The prediction for fatigue strength in very high cycle regime of high strength steel

Abstract

Background:

Fatigue strength is an important parameter of evaluating the material fatigue property. There are many prediction models for fatigue strength in the VHCF regime. Murakami’s model is extensively applied in numerous prediction models. The initiation mode of VHCF cracks in high strength steel is very different from the assumption in Murakami’s model that the internal inclusion is considered as a disc crack.

Objective:

We tried to analyze the effect of inclusion shape on the prediction by Murakami’s model and further examined the prediction effects of several generalized models.

Methods:

We reviewed the prediction models for fatigue strength in the VHCF regime. On the assumption that the fatigue crack which initiated from the internal inclusion is a circumferential crack around a spherical cavity and crack propagation is the critical condition, we modified Murakami’s model. The predicted values of Murakami’s model, the modified model and Liu’s model were compared with the test results of 28 groups of high strength steel in the literature.

Results:

The deviations between predictions by several generalized models and test results are mostly within 20%. The predicted values of four groups of high strength steel by the modified and Murakami’s models and six groups of high strength steel by Liu’s model have large deviations from the test results. The predicted fatigue strength values of Liu’s model are higher than that of the modified model, and the predicted values of the modified model are slightly higher than that of Murakami’s model. The mean value of fitting coefficient C of all types of high strength steel is closer to the coefficient of the modified model, and the sums of squares of coefficient deviations of the modified model is smaller than that of Murakami’s model.

Conclusions:

The effects of inclusion shape on the prediction of fatigue strength in the VHCF regime seem small. Murakami’s model, the modified model and Liu’s model coincide well with the test results in the literature. The prediction effect of the modified model is higher than that of Murakami’s model, and the prediction effect of Murakami’s model is higher than that of Liu’s model.

Keywords

Very high cycle fatigue internal inclusion prediction model modified model fatigue strength

1. Introduction

According to the category of fatigue cycles, there are three types of fatigue: low, high, and very high cycle. Cracks usually initiate from the surface in low and high cycle fatigue regimes, and from the internal inclusion in very high cycle fatigue (VHCF) regime [1–6]. Fatigue strength is an important parameter of evaluating the material fatigue property, and it plays a significant role in the study of the fatigue mechanism and design of long fatigue life. The fatigue strength can be obtained from fatigue tests or predicted by relevant theories and models. However, fatigue testing usually consumes considerable manpower, materials, and financial resources, so it is necessary to search for relevant prediction models for fatigue strength in the VHCF regime. Many studies have been done in this area and some valuable prediction models that predict very high cycle fatigue strength through the inclusion size have been proposed [7–11]. Murakami et al. [12–14] combined the parameters of fatigue strength, Vickers hardness and the square root of inclusion or defect projection area to give an equation for predicting the fatigue strength. Wang et al. [8] incorporated the fatigue life into Murakami’s model. Liu [9] proposed an expression to predict the fatigue strength of high strength steel in the VHCF regime according to the effect of hydrogen during forming a highlight area around the inclusion. Chapetti [10] proposed a simple model to estimate the fatigue strength in the VHCF regime, and the fatigue strength is obtained from the threshold for pure fatigue crack propagation and the inclusion size where the crack initiation takes place. Sun [11] proposed a model for predicting the fatigue strength of high strength steel with fish-eye mode failure based on the test results for the effect of inclusion size and stress ratio. To some extent, these empirical models reflect the fatigue mechanism in the VHCF regime. Murakami’s model is extensively applied in numerous prediction models for fatigue strength in the VHCF regime because there are few physical parameters and they are easily measured [12–14].

VHCF cracks in high strength steel usually initiate from the interface of matrix-inclusion due to debonding, and the inclusion shows no cracking [15]. This initiation mode is very different from the assumption in Murakami’s model that the internal inclusion is considered as a disc crack. In fact, Murakami conducted relevant tests and explained the equivalence of defect and crack with an identical projected area for the prediction of fatigue strength [16]. In this paper, we tried to analyze the effect of inclusion shape on the prediction by Murakami’s model and further examined the prediction effects of several generalized models. We took account of the effect of inclusion shape and modified Murakami’s model, and the predicted values of several generalized models were compared with the test results in the literature.

