A comparative study on combined high and low cycle fatigue life prediction model considering loading interaction

Abstract

Fatigue life estimation of aero-engine turbine components under combined high and low cycle fatigue (CCF) is of significance for guaranteeing the structural reliability during operation. According to the investigations on damage evolution process, a nonlinear damage accumulation method is proposed for life prediction under CCF loadings, and the interaction effect between high cycle fatigue (HCF) and low cycle fatigue (LCF) is considered by integrating the interaction factor and stress ratio of CCF. Furthermore, experimental results of alloys and turbine blades are utilized to validate the proposed method and conduct a comparative analysis among Miner’s rule and other two typical nonlinear cumulative damage methods under combined loading conditions. Comparative results demonstrate that the developed model holds better prediction robustness and accuracy than those of others.

Keywords

Combined high and low cycle fatigue turbine blades damage accumulation life prediction load interaction

Introduction

Fatigue failure is the major failure mode of aero-engine hot-end components, and it usually leads to serious consequences for the aircraft (Hu et al., 2016; Vantadori et al., 2020; Zhao et al., 2022; Zhou et al., 2022). During operation, hot-end components suffer from complicated loads, including low cycle fatigue (LCF) loads and high cycle fatigue (HCF) loads. Among them, LCF loads are induced by large-amplitude centrifugal force and thermal stress, and HCF loads are produced by the high-frequency vibrations during flight (Li et al., 2022; Liu et al., 2023a; Ma et al., 2022). Under the integration of HCF and LCF, the failure mode of hot-end components is named as the combined high and low cycle fatigue (CCF) (Patriarca et al., 2020; Zheng et al., 2013). Some existing methods, which only addressing pure HCF or LCF, cannot accurately estimate the fatigue life of components and effectively describe the fatigue behavior for the strength design (Yan et al., 2014; Yue et al., 2021a). Hence, investigations on fatigue life estimation and damage modelling in the CCF loading mode have been the prerequisite for the operational reliability of turbine components (Han et al., 2018; Liu et al., 2011).

Recently, fatigue life prediction under CCF loadings has been attached more attention. For the CCF life prediction methods, linear damage accumulation model (Miner’s rule) has achieved extensive application in practical engineering due to the simple expression. The predicted results by the typical rule, however, are always unreliable as it holds some inherent shortcomings, such as the ignored loading interaction effect and independent loading stress (Wang et al., 2022; Yue et al., 2023; Zhao et al., 2011).

In view of this, increasing fatigue life prediction models have been developed, and the predicted lives by nonlinear damage accumulation methods are supposed to be more accurate in the CCF loading mode. For instance, Trufyakov et al. (1981) performed multiple groups of CCF tests on different materials to obtain the introduced material parameter, and then a life prediction approach was built. Zhu et al. (2017) proposed a CCF life prediction model by separately quantifying the LCF damage, HCF damage as well as the coupled damage produced by the interaction among HCF/LCF. Through studying the experimental data under CCF loadings, Yue et al. (2021a) established a nonlinear cumulative damage model addressing the HCF-LCF interaction with a similar expression to Miner’s rule. Bai et al. (2021) developed a CCF life estimation method to consider coupled damage by blending a new interaction factor in the LCF damage. According to damage curve approach and linear damage rule, accounting for the HCF-LCF interaction effect, a damage accumulation model under CCF loadings was put forward for by Hou et al. (2023). Under the consideration of load interaction between HCF and LCF, Liu et al. (2023b) established a CCF life prediction approach by employing the equivalent damage principle. Ding et al. (2022) decomposed CCF loads into LCF, HCF and coupled damage to present a CCF life estimation method by using continuous damage mechanics. Through the observations of fatigue performance, Qiu et al. (2022) utilized two threshold parameters to propose a life estimation model under CCF loadings, and the comparative results demonstrated that the approach offered a higher prediction ability. Wang et al. (2023) proposed a life estimation method by investigating the experimental results under combined loading conditions. By conducting fatigue tests on titanium alloy, Zhao et al. (2021) proposed a modified CCF life prediction model based on Trufykov’s model. To estimate the CCF lifetime with higher prediction accuracy, a threshold-damage based approach was established by calculating the limit life of HCF (Yue et al., 2021b). Han et al. (2019) developed a cumulative damage method on the basis of the Kachanov-Lemaitre model, in which the nonlinear interaction among HCF/LCF was addressed.

