A practical model for efficient anti-fatigue design and selection of metallic materials: I. Model building and fatigue strength prediction

doi:10.1016/j.jmst.2020.08.038

Abstract

Abstract:

The high cost and low efficiency of fatigue tests are bottleneck problem for the anti-fatigue design of metallic materials. For this problem, a theoretical fatigue model is proposed in this study, the possible applications have also been discussed. Specific results would be introduced in two serial papers, in which the first paper focuses on the model building and the applications on fatigue strength prediction; the second paper put emphasis on the influencing factors of the model parameters and the applications on fatigue strength improvement. In this first paper, a theoretical model is proposed considering both the strength and plastic restrictions of fatigue strength. As the model builds up a brief relationship among yield strength (Y), tensile strength (T) and fatigue strength (F), it is named as the Y-T-F model. Through the verification with fatigue strength data covering various kinds of metallic materials and loading conditions, this Y-T-F model exhibits both generality and accuracy. With the Y-T-F model, the efficient fatigue strength prediction could be conducted by brief linear fitting and calculation, just through yield strength, tensile strength and several known fatigue strength data. Moreover, through its deduced Y-F model, the analytical formula of fatigue strength continuously changing with materials strengthening can be obtained, as well as the maximum value of fatigue strength and corresponding critical yield strength. In summary, the Y-T-F model would be useful for reducing the fatigue tests, thus providing new possibilities on the efficient anti-fatigue design and selection of metallic materials.

Key words: Yield strength, Tensile strength, Plasticity, Fatigue strength, 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: I. Model building and fatigue strength prediction[J]. J. Mater. Sci. Technol., 2021, 70: 233-249.

Figures/Tables 21

Fig. 1. Deviation from the linear trend of the tensile strength--fatigue strength relation and possible underlying mechanisms. (a) Typical tensile strength--fatigue strength relation of steels [9]. (b) Tensile strength--fatigue strength relations of various metallic materials with different fatigue damage mechanism transitions: (b1) 40CrNiMo [7]; (b2) Cu-5Al [6,28,29]; (b3) Cu-15Al [6,28,29]; (b4) QBe2 [7,41]. (c) Influences of strength and plasticity restrictions to the fatigue strength and the non-monotonic changing trend of tensile strength--fatigue strength relation caused by the transition of dominant restrictions.

Table 1 Tensile and fatigue properties of pure Cu and Cu-Al alloys with different microstructures.

Materials	Microstructures	Yield strength, σ_y (MPa)	Tensile strength, σ_b (MPa)	Hardening capacity, (σ_b-σ_y)/σ_b	Fatigue strength, σ_w (MPa)	Source/Ref.
Cu	CG	43	210	0.795	50	[28,29]
	UFG-FSP	302	333	0.093	120	[30]
	UFG-ECAP	374	422	0.114	100	[28,29]
	UFG-HPT	391	481	0.187	100	[28,29]
Cu-5at.%Al	CG	66	257	0.743	75	[28,29]
	FG-CR + A500	150	296	0.493	120	[6,18]
	UFG-CR + A425	213	321	0.336	155	[6,18]
	MG-CR + A350	295	361	0.183	160	[18]
	MG-CR + A325	356	418	0.148	160	[18]
	MG-CR + A300	533	595	0.104	170	[18]
	NG-CR	612	675	0.093	210	[18]
	NG-ECAP	467	573	0.185	170	[28,29]
	NG-HPT	613	776	0.210	150	[28,29]
Cu-11at.%Al	CG	82	337	0.757	100	[28,29]
	FG-CR + A500	226	448	0.496	190	[6]
	UFG-CR + A400	302	482	0.373	210	[6]
	NG-ECAP	619	770	0.196	190	[28,29]
Cu-15at.%Al	CG	90	396	0.773	110	[28,29]
	FG-CR + A500	310	547	0.433	250	[6]
	UFG-CR + A400	395	592	0.333	280	[6]
	NG-HPT	734	942	0.221	200	[28,29]

Fig. 2. Preliminary model construction in Cu and Cu-Al alloys. (a) Fatigue strength of Cu and Cu-Al alloys (Cu-5Al, Cu-11Al and Cu-15Al) changing with the average grain size (CG, FG, NG). (b) Trade-off relation between σy and (σb-σy)/σb, corresponding to strength and plasticity respectively. (c) The linear σw/σy--σy/σb relation of Cu and Cu-Al alloys, within an error about ±10 %. Details of the experimental results are in Ref. [6,28,29].