2. The review of prediction models for fatigue strength in VHCF regime

Many studies have been done in the prediction models for fatigue strength in VHCF regime and some valuable prediction models have been proposed [7–11].

Murakami et al. [12–14] proposed a model to predict the fatigue strength by linear elastic fracture mechanics. This model involved hardness, inclusion size, and fatigue strength. In Murakami’s model, a small flaw or inclusion is assumed as a disc crack in the infinite volume, and the crack propagation on the inclusion boundary as the critical condition. The fatigue strength of high strength steel with nonmetallic inclusions can be expressed as: $\begin{matrix} (1) & σ_{w} = \frac{C (HV + 120)}{{(\sqrt{A_{in}})}^{1 / 6}}, \end{matrix}$ where HV is the Vickers hardness of the matrix, in units of kgf/mm² and $\sqrt{A_{in}}$ is the square root of the inclusion projected area perpendicular to the direction of maximum principal stress, which is approximately equal to the inclusion diameter, in units of µm. C is the constant related to the inclusion position, for surface inclusion $C = 1.43$ , for subsurface inclusion $C = 1.41$ , and for internal inclusion $C = 1.56$ .

In the situation of a crack initiating from an internal inclusion, Eq. (1) can be written as: $\begin{matrix} (2) & σ_{w} = \frac{1.42 \times (HV + 120)}{a^{1 / 6}}, \end{matrix}$ where a is the radius of the inclusion, in units of µm.

Murakami’s model involving the stress ratio R can be written as: $\begin{matrix} (3) & σ = \frac{C (HV + 120)}{{(\sqrt{A_{in}})}^{1 / 6}} {(\frac{1 - R}{2})}^{α_{R}}, \end{matrix}$ where $α_{R} = 0.226 + HV \times 10^{- 4}$ .

Wang et al. [8] incorporated the number of cycles to failure into Murakami’s model and proposed: $\begin{matrix} (4) & σ = \frac{β_{m} (HV + 120)}{{(\sqrt{A_{in}})}^{1 / 6}} {(\frac{1 - R}{2})}^{α_{R}}, \end{matrix}$ where $β_{m} = 3.09 - 0.12 log N_{f}$ for interior inclusions or defects and $β_{m} = 2.79 - 0.108 log N_{f}$ for surface inclusions or defects for four low-alloy high-strength steels (42Cr–Mo4, Cr–Si(54SC6), Cr–Si(55SC7) and Cr–V(60CV2)).

Liu [9] proposed a model to predict the fatigue strength of high strength steel in VHCF regime based on the effect of hydrogen during formation of granular bright facet (GBF): $\begin{matrix} (5) & σ_{w} = \frac{2.7 {(HV + 120)}^{0.9375}}{{(\sqrt{A_{in}})}^{0.1875}} . \end{matrix}$

Akiniwa et al. [17] assumed that Paris relations was still valid for the fatigue crack propagation in fine granular area (FGA), and derived an approximate relation for the fatigue strength and the number of cycles to failure: $\begin{matrix} (6) & σ_{a} = \frac{2}{\sqrt{π}} {(\frac{2}{C_{A} (m_{A} - 2)})}^{\frac{1}{m_{A}}} {(\sqrt{A_{in}})}^{\frac{1}{m_{A}} - \frac{1}{2}} N_{f}^{- \frac{1}{m_{A}}} . \end{matrix}$

The parameters $m_{A} = 14.2$ , $C_{A} = 3.44 \times 10^{- 21}$ for bearing steel JIS SUJ2 with the tensile strength of 2316 MPa.

Chapetti et al. [10] showed a relation between FGA size, inclusion size and the number of cycles to failure in the form of $\sqrt{A_{FGA}} / \sqrt{A_{in}} = 0.25 N_{f}^{0.125}$ by fitting the experimental data of quenched and tempered JIS SUJ2, SCM435 and SNCM439 steels, and then proposed an expression to correlate the total fatigue life with the stress amplitude: $\begin{matrix} (7) & σ_{a} = 2.46 \frac{HV + 120}{{(\sqrt{A_{in}})}^{1 / 6}} N_{f}^{\frac{1}{48}}, \end{matrix}$ where $σ_{a}$ in MPa, HV in kgf/mm² and $A_{in}$ in µm².

