A practical model for efficient anti-fatigue design and selection of metallic materials: II. Parameter analysis and fatigue strength improvement

doi:10.1016/j.jmst.2020.08.037

Abstract

Abstract:

In the first paper, a Y-T-F model was proposed based on the restrictions of both strength and plasticity; the corresponding applications on the fatigue strength prediction have also been discussed. In this second paper, the emphasis will be put on the issues of fatigue strength improvement. Based on the primary form of the Y-T-F model, the parameters are further analyzed and quantified, to clarify the influences of various factors on fatigue strength. Firstly, the damage capacity C is proved to be sensitive to the elastic modulus E, which could change with the alloying components and nano-scaled grain boundaries; the increase of E would lead to the increasing C, thus increase the fatigue strength. Secondly, the microstructure characteristic coefficient a, as well as the yield strength σ_y and tensile strength σ_b in the crack initiation region could be influenced by the processing mode, grain size and microstructure uniformity of materials; the change of microstructure characteristics would affect the changing tendency of tensile strength--fatigue strength relation via varying the values of a, σ_y and σ_b. Thirdly, the damage weight coefficient ω is found to be a reflection of the fatigue strength declination induced by defects; the defect dimension D, the defect shape correlated stress concentration coefficient Kt, as well as the strengthening level of matrix materials σ_b are all corresponding factors. Quantified correlations between the above parameters and corresponding factors are comprehensively built up, hence obtaining the influences of either a single factor or multiple factors on fatigue strength. This further developed Y-T-F model would be helpful to clarify the direction of fatigue strength improvement, and contribute to the anti-fatigue design optimization of metallic materials.

Key words: Fatigue strength improvement, Alloying component, Elastic modulus, Microstructure, Characteristics, Defect, Anti-fatigue design

R. Liu, P. Zhang, Z.J. Zhang, B. Wang, Z.F. Zhang. A practical model for efficient anti-fatigue design and selection of metallic materials: II. Parameter analysis and fatigue strength improvement[J]. J. Mater. Sci. Technol., 2021, 70: 250-267.

Figures/Tables 21

Fig. 1. Influences of elastic modulus E on damage capacity C and fatigue strength. (a) The linear σw/σy--σy/σb relation shared by three different series of aluminum alloys (3003, 3004 and 5050, E = 71GPa) [8], with the separated σw--σb relations (the inserted figure). (b) Typical linear σw/σy--σy/σb relations of steels [1,[9], [10], [11], [12]], copper alloys [[13], [14], [15], [16], [17]], aluminum alloys [1,8,18] and magnesium alloys [1]; corresponding values of C have been marked out. (c) Linear relation of parameter C in the 1/C--E /E0 coordinate, E0 = 650GPa. (d) Calculated ranges of σw--E relations (substituted values of σy and σb have been marked out) and corresponding tested fatigue strength of steels and magnesium alloys.

Table 1 The typical values of elastic modulus and the parameters in the Y-T-F model for several series of metallic materials.

Series	Materials	E(GPa) [8]	C	ω	R-square	Source/Ref.
Steel	Carbon Steels & Alloy Steels	200	1.51	1.61	0.85	[1,9,10,11,12]
Cu Alloy	Cu-Al Alloys	125	1.26	0.87	0.97	[13,14,15,16,17]
Al Alloy	Wrought Al Alloys	70	1.16	0.85	0.71	[1,8,18]
Mg Alloy	Wrought Mg Alloys	45	1.11	0.79	0.6	[1]

Fig. 2. Influences of nano-crystallization on elastic modulus E, parameter C and fatigue strength. (a) The σw--σb relations of Cu and Cu-Al alloys [[14], [15], [16], [17]]; hollow points represent the data of nano-crystalline states. (b) Linear σw/σy--σy/σb relation of Cu and Cu-Al alloys, with the nano-crystalline data (hollow points) deviate downward; corresponding values of CCG and CNG have been marked out, which are in accordance with the change of E.

