can be predicted by six existing estimation methods [4,9–12] firstly, and then the fatigue limit is calculated using equation (6). The six existing estimation methods considered in this paper are listed in Table 4. In this table, E is the modulus of elasticity, σu is the ultimate tensile strength, HB is the Brinell hardness, σf is the true fracture strength, and εf is the true fracture ductility.Table 4
Approximation methods for Basquin’s fatigue properties.
Approximation methods
σ
f
′ (MPa)
b
Four-point correlation method (FPCM) [9]
1.12
σ
u
σ
f
/
σ
u
0.893
−
0.0792
−
0.179
lg
σ
f
/
σ
u
Universal slopes method (USM) [9]
1.9018
σ
u
–0.12
Modified universal slopes method (MUSM) [10]
E
×
0.623
σ
u
/
E
0.832
–0.09
Modified four-point correlation method (MFPCM) [11]
σ
u
1
+
ε
f
1
6
lg
0.16
σ
u
/
E
0.81
−
lg
σ
f
/
E
Medians method (MM) [12]
1.5
σ
u
–0.09
Hardness method (HM) [4]
4.25
HB
+
225
–0.09


## 3. Evaluation of the Estimation Methods

As analyzed above, the direct methods proposed by Roessle and Fatemi [4] give better predictions than the methods shown in References [1, 3]. Therefore, the direct methods proposed by Roessle and Fatemi [4] are evaluated by using the tested data of 171 steels. The predicted fatigue limit, σPre, is compared with the experimental one, σExp, in Figure 3. For comparison purpose, the same tested data are also used to evaluate the aforementioned six indirect methods (Figure 4). In these figures, the dotted lines, solid lines, and dashed-dotted lines represent factors of ±10%, ±20%, and ±30% scatter band, respectively. It can be seen that, for most of considered steels, the MFPCM-based indirect method tends to overestimate the fatigue limit, while the FPCM-based and USM-based indirect methods as well as the σu-based direct method tend to underestimate the fatigue limit. Both MUSM-based and HM-based indirect methods tend to overestimate the fatigue limit when the experimental fatigue limit is low (about the experimental fatigue limit less than 400 MPa), while underestimations are presented when the experimental fatigue limit is high (about the experimental fatigue limit larger than 400 MPa). As for the HB-based direct method, it may be said to provide the best estimation results among all the eight estimation methods.Figure 3
Comparison of the direct methods using the tested data from [6–8]: (a) HB-based and (b) σu-based.
(a)
(b)Figure 4
Comparison of the indirect methods using the tested data from [6–8]: (a) FPCM-based method, (b) MFPCM-based method, (c) USM-based method, (d) MUSM-based method, (e) MM-based method, and (f) HM-based method.
(a)
(b)
(c)
(d)
(e)
(f)Most of the work done in comparing both the direct and indirect methods use the scatter band criterion where the percentage of predicted fatigue limit versus experimental data falling within a scatter band is compared (Table5). Factors of ±10%, ±20%, and ±30% deviation from tested values are included in this table, along with the percentages of the data within each factor. It can be seen that the HB-based direct method has the most data with all deviation factors. For example, 86.5% of the predicted values are within a factor of ±20% of the experimental values based on the HB-based direct method. In contrast, the best results predicted by MM-based method among the considered indirect methods from the literature are 76.6% of the predicted fatigue limit within the same factor of experimental values.Table 5
Comparison of estimation methods in terms of evaluation values.
HB-based
σ
u-based
FPCM
MFPCM
USM
MUSM
MM
HM
±10%
0.544
0.450
0.440
0.102
0.327
0.392
0.427
0.392
±20%
0.865
0.772
0.687
0.361
0.608
0.754
0.766
0.766
±30%
0.977
0.965
0.892
0.512
0.877
0.889
0.918
0.895
S
D
0.155
0.170
0.199
0.353
0.240
0.182
0.167
0.187It should be noted here that one of the shortcomings of the scatter band criterion is that the trend of the correlation between prediction and experiment cannot be captured. Comparison of the HB-based andσu-based direct methods with experimental data using scatter band criterion is schematically shown in Figure 5. It is observed that both the HB-based and σu-based direct methods predict well within the scatter band and hence have to be considered as equal. However, the trend of the HB-based direct method with experimental data is closer than that of the σu-based direct method, which is not captured by the scatter band criterion. To overcome this drawback, the standard deviation criterion is introduced to compare different estimation methods:(7)SD=1np−1∑i=1nplnσPre−lnσExp2,where np is the number of experimental data. It can be seen that the bigger the SD is, the more scattered the predictions are. The comparisons using the standard deviation criterion are also tabulated in Table 5. It is shown that the HB-based direct method provides the best predictions.Figure 5
