
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12747-8
10.1016/j.heliyon.2024.e36716
e36716
Research Article
Residual fatigue life prediction based on a novel improved Manson-Halford model considering loading interaction effect
Liu Panglun ab
Zhang Jie a
Tang Haihong a
Duan Heng b
Jiang Bingyan jby@csu.edu.cn
a⁎
a State Key Laboratory of Precision Manufacturing for Extreme Service Performance, School of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, China
b AVIC Landing-Gear Advanced Manufacturing Co., Changsha, 410200, China
⁎ Corresponding author. jby@csu.edu.cn
22 8 2024
15 9 2024
22 8 2024
10 17 e3671623 5 2024
11 8 2024
21 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The classical nonlinear fatigue cumulative damage model Manson-Halford is widely used in fatigue life analysis and prediction, but this model lacks consideration for the effects of load interaction. At present, some studies have proposed new correction models to overcome the shortcomings of classical models. However, the problem with these models is that the parameters are difficult to determine, resulting in a cumbersome application process or the correction process is based only on a single parameter. Insufficient correction leads to large fluctuations in life prediction errors. To overcome the above problems, this article proposes a new correction method and establishes a new improved model. Unlike existing models, the newly improved model is based solely on S-N curve parameters and does not require additional material parameters. It fully considers the relationship between adjacent loads and the difference in fatigue life under different stress states, and dynamically modifies the classical model based on larger influence weights. By conducting multi-level fatigue tests on 300M steel, a new improved model was used to accurately predict its remaining fatigue life. Compared with the classical model, the prediction error was reduced by 11.75 %. Utilized fatigue test data from various other materials to predict the fatigue life of the improved model under different levels of stress loading. The results indicate that in the vast majority of cases, the improved model has the highest prediction accuracy among all compared models, with a minimum relative prediction error of only 3.79 % and small fluctuations in prediction error. Compared with the classical model, the maximum reduction in relative prediction error after model improvement reached 28.73 %, indicating the effectiveness and accuracy of the new improved model.

Keywords

Fatigue accumulation damage
Fatigue life prediction
Nonlinear damage model
Loading interaction effect
==== Body
pmc1 Introduction

In practical industrial production, when the failure damage of structural components occurs, most of them pertain to the condition of fatigue failure, and fatigue damage often tends to result in considerable economic losses and human casualties [1,2]. Fatigue failure is a complex analytical process because components are not subjected to one load action in the actual service environment, but to cyclic loads of variable amplitude [3,4]. To accurately predict the fatigue life of components, the exploration and selection of more suitable prediction models are essential.

Nowadays, the theory of fatigue cumulative damage can be roughly divided into linear cumulative damage theory, nonlinear cumulative damage theory, and various other modified theories [[5], [6], [7], [8], [9], [10], [11]]. The classic linear fatigue cumulative damage model is Miner rule [12], which is widely used in practical engineering because of its easy concept to understand, simple model structure, and low calculation cost. Classical nonlinear fatigue cumulative damage models include Marco-Starkey model [13], Corten-Dolan model [14], Manson-Halford model [15] et al. The classic models mentioned above were introduced early and widely used, but with the deepening of research on fatigue problems, more and more studies have found that load history and load interaction have a significant impact on the fatigue cumulative damage process, and ignoring these factors will lead to significant prediction errors [[9], [10], [11]]. To improve the prediction accuracy of fatigue life models, numerous new theories and models have emerged, such as fatigue cumulative damage correction models based on energy dissipation [16], quantifiable damage process indicators, material residual S-N curves and performance degradation parameters [[17], [18], [19]], and load block spectra [[20], [21], [22], [23]].

The Manson-Halford classic model has a significant flaw in that it does not consider the interaction between loads during the fatigue cumulative damage process, which results in poor accuracy in life prediction despite its ability to handle complex loading situations with a small number of parameters [22,24,25]. To overcome the above shortcomings, researchers have improved the traditional Manson Halford model by modifying the characteristic index in the equivalent process of fatigue cumulative damage and introducing consideration of load interaction effects. Yuan and Gao et al. [26,27] improved the model by determining and introducing the minimum value of the stress ratio between the front and rear levels during the equivalent process of cumulative damage, which to some extent improved the life prediction performance. Haghgouei et al. [28] used a ratio relationship that includes the fatigue life of the front and rear levels to modify the characteristic index term in the model. Yue and Gao et al. [22,29] directly adopted the ratio relationship between the two levels of stress before and after to further improve the model. The difference lies in the expression form of the relationship and its insertion position in the formula. Zhao et al. [30] considered introducing the influence of strength degradation during the cumulative damage process and established an improved cumulative damage model based on strength degradation.

The above research has revised the classic Manson Halford model, but still needs to improve the following shortcomings: (1) the universality of the model, which can meet the fatigue life prediction of most materials and multi-level complex stress loading, and the prediction error fluctuation of the model is small; (2) The applicability of a model refers to the calculation of the model being as simple and efficient as possible. After improvement, the predictive performance of the model is improved, but the structure is still concise, avoiding the need for additional testing to determine model parameters.