Although lots of CCF lifetime prediction methods have been presented by scholars for satisfactory prediction accuracy under the give loadings, the dispersion of life estimation for other materials under variable loading conditions may be great since the major damage accumulation theories are not established for CCF but single LCF or HCF. Hence, the predicted results by some models extended to CCF loadings may be overestimated or underestimated. Moreover, some modified life prediction methods may introduce material parameters determined by CCF tests, which limiting the applications under various loading conditions. In this analysis, a nonlinear damage accumulation method is established by using equivalent damage concept to gain the more acceptable and robust predictions, and the HCF-LCF interaction is considered by the new interaction factor without additional fitted parameters for convenient application.

The aim of this work is to propose a nonlinear cumulative damage method for CCF life estimation, and the effect of load interaction among HCF/LCF is addressed by the developed interaction factor. The current paper is arranged as follows: in the next section, the linear and nonlinear damage accumulation methods are briefly introduced under CCF loadings, and section ‘Proposed CCF life prediction model’ proposes an improved damage accumulation method under the consideration of load interaction effect among HCF/LCF loads. In section ‘Validation of the method’, the prediction accuracy of developed approach is validated by performing a comparative study on CCF test data of materials and turbine blades. In the final section, the conclusions are provided.

Linear and nonlinear damage accumulation methods

Turbine components are subjected to the complicated loads during the actual operation, and the damage accumulation is an essential issue for researchers to predict the fatigue lifetime (Hu and Wang, 2013; Li et al., 2018; Zhu et al., 2016, 2018b). Generally, the outfield load spectrum is simplified into the typical CCF load spectrum to conduct fatigue test in laboratory, in which HCF loads represented by triangular waves are superimposed on the top stage of each LCF load denoted by trapezoidal wave (Bai et al., 2022; Dungey and Bowen, 2004; Han et al., 2021; Li et al., 2022; Wen et al., 2022), as shown in Figure 1.

Figure 1.

CCF load spectrum.

From Figure 1, a CCF block comprises a LCF cycle and $n$ HCF cycles, and $n$ is the loading frequency of HCF and LCF. Moreover, the following expressions about HCF stress and maximum stress can be obtained, as

σ_{H} = Δ σ_{H} / 2

(1)

σ_{\max} = σ_{m} + σ_{H}

(2)

where

σ_{H}

is the stress amplitude of HCF,

σ_{m}

is the mean stress,

σ_{\max}

is the maximum stress of CCF.

The stress ratio of CCF can be expressed by

R = \frac{σ_{m} - σ_{H}}{σ_{m} + σ_{H}}

(3)

For the CCF damage accumulation, Miner’s rule is the most commonly adopted method, as

D = \sum_{i = 1}^{N} (\frac{n_{i}}{N_{f H, i}} + \frac{1}{N_{f L, i}})

(4)

where

n_{i}

is the number of loading cycles at HCF stress level

σ_{H, i}

N_{f H, i}

and

N_{f L, i}

are the fatigue life at HCF level

σ_{H, i}

and LCF stress level

σ_{L, i}

The fatigue failure of materials occurs once the cumulative damage reaches its damage critical value 1. For a CCF block, there are n HCF cycles and a LCF cycle, and then the fatigue life under CCF loadings can be derived by

N_{f} = \frac{1 + n}{\frac{n}{N_{f H}} + \frac{1}{N_{f L}}}

(5)

where

N_{f H}

and

N_{f L}

are the number of cycles to failure of HCF and LCF, respectively.

Miner’s rule has been frequently used in multiple fields; however, it just calculates the pure HCF and LCF damage without addressing the interaction effects between HCF and LCF, potentially leading to the overestimated fatigue lives (Yue et al., 2021a; Zhao et al., 2021; Zhu et al., 2017). To overcome the deficiency of linear cumulative damage method, lots of models have been developed for accurate life prediction under combined loading conditions.

Specially, a life prediction approach (named as T-K model) was proposed by Trufyakov and Koval’chuk (1982) through investigating the CCF tests of multiple materials under different loading conditions, and a material parameter was included to reveal the variable effect of different materials on fatigue lives, as

N_{f} = (1 + n) N_{f L} {(\frac{1}{n})}^{γ \frac{σ_{H}}{σ_{L}}}

(6)

According to the typical damage curve principle, Manson and Halford (1981) presented a nonlinear damage accumulation method, in which a damage exponent $q_{i}$ was introduced to account for the load sequence effect on fatigue life under variable amplitude loadings, as

{\begin{array}{l} D_{i} = (\frac{n_{i}}{N_{i}})^{q_{i}} \\ q_{i} = B N_{i}^{β} \end{array}

(7)

where

N_{i}

is the fatigue life at the stress level

σ_{i}

β

and

B

are material constants.