Fig. 3. Influences of alloying content on elastic modulus E, parameter C and fatigue strength. (a) The decreased parameter C with the increasing Al content in Cu-Al alloys, which is in accordance with the decreasing trend of E (the inserted figure). (b) Calculated curves of σw changing with Al content (substituted values of σy and σb have been marked out) and corresponding tested fatigue strength of Cu and Cu-Al alloys.

Fig. 4. The linear σb--σy relations and the influencing factors of slope a. (a) F138 steels produced by rolled and equal-channel angular pressing (ECAP) [21]; (b) 7075 Al alloys produced by cryorolled and room-temperature rolled processes [22]; (c) Cu and IF steel produced by the same ECAP and cold-rolled preparation process [23].

Fig. 5. Influences of preparation processes on the microstructure characteristic coefficient a and the tendency of fatigue strength. (a) The linear σb--σy relations of wrought Al alloys with different processing types and states [8]; corresponding values of a have been marked out. (b) A negative correlation between a and σ0. (c) The σw--σy curves with different tendencies influenced by the value of a·C. (H: strain-hardened state; T: solution heat-treated state).

Fig. 6. Microstructure morphologies of Cu-11at.%Al with rather uniform equiaxial grains: (a) ①-coarse grain (CG); (b) ②-fine grain (FG); (c) ③-ultra-fine grain (UFG); (d) ④-nano-grain (NG). Average grain sizes dˉ have been marked out.

Fig. 7. Influences of grain size on yield strength σy, tensile strength σb and fatigue strength σw in Cu-11at.%Al with rather uniform microstructures. (a) Hall-Petch relations between σy, σb and average grain size dˉ. (b) Linear σw/σy--σy/σb relation. (c) Calculated curve of σw changing with dˉ and corresponding tested fatigue strength data; for the rise-fall trend, there exists a critical grain size dcri corresponding to the maximum fatigue strength (the star mark).

Fig. 8. Microstructure morphologies and cracking behaviors of Cu-5at.%Al with various uniformity. (a) Typical microstructures of ① cold-rolled state (CR); ②③④ partially recrystallized states (cold-rolled and annealed at 300/325/350 °C, marked as CR + A300/325/350) and ⑤⑥ fully recrystallized states (cold-rolled and annealed at 425/500 °C, marked as CR + A425/500). (b) Typical initiation location of fatigue cracks: ① grain coarsening region; ② recrystallized region.

Fig. 9. Influences of grain size on yield strength σy, tensile strength σb and fatigue strength σw in Cu-5at.%Al with part of non-uniform microstructures (marked as hollow points). (a)(b) Hall-Petch relations between σy (a), σb (b) and average grain size dˉ; grain size distributions with the maximum grain sizes dmax and minimum grain sizes dmin have been marked out. (c) Original linear σw/σy--σy/σb relation. (d) Calculated curve of σw changing with dˉ and corresponding tested fatigue strength data; the data of non-uniform states (②③④ hollow points) derived from the curve. (e) Modified linear σw/σy--σy/σb relation, in which original parameters σy and σb are replaced by the localized properties σy-loc and σb-loc. (f) Modified curve of σw changing with dloc (dmax) and corresponding tested fatigue strength data; the original deviation of non-uniform data (②③④ hollow points) are diminished.

Fig. 10. Influences of defects on the damage concentration coefficient ω and fatigue strength. (a) Typical values of ω for aluminum alloys [1]; the ω of cast states is commonly larger than that of wrought states. (b) Linear σw/σy--σy/σb relations of low-carbon steels with or without inclusions [1,4]; the line turns downward with the increasing defect dimension, corresponding to an increasing ω.

Table 2 Fatigue properties of two series of low carbon steels (one with different tensile properties, another with different defect sizes).