Mayer et al. [18] pointed out that a relationship between the stress amplitude, the fatigue life and the inclusion size: $\begin{matrix} (8) & σ_{a} = c_{m}^{\frac{1}{n_{m}}} \frac{1}{{(\sqrt{A_{in}})}^{1 / 6}} N_{f}^{- \frac{1}{n_{m}}}, \end{matrix}$ where $n_{m} = 28.82$ , $c_{m} = 6.47 \times 10^{98}$ by fitting the fatigue data of specimen failed from interior inclusions for bainitic bearing 100Cr6 steel with tensile strength of 2387 MPa, and the dimension of stress amplitude is MPa and $A_{in}$ is µm².

Sun [11] proposed a model for predicting the fatigue strength of high strength steel with fish-eye mode failure based on the test results for the effect of inclusion size and stress ratio. In the model, the effect of inclusion size and stress ratio on fatigue strength is expressed as: $\begin{matrix} (9) & σ_{a} = C N_{f}^{l} a_{0}^{m} {(\frac{1 - R}{2})}^{α}, \end{matrix}$ where C, l, m and α are material parameters.

It is seen that, for several models (Eqs (1), (4), (5) and (7)) mentioned above, the parameters in models are determined, while for the other models (Eqs (6), (8) and (9)), several parameters in models are changing with types of materials. Although the parameters in Eqs (4) and (7) are determined, and relevant parameters are determined by fitting the fatigue data of specific materials. In a word, only Murakami’s model and Liu’s model are more generalized and can directly predict the fatigue strengths of different types of high strength steel. The prediction effects of Murakami’s model and Liu’s model are examined in the following part.

3. The modified model involving effects of inclusion shape

The phenomenon of VHCF cracks in high strength steel that initiate from the interface of matrix-inclusion due to the debonding and inclusion without cracking is in accordance with the usual initiation mode of internal cracks in high strength steel, which was mentioned by Tanaka et al. [15,19]. For example, the VHCF studies of high strength steel such as TT150, TT180 by Hong et al. [20], and FV520B-I by Zhang et al. [21] show the initiation character of internal cracks. The crack origins are shown in Figs 1 and 2. In Hong’s and Zhang’s studies, TT150 and FV520B-I were tested by the ultrasonic fatigue test method with a symmetrical tension–compression load, and TT180 was tested by the rotary bending fatigue test method.

Fig. 1.

VHCF crack origins of TT150 and TT180. (a) TT150 ( $σ = 860 MPa$ , $N_{f} = 2.81 \times 10^{6}$ ). (b) TT180 ( $σ = 808 MPa$ , $N_{f} = 1.79 \times 10^{7}$ ).

Fig. 2.

VHCF crack origins of FV520B-I. (a) Initiation from single inclusion ( $σ = 650 MPa$ , $N_{f} = 1.44 \times 10^{7}$ ). (b) Initiation from clusters ( $σ = 650 MPa$ , $N_{f} = 1.94 \times 10^{7}$ ).

The above initiation mode is very different from the assumption in Murakami’s model that assumes an internal inclusion as a disc crack. Therefore, Murakami’s model was modified based on the assumption that a VHCF crack initiates from the internal inclusion is a circumferential crack around the spherical cavity and the propagation of the crack is the critical condition.

Assume the inclusion after debonding as a spherical cavity in infinite volume, and the radius of a spherical cavity is a. There is a circumferential crack around the spherical cavity and the circumferential crack length is $λ a$ . The crack is under the action of the uniform tensile stress σ in the far distance, and the stress intensity factor $K_{I}$ at the crack front is [22]: $\begin{matrix} (10) & K_{I} = σ \sqrt{π λ a} \cdot F_{I} . \end{matrix}$

$F_{I}$ is the shape factor of a circumferential crack around the spherical cavity, and the polynomial fitting expression of $F_{I}$ is obtained when the Poisson ratio v is 0.3 according to the Table 2 in Section 4.1.4.7 of the literature [22]: $\begin{matrix} (11) & F_{I} = 2.56 λ^{4} - 7.503 λ^{3} + 8.492 λ^{2} - 4.866 λ + 2.297 . \end{matrix}$