This article proposes a new Manson Halford improved model that relies solely on the original S-N curve parameters of the material, without the need for additional testing. By comparing the stress levels and their corresponding fatigue life relationships before and after, the model can be dynamically modified during the cumulative damage equivalent process based on larger influence weights. The results of two-level and multi-level loading fatigue tests on various materials have verified the effectiveness and accuracy of the proposed new improved model. The birth of the research results in this paper is based on the analysis and improvement of the existing research, and the corresponding experiments are carried out to verify the results. Although this study presents novel ideas and valuable references for the improvement of the Manson-Halford model, it is undeniable that, as with most studies, this work has some limitations, mainly in the sample size of the data and the scope of the study.

The rest of the paper is as follows: The second part analyzes and summarizes the traditional Manson-Halford model and the existing improved model, and puts forward a new revision method and improved model; In the third part, multi-level fatigue test is carried out for 300M steel, and the test data of many materials are used to verify the effectiveness of the improved model. The fourth part draws the conclusions.

2 Analysis and research of improved models

2.1 Analysis of the classical Manson-Halford model

The Manson-Halford model is based on a crack growth model, which defines fatigue damage as the ratio of the instantaneous crack length to the final crack length [15], i.e:(1) Di=aiaf

where Di is the fatigue damage value under a certain stress loading; ai is the instantaneous crack length produced under a certain stress loading; af is the final crack length when fatigue damage occurs under a certain stress loading.

The Manson-Halford model defines the instantaneous crack length ai in Eq. (1) as:(2) ai=a0+(af−a0)(niNfi)23(Nfi)0.4

where, ni/Nfi is the load cycle ratio, defined as the ratio of the number of cycles under a certain stress loading to the fatigue life corresponding to that stress; a0 represents the initial crack length before loading; Nfi represents the fatigue life under a certain stress loading.

Substituting Eq. (2) into Eq. (1) gives:(3) Di=a0+(af−a0)(niNfi)23(Nfi)0.4af

Manson conducted extensive material tests and given the suggested value of 0.18 for the final crack length af15, Eq. (3) becomes:(4) Di=10.18(a0+(0.18−a0)(niNfi)23(Nfi)0.4)

Taking the two-level loading as an example, assume that it is first loaded with stress σ1 and cycled n1 times, at which time the fatigue damage caused by the first level stress is:(5) D1=10.18(a0+(0.18−a0)(n1Nf1)23(Nf1)0.4)

It should be noted that when the second level of stress loading is performed, the first level of stress has already caused damage to the member, and the damage caused by the first level of stress loading is equated to the second level of stress loading to obtain:(6) Deq,2=10.18(a0+(0.18−a0)(neq,2Nf2)23(Nf2)0.4)

where Deq, 2 is the equivalent damage value from the first level of stress to the second level of stress; neq, 2 is the equivalent loading cycle from the first level of stress to the second level of stress.

From the principle of fatigue-equivalent damage, it follows that:(7) D1=Deq,2

The equivalent load cycle ratio from the first level stress loading to the second level stress loading is obtained as:(8) neq,2Nf2=(n1Nf1)(Nf1Nf2)0.4

The Manson-Halford model assumes that fatigue damage occurs with a critical damage value of D = 1 and that the total fatigue cumulative damage can be expressed by the load-cycle ratio [15], then there is fatigue damage when fatigue damage occurs in the secondary stress loading condition:(9) neq,2Nf2+n2Nf2=(n1Nf1)(Nf1Nf2)0.4+n2Nf2=1

From Eq. (9), the residual fatigue life (expressed in terms of load-cycle ratio, same as later) under the second level loading after completion of the first level loading is:(10) n2Nf2=1−neq,2Nf2=1−(n1Nf1)(Nf1Nf2)0.4

Similarly, the total loss formed by the first two levels of stress loading is equated towards the third level of stress in the three-level stress loading state to obtain:(11) neq,3N=[(n1Nf1)(Nf1Nf2)0.4+n2Nf2](Nf2Nf3)0.4

Fatigue damage occurs when there is:(12) neq,3Nf3+n3Nf3=[(n1Nf1)(Nf1Nf2)0.4+n2Nf2](Nf2Nf3)0.4+n3Nf3=1

From Eq. (12), the remaining fatigue life under the third level of loading after completion of the first two levels is:(13) n3Nf3=1−neq,3Nf3=1−[(n1Nf1)(Nf1Nf2)0.4+n2Nf2](Nf2Nf3)0.4

By analogy, the criterion for the Manson-Halford model when fatigue damage occurs under multi-level stress loading conditions is obtained as:(14) {[(n1Nf1)α1,2+n2Nf2]α2,3+⋯+ni−1Nf(i−1)}αi−1,i+niNfi=1

where αi-1, i is the exponential term for the load-cycle ratio, in the original Manson-Halford model:(15) αi−1,i=(Nf(i−1)Nfi)0.4

From Eq. (14), the residual fatigue life of the last level under multi-level stress loading conditions is obtained:(16) niNfi=1−neq,iNfi=1−{[(n1Nf1)α1,2+n2Nf2]α2,3+⋯+ni−1Nf(i−1)}αi−1,i

in summary, from the analysis of the Manson-Halford model (hereinafter referred to as the M − H model), it can be seen that the M − H model in the process of fatigue cumulative damage, taking into account the impact of the stress loading sequence on the damage accumulation, to a certain extent, to improve the accuracy of the prediction of the remaining fatigue life. However, the classical M − H model still has obvious defects, that is, in the process of fatigue cumulative damage, it does not take into account the effect of the role of neighboring loading stresses(Referring to the two levels of stress applied sequentially during the multilevel stress loading).