Materials	Components	Yield strength, σ_y (MPa)	Tensile strength, σ_b (MPa)	Fatigue strength, σ_w (MPa)	Defect size, D (μm)	ω
Low carbon steel-1 [1]	Armco iron, 0.012C, 0.07Mn	103	284	191	~1	0.37
	Ingot iron, 0.015C, 0.03Mn	156	298	182
	Ingot iron, 0.014C, 0.015Mn	163	298	197
	Wrought iron, 0.017C, 0.122Si	204	319	204
	Wrought iron, 0.017C, 0.122Si	197	319	190
	Steel, 0.13C	304	465	248
	Steel, 0.15C, 0.66Mn	293	387	222
Low carbon steel-2 [4]	Annealed low carbon steel	~156	~298	181	60	0.41
				172	93	0.43
				157	136	0.47
				157	119	0.48
				147	185	0.51
				137	272	0.54
				142	298	0.53
				128	463	0.58
				118	681	0.63

Fig. 11. Illustration of the defect related influencing factors on fatigue strength. (a) Different forms of defects induced fatigue crack initiation [27]: (a1) scratch on surface; (a2) inclusion in subsurface. (b) Common features of defects: the origin of stress-strain concentration. (c) Main factors and the influences on stress concentration: (c1) defect dimension D, is related to stress distribution gradient; (c2) defect shape Kt, affects peak value of stress concentration; (c3) property of matrix material σy, σb that influences the plastic zone.

Fig. 12. Influences of defect dimension D on the damage concentration coefficient ω and fatigue strength. (a) Linear relations between ω/ω0 and D in double logarithm coordinate of various steels [4,28,[30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]]; (b) Linear relations between σw and D in double logarithm coordinate of these steels [4,28,[30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]]. The value of slope kω (1/6) in the model of Murakami has been marked out.

Table 3 Fatigue properties of various steels (divided into 4 types S1-S4 according to the hardness and strength) with different defect sizes.

Materials	Hardness,HV	Defect size, D (μm)	Fatigue strength, σ_w (MPa)	ω/ω₀	Type R-square	k
S10C(A) [ 30,31,32,33,34,35]	120	105	632	3.71	S1	k = 0.17 σ_b = 464 MPa
		134	316	2.91
		172	74	2.26
		181	60	2.15
		172	93	2.26
		157	136	2.48
		157	119	2.48
		147	185	2.65
		137	272	2.84
		142	298	2.74
		128	463	3.04
		118	681	3.30
		123	632	3.17
S30C(A) [36]	153	220	16	1.59	S1
		225	16	1.55
		199	32	1.76
		204	32	1.71
		208	32	1.68
		196	63	1.78
		140	316	2.50
		144	316	2.43
S35C(A) [37,38,39]	160	120	632	7.28	S1
		122	632	7.16
		126	632	6.93
		144	316	6.07
		127	474	6.88
		101	949	8.65
		152	409	5.75
		137	681	6.38
S45C(A1) [40,41,42]	180	280	16	1.70	S2	k = 0.19 σ_b = 569 MPa
		275	16	1.73
		245	32	1.94
		250	32	1.90
		245	32	1.94
		160	316	2.97
		151	316	3.14
S45C(A2) [30,31]	170	235	37	1.82	S2
		226	46	1.89
		226	68	1.89
		230	48	1.86
		211	74	2.03
		201	109	2.13
		226	60	1.89
		201	93	2.13
		196	93	2.18
		191	136	2.24
		201	119	2.13
		181	185	2.37
		172	272	2.49
		181	298	2.36
		157	463	2.72
		147	681	2.91
S50C(A) [43,44]	177	144	316	3.59	S2
		196	95	2.64
		160	379	3.23
		133	538	3.89
		123	791	4.20
S50C(T) [36,43]	319	209	316	4.39	S3	k = 0.23 σ_b = 1108 MPa
	378	252	316	3.64
	375	468	16	1.96
		478	16	1.92
		373	63	2.46
		252	316	3.64
YUS170 [45]	244	304	93	2.42	S3
		294	139	2.50
		265	185	2.78
S45C(Q) [46,47]	650	667	37	2.62	S4	k = 0.25 σ_b = 1843 MPa
		568	62	3.07
		559	93	3.12
		470	117	3.71
S45C(T) [46,47]	520	568	37	2.45	S4
		519	62	2.68
		421	117	3.31
		382	202	3.64
SUS603 [45]	355	530	37	1.97	S4
		441	74	2.37
		441	93	2.37
		432	139	2.42
		373	185	2.80
Maraging Steel [48]	720	686	19	2.00	S4
		677	37	2.03
		530	93	2.59
		441	185	3.12
		546	63	2.52
		454	95	3.03
60Si2CrV [28]	538	632	32	2.92	S4
	543	662	29	2.79
	565	560	31	3.30
	571	675	20	2.74
	562	750	16	2.46
	558	760	18	2.43

