A uniaxial tensile behavior based fatigue crack growth model

International Journal of Fatigue 131 (2020) 105324

Contents lists available at ScienceDirect

International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

A uniaxial tensile behavior based fatigue crack growth model a,⁎

a

a

b

a

S.C. Wu , C.H. Li , Y. Luo , H.O. Zhang , G.Z. Kang a b

T

State Key Lab. of Traction Power, Southwest Jiaotong University, Chengdu 610031, China State Key Lab. of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan 430074, China

ARTICLE INFO

ABSTRACT

Keywords: Fatigue crack propagation Monotonic tensile strength Low cycle fatigue Linear damage accumulation Plasticity-induced crack closure

Metallic components fail mostly due to fatigue cracks, and thus a suitable non-destructive detection is necessarily planned to ensure the operation safety. Among the requisite parameters, a fatigue crack growth (FCG) model might be the most important input, which is frequently acquired from well-defined and expensive fracture mechanics experiments. By correlating the uniaxial tensile properties (UTP) with cyclic plastic parameters and developing a unified fatigue crack closure factor U suitable for broad stress ratios, a novel FCG model of improved LAPS (iLAPS) was formulated in terms of linear damage accumulation and Rice-Kujawski-Ellyin field. Results show that the newly-developed iLAPS model can simulate the entire crack growth region as like modified NASGRO does. For open-reported structural materials available with both UTP and FCG data, it is found that current iLAPS predictions are well agreed with experimental results and are also better than original LAPS.

1. Introduction As the most essential input to predict the residual lifetime of engineering metallic materials, various fatigue crack growth (FCG) models have been proposed during the last few decades in terms of phenomenology and theory. In the early 1960s, Paris et al. [1] suggested that the FCG rate log(da/dN) was approximately linear with the stress intensity factor (SIF) range log(ΔK) during intermediate range and then was nominated for the well-known phenomenological Paris’ law. Besides such a famous model, by considering loading parameters, other phenomenological models have been proposed, such as models from Walker and Forman [2,3]. As opposed to that, based on the stress and strain field ahead of crack tip, some kinds of FCG models were popularly formulated in terms of theoretical derivation [4–7]. Generally, these field solutions can be divided into Hutchinson-RiceRosengred (HRR) field and Rice-Kujawski-Ellyin (RKE) field [7]. According to the test results and the theoretical analyses, the HRR field can describe the stress and strain at crack tip, which is controlled by Jintegral. On the contrary, the form of RKE is more concise and the required parameters can be obtained with low cycle fatigue (LCF) without requiring artificial adjusting. Among those theoretical models, Glinka [4] estimated the total failure cycles by bridging the SIF range with the local stress and strain fields, based on HRR singular field. By introducing the elastic-plastic notch stress field and Manson-Coffin relation, Glinka [5] also investigated the crack tip mean stress and resulting damage accumulation. Based on plastic strain energy (PSE), Shi et al.

⁎

[6] developed a FCG model (FCG-PSE) to tackle the crack propagation of structural materials only under a positive stress ratio (R). In view of considerable effect of crack closure or negative R on fatigue lifetime estimation, Wu et al. [7] furthermore proposed a long and physically short crack (LAPS) model based on standard LCF response in the framework of RKE field. The advantage of the LAPS over original FCG-PSE is that it is suitable for broad R range. Besides, the LAPS can also simulate the fatigue short crack behaviors particularly near the SIF threshold region. Nevertheless, foregoing FCG models are still expensive mainly because of requisite LCF properties or fracture mechanics tests. It is wellknown that the LCF parameters can be calculated in terms of classical Ramberg-Osgood and Coffin-Manson models. However, the calculated parameters are high cost and poor reliability. Researchers thus attempted to correlate the UTP from uniaxial tensile tests with LCF responses [8–10]. Among these models, a modified universal slope approach [8] was well proposed to estimate the parameters with true fracture ductility coefficient εf, elastic modulus E and tensile limit σb. To obtain better predictions, the LCF properties using different equations were also estimated for steel and aluminum, called as a uniform material law method. For steel, fatigue ductility coefficient εf′ was related with the ratio of σb to E. Oppositely, the parameter is a fixed value for aluminum or titanium [8]. Besides σb is reasonably estimated through the HB value, by such translation, LCF parameters can be related with HB [10]. From these investigations, various equations or models have been widely proposed for different types of engineering materials and

Corresponding author at: State Key Lab. of Traction Power, Southwest Jiaotong University, Chengdu 610031, China. E-mail address: [email protected] (S.C. Wu).

https://doi.org/10.1016/j.ijfatigue.2019.105324 Received 21 June 2019; Received in revised form 3 October 2019; Accepted 4 October 2019 Available online 14 October 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

then validated under different materials and loading [8–10], which usually confuses the understanding and application to evaluate the fatigue resistance. Since the emergence of crack closure, the cyclic plastic zone ahead of a crack is rarely equal to fatigue process zone. In general, the crack closure is controlled by three factors, including the surface roughness, plastic constraint and surface oxidation. For one specific material, a large amount of literatures have found that the crack closure factor U can be modeled as a function of stress ratio R [11–13]. Besides, Bachmann [14] and Chand [15] found that Kmax also can influence U for titanium or aluminum. However, these crack closure functions are different in terms of the material, loading, frequency and environments. In this paper, a linear damage accumulation (LDA) and LCF assumption based FCG model has been developed by bridging the LCF parameters with UTP under broad loading ratios and materials. The newly-proposed FCG model is termed as iLAPS that means a simple but relatively universal LAPS model. For those newly-designed materials without reliable LCF parameters, the iLAPS model can achieve an accurate FCG onlyby simple and rapid uniaxial tensile properties. To further validate the novel iLAPS model, open-published fatigue data sets are extensively collected and compared in details.

U 2c (E f )1 c ( K )2 da = 1 2+1 c dN 2 yc (1 + c + cn )

E

=2

1 c f yc

c + cn · rc [1 (1 + c + cn )

(

c

rc )1 + 1

c + cn

]

rc =

4 (n + 1)

2 yc

,

c

=

p (r

2 yc

p (r

+

+

c)

c)

=

=2

2 yc E

f

(2Nf )c

( ) rc r+ c

1 (1 + n )

(4)

in which Nf is the total cycles to failure. Assuming the damage per cycle can be expressed by D(r + ρc), then we can formulate as Nf = Dcri/D (r + ρc). According to Eq. (4), the following equation can be obtained as

D (r +

c)

= 2Dcri [

yc

E f]

1 c [r c

(r +

c )]

1 (c + cn )

(1)

D¯ =

rc

c

0

D (r +

c )dr

(rc

c)

(6)

As depicted in Fig. 1, the FCG rate can be integrated via N* = Dcri/D¯

da = dN

rc 0

c

D (r +

c )dr

Dcri

(7)

Combining Eqs. (6) and (7) can generate exactly the same equation as Eq. (1). It can be clearly concluded that the Dcri has limited influence on the FCG rate.

(2)

Unfortunately, foregoing FCG model in Eq. (1) does not include the crack closure effect frequently happening in many engineering components subjected to fatigue loading (see Fig. 1), inevitably producing an inaccurate FCG rate and lifetime prediction. To improve the prediction accuracy, this paper proposed a novel FCG model entitled as iLAPS model by introducing a generalized crack closure factor U:

2.2. Crack closure factor U It is well-known that the crack closure (such as plasticity-, oxideand roughness-induced) surely plays an important effect on residual life estimation of cracked engineering parts. Meanwhile, crack closure factor U has no unified expression for the majority of engineering

Uniaxial plastic zone Cyclic plastic zone

Cyclic plastic zone rc

Blunting radius size ρc

(5)

Giving that each step of crack extension equals to the cyclic plastic zone size rc in the growth direction, a unit average damage can thus be defined as

2

Kth 4 (n + 1)

(3)

It is generally argued that Eq. (1) is satisfied only at Dcri = 1, which means an assumed value of the materials in the nominal stress framework. To examine the effect of different Dcri, the following inferences are derived. It is assumed that r is the distance to a growing crack tip, and then the plastic strain range Δεp can be expressed by [16]

in which σyc is the cyclic yield stress, c is the Manson-Coffin exponent, n′ is the cyclic strain hardening exponent, N* is the cycles to penetrate through the rc-ρc zone in which the cyclic plastic zone size rc and critical blunting size ρc can be usually estimated by relating to the long fatigue crack growth threshold ΔK (ΔKth):

K2

2 c + cn

2.1. Critical damage Dcri

A LDA based FCG model [6] was firstly proposed at critical damage Dcri = 1: c

2+

Firstly, this section investigated the influence of critical damage sum Dcri and fatigue crack closure function U on FCG rate. To further improve the modeling, the LCF parameters given in Eq. (3) were tentatively related to uniaxial tensile strength. Finally, Section 3 will verify the iLAPS model by comparing its predictions with some open-reported data of relative materials in existing references.