Similar to Miner’s rule, equation (7) can be introduced to calculated the CCF damage, and the parameter $β$ is taken as 0.4, as

D = \sum_{i = 1}^{N} [{(\frac{1}{N_{f L, i}})}^{(\frac{N_{f L, i}}{N_{f H, i}})}^{^{^{^{0.4}}}} + \frac{n_{i}}{N_{f H, i}}]

(8)

Previous works indicated that the predicted results by equation (8) tended to be underestimated in the CCF loading mode (Yue et al., 2021b, 2024). In view of this, Yue et al. (2020) developed an improved damage accumulation approach (named as Yue model) for CCF life prediction to account for the HCF-LCF interaction by adopting damage curve concept, as

D = \sum_{i = 1}^{N} [\frac{1}{n_{i}} {(\frac{n_{i}}{N_{f L, i}})}^{(\frac{N_{f L, i}}{N_{f H, i}})}^{^{^{^{0.4 α_{e q}}}}} + \frac{n_{i}}{N_{f H, i}}]

(9)

where

α_{e q}

is the equivalent stress range ratio.

By investigating the fatigue damage of turbine blades consisting of LCF damage, HCF damage and coupled damage, Zhu et al. (2017) established a CCF life prediction method (Zhu model for short), in which the coupled damage term was integrated into the Miner’s rule to address the HCF-LCF interaction, as

N_{f} = {(\frac{1}{N_{LCF}} + \frac{n}{N_{HCF}} + \frac{1}{(1 + n) \log {(N_{HCF})}^{α}})}^{- 1}

(10)

Proposed CCF life prediction model

Recently, a damage model for correlating the experimental data was developed by utilizing an equivalent driving force concept. To take into account the load history effects, Zuo et al. (2015) built a fatigue damage accumulation method under variable amplitude loading by combining with the S-N curve. During fatigue process, the S-N curve is usually shown by a power function, as

C = σ^{m} N_{f}

(11)

where

σ

is the loading stress. m and C are the corresponding material constants fitted by the experimental data.

As depicted on fatigue driving force by Kwofie and Rahbar (2011), it was a function of the number of loading cycles, applied loading force together with the fatigue lifetime, and fatigue driving force at each loading stress condition was written by

σ_{D} = σ^{m} N_{f}^{\frac{n}{N_{f}}}

(12)

where

σ_{D}

is the fatigue driving stress.

Especially, when the number of loading cycles increases, the force $σ_{D}$ grows until it reaches the limiting value, i.e., fatigue strength constant $C$ . Hence, the fatigue life can be derived by using the principle of equivalent fatigue driving stress. In the case of two loading stress levels, the fatigue damage process can be well expressed by using the equivalent driving force method, and the predicted results were also found satisfactory (Zhu et al., 2018a, Zuo et al., 2015). However, that may be changed in the CCF loading mode due to multiples HCF cycles superimposed on the top of each LCF cycle, and the fatigue damage accumulation process is presented by the equivalent fatigue driving stress, as illustrated in Figure 2.

Figure 2.

Fatigue damage evolution process under CCF loadings.

From Figure 2, the fatigue driving force grows with the cumulative damage in a combined cycle block. The curve $\overset{⌢}{OAB}$ represents the nonlinear cumulative damage process of HCF driving stress $σ_{D H}$ , whereas the damage accumulation curve for LCF driving stress $σ_{D L}$ remains unchanged in a combined cycle block. That is to say, LCF damage is invariable, while HCF damage increases with the number of loading cycles. Based on the equivalent fatigue driving stress principle, the driving force at HCF stress level $σ_{D H}$ can be calculated by that under LCF loading. Consequently, the cumulative fatigue damage may deliver a deviation since the fatigue driving stress at LCF stress level is directly equivalent to that under HCF loading.

In this analysis, an improved nonlinear damage accumulation process with the number of combined cycle blocks $N_{B}$ in the CCF loading mode is presented in Figure 3. The curves $\overset{⌢}{OAB}$ and $\overset{⌢}{O A^{'} B}$ represent the nonlinear cumulative damage process for LCF driving stress $σ_{D L}$ and HCF driving stress $σ_{D H}$ . Seen from Figure 3, the nonlinear damage evolution curves are diverse due to the different stress amplitudes. Note that fatigue driving force $σ_{D L}$ under LCF loading is denoted by curve $\overset{⌢}{O A}$ with the damage growing from 0 to $\frac{N_{B}}{N_{f L}}$ , and followed by the equivalent transformation of fatigue driving stress at HCF stress level along with $\overset{⌢}{A A^{'}}$ . Thus, the LCF driving force $σ_{D L}$ can be obtained, as

σ_{D L} = σ_{L}^{m} N_{f L}^{\frac{N_{B}}{N_{f L}}}

(13)

Figure 3.