Table 3 Fatigue properties of various steels (divided into 4 types S1-S4 according to the hardness and strength) with different defect sizes.

Materials	Hardness,HV	Defect size, D (μm)	Fatigue strength, σ_w (MPa)	ω/ω₀	Type R-square	k
S10C(A) [ 30,31,32,33,34,35]	120	105	632	3.71	S1	k = 0.17 σ_b = 464 MPa
		134	316	2.91
		172	74	2.26
		181	60	2.15
		172	93	2.26
		157	136	2.48
		157	119	2.48
		147	185	2.65
		137	272	2.84
		142	298	2.74
		128	463	3.04
		118	681	3.30
		123	632	3.17
S30C(A) [36]	153	220	16	1.59	S1
		225	16	1.55
		199	32	1.76
		204	32	1.71
		208	32	1.68
		196	63	1.78
		140	316	2.50
		144	316	2.43
S35C(A) [37,38,39]	160	120	632	7.28	S1
		122	632	7.16
		126	632	6.93
		144	316	6.07
		127	474	6.88
		101	949	8.65
		152	409	5.75
		137	681	6.38
S45C(A1) [40,41,42]	180	280	16	1.70	S2	k = 0.19 σ_b = 569 MPa
		275	16	1.73
		245	32	1.94
		250	32	1.90
		245	32	1.94
		160	316	2.97
		151	316	3.14
S45C(A2) [30,31]	170	235	37	1.82	S2
		226	46	1.89
		226	68	1.89
		230	48	1.86
		211	74	2.03
		201	109	2.13
		226	60	1.89
		201	93	2.13
		196	93	2.18
		191	136	2.24
		201	119	2.13
		181	185	2.37
		172	272	2.49
		181	298	2.36
		157	463	2.72
		147	681	2.91
S50C(A) [43,44]	177	144	316	3.59	S2
		196	95	2.64
		160	379	3.23
		133	538	3.89
		123	791	4.20
S50C(T) [36,43]	319	209	316	4.39	S3	k = 0.23 σ_b = 1108 MPa
	378	252	316	3.64
	375	468	16	1.96
		478	16	1.92
		373	63	2.46
		252	316	3.64
YUS170 [45]	244	304	93	2.42	S3
		294	139	2.50
		265	185	2.78
S45C(Q) [46,47]	650	667	37	2.62	S4	k = 0.25 σ_b = 1843 MPa
		568	62	3.07
		559	93	3.12
		470	117	3.71
S45C(T) [46,47]	520	568	37	2.45	S4
		519	62	2.68
		421	117	3.31
		382	202	3.64
SUS603 [45]	355	530	37	1.97	S4
		441	74	2.37
		441	93	2.37
		432	139	2.42
		373	185	2.80
Maraging Steel [48]	720	686	19	2.00	S4
		677	37	2.03
		530	93	2.59
		441	185	3.12
		546	63	2.52
		454	95	3.03
60Si2CrV [28]	538	632	32	2.92	S4
	543	662	29	2.79
	565	560	31	3.30
	571	675	20	2.74
	562	750	16	2.46
	558	760	18	2.43

Fig. 13. Influences of defect shape (Kt) on the damage concentration coefficient ω and fatigue strength. (a) Linear relations between ω/ω0 and D in double logarithm coordinate of 45 steel and 40Cr steel [29], with the slop kω changing with Kt; a group data of low-carbon steel [4] with constant kω has also been included. (b) Linear relations between σw and D in double logarithm coordinate of these steels, with the slop -kω changing with Kt. (c) The natural exponential relations between kω and Kt of 45 steel and 40Cr steel; a reference line with kω = 1/6 (the model of Murakami) has been marked out, which is approximately the slope of the low-carbon steel (the inserted figure). (d) Calculated curves of σw changing with Kt and corresponding tested fatigue strength data of 45 steel and 40Cr steel; corresponding values of D have been marked out.