2. Improved fatigue crack growth model

rc da = dN N

Kth K

Process zone

Process zone

Roughness-induced crack closure

Plasticity-induced crack closure

Oxide induced crack closure

Fig. 1. Plastic behaviors and crack closure schematics of a growing crack tip. 2

International Journal of Fatigue 131 (2020) 105324

metallic materials. However, many researchers [17–19] have validated that the U can be directly correlated with the R. Therefore, a new crack closure factor U entitled as the New-Cod approach (NCA) is proposed, by improving the continuous Newman approach (CNA) and extended Codrington approach (ECA).

Crack closure factor U

1.0

2.2.1. Continuous Newman approach Based on extensive finite element simulations, a famous numerical approximation for the crack opening function f = Kop/Kmax was fitted as [17,19]

max(R; A0 + A1 R + A2 R2 + A3 R3) for R 0 A0 + A1 R for 2 R < 0 A0 2A1 for R < - 2

A 0 = (0.825 0.34 + 0.05 2)·[cos( A1 = (0.415 0.071 )·( max A2 = 1 A0 A1 A3 A3 = 2A0 + A1 1

max

f ) (1

(8)

(9)

U = 0.54 + 0.35R +

0 R 1 2 R<0

2

R

A ( ) = 0.45 + 0.27e B ( ) = 0.37 + 0.35e C ( ) = 0.20 0.67e

=

0.0

0.5

Kmax = (1 f h

1.0

0

-0.5

0.41

(14)

0.515

K R)

f

(15)

h

0

R

(16)

0.97

To verify the estimation accuracy when extending to R < 0, current Eq. (16) is compared with original Eq. (11) as shown in Fig. 3, showing the relative errors are less than 19%. Moreover, the results by Eq. (8) are just the reference values for the U, so the relative errors are acceptable and Eq. (16) can be used to calculate the U for R < 0.

Crack closure factor U

5

0.4

(13)

0

0.235

U = 0.45 + 0.37R + 0.20R2 ,

Relative error, %

Crack closure factor U

10

0.6

R

15

0.8

-1.0

-2 1.0

When η → ∞, U reaches the minimum value under plane stress condition:

(12)

1

Original PNA[18] Current CNA Relative error

-1.5

0.5

in which η is a non-dimensional parameter related to Kmax or ΔK and R. Giving that σf and the plate thickness (2h), η can be calculated by

To evaluate the prediction accuracy of Eq. (12), the detailed comparisons between two approaches are presented in Fig. 2. It is obviously found that the relative errors of two approaches are more than 10% rarely, especially the maximum value is less than 5% with R from −1 to 1, which proves that the proposed CNA can provide an excellent description to the relationship between the crack closure factor U and the stress ratio R.

0.2 -2.0

0.0

U (R, ) = A( ) + B ( ) R + C ( ) R2

(11)

1.0

-0.5

2.2.2. Extended Codrington approach Under the small-scale yield (ν = 0.3) and plane stress conditions, Codrington [20] suggested the following closure factor U, called as the Codrington approach (CA) suitable for positive R. This section extends the original CA to negative R entitled as ECA.

However, the segmented function, Eq. (11), is fairly inconvenient for engineering application. Therefore, a novel continuous closure function entitled as CNA can be obtained through fitting the data points of the formulation given above:

0.11R2 ,

4

0.4

Fig. 3. The prediction comparisons between original PNA and newly-proposed ECA.

f)

When the Poisson′s ratio ν = 0.3 and σmax/σf = 0.3, Newman [18] proposed a piecewise function to describe the crack closure factor under plane stress state based on Eq. (8), which was here called as the Piecewise Newman approach (PNA):

R)

8

0.6

Stress ratio R

(10)

0.52 + 0.42R + U= (0.52 0.1R) (1

12

0.2 -1.0

2 f )]1

R)

0.06R2

0.8

16

0

in which Kop and Kmax are the opening and maximum SIF, respectively. σmax is the maximum stress, σf is the flow strength [7]. α is the spatial constraint factor, A0, A1, A2 and A3 are the crack opening fitted parameters. The U is then derived as

U = (1

Current ECA[20] Relative error

19

0.8

20

Current ECA Current CNA Current NCA Error for ECA Error for CNA

16 12 8

0.6

Relative error, %

f=

Original PNA[18]

Relative error, %

S.C. Wu, et al.

4

0.4

0

-5 1.0

0.2 -1.0

Stress ratio R

-0.5

0.0

0.5

-2 1.0

Stress ratio R

Fig. 2. The prediction comparisons between original PNA and newly-formulated CNA.

Fig. 4. The crack closure prediction comparisons between different approaches. 3

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

2.2.3. New-Cod approach The predictions obtained by the novel ECA and CNA are displayed in Fig. 4. It can be clearly seen that the deviations are considerable. Thus, the selection of two approaches is relatively controversial in engineering applications. A unified New-Cod approach entitled as NCA can be developed by fitting the mean values of two functions.

1

R

Fatigue ductility coefficient n'

U = 0.49 + 0.36R + 0.16R2 ,

0.3

(17)

0.97

Fig. 4 shows the closure predictions by three approaches and the relative errors between the predictions by NCA and CNA or ECA. It is clearly found that the unified NCA can provide a relatively average data with the relative errors less than 12%. Therefore, Eq. (17) can be employed to depict the crack closure factor U.

0.2

0.1

0.4

Fig. 6. Correlation curve of the n′ vs. 1 − √(σyc/K′). Table 1 Prediction accuracy of the fatigue ductility coefficient εf′ Approaches

0.9–1.1 ( ± 10%)

0.8–1.2 ( ± 20%)

0.7–1.3 ( ± 30%)

Eq.(24) based on εf [35] This Eq.(26) with εf and σb/σ0.2

8.5% 23.0%

34.0% 70.2%

48.9% 87.2%

Table 2 Prediction accuracy by adjusting the Manson-Coffin stress exponent c.

(18)

The estimated results are based on above Eq. (18) and the limit of relative errors can be termed as K1′ and δK1′ = |(K1′ − K′)/ K′| as the comparison indicators, respectively. For the materials listed in Table A1, the errors are evidently unacceptable for some materials. For example, the computed δK1′ for SAE 1090 is significantly larger than 60% that cannot be accepted to evaluate the fatigue crack resistance. While the average of δK1′ is 20.77% and the standard deviation (STD) is 15.05% for these listed materials. As a result, this equation appears to show some limitations for some materials. However, a simple exponential relationship between K′ and K might be more expected when applying Eq. (18). Fig. 5 plots all data points to attempt to extract a similar equation. To examine the strength, the

Approaches

0.9–1.1 ( ± 10%)

0.8–1.2 ( ± 20%)

0.7–1.3 ( ± 30%)

The empirical value [35] Eq. (26) with εf, σ0.2/σb and σb/E

37.1% 51.4%

73.3% 79.0%

89.5% 96.2%

correlation coefficient r is assumed as 0.85. However, when sample size and significant level α are 76 and 1% (see Table A1), respectively, the required minimum value of correlation coefficient rα is 0.29 [27]. As a result, the relationship between the logarithmic variables is very close indicated by r ≫ rα and the hypothesis is supported. By solving for the cyclic strength coefficient K′, the following equation is derived (19)

K = 8K 0.72

Cyclic strength coefficient K', MPa

4000 3000 2000

For comparative purposes, the results of Eq. (19) and the limit of relative errors are denoted as K2′ and δK2′ = |(K2′ − K′)/K′|, respectively. For the listed materials in Table A1, the maximum value for δK2′, the average and the STD are 37.06%, 10.50% and 8.94%, respectively. In Fig. 5, the fitting curve and scatter bands with the factor of ± 3%, ± 5% deviation from predicted values are included. As can be clearly seen, 84.2% of the test data falls within the factor of ± 3%, while all the data falls within the factor of ± 5%. Thus Eq. (19) can provide accurate prediction to K′ for most of studied materials.

Experimental data Fitting curve deviation data within factor scatter bands ±3% 84.2% ±5% 100%

1000

400 400

1000

0.6

1-(σ yc/K')

2.3.1. Expression K′ A correspondence can be tentatively correlated between the cyclic strength coefficient K′ and uniaxial strength coefficient K. For this reason, two coefficients from open-published experimental results are collected and analyzed by [21]

1220

0.2

0.5

To acquire the FCG data of newly-developed materials, much efforts have been made to tentatively correlate the LCF parameters with simple UTP under uniaxial tensile conditions during the past decades [8,21]. Such equations are fairly complicated and diverse in the format. Based on open-reported data of metallic materials, this section aims to further refine those relationships. Finally, the predictions by newly-proposed iLAPS model will be compared with experiments and the predictions by other models.