2.2 Analysis of existing improved models

It has been shown that the fatigue cumulative damage process under multi-level stress states can be reflected more accurately by introducing an exponential term related to the neighboring stress level in the damage equivalence process [[31], [32], [33], [34]]. Based on the above ideas and research, in recent years, scholars in the industry have proposed a variety of improved M − H models [22,[27], [28], [29],35], striving to overcome the shortcomings existing in the traditional M − H model and improve the accuracy of fatigue life prediction. The more outstanding improved M − H models in recent years are briefly introduced, and the similarities and characteristics between the models are summarised.

2.2.1 Gao Huiying's model

In 2014, Gao Huiying et al. [27] considered the effect of interactions between neighboring loads by redefining the exponential parameter of the M − H model and proposed an improved M − H model. αi-1, i is the exponential term of the load-cycle ratio ni/Nfi, which becomes in the improved model:(17) αi−1,i=(Nf(i−1)Nfi)0.4⋅min{σi−1σi,σiσi−1}

Substituting Eq. (17) into Eq. (14) gives the criterion when fatigue damage occurs:(18) {[(n1Nf1)(Nf1Nf2)0.4⋅min{σ1σ2,σ2σ1}+n2Nf2](Nf2Nf3)0.4⋅min{σ2σ3,σ3σ2}+⋯+ni−1Nf(i−1)}(Nf(i−1)Nfi)0.4⋅min{σi−1σi,σiσi−1}+niNfi=1

Substituting into Eq. (16), the remaining fatigue life under the last level of loading is obtained:(19) niNfi=1−neq,iNfi=1−{[(n1Nf1)(Nf1Nf2)0.4⋅min{σ1σ2,σ2σ1}+n2Nf2](Nf2Nf3)0.4⋅min{σ2σ3,σ3σ2}+⋯+ni−1Nf(i−1)}(Nf(i−1)Nfi)0.4⋅min{σi−1σi,σiσi−1}

The authors compared the improved model with the conventional M − H model and the fatigue life prediction accuracy was improved.

2.2.2 Yue Peng's model

In 2020, in their study of the fatigue accumulation damage process of aero-engine rotors under mixed loading, Yue Peng et al. [29] proposed a new M − H model with an exponential term for the load cycle ratio ni/Nfi:(20) αi−1,i=(Nf(i−1)Nfi)0.4⋅(2σi2σi−1+σi)

The model takes into account the effects of load consequences and load interactions induced under mixed high and low-cycle fatigue loading.

2.2.3 Hadi Haghgouei's model

In 2020, Hadi Haghgouei et al. [28] investigated the fatigue cumulative damage of green agate rocks under two levels of loading, high and low, using the traditional M − H model and found that “the applicability of the M − H model depends on the fatigue life ratios of the specimens under the high and low loading conditions”, which is at variance with the above mentioned improved model that uses the modification of the model directly as a function of the ratio of the neighboring loads, and based on which the authors proposed a new exponential term for the load cycle ratio ni/Nfi:(21) αi−1,i=(Nf(i−1)Nfi)0.4−Ln(2)Ln(Nf(i−1)Nfi)

Using the improved M − H model, the authors carried out many tests to validate the model, which also showed that higher loading levels can have a significant effect on fatigue life.

2.2.4 Gao Kai's model

In 2021, Gao Kai et al. [22] analyzed the classic M − H model and several improved models, learned from predecessors' correction ideas, proposed to use the product of load amplitude ratio and fatigue life ratio instead of the originally modified exponent to obtain the following exponential term for the load-cycle ratio ni/Nfi, to construct the improved M − H model:(22) αi−1,i=(Nf(i−1)Nfi)0.4⋅(σiσi−1)

2.2.5 Similarity and characteristics of each model

The correction indices for each of these improved models are summarised as shown in Table 1.Table 1 Correction index terms of existing improved models.

Table 1Number	Improved models	Correction index term/αi−1,i	
1	Gao Huiying's model [27]	αi−1,i=(Nf(i−1)Nfi)0.4⋅min(σi−1σi,σiσi−1)	
2	Yue Peng's model [29]	αi−1,i=(Nf(i−1)Nfi)0.4⋅(2σi2σi−1+σi)	
3	Hadi Haghgouei's model [28]	αi−1,i=(Nf(i−1)Nfi)0.4−Ln(2)Ln(Nf(i−1)Nfi)	
4	Gao Kai's model [22]	αi−1,i=(Nf(i−1)Nfi)0.4⋅(σiσi−1)	

By analyzing the above-mentioned M − H improvement models, it can be found that there are some similarities between the models, i.e., all of them are corrected by correcting the exponent αi-1, i to improve the prediction accuracy of the models. Moreover, most of the models construct the new improved model by introducing the functional equation of the neighboring loads in the exponent, such as the above-mentioned Gao Huiying model and Yue Peng model, whereas the Haghgouei model employs the functional equation related to the fatigue life ratio, and Gao Kai model takes the product of the ratio of the neighboring loads and the fatigue life ratio as the new corrected exponent. On the surface, the biggest difference between the various correction models lies in the specific expression of the feature index (referring to the introduced correction factor) and the structural form (referring to where the correction factor is placed in the original formula), but in reality, small differences have a significant impact on the results of fatigue life prediction.