Fig. 3. Illustration of a restrictive relation between the two ratios of the Y-T-F model: σw /σy, which reflects the extent of fatigue damage localization; σy /σb, which reflects the resistance to deformation localization.

Fig. 4. Generality verification of the Y-T-F model: the σw/σy--σy/σb relation in different metallic materials. (a) Steels [9,[31], [32], [33], [34], [35], [36], [37], [38]]; (b) copper alloys [6,9,18,[28], [29], [30],[39], [40], [41]]; (c) aluminum alloys [9,42,43]; (d) magnesium alloys [9]; the data generally show good linear relations.

Fig. 5. Generality verification of the Y-T-F model: the σw/σy--σy/σb relation in different loading conditions. (a) Different fatigue lives Nf (105,106,107) [9,42]; (b) different loading ratios R (-1,0) [34,36,38]; (c) different loading directions (R.B.: rotating bending; Ax.: tension-compression) [9,31]; the data generally keep good linear relations despite the changing conditions.

Table 2 The parameter values of the Y-T-F model for some metallic materials.

Type	Materials	Loading condition	ω	C	R-square	Source/Ref.
Carbon Steel	Low carbon steels	R.B., R=-1, fatigue limit	0.30	0.91	0.96	[9,31,32,33]
Carbon Steel	Medium & high carbon steels	R.B., R=-1, fatigue limit	1.31	1.61	0.89	[9,31,32,33]
Alloy Steel	SPCC&SPRC-YJ Chang	Ax., R = 0, N_f=10⁷	0.63	1.50	0.99	[34]
	Si-Mn steel-Hankins	R.B., R=-1, N_f =10⁷	1.08	1.50	0.81	[33]
	En24& En36-Bardgett		0.76	1.33	0.94	[9,35]
	En56~ En58-Firth		0.94	1.38	0.83	[9]
Pre-strained Steel	TWIP Steel-B Wang	Ax., R=-1, N_f =10⁷	1.28	1.28	0.99	[36]
	TWIP Steel-YW Kim	Ax., R = 0.1, N_f = 2 × 10⁶	0.74	1.37	0.98	[37]
	SAE 1010 steel-MT Yu	Ax., R=-1, N_f = 4 × 10⁶	1.12	1.49	0.81	[38]
Cu Alloy	Cu-Al-XH An & R Liu & P. Xue	Ax., R=-1, N_f =10⁷	0.83	1.22	0.98	[6,18,28,29,30]
	Cu-Zn-ZJ Zhang		0.34	0.75	0.91	[39]
	Phosphor bronze-Anderson		0.50	1.14	0.96	[9,40]
	QBe2-JC Pang	Ax., R=-1, N_f =10⁷	0.56	0.99	0.93	[41]
Al Alloy	Wrought Al-Banbury	R.B., R=-1, N_f =10⁷	0.55	1.09	0.83	[9,42]
	Wrought Al-Bucks	R.B., R=-1, N_f =10⁷	0.56	1.03	0.82	[9]
	Wrought Al-ASM	R.B., R=-1, N_f = 5 × 10⁸	0.58	1.17	0.96	[43]
	Cast Al-Bucks	R.B., R=-1, N_f =10⁷	0.74	1.13	0.46	[9]
Mg Alloy	Mg 1-Magnesium Elektron	R.B., R=-1, N_f = 5 × 10⁷	0.84	1.13	0.78	[9]
	Mg 2-Magnesium Elektron		1.31	1.64	0.84	[9]
	Mg 3-J Stone		0.70	1.09	0.34	[9]

Fig. 6. Accuracy verification of the Y-T-F model: comparisons between the experimental tested fatigue strength (σw-experimental) and corresponding calculated value (σw-calculated) predicted by (a) Model-1 (Eq. (3)) [19]; (b) Model-2 (Eq. (4)) [20]; (c) Model-3 (Eq. (5)) [20]; (d) Y-T-F model (Eq. (2)). All the comparisons are based on the same group of experimental data for SAE-1141 steel, with the error bands of ±10 % displayed as dashed lines.