K =

deviation data within factor scatter bands ±10% 85.1% ±20% 98.9% ±30% 100%

0.05 0.1

2.3. Tensile based cyclic parameters

57K 0.545

Experimental data Fitting curve

2000

3000

2.3.2. Expression σyc In view of the effect of the tensile limit σb to yield stress σ0.2 of an assumed material [21], the cyclic yield strength or stress σyc can be related to σ0.2 by

4000

Strength coefficient K, MPa yc

Fig. 5. The correlation analysis of cyclic strength coefficient vs. uniaxial strength.

yc

4

= 0.75

0.2

= 0.0003

+ 82 2 0.2

0.15

0.2

(

b

0.2

> 1.2)

+ 526 (

b

0.2

1.2)

(20)

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Fig. 7. The flow chart to calculate the FCG rate based on UPT parameters.

2.3.3. Expression n′ With regard to the expression of n′, several functions have been developed from the UTP or hardness results. To enhance the prediction accuracy, by adopting the similar formula of the strain hardening coefficient n under uniaxial testing, an analog approach was thus proposed in this paper. For example, in an attempt performed by Hu et al. [28], the n can be extracted empirically in terms of yield stress σ0.2 and tensile limit σb as

Table 3 Basic UTP and LCF properties of open-reported engineering materials. Materials

E, GPa

RA, %

σb, MPa

σ0.2/σyc, MPa

K/K', MPa

7075-T6 [6,10] EA4T [7,17] SAE 1020 [10,21] A533-B1 [7,39] 8630 [7,40] SAE 1050 [7,21,35] E36 [5] 35NCD16 [41] 10Ni [42,43] Spring steel [44,45] G20Mn5 S38C (outer) S38C (transition) S38C (inner)

71 214 205 200 207 203 206 191 207 210 196 220 214 206

33 68 64 66.7 69 34 75 60 72 37.4 42 34 37 41

576 748 491 600 1178 829 550 2100 1441 1930 520 1126 1090 603

498/469 554/380 285/270 430/345 999/661 460/413 390/350 1690/1405 1317/1106 1810/1490 292/– 846/– 768/– 374/–

827/787 1164/892 933/941 998/1047 –/2267 1819/1206 –/1255 –/3580 –/2177 –/3190 –/748 2901/– 3672/– 1779/–

n=1

0.2

(21)

b

On the other hand, Zhang et al. found the n′ could be estimated by the σyc and K′ [21]. By solving the n′, the following equation is derived by

n =

0.37 lg(

yc

(22)

K)

Table 4 LCF and fracture mechanics parameters corresponding to listed materials in Table 3. Materials

n′

σf′, MPa

b

εf ′

c

σmax/σ0

α

R

ΔKth, MPa m1/2

7075-T6 [6,7] EA4T [7,46]

0.088 0.11

781 900

−0.045 −0.072

0.19 0.9

−0.52 −0.69

0.3 0.3

2 3

SAE 1020 [7,17] A533-B1 [1,7] 8630 [7] SAE 1050 [7,21,35] E36 [7,42] 35NCD16 [7,41] 10Ni [7,42,43] Spring steel [7,45] G20Mn5 S38C (outer) S38C (transition) S38C (inner)

0.18 0.165 0.195 0.253 0.21 0.15 0.109 0.148 0.12 /

815 869 1936 948 1194 3050 2019 2970 570 /

−0.114 −0.085 −0.121 −0.1 −0.12 −0.11 −0.08 −0.106 −0.06 /

0.25 0.32 0.42 0.17 0.6 0.58 0.54 0.62 0.11 /

−0.54 −0.52 −0.69 −0.50 −0.57 −0.76 −0.65 −0.71 −0.46 /

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

3 3 3 3 3 3 3 3 3 3

0.5 0.5 −1 0.7 0.1 0.5 0.1 0 0.1 0.1 0.1 0.1 0.1

1.98 9 14.5 7.8 7.7 10 3.3 5 5 5 2.05 9.96 20 15 29.1

5

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

10-1

da/dN, mm/cycle

10

(e)

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

-2

10-3 10-4 10-5

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

-3

10-4 10-5 10-6 10-7

10-6 10-7

10-2 10

da/dN, mm/cycle

(a)

7075-T6, R=0.5 1

10

10-8

100

A533-B1, R=0.1 1

10-2

10-4

10-2 10

da/dN, mm/cycle

da/dN, mm/cycle

10

(f)

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

-3

10-5 10-6 10-7

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

-3

10-4 10-5 10-6 10-7

8630, R=0.5

EA4T, R=0.5

10-8

10

10-8

100

1

10-3

10-2

10-5 10-6 10-7

10-4 10-5 10-6 10-7

EA4T, R=-1

10-8

20

40

10-8

60

SAE 1050, R=0.1 1

10

(h) 10-2

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-5

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-3

da/dN, mm/cycle

da/dN, mm/cycle

10-4

100

∆ K, MPa·m1/2

∆ K, MPa·m1/2

(d)

100

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-3

da/dN, mm/cycle

da/dN, mm/cycle

(g)

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-4

10

∆ K, MPa·m1/2

∆ K, MPa·m1/2

(c)

100

∆ K, MPa·m1/2

∆ K, MPa·m1/2

(b)

10

10-6

10-4 10-5 10-6

10-7

10-7 10-8

SAE 1020, R=0.7 5

10

15

10-8

20

E36, R=0 1

10

100

∆ K, MPa·m1/2

∆ K, MPa·m1/2

Fig. 8. Comparisons of experiments with predictions: (a) 7075-T6; (b) EA4T (R = 0.5); (c) EA4T (R = −1); (d) SAE 1020; (e) A533-B1; (f) 8630; (g) SAE 1050; (h) E36; (i) 35NCD16; (j) 10Ni; (k) Spring steel; (l) railway bogie frame G20Mn5; (m) railway axle S38C.

6

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

(i) 10-2 -3

10-4 10-5 10

-6

10

-7

10-8

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-2

da/dN, mm/cycle

da/dN, mm/cycle

10

(k) 10-1

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

10-3 10-4 10-5 10-6 10-7

35NCD16, R=0.1 1

10

10-8

100

Spring steel, R=0.1 1

10

∆ K, MPa·m1/2 10-1

da/dN, mm/cycle

10-2 10

(l)

Experimental data iLAPS Ref. [19] Ref. [6] LAPS[7]

-3

10-4 10-5 10-6

Experimental data iLAPS Ref. [9] Ref. [4] LAPS[5]

10-3 10-4 10-5 10-6

10-7 10-8

10-1 10-2

da/dN, mm/cycle

(j)

100

∆ K, MPa·m1/2

10Ni, R=0.1 1

10

10-7

100

G20Mn5, R=0.1 5

10

100

∆ K, MPa·m1/2

(m) 10

∆ K, MPa·m

-2

Experimental data of transition layer Experimental data of outer layer

10-3

da/dN, mm/cycle

1/2

Experimental data of inner layer iLAPS at transition layer iLAPS at outer layer iLAPS at inner layer

10-4 10-5 10-6 10-7 10-8 10-9

S38C, R=0.1

1

10

100

∆ K, MPa·m1/2 Fig. 8. (continued)

To validate Eq. (22), the limit of relative errors is defined as n1′ and δn1′. As shown in Table A2, the maximum value of δn1′ is about 40%, the average of δK1′ is 1.79% and the standard deviation (STD) is 4.83% for these listed materials, thus giving a larger deviation. It appears that a more proper expression of n′ is required. It appears that a more proper expression of n′ is required. According to Eq. (21), it might be more practical to suppose that the n′ can be varied linearly by 1−√(σyc/K′). To verify such hypothesis, a plot of n′ vs. 1−√(σyc/K′) is provided in Fig. 6. During fitting the linearity relationship, the correlation coefficient r is set to be 0.98. When the sample size n = 93 and the significance level α = 0.01, i.e., the confidence level γ = 1 − α = 0.99, the minimum value of correlation coefficient rα = 0.27 [27]. Obviously, the correlation of two parameters is significantly large for r ≫ rα, thus the hypothesis is supported by

Table 5 The relative errors for different methods. The materials

7075-T6 EA4T (R = 0.5) EA4T (R = -1) SAE 1020 A533-B1 8630 SAE 1050 E36 steel 35NCD16 10Ni Spring steel G20Mn5

Maximum value

Average value

LAPS

FCG-LDA

iLAPS

LAPS

FCG-LDA

iLAPS

92.6% 94.2% 78.2% 86.8% 56.3% 90.8% 99.4% 96.4% 97.4% 73.1% 43.2% 99.2%

90.1% 79.9% 82.5% 91.0% 54.9% 89.6% 99.5% 97.6% 77.2% 72.0% 29.4% 88.9%

84.6% 72.3% 62.8% 78.2% 58.9% 67.8% 98.5% 97.6% 64.4% 23.2% 36.3% 83.2%

39.6% 85.3% 32.1% 51.7% 22.9% 84.3% 74.3% 37.7% 93.3% 66.7% 30.4% 38.1%

65.8% 52.2% 43.9% 66.3% 18.0% 82.4% 68.3% 37.4% 41.7% 61.1% 15.9% 51.3%

37.2% 42.4% 20.0% 29.2% 21.8% 42.4% 42.5% 36.4% 29.4% 12.4% 8.4% 25.7%

n = 0.65(

yc

K)