Improved nonlinear damage evolution process under CCF loadings.

The equivalent driving force under HCF loading ${\tilde{σ}}_{D H}$ can be expressed by

{\tilde{σ}}_{D H} = σ_{H}^{m} N_{f H}^{\frac{{\tilde{n}}_{H}}{N_{f H}}}

(14)

where

{\tilde{n}}_{H}

denotes the equivalent cycles at HCF stress level.

Based on the equivalent principle of fatigue driving force, ${\tilde{σ}}_{D H}$ is equal to LCF driving stress $σ_{D L}$ , as

σ_{L}^{m} N_{f L}^{\frac{N_{B}}{N_{f L}}} = σ_{H}^{m} N_{f L}^{\frac{{\tilde{n}}_{H}}{N_{f L}}}

(15)

Assuming that $n N_{B}$ is the residual cycles under HCF loading, the final HCF driving force can be considered as that calculated by ${\tilde{n}}_{H}$ and $n N_{B}$ cycles at HCF loading along with the nonlinear cumulative damage curve $\overset{⌢}{O A^{'} B}$ , as

σ_{D H} = σ_{H}^{m} N_{f H}^{(\frac{n N_{B}}{N_{f H}} + \frac{{\tilde{n}}_{H}}{N_{f H}})}

(16)

Equation (16) can be rewritten by integrating with equation (15), as

σ_{D H} = σ_{H}^{m} N_{f H}^{\frac{{\tilde{n}}_{H}}{N_{f H}}} N_{f H}^{\frac{n N_{B}}{N_{f L}}} = σ_{L}^{m} N_{f L}^{\frac{N_{B}}{N_{f L}}} N_{f H}^{\frac{n N_{B}}{N_{f H}}}

(17)

HCF driving force increases with the cumulative damage raising from 0 to $\frac{n N_{B}}{N_{f H}}$ , and fatigue failure occurs upon finial HCF driving force reaching the critical value $C$ . Then,

C = σ_{L}^{m} N_{f L}^{\frac{N_{B}}{N_{f L}}} N_{f H}^{\frac{n N_{B}}{N_{f H}}}

(18)

Combining with the S-N curve, equation (18) leads to

\frac{C}{σ_{L}^{m}} = N_{f L}^{\frac{N_{B}}{N_{f L}}} N_{f H}^{\frac{n N_{B}}{N_{f H}}}

(19)

Furthermore, equation (19) can be rewritten by taking the logarithm of both sides, as

\frac{N_{B}}{N_{f L}} \ln N_{f L} + \frac{n N_{B}}{N_{f H}} \ln N_{f H} = \ln N_{f L}

(20)

Thus, similar to Miner’s rule, the cumulative damage model can be obtained by

\frac{N_{B}}{N_{f L}} + \frac{n N_{B}}{N_{f H}} \cdot \frac{\ln N_{f H}}{\ln N_{f L}} = 1

(21)

Note from that HCF-LCF interaction effect exerts nonnegligible influence on the CCF life (Priyanka and Andrew, 2015; Schweizer et al., 2011; Wang et al., 2023), and it can be greatly altered with the LCF damage produced by major temperature load and centrifugal force, together with the interaction factor controlled by the stress ratio of CCF. To address the load interaction effect on cumulative damage, the various interaction factors were used for fatigue life estimation under variable amplitude loadings (Gao et al., 2021; Lv et al., 2015; Zhu et al., 2018a, 2018c). In this regard, the damage accumulation model in equation (21) can be modified by introducing the LCF life and interaction factor to consider the HCF-LCF interaction effect under CCF loadings, as

\frac{N_{B}}{N_{f L}} + \frac{n N_{B}}{N_{f H} \log {(N_{f L})}^{- ω} \cdot \log {(N_{f L})}^{ω} \frac{\ln N_{f L}}{\ln N_{f H}}} = 1

(22)

where

\log {(N_{LCF})}^{- ω} \cdot \log {(N_{LCF})}^{ω}

is equal to one from the point view of mathematics. It is worth noting that

\log {(N_{f L})}^{ω}

is great than but closed to one for interaction factor

0 < w < 1

in the CCF loading mode, and the ratio of logarithm LCF life and HCF life

\frac{\ln N_{f L}}{\ln N_{f H}}

is less than 1. In this analysis,

\log {(N_{f L})}^{ω} \frac{\ln N_{f L}}{\ln N_{f H}}

is assumed to be approximately equal to unity.