Table 3 Comparisons of the fatigue strength calculation results based on the Y-T-F model and another three models (Data of SAE-1141 steel).

States	σ_y (MPa)	σ_b (MPa)	σ_w-Exp (MPa)	σ_w-Mod1 (MPa)	Error-Mod1 (%)	σ_w-Mod2 (MPa)	Error-Mod2 (%)	σ_w-Mod3 (MPa)	Error-Mod2 (%)	σ_w-YTF (MPa)	Error-YTF (%)
A1	457	771	286	358	25.17	262	-8.39	281	-1.75	299	4.55
A2	814	925	433	420	-3.00	353	-18.48	332	-23.33	421	-2.77
A3	418	695	276	324	17.39	251	-9.06	254	-7.97	272	-1.45
A4	602	802	342	367	7.31	293	-14.33	289	-15.50	348	1.75
A5	450	725	287	332	15.68	274	-4.53	262	-8.71	288	0.35
A6	610	797	332	369	11.14	296	-10.84	290	-12.65	349	5.12
A7	493	789	296	355	19.93	293	-1.01	281	-5.07	315	6.42

Fig. 7. Accuracy verification of the Y-T-F model: comparisons between the experimental tested fatigue strength (σw-experimental) [6,9,18,[31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]] and the Y-T-F model predicted results (σw-calculated), which covers various metallic materials and loading conditions; the error bands of ±10% displayed as dashed lines.

Fig. 8. Illustration of the main steps of fatigue strength prediction by the Y-T-F model. (a) Step 1. tensile test; (b) step 2. selective fatigue test; (c) step 3. linear fitting; (d) step 4. fatigue strength calculation.

Fig. 9. Examples for fatigue strength prediction by the Y-T-F model. (a) Pre-stretched TWIP steel [36]; (b) P/M diffusion alloyed-steel [43]; (c) 1060-H1x Al alloy [43]; (d) 5052-H3x Al alloy [43]. (a1, b1, c1, d1) the σw/σy--σy/σb relations; (a2, b2, c2, d2) comparisons between the experimental tested fatigue strength (σw-experimental) and corresponding calculated data (σw-calculated), with the error bands of ±10 %. Filled points: experimental data; hollow points: calculated data; half-filled points: verified data.

Table 4 Cases of fatigue strength prediction based on the Y-T-F model.

Materials	States	σ_y (MPa)	σ_b (MPa)	σ_w-Exp (MPa)	YTF parameters	σ_w-YTF (MPa)	Error (%)
Fe-30Mn-0.9C TWIP steel [36]	Pre-0%	350	960	250	C = 1.27, ω = 1.27	250	--
	Pre-30 %	940	1200	350		361	3.26
	Pre-60 %	1370	1460	360		360	--
	Pre-70 %	1570	1610	380		367	-3.51
P/M diffusion alloyed-steel [43]	FD-0205-45	360	470	170	C = 1.33, ω = 0.94	170	--
	FD-0205-50	390	540	200		200	0.07
	FD-0205-55	420	610	220		229	4.00
	FD-0205-60	460	690	260		260	--
1060 Al alloy [43]	1060-O	30	70	20	C = 1.67, ω=1.87	20	--
	1060-H12	75	85	30		32	5.93
	1060-H14	90	95	35		35	--
	1060-H16	105	110	45		40	-10.16
	1060-H18	125	130	45		48	5.92
5052 Al alloy [43]	5052-O	90	195	110	C = 1.20, ω = 0.60	110	--
	5052-H32	195	230	115		114	-1.13
	5052-H34	215	260	125		133	6.24
	5052-H36	240	275	130		130	--
	5052-H38	255	290	140		135	-3.29

Fig. 10. Linear relations between tensile strength σb and yield strength σy. (a) Cu and Cu-Al alloys [6,51]; (b) engineering materials including 304 stainless steel [52], Cu-Nb alloy [53] and SAE 1010 steel [38].