1.34

yc

K + 0.74

(23)

Note that in Table A2, theoretical results are denoted as n2′ and the 7

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Relative error, %

(a)

20

σ0.2, the expression of εf′ can be revised by the σb/σ0.2 ratio. Table A3 lists the UTP and LCF properties of various materials. Meanwhile, the true fracture ductility coefficient εf is calculated from the reduction in area %RA by the following equation [9]

7075-T6[6] EA4T[46] SAE1020[17] A533-B1[1] 8630[7] 1050[33] E36[42] 35NCD16[41] 10Ni[42] spring steel[45] G20Mn5

10

f

0.1

0.2

0.3

f

Relative error, %

30

0.4

7075-T6[6] EA4T[46] SAE1020[17] A533-B1[1] 8630[7] 1050[33] E36[42] 35NCD16[41] 10Ni[42] spring steel[45] G20Mn5

20

10

1000

2000

3000

4000

5000

The cyclic strength coefficient K' Fig. 9. Detailed comparisons of experimental n′ and K′ with the theoretical values.

e

0.6 f

0.2

1.83

(26)

b

0.0132 1 log 3 1.91 = 10

b·log (4 × 104) + log

e

1 1 log 3 4

3 4 f

(27)

2.5 b (1 + f ) E

(28)

in which b is the fatigue strength exponent, Δεe* is the strain on the elastic line at 104 cycles. Based on Eqs. (27) and (28), the Manson-Coffin stress exponent c can be estimated from the true fracture ductility coefficient εf and the ratio of tensile limit σb to the elastic modulus E. In addition, the effect of σb/σ0.2 is considered for the c in this paper. Within each group in Table A4, the data of materials published in the open literature are listed. To calculate the Manson-Coffin stress exponent c, the following relation can be used by fitting the data of c, εf, log(σb/E) and log(σ0.2/ σb) as listed in Table A4:

2.3.4. Expression εf′ Generally, the fatigue ductility coefficient εf′ can be linearly related by true fracture ductility coefficient εf [8,10,35] and the relationship can be expressed as

= 0.7579

= 0.02(100 f )0.82

c=

limit of prediction relative errors is defined as δn2′. It can be obviously found that the maximum value of δn2′ is 29.14%, the mean value of δn2′ is 2.75% and the STD is 4.05% or so. To validate Eq. (23), testing data in Table A2 is plotted in Fig. 6, in which theoretical results in termed of scatter bands via the factors of ± 10%, ± 20% and ± 30% are also shown. It can be clearly observed that 85.1% of experimental data are within a factor of ± 10% and 98.9% is covered with a factor of ± 20%. Consequently, the proposed new equation can effectively predict the n′ with a fairly high degree of accuracy.

f

(25)

2.3.5. Expression c It is normally known that the Manson-Coffin stress exponent c is about −0.6 without combining material properties [8,10]. To validate this point, the empirical value is set as c1 and compared with the testing data. At the same time, the δc1′ is defined as the limit of relative errors to illustrate the deviation. From Table A4, the maximum value of δc1′ is above 50% and the average is 20% or so. It can be seen that the estimation of the c is too roughly by the experience and the material properties should be adopted for the prediction. Studies have shown that the c is related to εf, σb and the elastic modulus E [10]. In the fourpoint correlation approach, the Manson-Coffin stress exponent c can be calculated by

0 -5 500

100 \% RA

Theoretical results from Eq. (26) are denoted as εf1′ and the limit of relative errors is defined as δεf2′. As shown in Table A3, the maximum value of δεf2′ is 38.14%, the mean value of δεf2′ is 17.47%. Meanwhile, the fatigue ductility coefficients εf′ predicted from Eq. (24) and Eq. (26) are compared in Table 1. The percentages of data points within the different deviation factors, including ± 10%, ± 20% and ± 30%, are also listed. It can be seen that the proposed approach contains the most data for all deviation factors. For example, when the factor is ± 20% of the theoretical results, 70.2% of the test data is included based on Eq. (26), but the coverage rate is only 34% based on Eq. (24). Therefore, Eq. (26) can accurately predict the fatigue ductility coefficient εf′.

The cyclic strain harding coefficient n'

(b)

100

By the nonlinear fitting, the following equation can be derived as

0

-5 0.0

= ln

c = - 0.96 - 0.091log(100 f ) - 0.22log

b

E

- 0.13log

0.2 b

(29)

The calculated result is denoted as c2 and the limit of relative error is defined as δc2. As shown in Table A4, the maximum value of δc2 is 39.28%, the mean value is 11.44% and the STD is 8.36% or so. The new approach along with the empirical value is also compared in Table 2. From the comparisons, we can basically draw the conclusion that the proposed new approach can provide better prediction accuracy. In summary, the iLAPS model can be modeled by using the estimated LCF parameters, the translation process to include the crack

(24)

Assuming that prediction results can be denoted by εf1′ and the limit of relative errors are defined as δεf1′. It can be clearly found from Table A3 that the limit of relative error δεf1′ for the 1038 steel is more than 200% and the average value is about 60%. As a result, the prediction of the εf′ should be improved. Considering the influence of both σb and

8

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

closure factor U and the relationship between the UTP and LCF can be concisely described in Fig. 7. As shown in the figure, based on the proposed theory, the da/dN can be estimated with UTP.

the newly-proposed iLAPS can still achieve a satisfactory prediction. Detailed comparisons from Fig. 9 clearly indicate the newly-developed iLAPS model appears not to be sensitive to the cyclic hardening or cyclic softening effects. One of the reason is that the present iLAPS model is formulated from those uniaxial tensile parameters that have intrinsically excluded the cyclic hardening or softening effect included in original FCG-LDA and LAPS [6,7]. On the other hand, current iLAPS model also achieves better FCG predictions than well modified NASGRO equation, which frequently requires complex and expensive fracture mechanics experiments.

3. Model validation In this section, the newly-developed iLAPS model is thoroughly validated by calculating the FCG curve for widely used engineering materials under different loading conditions and specimens and then compared with Eq. (1) [6], original LAPS [7] and modified NASGRO models [19], respectively. Open-reported and our results are collected as listed in Tables 3 and 4, in which σf′ is the fatigue strength coefficient. It should be noted that the long fatigue crack growth threshold ΔKth is in Tables 4 actually obtained under different loading conditions and specimen geometry. Predicted results are completely plotted in Fig. 8 for all 12 types of metallic materials. Note that in Tables 3 and 4, S38C used in China high-speed railway axle is employed because this grade steel has a surface hardened layer ranged from 2 mm to 4 mm, which brings a huge challenge to the scientists to acquire the fatigue cracking resistance and cyclic plastic strain parameters if standard specimens are adopted. To compare the prediction accuracy for iLAPS, LAPS and FCG-LDA, the relative errors between test data and prediction result are calculated as shown in Table 5. As shown in Table 5, it is clearly observed that current iLAPS model provides better FCG predictions than other famous models, such as LAPS model [19]. Similar to FCG-LDA and LAPS models, the uniaxial tensile behaviors based FCG model can well simulate the entire FCG region by correlating LCF with uniaxial tensile properties. Specially, the novel iLAPS model can provide a prospective FCG curve for the surface induction hardened S38C axle steel compared with experimental results, as illustrated in Fig. 8(m). In view of discrete FCG data of strengthened S38C materials due to difficult sampling and residual stress release [47], current iLAPS model still shows a huge advantage to effectively predict the FCG. Nevertheless, the significant deviation happens between those wellrecognized models with current iLAPS model when predicting the FCG curve, as shown in Fig. 8(f)–(h). For example, the errors between LAPS and iLAPS might be from different adaptation mechanisms when obtaining the LCF parameters. By integrating the Manson-Coffin equation and cyclic stress-strain response, foregoing n′ and K′ can be extracted by,

n = b c, K =

f

(

f)

b c

4. Conclusions The newly proposed model carried out some substantial improvements over past models (such as original FCG-LDA and LAPS together with well-known NASGRO) to produce the residual life prediction, which required basic material parameters only from the uniaxial tensile test and the threshold ∆Kth. This can be accomplished by unifying the crack closure factor U and also relating the LCF parameters to the UTP, regardless of critical damage accumulation value in the framework of nominal stress. The prediction results are strictly compared with the test data and a good agreement is found for a few engineering materials tested with different specimen geometry and loading. The iLAPS model can effectively capture the crack closure exactly with concise expression, which is a universal phenomenon for loaded engineering structures. Furthermore, the new model is a relatively cheap and simple form for the damage tolerant design primarily by adopting the UTP. The following conclusions are thus drawn as: (1) The fatigue damage sum Dcri in the framework of nominal stress has little influence on the FCG predicted by the iLAPS model. (2) Current iLAPS can effectively predict the FCG rate curve at the entire cracking region, and the universal crack closure factor is expressed as U=0.49+0.36R+0.16R2, typically when R varies from -1 to 0.97. (3) For most metallic materials, the cyclic plasticity behaviors can be effectively related with uniaxial tensile parameters, which derive a phenomenal FCG model without requiring cyclic hardening or cyclic softening parameters. (4) Compared with standard FCG-LDA and LAPS together with modified NASGRO and experiments, current iLAPS model appears to achieve better FCG estimations regardless of material types and loading ratios.