According to the test results, the threshold of stress intensity factor range $Δ K_{th}$ (MPa · m^1/2) for steel with an internal crack is concluded as [12,13]: $\begin{matrix} (12) & Δ K_{th} = 2.77 \times 10^{- 3} (HV + 120) {(\sqrt{π {(a + λ a)}^{2}})}^{1 / 3} . \end{matrix}$

Assume that the stress intensity factor range $Δ K$ as the driving force equals $2 K_{I}$ for symmetrical fatigue, and $Δ K$ is equal to $Δ K_{th}$ as the resistance: $\begin{matrix} (13) & Δ K = 2 K_{I} = Δ K_{th} . \end{matrix}$

Then the stress amplitude which the specimen suffered is equal to the fatigue strength, and the fatigue strength is determined by Eqs (10)–(13): $\begin{matrix} (14) & σ_{w} = \frac{0.95 \times (HV + 120) {(1 + λ)}^{1 / 3}}{F_{I} \sqrt{λ} a^{1 / 6}}, \end{matrix}$ where a is equal to the radius of the inclusion, in units of µm. Taking FV520B-I as an example, the evaluation results of Eq. (14) were investigated. Substituting the Vickers hardness 380 kgf/mm² and the maximum inclusion radius 8.2 µm on fatigue fractures into the Eq. (14) [21], the estimations of FV520B-I fatigue strength with different values of λ are shown in Table 1.

Table 1

FV520B-I fatigue strength estimations with different values of λ

	λ

	0.05	0.1	0.2	0.4	0.6	0.8	1.0
$σ_{w}$ (MPa)	728	578	492	453	441	431	428

The fatigue strength greatly varies with different values of λ in Table 1, so the determination of λ is important to estimate the fatigue strength.

When the size of the spherical cavity is constant, the value of λ depends on the length of the circumferential crack around the spherical cavity. It is assumed that λ is determined by the fatigue crack initiation size and the crack size in the initiation process does not reach the threshold size of crack propagation. Then the crack in the initiation process is not propagated. So through estimating the initiation size of a fatigue crack, the characteristic parameter λ can be determined and the modified model is obtained. For the initiation mechanism of fatigue crack, Dowling [23] assumed that a fatigue crack initiating from the notch root is under the control of the local stress strain field at the beginning stage, and it changes to be under the control of the nominal stress strain field as the crack propagates to a characteristic length, and that the fatigue crack initiation size is defined as the size when the crack is at the transition demarcation point. On the premise of the definition of fatigue crack initiation by Dowling, we analyzed the situation of a circumferential crack around a spherical cavity. When the circumferential crack is short, the crack is under the control of the local stress strain field caused by the spherical cavity, and $K_{I}$ can be computed by Eq. (10). The crack is mainly under the control of the nominal stress strain field as the circumferential crack continually propagates to a characteristic length, and the effect of the spherical cavity on crack propagation can be ignored. In this situation, we can assume that the circumferential crack around the spherical cavity is a circular crack in infinite volume, and $K_{I}$ is [24]: $\begin{matrix} (15) & K_{l} = 2 σ \sqrt{\frac{(λ + 1) a}{π}} . \end{matrix}$

The crack initiation size is determined by the intersection of Eqs (10) and (15) [23], and then we can determine the characteristic parameter $λ = 0.164$ .

The characteristic parameter λ is determined by the inclusion shape and λ equals 0.164 for the spherical inclusion.

Then Eq. (14) can be simplified as: $\begin{matrix} (16) & σ_{w} = \frac{1.45 \times (HV + 120)}{a^{1 / 6}} . \end{matrix}$

The expression of the modified model is similar to Murakami’s model, and the coefficient of the modified model is slightly higher than that of Murakami’s model.

4. The comparison of model predictions and test results

4.1. VHCF test results of relevant high strength steel

The results of VHCF tests and other relevant tests of 28 groups of high strength steel were found in the literature [8,19,21,25–30], as shown in Table 2.