Fig. 11. Examples of the Y-T-F model derived Y-F relation. (a) QBe alloy [41] with a?C<1; (b) Al alloy [9,42] with a?C≈1; (c) SPCC steel [34] with a?C>1. (a1, b1, c1) the linear σw/σy--σy/σb relations; (a2, b2, c2) the linear σb--σy relations; (a3, b3, c3) the σw--σy trends.

Fig. 12. Different changing trends of the Y-F relation with the value of a?C. (a) Y-F tendency changing with the value of C; (b) Y-F tendency changing with the value of a; (c) values of C, a and a?C changing with Al content in Cu and Cu-Al alloys; (d) Y-F relations of Cu and Cu-Al alloys (a?C<1), and the virtual trends when a?C=1 and a?C>1.

Fig. 13. Illustrations of the formation of different Y-F trends. (a) a?C<1, the rise and fall trend; (b) a?C=1, rise up to the horizontal asymptote; (c) a?C>1, rise monotonically till the up limit strength.

Fig. 14. Examples for fatigue strength prediction through the Y-F relation. (a) SAE-1141 steel [49]; (b) P/M Fe-Ni and Ni-steel [43]; (c) 202x-Tx Al alloy [43]; (d) 3004-H3x Al alloy [43]. (a1, b1, c1) the linear σw/σy--σy/σb relations; (a2, b2, c2) the linear σb--σy relations; (a3, b3, c3) the σw--σy trends, with the error bands of ±10 %. Filled points: experimental data; hollow points: calculated data; half-filled points: verified data.

Table 5 Cases of fatigue strength prediction based on the Y-F model.

Materials	States	σ_y (MPa)	σ_b (MPa)	σ_w-Exp (MPa)	YF parameters	σ_w-YF (MPa)	Error (%)
SAE-1141 steel [49]	A1	457	771	286	C = 1.96, ω = 2.09, a = 0.58, σ₀ = 449 MPa	289	0.95
	A2	814	925	433		419	-3.12
	A3	418	695	276		271	-1.75
	A4	602	802	342		347	1.60
	A5	450	725	287		286	-0.48
	A6	610	797	332		350	5.56
	A7	493	789	296		304	2.77
P/M Fe-Ni / Ni-steel [43]	FN-0208-30	240	310	110	C = 1.49, ω = 1.55, a = 2.40, σ₀ = 266 MPa	--	--
	FN-0208-35	280	380	140		144	2.56
	FN-0208-40	310	480	170		167	-1.66
	FN-0208-45	340	550	190		--	--
	FN-0208-50	380	620	220		220	-0.14
2024/2025 Al alloy [43]	2024-O	75	185	90	C = 0.85, ω = 0.37, a = 1.14, σ₀ = 99.5 MPa	--	--
	2024-T3	345	485	140		141	0.46
	2024-T4	325	470	140		--	--
	2024-T361	395	495	125		142	13.22
	2025-T6	255	400	125		136	8.71
3004 Al alloy [43]	3004-H32	170	215	105	C = 1.21, ω = 0.67, a = 0.75, σ₀ = 87.5 MPa	--	--
	3004-H34	200	240	105		108	3.12
	3004-H36	230	260	110		--	--
	3004-H38	250	285	110		110	0.44

Fig. 15. Examples for the maximum fatigue strength prediction through the Y-F relation (a?C<1). Red line: the predicted Y-T relation; blue line: the trade-off σy--(σb-σy)/σb relation; yellow point: the maximum value of fatigue strength σwmax (point on red line) and corresponding critical yield strength σycri (point on blue line). (a) Cu; (b) Cu-5at.%Al; (c) Cu-11at.%Al; (d) Cu-15at.%Al; detailed information of the data are in Ref. [6,30,51].

Fig. 16. Flowchart of fatigue strength prediction methods by the Y-T-F model and the Y-F relation.

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