(30)

It is well-known that Eq. (30) is commonly satisfied so long as the stabilizing process is relatively short. For some materials whose cyclic hardening or softening is considerable, however, the hypothesis might be no longer reasonable. To reveal the influence of cyclic hardening or softening, the relative errors (see Fig. 9) can be estimated by comparing experimental values with theoretical Eq. (30). It is clearly observed from Fig. 9 that the smaller the relative errors, the less significant of two parameters to contribute to the FCG model. For the materials of A533-B1, Spring steel, E36 and 35NCD16, current iLAPS model provides exactly the same predictions to original FCG-LDA one. However, with the increase of relative errors, taking for 8630, SAE 1050 and 10Ni steels for examples, both original FCG-LDA and LAPS based on LCF response considerably deviate from current iLAPS model. By contrast,

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Sincere thanks are given to the National Natural Science Foundation of China (11572267), the Open Project of State Key Laboratory of Traction Power (2018TPL_T03) and the Major Systematic Project of China Railway Corporation (P2018J003).

Appendix A See Tables A1–A4.

9

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Table A1 Theoretical predictions of the cyclic strength coefficient K′. The materials

K

K′

K1′

δK1′, %

K2 ′

δK2′, %

1020 [21] 1038 [21] 1541 [21] 1141Nb [21] 1141Nb [21] 1050M [21] 1151V [21] 10V45 [21] 10V45 [21] 1141V [21] 1141V [21] 1141V [21] 1141AL [21] 1141AL [21] 5150 [21] 1090 [21] 1090 [21] 1090M [21] 1090 [21] 1090M [21] 15V41 [21] 4620 [21] 4620 [21] 8620 [21] 8620 [21] 8620 [21] 8620 [21] 9310 [21] 9310 [21] 41B17M [21] 15B35 [21] 8822 [21] 8822 [21] 5140 [21] 10B21 [21] 4140 [21] 4140 [21] 4140 [21] 4140 [21] 4140 [21] 20MnCr5 [21] 20MnCr5 [21] 20MnCr5 [21] 1050N [8] 1051QT [8] 1045N [8] Inconel 718 [8] 304 SS [8] Haynes 188 [8] AISI 316 [22] 304 steel [23] FeE 460 [24] RN [25] FP [25] SAE 1038 [9] SAE 1038 [9] SAE 1038 [9] SAE 1045 [26] SAE 1050 [9] SAE 1050 [9] SAE 1050 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9] SAE 1141 [9] SAE 1541 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9]

933 1159 1165 1287 1199 1819 1346 1456 1520 1321 1379 1244 1394 1205 1630 1780 1576 2273 1913 2202 1670 1448 1200 1624 2132 2276 2011 1136 1796 1031 1173 1074 2175 1276 1248 1303 1911 1680 1499 1460 1477 1762 1816 1455 1461 1185 1910 1210 512 641 1210 477 1936 1315 1106 1186 1183 1370 1819 1313 2837 1765 1980 1895 2757 1535 1379 1576 1394 1205 1287 1199 1321 1244

1171 1393 1400 1448 1254 1987 1473 1642 1691 1467 1441 1280 1515 1260 1485 1877 1964 1943 2163 2005 1575 1824 1213 1872 2078 2137 2029 1035 2098 1029 1094 1154 2055 1329 1109 1460 1614 1306 1696 1617 1930 2062 2231 1480 1558 1258 1564 1660 891 904 1660 508 1625 1021 1340 1420 1330 1480 1673 1292 3538 1611 1663 1873 1835 1653 1441 1416 1515 1277 1448 1254 1467 1270

1148 1446 1453 1602 1495 2188 1672 1799 1870 1643 1711 1551 1728 1503 1990 2148 1932 2628 2283 2562 2033 1790 1497 1984 2496 2631 2379 1417 2164 1281 1463 1337 2537 1589 1555 1621 2281 2043 1847 1803 1822 2129 2185 1797 1804 1478 2280 1509 488 710 1509 423 2306 1636 1379 1479 1476 1700 2188 1633 3122 2132 2349 2265 3055 1887 1711 1932 1728 1503 1602 1495 1643 1551

1.93 3.78 3.79 10.65 19.25 10.11 13.51 9.53 10.60 11.97 18.70 21.13 14.05 19.27 34.01 14.43 1.64 35.25 5.54 27.78 29.07 1.89 23.38 5.96 20.11 23.10 17.25 36.88 3.16 24.48 33.74 15.88 23.44 19.57 40.24 11.05 41.31 56.46 8.89 11.51 5.59 3.26 2.07 21.45 15.80 17.50 45.77 9.10 45.25 21.45 9.10 16.71 41.88 60.19 2.87 4.18 10.95 14.86 30.78 26.41 11.76 32.36 41.26 20.92 66.47 14.14 18.70 36.42 14.05 17.68 10.65 19.25 11.97 22.09

1055 1270 1275 1389 1307 1866 1443 1543 1601 1420 1473 1349 1487 1313 1699 1832 1651 2257 1948 2197 1735 1536 1308 1694 2137 2260 2033 1248 1846 1149 1283 1190 2174 1379 1353 1403 1946 1744 1582 1547 1562 1816 1863 1542 1547 1294 1945 1317 710 835 1318 675 1848 1399 1236 1299 1297 1441 1767 1398 2432 1729 1878 1820 2383 1564 1448 1594 1459 1314 1378 1310 1404 1345

9.90 8.84 8.90 4.10 4.23 6.09 2.05 6.04 5.35 3.21 2.21 5.38 1.87 4.18 14.42 2.41 15.94 16.16 9.94 9.56 10.13 15.81 7.83 9.52 2.83 5.74 0.20 20.60 12.02 11.66 17.27 3.10 5.78 3.72 21.97 3.88 20.58 33.50 6.74 4.36 19.07 11.94 16.48 4.19 0.68 2.87 24.39 20.64 20.29 7.68 20.60 32.85 13.72 37.06 7.79 8.51 2.49 2.62 5.62 8.19 31.25 7.33 12.93 2.84 29.85 5.38 0.49 12.56 3.68 2.91 4.84 4.43 4.30 5.87

10

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Table A2 Theoretical predictions of the cyclic strain hardening coefficient n′. The materials

σyc, MPa

K′, MPa

n′

n1′

δn1′, %

n2′

δn2′, %

5150 [21] 1050M [21] 1038 [21] 4620 [21] 1020 [21] 1090 [21] 1141AL [21] 1151V [21] 1090M [21] 1541 [21] 10v45 [21] 1141V [21] 15v41 [21] 1141V [21] 1141Nb [21] 10V45 [21] 1090 [21] 1090 [21] 4620 [21] 8620 [21] 8620 [21] 8620 [21] 20MnCr5 [21] 1141V [21] 1090 [21] 1141Nb [21] 8620 [21] 1141V [21] 1090M [21] 20MnCr5 [21] 20MnCr5 [21] 9310 [21] 4140 [21] 8822 [21] 4140 [21] 9310 [21] 4140 [21] 5140 [21] 15B35 [21] 4620 [21] 4140 [21] 4140 [21] 8822 [21] 4140 [21] 41B17M [21] 10B21 [21] 304 steel [23] 45 steel [29] 6063 [30] A356-T6 [30] 6063 [30] AlMg4.5Mn [30] 5456-H311 [30] 7475-T761 [30] 7075-T7351 [30] 2014-T6 [30] 2024-T3 [30] CK45 [31] AISI 303 [31] 42CrMo4 [31] S460N [32] 347 steel [32] 1050N [33] 1050QT [33] 1045HR [33] 16MnR [33] AISI 304 [33] A533B [33] AW-5083 [34] RN [25] SQ [25] FP [25] SAE 1045 [26] L290GA [26]

497 413 353 353 317 618 424 456 637 421 436 447 712 481 405 616 535 602 603 580 705 601 567 584 614 481 927 487 730 976 613 798 851 1095 670 616 856 702 645 667 852 895 644 911 611 756 326 346 254 291 161 318 377 468 388 449 429 340 310 640 457 270 398 680 369 335 326 406 341 494 580 394 379 300