Table 4 Fatigue properties of two steels (normalized 45 steel and quenched and tempered 40Cr steel) with U-shape notches of various sizes and shapes.

Materials	Notch size, D (μm)	Notch shape,Kt	Fatigue strength, σ_w (MPa)	ω/ω₀	k
Normalized45 steel(σ_b = 660 MPa) [29]	~1	1	318	1	0
	170	4.62	144	2.21	0.154
	221	5.14	131	2.43	0.164
	315	5.76	108	2.94	0.188
	236	3.49	155	2.05	0.131
	309	3.85	148	2.15	0.133
	442	4.38	114	2.79	0.168
	376	2.35	201	1.58	0.077
	511	2.53	180	1.77	0.092
	749	2.68	154	2.06	0.109
	528	1.78	234	1.36	0.049
	724	1.84	221	1.44	0.055
	1128	1.92	206	1.54	0.061
	1304	1.34	286	1.11	0.015
	1346	1.35	277	1.15	0.019
	1750	1.36	277	1.15	0.019
Quenched and tempered 40Cr steel (σ_b = 1945 MPa) [29]	~1	1	865	1	0
	170	4.62	322	2.69	0.193
	221	5.14	280	3.09	0.209
	315	5.76	240	3.61	0.223
	236	3.49	373	2.32	0.154
	309	3.85	335	2.58	0.165
	442	4.38	285	3.04	0.183
	376	2.35	489	1.77	0.096
	511	2.53	433	2	0.111
	749	2.68	386	2.24	0.122
	528	1.78	592	1.46	0.060
	724	1.84	558	1.55	0.067
	1128	1.92	524	1.65	0.071
	1304	1.34	733	1.18	0.023
	1346	1.35	752	1.15	0.019
	1750	1.36	739	1.17	0.021

Fig. 14. Influences of matrix material (σb) on the damage concentration coefficient ω and fatigue strength. (a) Linear relations between σw and D in double logarithm coordinate of the four series of steels S1-S4 [4,28,[30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]] (classified according to the strength of matrix materials σb), with the slop -kω changing with σb. (b) The natural exponential relations between kω and σb of the S1-S4 steels; data of 45 steel and 40Cr steel with various Kt are also included. (c) Correlations among Kt and the two parameters kσb and σb0: a natural exponential kσb--Kt relation, and a weak correlation between σb0 and Kt (the insert figure). (d) Calculated curves of σw changing with σb and the corresponding tested fatigue strength data of 45 steel; the corresponding values of D and Kt have been marked out.

Fig. 15. Illustration of the defect related influences considered in the Y-T-F model. (a) Comparisons of the fatigue models [4,[49], [50], [51], [52]] considering the effect of defects, and the various values of kω changing with defect size; (b) influences of defect shape and the quantified relation between kω and Kt in the Y-T-F model; (c) influences of matrix material and the quantified relation between kω and σb in the Y-T-F model.

Fig. 16. Comprehensive influences of defect related factors on fatigue strength. (a) Calculated curves of σw changing with σb under a certain Kt and various of D. (b) Calculated curves of σw changing with σb under a certain D and various of Kt. Parameter values have been marked out, as well as the peak values of the curves (star marks).

Fig. 17. Illustration of possible approaches for fatigue strength improvement. (a) Three series of parameters in the Y-T-F model: the component related damage capacity C; the microstructure related parameters a, σy and σb; and the defect related damage concentration coefficient ω; possible changing trends of the parameters for fatigue strength improvement are marked out by arrows. (b) Trends of fatigue strength changing with the above three types of factors, and detailed approaches for fatigue strength improvement: (b1) increase elastic modulus (increase C); (b2) balance microstructures (change a, σy and σb); (b3) reduce defect damage (decrease ω).