Hence, the nonlinear damage accumulation model can be derived by

\frac{N_{B}}{N_{f L}} + \frac{n N_{B}}{N_{f H} \log {(N_{f L})}^{- ω}} = 1

(23)

Furthermore, an equivalent stress range ratio $α_{e q} = Δ σ_{H} / σ_{\max}$ was considered as the interaction factor to account for the HCF-LCF interaction (Yue et al., 2020). In general, the fatigue lives can be calculated by S-N curve for the given stress amplitude, which are greatly influenced by the stress ratio of fatigue tests. Specially, the available studies have suggested that stress ratio possesses a considerable effect on the HCF-LCF interaction under CCF loadings, and the corresponding damage could raise with the increasing stress ratio of CCF (Yan et al., 2014; Yue et al., 2021b; Zhao et al., 2021). In this regard, a new interaction factor is presented by integrating stress ratio of CCF $R$ into equivalent stress range ratio, as

w = \frac{Δ α_{H}}{σ_{\max}} R

(24)

Thus, a new life prediction model, addressing the HCF-LCF interaction effect, is proposed by introducing the interaction factor determined by the equivalent stress range ratio $α_{e q}$ and stress ratio of HCF $R$ , as

N_{f} = (n + 1) / [\frac{1}{N_{f L}} + \frac{n}{N_{f H} \log {(N_{f L})}^{- \frac{Δ α_{H}}{σ_{\max}} R}}]

(25)

According to equation (25), CCF life can be calculated by the nonlinear fatigue accumulation damage method, and it describes the nonlinearity of cumulative damage as well as sufficiently considers the effect of HCF-LCF interaction under CCF loading conditions.

Validation of the method

In this section, using five experimental datasets from published CCF test results of alloys, including TC11 (Zhao et al., 2021), TC4 (Yue et al., 2021b), DZ22 (Wang and Li, 1995), LY12-CZ AL (Wang and Shao, 1998), the developed method is validated. Considering the predicted errors of Manson-Halforld model and Zhu model under small cyclic frequency ratios, Yue model yielded better prediction accuracy in the CCF loading mode (Yue et al. 2020). Accordingly, the predicted results by Miner’s rule, T-K model, Yue model are utilized to compared with those of the proposed.

The experimental data for different materials were acquired under different CCF loadings. For TC11 alloy, the LCF, HCF tests were performed to study the fatigue performance and two groups of CCF tests with the loading frequency of $1000$ were carried out under different mean stress at 800 MPa and 750 MPa (Zhao et al., 2021). For the LCF tests of TC4 alloy, four specimens were utilized to calculate the medium life, and the LCF stress was set as 800 MPa with a stress ratio $R$ of $0.1$ . The CCF tests were designed at a same LCF load, while the HCF loads with the loading frequency of 5 HZ varied from 40 MPa to 400 MPa (Yue et al., 2021b). For DZ22 alloy, the CCF tests were performed on MTS800 fatigue test machine, and a high temperature resistance furnace was used to achieve high temperature loading. In the CCF loading mode, the specimens were loaded with different LCF loads of 736 MPa and 687 MPa at different temperature of 900°C, 850°C and 760°C. The cyclic frequency of LCF was 1.5/min, and the vibration frequency was set as 40 HZ (Wang and Li, 1995). For LY12-CZ Al alloy, LCF stress was denoted by trapezoidal wave with loading frequency of 0.5 HZ, and triangular wave was used to represent the HCF loads with cyclic frequency of 5 HZ. The loaded HCF stresses in CCF tests were designed at the 20% with those of LCF loads (Wang and Shao, 1998).

According to the expression of T-K model, the material parameter $γ$ should be determined. In this study, $γ$ is taken as 1.6 for TC11 alloy, TC4 alloy and DZ22 alloy, and 1.3 for LY12-CZ Al alloy (Zhu et al., 2017). Accordingly, the comparisons between experimental results and predictions by the proposed method, Miner’s rule, T-K model and Yue model are presented in Figure 4.

Figure 4.

Comparison between tested data and predicted results of the four models of (a) TC1 alloy (b) TC4 alloy (c) DZ22 alloy under different LCF stress (d) DZ22 alloy under different temperature (e) LY12-CZ Al alloy.

Furthermore, aero-engine hot-end components always experience CCF loadings induced by high temperature loads and centrifugal loads as well as vibration loads. Hence, two datasets of experimental results of turbine blades (Chen and Yan, 2014; Yan et al., 2011) were introduced to validate the applicability and accuracy of the developed life prediction model in practical engineering. In the CCF loading mode of turbine blades, LCF stress caused by centrifugal force was implemented by a tensile testing device, while HCF loads produced by the vibration were applied by adopting an electromagnetic exciter. Specially, the test temperature was designed to be a constant. Additional details can be referred in Chen and Yan (2014) and Yan et al. (2011). Figure 5 presents the comparisons of turbine blades among different models.