Table 2
The test results of 28 groups of high strength steel

Serial number Steel type References Tensile strength $R_{m}$ /MPa Vickers hardness HV/kgf · mm⁻² Fatigue strength $σ_{w}$ /MPa Analyze range/cycles Specimen diameter d/mm Inclusion diameter $d_{in}$ /µm

1 60Si2CrV-1 [19] 1750 538 632 10⁹ 3 32

2 60Si2CrV-2 [19] 1804 543 662 10⁹ 3 29

3 60Si2CrV-3 [19] 1945 565 560 10⁹ 3 31

4 60Si2CrV-4 [19] 1955 571 675 10⁹ 3 20

5 60Si2CrV-5 [19] 1925 562 750 10⁹ 3 16

6 60Si2CrV-6 [19] 1954 558 760 10⁹ 3 18

7 60Si2Mn-1 [19] 1732 511 621 10⁹ 3 30

8 60Si2Mn-2 [19] 2182 611 630 10⁹ 3 38

9 60Si2Cr [19] 1753 513 645 10⁹ 3 27

10 GCr15-1 [19] 1785 703 788 10⁹ 3 11

11 GCr15-2 [19] 1700 708 746 10⁹ 3 22

12 NHS1 [19] 2025 600 715 10⁹ 3 11

13 40CrNiMo [19] 1820 540 610 10⁹ 3 7

14 50CrV4-1 [19] 1680 506 632 10⁹ 3 32

15 50CrV4-2 [19] 1750 519 720 10⁹ 3 6.2

16 54SiCrV6 [19] 1729 515 722 10⁹ 3 3.5

17 54SiCr6 [19] 1743 500 720 10⁹ 3 1

18 0.46Carbon [25] 2150 660 490 5 × 10⁸ 7 71.9

19 SCM435 [26] 1828 561 560 10⁸ 7 55.6

20 SCM435H [27] 1950 564 1050 10⁸ 4 10

21 SNCM439 [28] 1955 598 750 10¹⁰ 3 50

22 SUP7 [29] 1423 441 540 10⁸ 6 35

23 SUP7 [29] 1730 528 580 10⁸ 6 34

24 SUP10M [8] 1923 590 862 0.2 × 10⁸ 2.5 28.4

25 SUP12 [19] 1825 604 771 10⁹ 3 17

26 SUP12 [30] 1720 516 640 10⁸ 6 16.6

27 SWOSC-V [30] 1764 528 720 10⁸ 4.5 10.7

28 FV520B-I [21] 1170 380 550 10⁹ 3 16.4

Serial number	Steel type	References	Tensile strength $R_{m}$ /MPa	Vickers hardness HV/kgf · mm⁻²	Fatigue strength $σ_{w}$ /MPa	Analyze range/cycles	Specimen diameter d/mm	Inclusion diameter $d_{in}$ /µm
1	60Si2CrV-1	[19]	1750	538	632	10⁹	3	32
2	60Si2CrV-2	[19]	1804	543	662	10⁹	3	29
3	60Si2CrV-3	[19]	1945	565	560	10⁹	3	31
4	60Si2CrV-4	[19]	1955	571	675	10⁹	3	20
5	60Si2CrV-5	[19]	1925	562	750	10⁹	3	16
6	60Si2CrV-6	[19]	1954	558	760	10⁹	3	18
7	60Si2Mn-1	[19]	1732	511	621	10⁹	3	30
8	60Si2Mn-2	[19]	2182	611	630	10⁹	3	38
9	60Si2Cr	[19]	1753	513	645	10⁹	3	27
10	GCr15-1	[19]	1785	703	788	10⁹	3	11
11	GCr15-2	[19]	1700	708	746	10⁹	3	22
12	NHS1	[19]	2025	600	715	10⁹	3	11
13	40CrNiMo	[19]	1820	540	610	10⁹	3	7
14	50CrV4-1	[19]	1680	506	632	10⁹	3	32
15	50CrV4-2	[19]	1750	519	720	10⁹	3	6.2
16	54SiCrV6	[19]	1729	515	722	10⁹	3	3.5
17	54SiCr6	[19]	1743	500	720	10⁹	3	1
18	0.46Carbon	[25]	2150	660	490	5 × 10⁸	7	71.9
19	SCM435	[26]	1828	561	560	10⁸	7	55.6
20	SCM435H	[27]	1950	564	1050	10⁸	4	10
21	SNCM439	[28]	1955	598	750	10¹⁰	3	50
22	SUP7	[29]	1423	441	540	10⁸	6	35
23	SUP7	[29]	1730	528	580	10⁸	6	34
24	SUP10M	[8]	1923	590	862	0.2 × 10⁸	2.5	28.4
25	SUP12	[19]	1825	604	771	10⁹	3	17
26	SUP12	[30]	1720	516	640	10⁸	6	16.6
27	SWOSC-V	[30]	1764	528	720	10⁸	4.5	10.7
28	FV520B-I	[21]	1170	380	550	10⁹	3	16.4