2.3 The new Manson-Halford improved model

This paper is inspired by the above research in the process of studying the nonlinear fatigue cumulative damage model. At the same time, the following new viewpoints are proposed, which are different from the existing models [15,22,[27], [28], [29],35]. This paper argues that when constructing a new modified M − H model, we should focus on the following considerations:(1) The applicability and prediction accuracy of the improved model depend not only on the adjacent load relationship, but also on the fatigue life relationship of the material under high and low stress loading.

(2) In the process of fatigue cumulative damage, the greater the difference between adjacent loads and the corresponding fatigue life, the more significant the impact on the residual fatigue life, and the greater value between the two should be taken into account in the revision of the cumulative damage model.

Based on this, this paper has carried out a lot of model construction attempts and data validation, and finally proposed the following new correction index:(23) αi−1,i=(Nf(i−1)Nfi)0.4⋅max{σi−1σi,Ln(Nfi)Ln(Nf(i−1))}

in the formula, max{x,y} indicates that the larger value is taken in x, y. The new correction index takes into account both the neighboring load ratio and the fatigue life ratio under different stress states, which can be weighed according to the specific loading conditions, and the larger value is chosen to complete the correction of the M − H model.

Substituting Eq. (23) into Eq. (9), the criterion when fatigue damage occurs in two-level stress loading conditions is obtained as:(24) neq,2Nf2+n2Nf2=(n1Nf1)(Nf1Nf2)0.4⋅max{σ1σ2,Ln(Nf2)Ln(Nf1)}+n2Nf2=1

Substituting Eq. (23) into Eq. (10), the remaining fatigue life under the second level loading after completion of the first level loading is obtained as:(25) n2Nf2=1−neq,2Nf2=1−(n1Nf1)(Nf1Nf2)0.4⋅max{σ1σ2,Ln(Nf2)Ln(Nf1)}

Substituting Eq. (23) into Eq. (14), the criterion of the improved M − H model, when fatigue damage occurs under multi-level stress loading conditions, is obtained as:(26) {[(n1Nf1)(Nf1Nf2)0.4⋅max{σ1σ2,Ln(Nf2)Ln(Nf1)}+n2Nf2](Nf2Nf3)0.4⋅max{σ2σ3,Ln(Nf3)Ln(Nf2)}+⋯+ni−1Nf(i−1)}(Nf(i−1)Nfi)0.4⋅max{σi−1σi,Ln(Nfi)Ln(Nf(i−1))}+niNfi=1

Substituting Eq. (23) into Eq. (16), the residual fatigue life under the multi-level stress loading condition with the last level of loading is obtained as:(27) niNfi=1−neq,iNfi=1−{[(n1Nf1)(Nf1Nf2)0.4⋅max{σ1σ2,Ln(Nf2)Ln(Nf1)}+n2Nf2](Nf2Nf3)0.4⋅max{σ2σ3,Ln(Nf3)Ln(Nf2)}+⋯+ni−1Nf(i−1)}(Nf(i−1)Nfi)0.4⋅max{σi−1σi,Ln(Nfi)Ln(Nf(i−1))}

Take the two-level stress loading case as an example, the fatigue cumulative damage process of the improved model is shown in Fig. 1, (a) depicting the cumulative damage process from high to low loading and (b) depicting the cumulative damage process from low to high loading.Fig. 1 Schematic of damage curve from two-level loading.

Fig. 1

In Fig. 1, the vertical coordinate represents the accumulated fatigue damage caused to the member under stress loading; the horizontal coordinate represents the fatigue life corresponding to different damages, expressed in terms of load cycle ratio.

For the two-level stress loading state, the fatigue cumulative damage process of the improved M − H model can be expressed as:(28) n1Nf1+n2Nf2=n1Nf1+(1−neq,2Nf2)=1+(n1Nf1−neq,2Nf2)

From Fig. 1 and Eq. (28), when the load loading sequence is from high to low, then neq,2/Nf2>n1/Nf1, so there is:(29) 1+(n1Nf1−neq,2Nf2)<1+(n1Nf1−n1Nf1)=1

This indicates that the cumulative damage value of fatigue from high to low loads is less than the critical value of 1. This is reflected in the equivalent loss curve, i.e., the Q-W-E-R′ process in Fig. 1(a), which is because high stresses make cracks in a member more easily, and then at low stresses the cracks undergo expansion more quickly until damage occurs.