Figure 5.

Comparison of predictions with experimental results for (a) turbine blades and (b) turbine blades at 530°C.

To reveal the prediction errors of different methods, the statistical analysis is performed via calculating the difference of logarithmic predicted life and logarithmic experimental life, and prediction errors among proposed method, Miner’s rule, T-K model and Yue model are compared, as illustrated in Figure 6.

Figure 6.

Model prediction errors for different alloys and turbine blades.

As illustrated from Figures 4 to 6, the predictions by the proposed model tend to be more consistent with the experimental results, and 52 out 53 predictions lie within two life factors. Moreover, the statistical mean error of the developed model, accounting for the overall performance, is smaller than those of others and approaches to zero with lesser dispersion. For Miner’s rule, the prediction results are more nonconservative under CCF loadings except for LY12-CZ Al alloy. For T-K model, the predictions for different alloys and turbine blades show a great dispersion due to the introduction of the material parameter $γ$ . The estimation results of TC4 and DZ22 alloy at 850°C are underestimated, while nonconservative predictions are exhibited for others. In addition, Yue model provides an acceptable estimated result apart from the materials of TC4 and LY12-CZ Al under small ratios of loading frequency.

Furthermore, the relative error (MRE) of the experimental lives and the predictions is utilized to compare the estimation ability of these four methods (Li et al., 2022), as

MRE = \frac{1}{k} \sum_{i = 1}^{k} | \frac{N_{f t} - N_{f p}}{N_{f t}} |

(26)

where

N_{f t}

is the experimental life,

N_{f p}

is the predicted value of the model, and k is the number of tested samples.

The MRE value of different materials for the four models can be obtained by using equation (26), as illustrated in Figure 7. For the seven experimental datasets, the developed model presents a better prediction ability with smaller mean values of MRE under CCF loadings. Specially, for Yue model, the values of MRE for TC11 and DZ22 at 850°C is lesser than those of other models, while the predicted error for TC4, DZ22 at LCF stress range of 736 MPa and LY12-CZ Al are larger. Accordingly, the proposed method shows the higher prediction accuracy and estimation performance under CCF loading conditions compared to other methods.

Figure 7.

Mean relative errors by Miner’s rule, T-K model, Yue model and proposed model for different alloys and components.

Conclusion

In this study, a nonlinear damage accumulation model accounting for interaction effect between HCF and LCF loads is developed to perform a comparative analysis in the CCF loading mode. Experimental datasets of five materials and two turbine blades are utilized to compare the predictions by the Miner’s rule, T-K model, Yue model and proposed one. The main conclusions are listed as follows:

Through investigating the damage evolution process under CCF loadings, a nonlinear cumulative damage method was established for CCF life prediction, in which the interaction effect among HCF/LCF loads was fully addressed by integrating the interaction factor and stress ratio of CCF.

To validate the prediction performance of developed method, seven datasets of alloys and turbine blades were employed under CCF loading conditions. Based on the comparative study, the proposed model was capable to estimate the CCF life with a satisfactory accuracy.

Considering the overall prediction accuracy and robustness, the proposed model was higher than Yue model, especially for the small loading frequency in the CCF loading mode. For the Miner’s rule, the predicted results tended to be overestimated, while the application of T-K model was limited due to the introduced material parameter.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present work was supported by Natural Science Foundation of China (Grant No. 51975271) and Natural Science Foundation of China (Grant No. 12402092)

ORCID iD

Peng Yue

References

Bai

Huang

, et al. (2021) A modified damage accumulation model for life prediction of aero-engine materials under combined high and low cycle fatigue loading. Fatigue & Fracture of Engineering Materials & Structures 44(11): 3121–3134.

Bai

Huang

, et al. (2022) A probabilistic combined high and low cycle fatigue life prediction framework for the turbine shaft with random geometric parameters. International Journal of Fatigue 165: 107218.

Chen

Yan

(2014) Combined low and high cycle fatigue tests on full scale turbine blades. Proceedings of ASME Turbo Expo 2014: Turbine Technical Conference and Exposition., GT2014–26569.

Ding

Yan

Guo

, et al. (2022) A combined low- and high-cycle life prediction model considering the closure effect of micro-defects. Fatigue & Fracture of Engineering Materials & Structures 45(7): 2058–2071.