1485 1987 1393 1393 1171 2163 1515 1473 1943 1400 1642 1467 1575 1441 1448 1691 1877 1964 1824 1888 2078 1872 1930 1901 1945 1254 2137 1280 2005 2231 2062 2098 1617 2055 1460 1035 1696 1329 1094 1213 1306 1591 1154 1614 1029 1109 1660 881 384 430 245 693 636 675 695 704 843 1206 2450 1420 1115 1329 1480 1558 1258 1106 1660 827 544 1625 1021 1021 1480 981

0.176 0.253 0.221 0.221 0.210 0.202 0.205 0.189 0.179 0.193 0.213 0.191 0.128 0.177 0.205 0.162 0.202 0.190 0.178 0.190 0.174 0.183 0.197 0.190 0.186 0.154 0.134 0.154 0.163 0.133 0.195 0.156 0.103 0.101 0.125 0.083 0.110 0.103 0.085 0.096 0.069 0.093 0.094 0.092 0.084 0.062 0.287 0.140 0.067 0.063 0.068 0.125 0.084 0.059 0.094 0.072 0.109 0.200 0.350 0.120 0.161 0.244 0.223 0.123 0.208 0.186 0.287 0.130 0.075 0.173 0.102 0.154 0.221 0.137

0.176 0.252 0.221 0.221 0.210 0.201 0.205 0.188 0.179 0.193 0.213 0.191 0.128 0.176 0.205 0.162 0.202 0.190 0.178 0.190 0.174 0.183 0.197 0.190 0.185 0.154 0.134 0.155 0.162 0.133 0.195 0.155 0.103 0.101 0.125 0.083 0.110 0.103 0.085 0.096 0.069 0.092 0.094 0.092 0.084 0.062 0.262 0.150 0.066 0.063 0.067 0.125 0.084 0.059 0.094 0.072 0.109 0.203 0.332 0.128 0.143 0.256 0.211 0.133 0.197 0.192 0.262 0.114 0.075 0.191 0.091 0.153 0.219 0.190

0.06 0.22 0.19 0.19 0.01 0.34 0.18 0.31 0.11 0.04 0.04 0.02 0.33 0.39 0.13 0.17 0.15 0.01 0.08 0.18 0.17 0.24 0.09 0.18 0.39 0.02 0.16 0.83 0.40 0.11 0.04 0.43 0.14 0.16 0.13 0.46 0.12 0.43 0.12 0.11 0.53 0.60 0.29 0.11 0.29 0.69 8.87 7.27 0.87 0.41 0.79 0.14 0.04 0.25 0.35 0.38 0.42 1.73 5.09 6.72 10.98 4.96 5.36 8.31 5.25 3.18 8.87 12.06 0.07 10.60 10.91 0.65 0.95 38.97

0.180 0.243 0.219 0.219 0.210 0.203 0.206 0.192 0.183 0.196 0.213 0.194 0.132 0.181 0.206 0.167 0.203 0.193 0.182 0.193 0.178 0.186 0.199 0.193 0.189 0.159 0.139 0.160 0.167 0.137 0.197 0.160 0.104 0.102 0.129 0.081 0.112 0.104 0.082 0.096 0.062 0.092 0.093 0.091 0.081 0.053 0.250 0.155 0.059 0.054 0.060 0.129 0.081 0.049 0.093 0.067 0.111 0.205 0.295 0.132 0.148 0.246 0.211 0.138 0.199 0.195 0.250 0.117 0.070 0.194 0.090 0.158 0.218 0.193

2.31 3.92 0.94 0.94 0.11 0.43 0.35 1.35 2.31 1.39 0.09 1.47 2.83 1.97 0.39 3.14 0.60 1.56 2.20 1.40 2.32 1.78 1.01 1.40 1.47 3.21 3.47 3.99 2.59 3.19 1.19 2.77 1.27 0.97 3.19 3.02 1.85 0.61 3.10 0.02 11.53 1.49 0.88 1.12 3.66 17.87 14.78 9.85 13.61 16.34 12.68 3.19 3.23 20.46 0.96 8.18 1.41 2.30 18.57 9.26 8.55 0.87 5.63 10.72 4.40 4.47 14.78 11.00 6.98 10.91 13.66 2.61 1.57 29.14

(continued on next page) 11

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Table A2 (continued) The materials

σyc, MPa

K′, MPa

n′

n1′

δn1′, %

n2′

δn2′, %

SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE SAE

424 591 405 481 447 487 481 342 358 364 424 469 427 523 1796 545 730 627 645 615

1515 1277 1448 1254 1467 1270 1441 1340 1420 1330 1416 950 1673 1292 3538 1611 1663 1873 1835 1653

0.205 0.124 0.205 0.154 0.191 0.154 0.177 0.220 0.222 0.208 0.194 0.114 0.220 0.146 0.109 0.174 0.133 0.176 0.168 0.159

0.205 0.124 0.205 0.154 0.191 0.154 0.176 0.219 0.221 0.208 0.194 0.113 0.219 0.145 0.109 0.174 0.132 0.176 0.168 0.159

0.18 0.16 0.13 0.02 0.02 0.01 0.39 0.25 0.27 0.10 0.12 0.51 0.26 0.46 0.05 0.09 0.53 0.09 0.00 0.08

0.206 0.128 0.206 0.159 0.194 0.159 0.181 0.218 0.220 0.209 0.196 0.116 0.218 0.150 0.111 0.179 0.137 0.180 0.173 0.164

0.35 2.84 0.39 3.21 1.47 3.23 1.97 0.91 1.09 0.36 1.19 1.81 0.91 2.91 1.82 2.55 2.77 2.30 2.75 3.01

1141 1141 1141 1141 1141 1141 1141 1038 1038 1038 1541 1541 1050 1050 1050 1090 1090 1090 1090 1090

[9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9]

Table A3 Theoretical predictions of the fatigue ductility coefficient εf′. The materials

RA, %

σb, MPa

σ0.2, MPa

εf ′

εf1′

δεf1′, %

εf2′

δεf2′, %

API 5DS135 [10] API 5LGr.B [10] SAE 1020 [10] SAR 60 [10] 7075-T6 [10] 1020 [35] 1045 [35] 1045 [35] 1038 [35] 1038 [35] 1141 [35] 1141 [35] 1141 [35] 1141 [35] 1144 [35] 1144 [35] A538B [35] 1541F [35] 1541F [35] A538C [35] RQC-100(b) [35] RQC-100(b) [35] 10B62 [35] 4130 [35] 4130 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4340 [35] 4340 [35] 4340 [35] 5160 [35] 52100 [35] 9262 [35] 9262 [35] 525C [35] SCM435 [35] FeE 460 [24] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1038 [9] SAE 1038 [9]

60 60 54 40 11 62 65 51 54 53 54 58 57 53 33 25 56 49 60 55 43 67 38 67 55 48 42 37 47 20 43 38 57 42 11 33 32 52 66 65 57 53 54 49 58 54 53

1175 423 491 620 576 440 725 1450 582 652 802 797 771 695 930 1035 1860 951 889 2000 940 930 1640 895 1425 1415 1760 1930 1550 2035 825 1470 1240 1670 2015 1000 1565 508 951 670 771 695 802 725 797 582 652

1033 294 285 540 498 262 634 1365 331 359 602 610 457 418 717 1020 1793 889 786 1931 896 883 1510 779 1358 1379 1586 1862 1448 1896 634 1372 1172 1531 1924 786 1379 280 795 467 457 418 602 450 610 331 359

0.49 0.36 0.25 0.45 0.12 0.41 1.00 0.60 0.31 0.20 0.36 0.53 0.26 0.26 0.32 0.27 0.80 0.68 0.93 0.60 0.66 0.66 0.32 0.92 0.89 0.45 0.40 0.60 0.50 0.20 0.45 0.48 0.73 0.40 0.18 0.41 0.38 0.22 1.00 0.52 0.26 0.26 0.36 0.43 0.53 0.31 0.20

0.72 0.72 0.65 0.51 0.21 0.74 0.78 0.62 0.65 0.64 0.65 0.70 0.69 0.64 0.44 0.36 0.67 0.60 0.72 0.66 0.54 0.81 0.49 0.81 0.66 0.59 0.53 0.48 0.58 0.31 0.54 0.49 0.69 0.53 0.21 0.44 0.43 0.63 0.79 0.78 0.69 0.64 0.65 0.60 0.70 0.65 0.64

46.73 99.72 160.40 12.44 74.17 81.22 22.00 3.17 110.68 216.83 80.33 30.34 166.54 142.42 36.88 32.96 15.88 12.06 22.69 10.33 18.79 22.12 52.19 12.39 25.62 30.44 31.50 20.50 15.40 54.00 19.11 1.46 6.16 31.50 16.11 6.83 12.63 191.67 20.38 49.14 166.54 142.42 80.33 39.07 30.34 110.68 216.83