When the load loading order is from low to high, then neq,2/Nf2<n1/Nf1, so there is:(30) 1+(n1Nf1−neq,2Nf2)>1+(n1Nf1−n1Nf1)=1

This indicates that the cumulative damage value of fatigue from low to high loads is greater than the critical value of 1, which is reflected in the equivalent loss curve, i.e., the Q-E-W-R′ process in Fig. 1(b), which is due to the “workout effect” produced by the material at low stresses, which in turn slows down the formation of fatigue cracks and the damage process.

Many experiments have confirmed the influence of different loading sequences on the cumulative fatigue damage results [7,17,20,21,36]. The derivation of Eq. (24) to Eq. (27) is all based on the conversion process of equivalent fatigue damage. The new improved model can fully consider the influence of loading sequence and load interaction in the cumulative fatigue damage process, and verify the consistency between its behavior and experimental phenomena in the cumulative damage process.

3 Fatigue test and example analysis

To verify the validity and accuracy of the new M − H improved model proposed in this paper, fatigue life prediction calculations were carried out for a variety of different materials under two-level stress loading and multi-level stress loading, respectively. The Relative Prediction Error (RPE) and the Mean Relative Prediction Error (MRPE) were selected as the evaluation indexes of the model prediction accuracy, and the calculation formulas are as follows:(31) RPE=|Theoreticalpredictedvalue−ExperimentaltestvalueExperimentaltestvalue|×100%MRPE=1n(∑inRFE)

For this section of the work it should be declared that since the Miner criterion [12] is a classical linear fatigue cumulative damage model and the M − H model [15] is a nonlinear fatigue cumulative damage model, and since the Miner criterion does not take into account either the influence of the loading sequence or the influence of the load interaction effect, it has lower prediction accuracy compared with other models, a statement that has been verified by numerous models [11,[21], [22], [23],33,34]. Therefore, the new improved model predictions are only compared and analyzed with the traditional M − H model and four other excellent models (Gao Huiying's model, Yue Peng's model, Hadi Haghgouei's model, and Gao Kai's model, which are denoted by the abbreviations of the respective authors' names in the subsequent graphs and tables) in the subsequent contents.

3.1 Multilevel fatigue test and life prediction of 300M steel

300M steel with high strength, high fracture toughness and excellent fatigue properties and other excellent comprehensive performance characteristics, is currently the most widely used aircraft landing gear material.

Multi-level fatigue test was carried out for 300M steel material, which was derived from the forging of the outer cylinder of a civil landing gear. Before the test, Thermo Fisher plasma mass spectrometer was used to determine the main element content of 300M steel, as shown in Table 2.Table 2 The main chemical composition of 300M steel material.

Table 2Main chemical composition and mass fraction/%	
C	Si	Mn	Cr	Ni	Mo	V	
0.43	1.70	0.82	0.86	1.83	0.39	0.07	

MTS material testing machine was used to measure the tensile mechanical properties of 300M steel at room temperature, as shown in Table 3.Table 3 Tensile mechanical properties of 300M steel.

Table 3Performance indicators and test results	
Ultimate tensile strength σb/MPa	Yield strength σs/MPa	Elongation after breaking δ/%	Shrinkage of section ψ/%	
1960	1670	11	39	

The specimens used for fatigue testing were designed and manufactured in accordance with the Chinese aviation industry standard document: HB 5287-1996. The sample design and physical object are shown in Fig. 2, Fig. 3.Fig. 2 Dimensions diagram of 300M steel fatigue sample.

Fig. 2

Fig. 3 300M steel fatigue sample physical diagram.

Fig. 3

The high frequency fatigue testing machine produced by China Changchun Qianbang Testing Equipment Co., LTD., model QBG, is used as the test equipment, as shown in Fig. 4.Fig. 4 Electromagnetic resonance high frequency fatigue testing machine.

Fig. 4

Fatigue test stress ratio R = −1, that is, constant amplitude symmetric cyclic tensile and compression fatigue test. Test loading frequency 100–120 Hz, sample stress concentration coefficient Kt = 1.

The fatigue test of 300M steel under two-level stress loading was carried out, and the stress levels selected were σ1 = 1050 MPa and σ2 = 950 MPa. The reason for choosing these two stress levels is: 1) Fatigue testing shows that as the stress level decreases and approaches the fatigue limit, the dispersion of fatigue test data gradually increases. Therefore, choosing a higher stress level can reduce the dispersion of test data to a certain extent. 2) Secondly, in addition to the requirements of aviation industry standards, the design of this fatigue test also referred to the fatigue test results of 300M steel in the 2nd edition of the China Aviation Materials Manual. Finally, two levels of stress were selected as 1050 MPa and 950 MPa.

The basic process of the test is as follows: first at the first level of stress, load the 300M steel fatigue sample, stop after reaching the specified number of cycles n1, and then load the original sample at the second level of stress until the fatigue failure and fracture of the sample, record the life n2.Before carrying out multilevel fatigue test, it is necessary to determine the fatigue life corresponding to each stress level. Under the selected stress levels, three samples were tested respectively, and the test results were shown in Table 4.Table 4 Single level stress loading fatigue test results of 300M steel.