Dungey

Bowen

(2004) The effect of combined cycle fatigue upon the fatigue performance of Ti-6Al-4V fan blade material. Journal of Materials Processing Technology 153–154(22): 374–379.

Gao

Liu

Tang

(2021) An improved Manson-Halford model for multi-level nonlinear fatigue life prediction. International Journal of Fatigue 151: 106393.

Han

, et al. (2018) Experimental and numerical study of fatigue damage assessment under combined high and low cycle loading. Shock and Vibration 2018(1): 12.

Han

Huang

Yan

, et al. (2019) Combined high and low cycle fatigue life prediction model based on damage mechanics and its application in determining the aluminized location of turbine blade. International Journal of Fatigue 127: 120–130.

Han

Wang

Zhang

, et al. (2021) Competitive cracking behavior and microscopic mechanism of Ni-based superalloy blade respecting accelerated CCF failure. International Journal of Fatigue 150: 106306.

10.

Hou

Wang

Guo

, et al. (2023) Combined high and low cycle fatigue life prediction model based on damage curve method considering loading interaction effect. International Journal of Damage Mechanics 32(2): 185–203.

11.

Wang

(2013) Combined fatigue experiments on full scale turbine components. Aircraft Engineering and Aerospace Technology 85(1): 4–9.

12.

Meng

Liu

, et al. (2016) Experimental investigation of fatigue crack growth behavior of GH2036 under combined high and low cycle fatigue. International Journal of Fatigue 85: 1–10.

13.

Kwofie

Rahbar

(2011) An equivalent driving force model for crack growth prediction under different stress ratios. International Journal of Fatigue 33(9): 1199–1204.

14.

Huang

H-Z

Y-F

, et al. (2018) Physics of failure-based reliability prediction of turbine blades using multi-source information fusion. Applied Soft Computing 72: 624–635.

15.

Wang

Fan

, et al. (2022) A life prediction method and damage assessment for creep-fatigue combined with high-low cyclic loading. International Journal of Fatigue 161: 106923.

16.

Song

Bai

(2022) Deep learning regression-based stratified probabilistic combined cycle fatigue damage evaluation for turbine bladed disks. International Journal of Fatigue 159: 106812.

17.

Shi

, et al. (2022) Combined tensile and bending fatigue behavior and failure mechanism of a blade-like specimen at elevated temperature. International Journal of Fatigue 164: 107163.

18.

Liu

Chen

(2011) Experimental investigation on low and high cycle combined fatigue performance of TC11 titanium alloy. Gas Turbine Experiment and Research 24(2): 23–26.

19.

Liu

Wang

Shen

, et al. (2023a) On the consideration of loading interaction in combined high and low cycle fatigue life prediction. Fatigue & Fracture of Engineering Materials & Structures 46(7): 2651–2661.

20.

Liu

Wang

Shen

, et al. (2023b) A nonlinear cumulative fatigue damage life prediction model under combined cycle fatigue loading considering load interaction. International Journal of Fatigue 177: 107972.

21.

Huang

Zhu

, et al. (2015) A modified nonlinear fatigue damage accumulation model. International Journal of Damage Mechanics 24(2): 168–181.

22.

Yue

, et al. (2022) Reliability-based combined high and low cycle fatigue analysis of turbine blades using adaptive least squares support vector machines. Structural Engineering and Mechanic 83(3): 293–304.

23.

Manson

Halford

(1981) Practical implementation of the double liner damage rule and damage curve approach for treating cumulative fatigue damage. International Journal of Fracture 17(2): 169–192.

24.

Patriarca

Beretta

Foletti

, et al. (2020) A probabilistic framework to define the design stress and acceptable defects under combined-cycle fatigue conditions. Engineering Fracture Mechanics 224: 106784.

25.

Priyanka

Andrew

(2015) Aeromechanical modeling of rotating fan blades to investigate high-cycle and low-cycle fatigue interaction. Journal of Engineering for Gas Turbines and Power 137: 1–11.

26.

Qiu

Cui

Zhang

, et al. (2022) A dual-threshold modelling approach for fatigue life prediction under combined high and low cycle fatigue. International Journal of Fatigue 164: 107110.

27.

Trufyakov

V I

Koval’chuk

V S

(1982) Determination of life under two-frequency loading: Report 2: Proposed method. Strength of Materials 14(10): 1303–1308.

28.

Schweizer

Seifert

Nieweg

, et al. (2011) Mechanisms and modeling of fatigue crack growth under combined low and high cycle fatigue loading. International Journal of Fatigue 33(2): 194–202.

29.

Vantadori

Carpinteri

Ronchei

, et al. (2020) Lifetime estimation for 316 stainless steel specimens by using a critical plane approach. Procedia Structural Integrity 26: 106–112.