0.65 0.42 0.27 0.39 0.12 0.33 0.72 0.60 0.26 0.24 0.42 0.48 0.29 0.28 0.26 0.31 0.70 0.56 0.65 0.69 0.50 0.87 0.41 0.74 0.67 0.59 0.44 0.44 0.54 0.23 0.34 0.42 0.69 0.46 0.14 0.27 0.32 0.23 0.68 0.47 0.29 0.28 0.42 0.27 0.48 0.26 0.24

32.24 16.94 6.00 12.44 4.17 19.02 28.30 0.50 17.48 16.34 17.17 9.93 14.40 4.55 19.38 14.07 12.50 17.35 29.68 14.33 23.79 32.12 29.06 19.13 24.72 31.78 10.50 27.17 7.00 13.00 24.67 11.88 5.34 14.00 23.33 34.88 16.05 6.48 32.23 9.37 14.40 4.55 17.17 38.14 9.93 17.48 16.34

12

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Table A4 Theoretical predictions of the Manson-Coffin stress exponent c. The materials

E, GPa

εf , %

σb, MPa

σ0.2, MPa

c

c1

δc1, %

c2

δc2, %

API 5DS135 [10] API 5LGr.B [10] API 5LX60 [10] SAE 1020 [10] SAR 60 [10] 7075-T6 [10] 1038 [35] 1038 [35] 1038 [35] 1050 [35] 1050 [35] 1090 [35] 1090 [35] 1090 [35] 1141 [35] 1141 [35] 1141 [35] 1141 [35] 1141 [35] 1141 [35] 1141 [35] 1541 [35] 1541 [35] A538A [35] A538B [35] 1541F [35] 1541F [35] A538C [35] H-11 [35] RQC-100 [35] RQC-100 [35] 10B62 [35] 1005-1009 [35] 1005-1009 [35] 1005-1009 [35] 1005-1009 [35] 1015 [35] 1020 [35] 1040 [35] 1045 [35] 1045 [35] 1045 [35] 1045 [35] 1045 [35] 1045 [35] 4130 [35] 4130 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4142 [35] 4340 [35] 4340 [35] 4340 [35] 5160 [35] 52100 [35] 9262 [35] 9262 [35] 9262 [35] 950C [35] 950X [35] 950X [35] 980X [35] 1144 [35] 1144 [35] 950C [35] SNCM630 [35] SNCM439 [35] 525C [35] 545C [35]

200 208 198 205 205 72 201 219 219 211 203 203 217 203 217 214 215 220 216 227 220 205 205 185 185 206 206 180 205 205 205 195 205 205 200 200 205 205 200 200 200 205 205 205 205 220 200 200 205 200 200 205 205 200 200 195 200 195 195 205 205 195 200 205 205 205 195 195 200 205 196 208 209 206

60 60 46 54 40 11 54 53 67 50 34 14 22 14 54 49 58 47 57 59 53 55 42 67 56 49 60 55 33 43 67 38 73 66 64 80 68 62 60 65 51 59 55 51 41 67 55 29 48 42 37 35 27 47 20 43 38 57 42 11 14 33 32 69 65 72 68 33 25 64 49 37 52 39

1175 423 533 491 620 576 582 652 649 821 829 1090 1147 1251 802 725 797 789 771 925 695 783 906 1515 1860 951 889 2000 2585 940 930 1640 360 470 415 345 415 440 620 725 1450 1345 1585 1825 2240 895 1425 1060 1415 1760 1930 1930 2240 1550 2035 825 1470 1240 1670 2015 925 1000 1565 565 440 530 695 930 1035 565 1100 1050 508 798

1033 294 457 285 540 498 331 359 410 465 460 735 650 760 602 450 610 493 457 814 418 475 475 1482 1793 889 786 1931 2034 896 883 1510 269 448 400 262 228 262 345 634 1365 1276 1517 1689 1862 779 1358 1048 1379 1586 1862 1724 1689 1448 1896 634 1372 1172 1531 1924 455 786 1379 324 345 331 565 717 1020 315 951 950 280 590

−0.73 −0.55 −0.53 −0.53 −0.62 −0.75 −0.48 −0.44 −0.46 −0.51 −0.50 −0.50 −0.60 −0.64 −0.51 −0.53 −0.56 −0.58 −0.46 −0.51 −0.46 −0.55 −0.56 −0.62 −0.71 −0.65 −0.65 −0.75 −0.74 −0.69 −0.69 −0.56 −0.43 −0.51 −0.41 −0.39 −0.64 −0.51 −0.57 −0.66 −0.70 −0.68 −0.69 −0.68 −0.60 −0.63 −0.69 −0.51 −0.75 −0.73 −0.76 −0.61 −0.76 −0.75 −0.77 −0.54 −0.60 −0.62 −0.57 −0.56 −0.47 −0.60 −0.65 −0.59 −0.54 −0.61 −0.53 −0.58 −0.53 −0.61 −0.82 −0.80 −0.46 −0.56

−0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60

17.81 9.09 13.21 13.21 3.23 20.00 25.00 36.36 30.43 17.19 19.52 20.97 0.00 6.54 18.11 13.42 8.11 3.45 29.31 16.73 29.87 9.49 7.72 3.23 15.49 7.69 7.69 20.00 18.92 13.04 13.04 7.14 39.53 17.65 46.34 53.85 6.25 17.65 5.26 9.09 14.29 11.76 13.04 11.76 0.00 4.76 13.04 17.65 20.00 17.81 21.05 1.64 21.05 20.00 22.08 11.11 0.00 3.23 5.26 7.14 27.66 0.00 7.69 1.69 11.11 1.64 13.21 3.45 13.21 1.64 27.10 25.09 31.00 6.38

−0.64 −0.53 −0.55 −0.53 −0.56 −0.59 −0.55 −0.55 −0.57 −0.57 −0.55 −0.55 −0.56 −0.56 −0.59 −0.56 −0.59 −0.56 −0.57 −0.61 −0.56 −0.58 −0.57 −0.69 −0.7 −0.61 −0.62 −0.7 −0.68 −0.61 −0.63 −0.66 −0.54 −0.57 −0.56 −0.54 −0.53 −0.53 −0.56 −0.6 −0.66 −0.66 −0.67 −0.68 −0.68 −0.62 −0.66 −0.6 −0.65 −0.66 −0.67 −0.66 −0.65 −0.66 −0.64 −0.59 −0.64 −0.65 −0.66 −0.62 −0.52 −0.59 −0.64 −0.56 −0.55 −0.56 −0.6 −0.59 −0.59 −0.55 −0.63 −0.61 −0.53 −0.57

12.33 3.64 3.77 0 9.68 21.33 14.58 25.00 23.91 11.33 9.56 10.89 6.67 12.77 16.14 5.86 6.31 3.45 22.84 18.68 21.21 5.84 2.33 11.29 1.41 6.15 4.62 6.67 8.11 11.59 8.70 17.86 25.58 11.76 36.59 38.46 17.19 3.92 1.75 9.09 5.71 2.94 2.90 0.00 13.33 1.59 4.35 17.65 13.33 9.59 11.84 8.20 14.47 12.00 16.88 9.26 6.67 4.84 15.79 10.71 10.64 1.67 1.54 5.08 1.85 8.20 13.21 1.72 11.32 9.84 23.45 23.85 15.72 1.06

(continued on next page) 13

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al.

Table A4 (continued) The materials

E, GPa

εf , %

σb, MPa

σ0.2, MPa

c

c1

δc1, %

c2

δc2, %

SFNCM85S [35] SF60 [35] SCM435 [35] SCM440 [35] SNCM630 [36] 304 steel [23] 42CrMo [37] 1Cr18Ni9Ti [38] CK45 [31] FeE 460 [24] Haynes 188 [8] S460N [32] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1141 [9] SAE 1038 [9] SAE 1038 [9] SAE 1038 [9] SAE 1541 [9] SAE 1541 [9] SAE 1050 [9] SAE 1050 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9] SAE 1090 [9]

201 208 210 204 196 183 210 193 206 206 170 206 216 227 220 217 214 215 220 201 219 219 205 205 211 203 203 203 217 203 203

66 53 66 36 49 80 62 75 50 65 55 65 57 59 53 54 49 58 47 54 53 67 55 42 50 34 14 25 22 14 38

825 820 951 1000 1103 650 955 605 660 670 490 670 771 925 695 802 725 797 789 582 652 649 783 906 821 829 1090 1388 1147 1251 1124

565 580 795 846 951 325 868 310 410 467 268 467 457 814 418 602 450 610 493 331 359 410 475 475 465 460 735 950 650 760 765

−0.52 −0.44 −0.71 −0.65 −0.82 −0.40 −0.85 −0.63 −0.44 −0.60 −0.73 −0.60 −0.46 −0.51 −0.46 −0.51 −0.53 −0.56 −0.58 −0.48 −0.44 −0.46 −0.55 −0.56 −0.51 −0.50 −0.50 −0.78 −0.60 −0.64 −0.68