Schematic showing the limitation in comparing two different methods using scatter band criterion.In order to have a better quantitative comparison, the comparisons for different estimation methods are also given in the format of fitted probability density functions (PDF) for the predicted errors. For the statistical study, each data set is sorted in ascending order, and then each data point is associated with its mean rank. After that, each data set is fitted using the selected probability distribution. Here, the log-logistic distribution is selected based on the recommendation of Meggiolaro and Castro [12]. The cumulative distribution function of the log-logistic distribution function is given by the following equation [13](8)Fx=xpap+xp,x>0,a>0,p≥1,where p is a shape parameter and a is a scale parameter. The corresponding probability density function (PDF) of the log-logistic distribution is given by [13](9)fx=papxp−1ap+xp2,x>0,a>0,p≥1.The cumulative probabilityPf of estimated data is evaluated by the mean rank method as(10)Pf=iI+1,where i is the rank number and I is the total number of considered steels. Symbols show the σPre/σExp ratio of the considered eight estimation methods (Figure 6). Optimum values of the parameters a and p are obtained by applying the least squares method [14]. The curvilinear lines indicate the results of fitting (Figure 6). The optimum values of a and p are also presented in Figure 6. Then, the best-fitted distributions of different estimation methods are shown in Figure 7. It is clearly shown that the FPCM-based and USM-based indirect methods as well as the σu-based direct method provide underestimations, while the MFPCM-based indirect method provides overestimation. The HB-based direct method provides reasonable predictions while somewhat underestimations are shown in Figure 7. In contrast to the considered estimation methods, the HB-based direct method on the basis of Brinell hardness provides the best predictions for the fatigue limits of most of steels.Figure 6
The estimates of the considered estimation methods (the curvilinear lines show the fit of log-logistic distribution): (a) HB-based method, (b)σu-based method, (c) FPCM-based method, (d) MFPCM-based method, (e) USM-based method, (f) MUSM-based method, (g) MM-based method, and (h) HM-based method.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 7
Probability density functions of the considered estimation methods.Although the HB-based direct method presents the best predictions among the considered estimation methods, it is known that, for advanced steels, the predictability of the fatigue limit based on monotonic properties often fails. The failed predictions may lie in the high notch sensitivity since a lot of the advanced steels belong to the high-strength materials. The inclusions act as micronotches in the material which can generate fatigue crack nuclei, especially in high-strength material [15]. Because of the high-yield strength of these materials, they have a high notch sensitivity, even for micronotches [15]. Therefore, in high-strength steel, inclusions are more harmful because of the higher notch sensitivity. Then, the predictability of the fatigue limit based on monotonic properties for advanced steels is more challengeable, which is needed to be studied further.


## 4. Conclusions

(1)
The Brinell hardness, HB, can correlate the fatigue limit well. The fatigue limit can be estimated well by the HB-based direct method for most of considered steels. In the absence of HB, theσu-based direct method is recommended to calculate the fatigue limit.
(2)
In contrast to the direct methods, all the indirect methods are not only relatively complicated to use but also more scattered in predictions.Finally, it is important to point out that HB-based direct method is not physical law although it correlates the fatigue limit of many steels well. It is still an empirical equation. Therefore, such estimates should not be used to replace experiments. Whenever possible, the fatigue limit should be obtained from the tested data. These estimates, even the HB-based direct method, are only admissible during the first stage of design.



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*Source: 4098623-2019-07-01.xml*