Table 4Serial number	Stress amplitude/MPa	Fatigue life	Mean fatigue life/Ni	
1	σ1 = 1050	5.70E+4	5.47E+4	
2	5.20E+4	
3	5.50E+4	
4	σ2 = 950	6.27E+5	6.69E+5	
5	4.75E+5	
6	9.05E+5	

According to the results in Table 4, the mean fatigue life of 300M steel fatigue sample under different stress levels is: N1 = 5.47E+4(corresponding to σ1 = 1050 MPa), N2 = 6.69E+5 (corresponding to σ2 = 950 MPa).

Two-level fatigue test were carried out in different loading sequences from high load to low load and from low load to high load, and the test results and load cycle ratio ni/Ni corresponding to each level were shown in Table 5.Table 5 Two level stress loading fatigue test results of 300M steel.

Table 5Loading sequence/MPa	Load cycle n1	cycle ratio n1/N1	Load cycle n2	cycle ratio n2/N2	
1050–950	2.00E+4	0.3656	8.30E+4	0.1241	
3.50E+4	0.6399	9.00E+4	0.1345	
950–1050	5.00E+4	0.0747	9.30E+4	1.7002	
7.50E+4	0.1121	7.20E+4	1.3163	

The test results show that the higher load of 300M steel will reduce the fatigue life corresponding to the subsequent low load, and the total cumulative fatigue damage is less than 1. When the load is from low to high, the lower load will increase the fatigue life corresponding to the subsequent high load, the cumulative fatigue damage value is greater than 1, and even the remaining fatigue life under high load will be greater than the corresponding life of the original material (that is, when it is not subjected to low load).

According to the fatigue test results of 300M steel under two-level stress loading, the residual fatigue life of 300M steel under the second level load is predicted by the new improved model proposed in this paper. The predicted results were compared with M − H model and other four fatigue cumulative damage models, and the results were shown in Table 6.Table 6 Prediction of residual fatigue life of 300M steel under two-level stress loading.

Table 6Stress/MPa	Test Value	M-H	GHY	YP	HH	GK	Proposed	
n2/Nf2	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	
High-Low
1050 → 950	0.1241	0.3090	149.03 %	0.3341	169.25 %	0.4167	235.90 %	0.1687	35.98 %	0.2842	129.07 %	0.2545	105.11 %	
0.1345	0.1513	12.44 %	0.1651	22.71 %	0.2128	58.17 %	0.0787	41.48 %	0.1379	2.51 %	0.1222	9.17 %	
Low-High
950 → 1050	1.7002	0.9991	41.23 %	0.9984	41.28 %	0.9950	41.48 %	0.9707	42.91 %	0.9996	41.21 %	0.9984	41.28 %	
1.3163	0.9974	24.22 %	0.9956	24.37 %	0.9885	24.90 %	0.9491	27.89 %	0.9986	24.13 %	0.9965	24.37 %	
MRPE			56.73 %		64.40 %		90.11 %		37.06 %		49.23 %		44.98 %	

According to the prediction results and errors of each model in Table 6, the prediction results of residual fatigue life under high to low loading sequence are shown in Fig. 5, and the results under low to high loading sequence are shown in Fig. 6. The comparison of mean relative prediction errors of each model is shown in Fig. 7.Fig. 5 Prediction results of residual fatigue life of 300M steel under high to low loading.

Fig. 5

Fig. 6 Prediction results of residual fatigue life of 300M steel under low to high loading.

Fig. 6

Fig. 7 The mean relative prediction error of 300M steel under two-level loading.

Fig. 7

As can be seen from Fig. 5, Fig. 6, Fig. 7, for 300M steel material, the difference of prediction effect of each model is obvious in the loading order from high to low. The new improved model proposed in this paper, as well as the HH model and the GK model, have the highest accuracy in the prediction of residual fatigue life, and the prediction effect ranks among the top three. In the second prediction result, the minimum relative prediction error of the new improved model is only 9.17 %. Under the loading order from low to high, the difference of prediction effect of each model is small. The prediction results of HH model and YP model are basically the same except that there is a certain deviation in the prediction results.

As 300M steel is a special high-strength steel, it is extremely sensitive to the influencing factors of fatigue life, and the defect distribution is uncertain, resulting in a large dispersion of fatigue test data. Therefore, it is difficult to predict the fatigue life of 300M steel. From the residual fatigue life prediction results of each model, it can be found that the prediction error will fluctuate to a certain extent for different loading sequence and different sample results. Considering the average relative prediction error of all loading sequences, the accuracy of the new improved model proposed in this paper ranks second among the six comparison models, and the average relative prediction error is 44.98 %, which is 11.75 % lower than that of the original M − H model, and the accuracy is significantly improved, showing effective and accurate prediction performance.

3.2 Model validation of other materials

3.2.1 Fatigue life prediction of C35 steel

C35 steel is a high-quality carbon structural steel that is widely used in the manufacture of levers, pivots, pins, bolts, nuts, etc. in machinery. Table 7 shows the fatigue test data and model prediction results of C35 steel under two-level loading. The experimental test data in the table are derived from Ref. [32].Table 7 Two-level loading fatigue life prediction of C35 steel.