Note: The inclusion diameters of the 1st–17th and the 25th group specimens are the average size measured by the metallurgical method. The inclusion diameters of the 18th–24th and the 26th–28th group specimens are the maximum size on fatigue fractures.

Figure 3 shows the relationship between tensile strength $R_{m}$ and Vickers hardness HV in 28 groups of high strength steel. There is an approximate linear relationship of tensile strength and Vickers hardness of most groups of high strength steel. The locations of the two groups of steel were far from the fitting line when the values of Vickers hardness were larger than 700.

Generally, the axial fatigue limit of smooth specimens ( $σ_{w}$ ) has a good linear relationship with the Vickers hardness (HV), or tensile strength ( $R_{m}$ ) for low or medium strength steel whose Vickers hardness (HV) is less than or equal to 400 [16]: $\begin{array}{l} (17) & σ_{w} \approx 1.6 HV (HV ⩽ 400), \\ (18) & σ_{w} \approx 0.5 R_{m} (R_{m} ⩽ 1200 MPa) . \end{array}$

Figure 4 shows the relationships between fatigue strength $σ_{w}$ and Vickers hardness HV, and tensile strength $R_{m}$ in 28 groups of high strength steel. The solid lines represent the empirical Eqs (17) and (18), respectively. In Fig. 4, it can be seen that the fatigue strength values of most high strength steel are much lower than the empirical equation values, which is obviously different from the characteristic of fatigue limit of low or medium strength steel and maybe caused by the subsurface fatigue failure mode in the VHCF regime. The test value of the 20th group, SCM435H, is higher than the empirical equation values and the test value of the 28th group, FV520B-I, is slightly lower than the empirical equation values. We regard that the relatively low cycles of SCM435H and the relatively low Vickers hardness of FV520B-I result in the situation abnormal of relevant points in Fig. 4.

Fig. 3.

The relationship between tensile strength and Vickers hardness.

Fig. 4.

The relationships between fatigue strength and Vickers hardness and Tensile strength (the numbers of test points in the figure are the serial numbers in Table 2). (a) The relationship with Vickers hardness. (b) The relationship with Tensile strength.

4.2. The comparison of model predictions and test results

We predicted the fatigue strength values in the VHCF regime of 28 groups of high strength steel by Murakami’s model, the modified model and Liu’s model. The predicted values are compared with the test results as shown in Figs 5 and 6, respectively.

Though there are some differences in chemical composition, mechanical properties, cycles, specimen diameter, inclusion detection method, and inclusion size among above 28 groups of high strength steel. Figure 5 shows that both of the modified model and Murakami’s model adequately predict the fatigue strength values in the VHCF regime of high strength steel and the deviations between predictions and test results are mostly within 20%. The predicted values of the 17th group, 54SiCr6, the 20th group, SCM435H, the 21th group, SNCM439, and the 24th group, SUP10M, by the modified and Murakami’s models have large deviations from the test results. For the 54SiCr6 specimen, its inclusion diameter is less than 1 µm, which belongs to ultraclean steel, and its fatigue cracks mostly initiate from the surface in the VHCF regime, so there are some big deviations between the predicted values of the modified or Murakami’s model based on cracks initiating from internal inclusion and the test results. For SCM435H, Fig. 4(b) shows that the fatigue strength is 1050 MPa which is more than $R_{m} / 2$ , so the value is very different from the usual fatigue strength range of high strength steel and results in great deviations between the model predictions and test results. For SUP10M, the 2 × 10⁷ cycles is far less than 10⁹ cycles of the usual VHCF research range. The test result will decrease as the cycles increase, then the deviations between model predictions and test results will decrease and the prediction accuracies of models will improve.