−0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60 −0.60

14.94 36.67 15.25 7.69 27.10 49.25 29.58 4.76 36.36 0.84 17.81 0.84 29.31 16.73 29.87 18.11 13.42 8.11 3.27 24.74 36.36 30.43 9.49 7.72 17.19 19.52 20.97 22.78 0.00 6.54 12.15

−0.6 −0.59 −0.62 −0.6 −0.61 −0.52 −0.62 −0.51 −0.51 −0.55 −0.49 −0.55 −0.52 −0.6 −0.51 −0.56 −0.52 −0.57 −0.52 −0.49 −0.49 −0.53 −0.53 −0.51 −0.52 −0.5 −0.51 −0.56 −0.51 −0.51 −0.56

14.94 34.40 12.43 7.69 25.88 29.35 27.23 19.05 15.91 7.56 32.88 7.56 12.07 16.73 10.39 10.24 1.70 2.70 10.50 1.87 11.36 15.22 3.28 8.44 1.56 0.40 2.82 27.93 15.00 20.56 18.01

Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijfatigue.2019.105324.

tolerance for railway axles in China. Int J Fatigue 2016;93(1):64–70. [18] Meggiolaro MA, Castro JTP. On the dominant role of crack closure on fatigue crack growth modeling. Int J Fatigue 2003;25(9):843–54. [19] Maierhofer J, Pippan R, Ganser HP. Modified NASGRO equation for physically short cracks. Int J Fatigue 2014;59:200–7. [20] Codrington J, Kotousov A. A crack closure model of fatigue crack growth in plates of finite thickness under small-scale yielding conditions. Mech Mater 2009;41(2):165–73. [21] Lopez Z, Fatemi A. A method of predicting cyclic stress-strain curve from tensile properties for steels. Mater Sci Eng A 2012;556:540–50. [22] Jen YM, Wang WW. Crack initiation life prediction for solid cylinders with transverse circular holes under in-phase and out-of-phase multiaxial loading. Int J Fatigue 2005;27(5):527–39. [23] Socie D. Multiaxial fatigue damage models. J Eng Mater Technol 1987;109(4):293–8. [24] Sonsino CM. Influence of load and deformation-controlled multiaxial tests on fatigue life to crack initiation. Int J Fatigue 2001;23(2):159–67. [25] Chakraborti PC, Mitra MK. Microstructural response on the room temperature low cycle fatigue behaviour of two high strength duplex ferrite-martensite steels and a normalised ferrite-pearlite steel. Int J Fatigue 2006;28(3):194–202. [26] Seweryn A, Buczyński A, Szusta J. Damage accumulation model for low cycle fatigue. Int J Fatigue 2008;30(4):756–65. [27] Bewick V, Cheek L, Jonathan B. Statistics review 7: correlation and regression. Crit Care 2003;7:451–9. [28] Hu ZZ, Cao SZ. Relationship between fatigue notch factor and strength. Eng Fract Mech 1994;48(1):127–36. [29] Shang DG, Wang DJ, Yao WX. A simple approach to the description of multiaxial cyclic stress-strain relationship. Int J Fatigue 2000;22(3):251–6. [30] Fatemi A, Plaseied A, Khosrovaneh AK, Tanner D. Application of bi-linear log-log SN model to strain-controlled fatigue data of aluminum alloys and its effect on life predictions. Int J Fatigue 2005;27(9):1040–50. [31] Reis L, Li B, Freitas MD. Crack initiation and growth path under multiaxial loading in structural steels. Int J Fatigue 2009;31(11):1660–8. [32] Hoffmeyer J, Döring R, Seeger T, Vormwald M. Deformation behaviour, short crack growth and fatigue lives under multiaxial nonproportional loading. Int J Fatigue 2006;28(5):508–20. [33] Li J, Li CW, Qiao YJ, Zhang ZP. Fatigue life prediction for some metallic materials

References [1] Paris P, Erdogan F. A critical analysis of crack propagation laws. J Basic Eng 1963;85(4):528–33. [2] Pippan R, Hohenwarter A. Fatigue crack closure: a review of the physical phenomena. Fatigue Fract Eng M 2017;40(4):471–95. [3] Forman RG, Kearney VE, Engle RM. Numerical analysis of crack propagation in cyclic-loaded structure. J Fluid Eng 1967;89(3):459–63. [4] Glinka G. A cumulative model of fatigue crack growth. Int J Fatigue 1982;4(2):59–67. [5] Glinka G. A notch stress-strain analysis approach to fatigue crack growth. Eng Fract Mech 1985;21(2):245–61. [6] Shi KK, Cai LX, Chen L, Wu SC, Bao C. Prediction of fatigue crack growth based on low cycle fatigue properties. Int J Fatigue 2014;61:220–5. [7] Wu SC, Xu ZW, Yu C, Kafka OL, Liu WK. A physically short fatigue crack growth approach based on low cycle fatigue properties. Int J Fatigue 2017;103:185–95. [8] Shamsaei N, Mckelvey SA. Multiaxial life predictions in absence of any fatigue properties. Int J Fatigue 2014;67:62–72. [9] Roessle ML, Fatemi A. Strain-controlled fatigue properties of steels and some simple approximations. Int J Fatigue 2000;22(6):495–511. [10] Meggiolaro MA, Castro JTP. Statistical evaluation of strain-life fatigue crack initiation predictions. Int J Fatigue 2004;26(5):463–76. [11] Zhu ML, Xuan FZ, Tu ST. Effect of load ratio on fatigue crack growth in the nearthreshold regime: a literature review, and a combined crack closure and driving force approach. Eng Fract Mech 2015;141:57–77. [12] Schijve J. Some formulas for the crack opening stress level. Eng Fract Mech 1981;14(3):461–5. [13] Kumar R, Singh K. Influence of stress ratio on fatigue crack growth in mild steel. Eng Fract Mech 1995;50(3):377–84. [14] Bachmann V, Munz D. Crack closure in fatigue of a titanium alloy. Int J Fracture 1975;11(4):713–6. [15] Chand S, Garg SBL. Crack closure studies under constant amplitude loading. Eng Fract Mech 1983;18(2):333–47. [16] Ellyin F. Fatigue Damage, Crack Growth and Life Prediction. London: Chapman & Hall; 1997. [17] Wu SC, Zhang SQ, Xu ZW, Kang GZ, Cai LX. Cyclic plastic strain based damage

14

International Journal of Fatigue 131 (2020) 105324

S.C. Wu, et al. under constant amplitude multiaxial loading. Int J Fatigue 2014;68:10–23. [34] Hertel O, Vormwald M. Short-crack-growth-based fatigue assessment of notched components under multiaxial variable amplitude loading. Eng Fract Mech 2011;78(8):1614–27. [35] Genel K. Application of artificial neural network for predicting strain-life fatigue properties of steels on the basis of tensile tests. Int J Fatigue 2004;26(10):1027–35. [36] Han C, Chen X, Kim KS. Evaluation of multiaxial fatigue criteria under irregular loading. Int J Fatigue 2002;24(9):913–22. [37] Chen X, Gao Q, Sun XF. Damage analysis of low-cycle fatigue under non-proportional loading. Int J Fatigue 1994;16(3):221–5. [38] Chen X, AN K, Kim KS. Low-cycle fatigue of 1Cr-18Ni-9Ti stainless steel and related weld metal under axial, torsional and 90° out-of-phase loading. Fatigue Fract Eng Mater Struct 2004;27(6):439–48. [39] Mashayekhi M, Ziaei-Rad S. Identification and validation of a ductile damage model for A533 steel. J Mater Process Technol 2006;177(1):291–5. [40] Stephens RI, Chung JH, Fatemi A, Lee HW, Lee SG, Vaca-Oleas C, et al. Constant and variable amplitude fatigue behavior of five cast steels at room temperature and 45

°C. J Eng Mater Technol 1984;106(1):25–37. [41] Glinka G, Robin C, Pluvinage G, Chehimi C. A cumulative model of fatigue crack growth and the crack closure effect. Int J Fatigue 1984;6(1):37–47. [42] Pandey KN, Chand S. An energy based fatigue crack growth model. Int J Fatigue 2003;25(8):771–8. [43] Barsom JM. Fatigue-crack propagation in steels of various yield strengths. J Eng Ind 1971;93(4):1190–6. [44] Li DM, Kim KW, Lee CS. Low cycle fatigue data evaluation for a high-strength spring steel. Int J Fatigue 1997;19(8):607–12. [45] Li DM, Nam WJ, Lee CS. An improvement on prediction of fatigue crack growth from low cycle fatigue properties. Eng Fract Mech 1998;60(4):397–406. [46] Wu SC, Liu YX, Li CH, Kang GZ, Liang SL. On the fatigue performance and residual life of intercity railway axles with inside axle boxes. Eng Fract Mech 2018;197:176–91. [47] Wu SC, Xu ZW, Liu YX, Kang GZ, Zhang ZX. On the residual life assessment of highspeed railway axles due to induction hardening. Int J Rail Transportation 2018;6(4):218–32.

15