30.

Wang

(1995) Study on complex fatigue properties of direction alloy solidified high temperature DZ22 alloy. Physics Examination and Testing 2: 1–7.

31.

Wang

Liu

Chen

, et al. (2022) Prediction and evaluation of fatigue life via modified energy method considering surface processing. International Journal of Damage Mechanics 31(3): 426–443.

32.

Wang

Shao

(1998) Contrast between low cycle fatigue and high-low cycle complex fatigue behavior of metal materials. International Conference on Mechanical Testing of Materials and Structures

33.

Wang

Hou

Guo

, et al. (2023) A miner’s rule based fatigue life prediction model for combined high and low cycle fatigue considering loading interaction effect. Fatigue & Fracture of Engineering Materials & Structures 46(12): 4525–4540.

34.

Wen

Fei

Ahn

, et al. (2022) Accelerated damage mechanisms of aluminized superalloy turbine blades regarding combined high-and-low cycle fatigue. Surface & Coatings Technology 451: 129408.

35.

Yan

Qi Mj Deng

Chen

, et al. (2014) Investigation on material’s fatigue property variation among different regions of directional solidification turbine blades – Part II: Fatigue tests on bladelike specimens. Journal of Engineering for Gas Turbines & Power 136(10): 102503.

36.

Yan

Sun

Deng

, et al. (2011) Experimental study on fatigue curve law of turbine blade under combined high and low cycle loading. Journal of Aerospace Power 26(8): 1824–1829.

37.

Yue

Zhou

, et al. (2020) A fatigue damage accumulation model for reliability analysis of engine components under combined cycle loadings. Fatigue & Fracture of Engineering Materials & Structures 43(8): 1880–1892.

38.

Yue

Zhou

, et al. (2021a) Dynamic fatigue reliability analysis of turbine blades under the combined high and low cycle loadings. International Journal of Damage Mechanics 30(6): 825–844.

39.

Yue

Huang

, et al. (2021b) Threshold damage-based fatigue life prediction of turbine blades under combined high and low cycle fatigue. International Journal of Fatigue 150(6): 106323.

40.

Yue

Dai

, et al. (2023) Probabilistic framework for reliability analysis of gas turbine blades under combined loading conditions. Structures 55: 1437–1446.

41.

Yue

Dong

, et al. (2024) A modified nonlinear cumulative damage model for combined high and low cycle fatigue life prediction. Fatigue & Fracture of Engineering Materials & Structures 47(4): 1300–1311.

42.

Zhao

Liu

(2022) An improved fatigue accumulation damage model based on load interaction and strength degradation. International Journal of Fatigue 156: 106636.

43.

Zhao

Chen

(2011) Fatigue life predictions under high and low combined cycle loading. Journal of Mechanical Strength 33(4): 629–632.

44.

Zhao

Wang

, et al. (2021) Prediction of combined cycle fatigue life of TC11 alloy based on modified nonlinear cumulative damage model. Chinese Journal of Aeronautics 34(7): 73–84.

45.

Zheng

Engler-Pinto

, et al. (2013) Modeling of fatigue damage under superimposed high-cycle and low-cycle fatigue loading for a cast aluminum alloy. Materials Science and Engineering: A 560: 792–801.

46.

Zhou

Huang

, et al. (2022) A framework for fatigue reliability analysis of high-pressure turbine blades. Annals of Operations Research 311(1): 489–505.

47.

Zhu

Lei

Huang

, et al. (2017) Mean stress effect correction in strain energy-based fatigue life prediction of metals. International Journal of Damage Mechanics 26(8): 1219–1241.

48.

Zhu

Yue

, et al. (2017) A combined high and low cycle fatigue model for life prediction of turbine blades. Materials 10(7): 698.

49.

Zhu

Hao

Correia

, et al. (2018a) Nonlinear fatigue damage accumulation and life prediction of metals: a comparative study. Fatigue & Fracture of Engineering Materials & Structures 42(6): 1271–1282.

50.

Zhu

Liu

Peng

, et al. (2018b) Computational-experimental approaches for fatigue reliability assessment of turbine bladed disks. International Journal of Mechanical Sciences 142–143: 502–517.

51.

Zhu

Yue

Correia

, et al. (2018c) Strain energy-based fatigue life prediction under variable amplitude loadings. Structural Engineering and Mechanics 66(2): 151–160.

52.

Zuo

Huang

Zhu

, et al. (2015) Fatigue life prediction under variable amplitude loading using a non‐linear damage accumulation model. International Journal of Damage Mechanics 24(5): 767–784.