Table 7Stress/MPa	Test Value	M-H	GHY	YP	HH	GK	Proposed	
n2/Nf2	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	PRE	RPE	
High-Low
353 → 275	0.4580	0.5451	19.01 %	0.6315	37.88 %	0.7169	56.52 %	0.3255	28.93 %	0.4586	0.13 %	0.4406	3.79 %	
0.2810	0.3776	34.38 %	0.4517	60.76 %	0.5322	89.39 %	0.2111	24.88 %	0.3088	9.91 %	0.2951	5.03 %	
0.1090	0.2111	93.65 %	0.2596	138.13 %	0.3160	189.94 %	0.1118	2.56 %	0.1686	54.72 %	0.1604	47.20 %	
0.0540	0.0937	73.54 %	0.1173	117.14 %	0.1459	170.09 %	0.0480	11.10 %	0.0738	36.65 %	0.0700	29.65 %	
Low-High
275 → 353	1.1200	0.9988	10.82 %	0.9951	11.16 %	0.9951	11.15 %	0.9655	13.80 %	0.9998	10.73 %	0.9957	11.10 %	
1.0300	0.9826	4.60 %	0.9591	6.88 %	0.9595	6.84 %	0.8682	15.71 %	0.9945	3.45 %	0.9623	6.58 %	
0.8500	0.8682	2.14 %	0.7979	6.13 %	0.7988	6.02 %	0.6370	25.06 %	0.9258	8.92 %	0.8057	5.21 %	
MRPE			34.02 %		54.01 %		75.71 %		17.43 %		17.79 %		15.51 %	

As can be seen from the calculation results in Table 7 and Fig. 9, the Mean Relative Prediction Error (MRPE) of the model in this paper is 15.51 %, which is the smallest among all comparison models, and compared with the traditional M − H model, the error decreases the most, by 18.51 %. MRFE of M − H model, GHY model, YP model, HH model, and GK model were 34.02 %, 54.01 %, 75.71 %, 17.43 %, and 17.79 %, respectively.

As can be seen from Fig. 8, in the order of load from high to low, the YP model and GHY model have the worst prediction effect, with their relative prediction errors both exceeding 25 % and falling outside the error range of 50 % many times. The prediction effect of the classical M − H model is better than that of the YP model and GHY model. The first three models with good prediction performance are the model proposed in this paper, the GK model, and the HH model, among which the relative prediction error of the model in this paper is mostly stable within 30 %, and the minimum value is only 3.79 %, and the prediction result is close to the test value. Under the loading order from low to high, the HH model has a large prediction error, close to the 25 % error band. The prediction error of the model in this paper is in the middle, and the prediction effect is close to that of other models.Fig. 8 Predicted cycle ratio of C35 steel under two-level loading.

Fig. 8

Fig. 9 The mean relative prediction error of C35 steel under two-level loading.

Fig. 9

The results show that the proposed model can effectively predict the residual fatigue life of C35 steel under different loading sequences.

In practice, most of the members are usually subjected to more complex multi-level stress loading rather than the two-level stress loading condition, and the fatigue test data of the four-level and five-level stress loading were used to validate and compare the improved models, respectively.

3.2.2 Fatigue life prediction of 6082-T6 aluminum alloy

Table 8 show the four-level stress fatigue test data and model prediction results of 6082-T6 aluminum alloys under random loading sequences. The experimental test data in the table are derived from Ref. [37].Table 8 Four-level random loading fatigue life prediction of 6082-T6 aluminum alloy.

Table 8Stress/MPa	Test Value	M-H	GHY	YP	HH	GK	Proposed	
ni/Nfi	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	
280	0.2217	0.1775	61.40 %	0.2091	90.09 %	0.2446	122.40 %	0	100 %	0.1434	30.39 %	0.1459	32.67 %	
305	0.2882	
260	0.1453	
240	0.1100	

As can be seen from Table 8 and Fig. 10, for the four-level stress loading of 6082-T6 aluminum alloy, the prediction accuracy of the GK model is the highest under random sequential loading, with a prediction error of 30.39 %, and the predicted value is the closest to the experimental value, followed by the improved model in this paper, with a prediction error of 32.67 %, ranking second among all models. Ranking third is the classical M − H model, whose prediction error is 61.40 %, and the other models have large prediction errors. The HH model has fatigue cumulative loss failure before the fourth level of stress loading.Fig. 10 Predicted cycle ratio of 6082-T6 aluminum alloy under four-level random loading.

Fig. 10

By summarizing the fatigue life prediction results of 6082-T6 aluminum alloy, it can be found that among the six comparison models, the prediction accuracy of the improved model proposed in this paper ranks the third in the loading order from high to low, the second in the loading order from low to high, and the second in the random loading order. In most cases, it is better than the M − H model and other prediction models, which shows that the proposed model is effective and accurate for predicting the residual fatigue life of 6082-T6 aluminum alloy under four stress loading and different loading sequences.

3.2.3 Fatigue life prediction of 41Cr4 alloy steel

Table 9 shows the five-level stress fatigue test data and model prediction results of 41Cr4 alloy steel. The experimental test data in the table are derived from Ref. [17].Table 9 Five-level high-low loading fatigue life prediction of 41Cr4 alloy steel.