Fig. 5.

Comparison of Murakami’s model and the modified model predictions and test results (the numbers of test points in the figure are the serial numbers in Table 2).

Fig. 6.

Comparison of Liu’s model predictions and test results (the numbers of test points in the figure are the serial numbers in Table 2).

In above 28 groups of high strength steel, there are 10 groups test results of high strength steel that are lower than the values predicted by Murakami’s model, and 18 groups are higher than the values predicted by Murakami’s model. In general, there is a tendency that the predictions of Murakami’s model are lower than the test result. The expression of the modified model is similar to the expression of Murakami’s model, and the coefficient of the modified model is slightly higher than that of Murakami’s model. So the predicted value of the modified model is slightly higher than that of Murakami’s model. In order to more effectively compare model predictions with test results, the test results of relevant high strength steel were fitted according to the expression $C = σ_{w} a^{1 / 6} / (HV + 120)$ , and the coefficient C are shown in Table 3.

From Table 3, it can be seen that the mean value of fitting coefficients is 1.47 which is closer to the coefficient of modified model 1.45. We further computed the sums of squares of deviations between fitting coefficients and both model coefficients, and the sums of squares of deviations are 1.25 and 1.30 for the modified model and Murakami’s model respectively. The prediction values of the modified model seem closer to the test results of relevant high strength steel.

Figure 6 shows that Liu’s model adequately predict the fatigue strength values in the VHCF regime of high strength steel and the deviations between predictions and test results are mostly within 20% too. The more predicted values of the 13th group, 40CrNiMo, the 17th group, 54SiCr6, the 18th group, 0.46Carbon, the 20th group, SCM435H, the 21th group, SNCM439, and the 24th group, SUP10M, by Liu’s model have large deviations from the test results. The prediction effect of Liu’s model is lower than that of Murakami’s model and the modified model. And the relationships of predicted values by three generalized models and test results are shown in Fig. 7. The variation trends of predicted values by three models are similar to that of test results. The predicted values of Liu’s model are higher than that of the modified model, and the predicted values of the modified model are higher than that of Murakami’s model.

Table 3

Fitting coefficient C of relevant high strength steel test results

	Steel type

	60Si2CrV-1	60Si2CrV-2	60Si2CrV-3	60Si2CrV-4	60Si2CrV-5	60Si2CrV-6	60Si2Mn-1
C	1.53	1.56	1.29	1.43	1.56	1.62	1.55

	Steel type

	60Si2Mn-2	60Si2Cr	GCr15-1	GCr15-2	NHS1	40CrNiMo	50CrV4-1
C	1.41	1.57	1.27	1.34	1.32	1.14	1.60

	Steel type

	50CrV4-2	54SiCrV6	54SiCr6	0.46Carbon	SCM435	SCM435H	SNCM439
C	1.36	1.25	1.04	1.14	1.43	2.01	1.79

	Steel type

	SUP7	SUP7	SUP10M	SUP12	SUP12	SWOSC-V	FV520B-I
C	1.55	1.44	1.89	1.52	1.43	1.47	1.56

Fig. 7.

Comparison of model predictions and test results.

5. Conclusions

Murakami’s model and Liu’s model are more generalized and can directly predict the fatigue strengths in VHCF regime of different types of high strength steel. The effects of inclusion shape on the prediction of fatigue strength in the VHCF regime seem small. Murakami’s model, the modified model and Liu’s model coincide well with the test results in the literature. The prediction effect of Liu’s model is lower than that of Murakami’s model and the modified model. The predicted values of Liu’s model are higher than that of the modified model, and the predicted values of the modified model are higher than that of Murakami’s model. The predicted values of the modified model seem closer to the test results.

Footnotes

Acknowledgements

The authors gratefully acknowledge the financial support provided by the National Key Basic Research and Development Program (973 Program 2011CB013401). Thanks to Dr. Edward C. Mignot, Shandong University, for linguistic advice.

Conflict of interest

The authors have no conflict of interest to report.

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