References 52

[1]	P.G. Forrest, Fatigue of Metals, Pergamon Press, Oxford 1962.
[2]	A.S. Argon, Strengthening Mechanisms in Crystal Plasticity, Oxford University Press, New York 2008.
[3]	S. Suresh, Fatigue of Materials, Cambridge University Press 1998.
[4]	Y. Murakami, Oxford 2002.
[5]	M. Meyers, K. Chawla, Mechanical Behavior of Materials (Second Edition), Cambridge University Press 2009.
[6]	J.C. Pang, S.X. Li, Z.G. Wang, Z.F. Zhang, Nucl. Eng., 37(2014), pp. 958-976.
[7]	R. Liu, P. Zhang, B. Wang, Y.Z. Tian, Z.J. Zhang, Z.F. Zhang, J. Mater. Sci. Technol. (2020), in press.
[8]	J.R. Davis, USA 1998.
[9]	H.J. Gough, London 1924.
[10]	H.J. Gough, H.V. Pollard, W.J. Clenshaw, Aero. Res. Council R. (1951), p. 2522.
[11]	G.A. Hankins, D. Hanson, G.W. Ford, J. Iron Steel Inst., 114(1926), p. 265.
[12]	W.E. Bardgett, Iron Steel, 29(1956), p. 392.
[13]	R. Liu, Y.Z. Tian, Z.J. Zhang, P. Zhang, Z.F. Zhang, Mater. Sci. Eng. A, 702(2017), pp. 259-264. DOI URL
[14]	X.H. An, S.D. Wu, Z.G. Wang, Z.F. Zhang, Acta Mater., 74(2014), pp. 200-214. DOI URL
[15]	X.H. An, Q.Y. Lin, S.D. Wu, Z.F. Zhang, Mater. Res. Lett., 3(2015), pp. 135-141. DOI URL
[16]	R. Liu, et al., Acta Mater., 144(2018), pp. 613-626. DOI URL
[17]	P. Xue, et al., Sci.China Mater., 59(2016), pp. 531-537.
[18]	Anon, Fatigue Properties of Some Noral Wrought Aluminium Alloys, Aluminium Labs. Ltd. (1952), Research Bulletin No. 1.
[19]	M.F. Ashby, Materials Selection in Mechanical Design, (4th ed.), Butterworth-Heinemann, Elsevier Ltd., Burlington 2011.
[20]	P. Sharma, S. Ganti, J. Mater. Res., 18(2003), pp. 1823-1826. DOI URL
[21]	N.S. De Vincentis, M.C. Avalos, A.M. Kliauga, V.L. Sordi, N. Schell, H.G. Brokmeier, R.E. Bolmaro, Deformation analysis on F138 austenitic stainless steel: ECAE and rolling, 6th International Conference on Nanomaterials by Severe Plastic Deformation 2014.
[22]	P. Das, R. Jayaganthan, T. Chowdhury, I.V. Singh, Mater. Sci. Eng. A, 528(2011), pp. 7124-7132. DOI URL
[23]	A.A. Gazder, S.S. Hazra, C.F. Gu, W.Q. Cao, C.H.J. Davies, E.V. Pereloma, Mechanical, microstructure and texture properties of interstitial-free steel and copper subjected to equal channel angular extrusion and cold-rolling, 15th International Conference on the Strength of Materials 2010.
[24]	E.O. Hall, Proc. Phys. Soc. Lond. B, 64(1951), p. 747. DOI URL
[25]	N.J. Petch, J. Iron Steel Inst., 174(1953), pp. 25-28.
[26]	Y. Wang, M. Chen, F. Zhou, E. Ma, Nature, 419(2002), pp. 912-915. DOI URL
[27]	J.C. Pang, S.X. Li, Z.G. Wang, Z.F. Zhang, Mater. Sci. Eng. A, 564(2013), pp. 331-341. DOI URL