Table 9Stress/MPa	Test Value	M-H	GHY	YP	HH	GK	Proposed	
ni/Nfi	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	PRE	RFE	
350	0.0008	0.4397	24.90 %	0.4948	40.57 %	0.5667	60.99 %	0.0321	90.87 %	0.3186	9.49 %	0.3755	6.68 %	
332	0.0048	
298	0.0474	
254	0.2137	
201	0.3520	

As can be seen from Table 9 and Fig. 11, among the fatigue life prediction results of various models for 41Cr4 alloy steel under grade 5 stress load, the predicted value of the improved model in this paper is the closest to the experimental value, and the prediction accuracy is the highest among all comparison models, with a prediction error of only 6.68 %, which is 18.22 % higher than that of the traditional M − H model. The prediction error of other models is large, except for the GK model, all of them exceed 20 %, and the maximum error reaches 90.87 %.Fig. 11 Predicted cycle ratio of 41Cr4 alloy steel under five-level loading.

Fig. 11

According to the data from Table 6, Table 7, Table 8, Table 9, the total mean relative prediction error (Loading order is not differentiated) of each model for the remaining fatigue life of four different materials was calculated, and the results were summarised in Table 10.Table 10 The total mean relative prediction error of each model for different materials.

Table 10Materials	M-H	GHY	YP	HH	GK	Proposed	
300M steel	56.73 %	64.40 %	90.11 %	37.06 %	49.23 %	44.98 %	
C35 steel	34.02 %	54.01 %	75.71 %	17.43 %	17.79 %	15.51 %	
6082-T6 aluminum alloy	61.40 %	90.09 %	122.40 %	100.00 %	30.39 %	32.67 %	
41Cr4 alloy steel	24.90 %	40.57 %	60.99 %	90.87 %	9.49 %	6.68 %	

According to the statistical results in Table 10, the new improved model proposed in this paper has an average error range of 6.68 %–44.98 % for fatigue life prediction of four materials, with a fluctuation amplitude of 38.3 % (the difference between the maximum and minimum errors). The error fluctuation amplitudes of the M − H original model, GHY model, YP model, HH model, and GK model are 36.5 %, 49.52 %, 61.41 %, 82.57 %, and 39.74 %, respectively. The top three comparison models with the smallest error fluctuation are (from small to large): M − H original model, the model proposed in this paper, and GK model. From the specific prediction results, although the error fluctuation of the model proposed in this article is slightly larger than that of the M − H original model, the average error of the fatigue life prediction results of the four materials proposed in this article is all smaller than that of the M − H original model, and overall it is also better than the GK model and other models. This proves that the model proposed in this article fully reflects the influence of loading sequence and load interaction during multi-level stress loading, and has higher sensitivity and adaptability in different states, making the prediction of fatigue life more accurate.

4 Conclusions

Aiming at the existing nonlinear fatigue cumulative damage model under complex stress state with low prediction accuracy, large error fluctuation, and inefficient calculation and application, based on the analysis of the existing models and common correction methods, a new model correction method is proposed and a new improved M − H model is established. The fatigue test data under many different materials and complex multilevel stress loading states are used for verification and comparative analysis, and the main conclusions are as follows:(1) Different from the existing improved model, this paper considers the neighboring load relationship and the fatigue life difference under different stresses when further refining the correction index of the M − H model, and the improved model more fully considers the influence of load order and load interaction effect on the fatigue life prediction.

(2) The accuracy and validity of the improved model in this paper are verified by using fatigue test data of four different materials and multiple complex stress loading states, and compared with the traditional M − H model and a variety of existing improved models, and the data results show that the improved model has a better prediction accuracy, and it is superior to the other models in the vast majority of cases.

(3) It's worth noting that the study has two major limitations. The first is the data limitation of this study. In the process of this study, the verification data mainly comes from the collection of data from existing studies, and then the experiment is conducted. For the data obtained through collection, there may be certain errors or subjectivity of the researchers. The second is the scope of the study. This study only pays special attention to the revision of the model from the perspective of load and life relationship, and focuses on high-cycle fatigue test data, lacking the verification of strain fatigue and emerging materials. In the future, we can consider in-depth research from the above aspects to explore a more comprehensive theoretical framework.

Data availability statement

Data sharing is applicable to this article. Data will be made available on request.

Funding information

Civil Aircraft Special Project of the Ministry of Industry and Information Technology of China: Fatigue Life Assessment Technology for Landing Gear Structures, Project Number: JZ025-GJJS02-01 .

CRediT authorship contribution statement

Panglun Liu: Writing – review & editing, Resources, Funding acquisition, Data curation. Jie Zhang: Writing – original draft, Validation, Conceptualization. Haihong Tang: Visualization, Validation, Formal analysis. Heng Duan: Resources, Data curation. Bingyan Jiang: Supervision, Resources, Project administration, Funding acquisition.

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.

Acknowledgment

The research is financially supported by Civil Aircraft Special Project of the 10.13039/501100006579 Ministry of Industry and Information Technology of China: Fatigue Life Assessment Technology for Landing Gear Structures, Project Number: JZ025-GJJS02-01 .
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