[28]	S.X. Li, Int. Mater. Rev., 57(2012), pp. 92-114. DOI URL
[29]	S.B. Zhao, Handbook of Anti-fatigue Design, China Machine Press, Beijing 2015.
[30]	Y. Murakmi, M. Endo, Trans. Jpn. Soc. Mech. Eng. A, 49(1983), pp. 127-136. DOI URL
[31]	Y. Murakami, S. Fukuda, T. Endo, Trans. Jpn. Soc. Mech. Eng. Ser. I, 44(1978), pp. 4003-4013.
[32]	H. Nisitani, M. Kage, Trans. Jpn. Soc. Mech. Eng. Ser. I, 39(1973), pp. 2005-2012.
[33]	H. Ohba, Y. Murakami, T. Endo, Trans. Jpn. Soc. Mech. Eng. A, 49(1983), pp. 901-910. DOI URL
[34]	H. Nisitani, S. Nishida, Trans. Jpn. Soc. Mech. Eng. Set I, 35(1969), pp. 2310-2315.
[35]	H. Nisitani, Y. Murakami, Trans. Jpn. Soc. Mech. Eng. Ser. I, 35(1969), pp. 1389-1396.
[36]	H. Nisitani, Correlation between notch sensitivity of a material and its non-propagating crack, under rotating bending stress, Proc. 1st Int. Conf. Mech. Behav. Mater., Kyoto, I(1972), pp. 312-322.
[37]	H. Nisitani, K. Kawano, Non-propagating crack and crack strength of shafts with a shoulder fillet subjected to rotary bending, Proc. 11th Jpn. Congr. Mater. Res.-Metallic Mater(1968), pp. 49-51.
[38]	H. Kobayashi, H. Nakazawa, Trans. Jpn. Soc. Mech. Eng. Ser. I, 35(1969), pp. 1856-1863.
[39]	H. Nisitani, K. Kawano, Trans. Jpn. Soc. Mech. Eng., Ser. I, 37(1971), pp. 1492-1496.
[40]	H. Nisitani, M. Endo, Trans. Jpn. Soc. Mech. Eng. A, 51(1985), pp. 1008-1016. DOI URL
[41]	H. Nisitani, M. Endo, Trans. Jpn. SOC. Mech. Eng. A, 51(1985), pp. 784-789. DOI URL
[42]	H. Nisitani, M. Endo, ASTM STP, 924(1988), pp. 136-153.
[43]	H. Nisitani, I. Chishiro, Trans. Jpn. Soc. Mech. Eng. Ser. I, 40(1974), pp. 41-52.
[44]	H. Kobayashi, H. Nakazawa, Trans. Jpn. Soc. Mech. Eng. Ser. I, 36(1970), pp. 1789-1798.
[45]	S. Nishida, Private Communication, 1985.
[46]	Y. Murakami, H. Kawano, T. Endo, Trans. Jpn. Soc. Mech. Eng. A, 45(1979), pp. 1479-1486. DOI URL
[47]	Y. Murakami, T. Endo, The effects of small defects on the fatigue strength of hard steels, Proc. Int. Conf. Fatigue’ 81, Materials Experimentation and Design, Warwick Univ. (1981), pp. 431-440.
[48]	Y. Murakami, M. Abe, T. Kiyota, Trans. Jpn. Soc. Mech. Eng. A, 53(1987), pp. 1482-1491. DOI URL
[49]	Y. Murakami, M. Endo, Int. J. Fatigue, 16(1994), pp. 163-182. DOI URL
[50]	N.E. Frost, K.J. Marsh, L.P. Peek, Metal Fatigue, Oxford University Press, Oxford 1974.
[51]	H. Kobayashi, H. Nakazawa, Trans. Jpn. Soc. Mech. Eng., 36(1969), p. 1856.
[52]	M.H. El Haddad, T.H. Topper, K.N. Smith, Eng. Fract. Mech., 11(1979), pp. 573-584. DOI URL