Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 7136846 | https://doi.org/10.1155/2021/7136846

Haoran Li, Jiadong Wang, Juncheng Wang, Ming Hu, Yan Peng, "Continuum Damage Mechanics Approach for Modeling Cumulative-Damage Model", Mathematical Problems in Engineering, vol. 2021, Article ID 7136846, 12 pages, 2021. https://doi.org/10.1155/2021/7136846

Continuum Damage Mechanics Approach for Modeling Cumulative-Damage Model

Academic Editor: José António Fonseca de Oliveira Correia
Received10 Apr 2021
Revised28 Apr 2021
Accepted15 May 2021
Published02 Jun 2021

Abstract

In this study, we propose a novel cumulative-damage model based on continuum damage mechanics under situations where the mechanical components are subjected to variable loading. The equivalent completely reversed stress amplitude accounting for the effect of mean stress, stress gradients, loading history, and additional hardening behavior related to nonproportional loading paths on high-cycle fatigue under variable loading is elaborated. The effect of mean stress, stress gradients, loading history, and additional hardening behavior related to nonproportional loading paths is considered by averaging the superior limit of the intrinsic damage dissipation work in the critical domain. We developed a novel cumulative-damage model by introducing the equivalent completely reversed stress amplitude into the damage-evolution model. For better comparison, existing cumulative-damage models, including the Palmgren–Miner law, corrected Palmgren–Miner law, Morrow’s plastic work interaction rule, and Wang’s rule, were employed to predict the fatigue life under variable loading. The proposed model performed better, considering the error scatter band obtained by plotting the predicted and experimental fatigue life on the same coordinate system. The model precisely predicts fatigue life under variable loading and easily identifies its material constants.

1. Introduction

Fatigue failure should be considered in the engineering design to ensure safety and reliability during service life [1]. Fatigue failure assessment plays an important role in the design of engineering structures since it ensures the safety of engineering structures during their service lives. Since its initiation by Wohler in 1860, despite enormous efforts, reliable and consistent fatigue models applicable to complex loading history are still under development [2]. Substantial engineering structures in the industry fail by fatigue.

Engineering structures in the industry are usually subjected to variable loading. Variable loading, including variable amplitude and loading paths, is related to the fatigue life of engineering structures. Comparing the fatigue on constant loading, the fatigue on variable loading is always additionally investigated for the interactive effect associated with two adjacent load steps on fatigue life. For constant-amplitude and path loading, several researchers have proposed various fatigue criteria and described the effect of additional hardening behavior related to nonproportional loading paths [3], mean stress [4], and stress gradients [5, 6] on high-cycle fatigue. In general, those approaches can be roughly categorized into three, namely, strain energy method [7], critical plane approach [8, 9], stress-field intensity [10]. Stress gradients, well known as a factor affecting fatigue strength of metals, can be predicted by the theory of stress-field intensity since it was initially proposed aiming to the damage domain. Qylafku [11], Taylor [12], and Zeng [13] employed the stress-field intensity concept to model several fatigue criteria. Despite the significant efforts made on stress-field intensity methods, there is no general consensus as to the suitability of various loading situations, e.g., nonproportional loading. For high-cycle fatigue under variable loading, it has been reported that loading history significantly affects fatigue life, and several damage-accumulation models have been proposed. Despite the significant efforts made on damage-accumulation models, none has gained widespread acceptance. Among them, Miner’s damage-accumulation rule remains the most commonly adopted in practice [14]. Zhao [15] proposed a corrected Miner’s damage-accumulation rule to improve the accuracy of prediction, while life predictions’ accuracy has still been found unsatisfactory [16]. Drawbacks associated with the rules include their inability to express definite effects of the loading sequence on fatigue life [17]. Some other researchers, including Morrow et al. [18], Wang et al. [19], and Zhu et al. [20] established damage-accumulation models aiming at definite effects. Wang’s rule was validated by multiaxial variable loading tests. Notably, damage parameters and fatigue life under constant-amplitude loading were employed to model Morrow’s plastic work interaction rule and Wang’s rule. Zhu et al. focused on isodamage lines-based methods and proposed a new nonlinear fatigue damage-accumulation model. Nevertheless, Xia et al. [21] reported that either damage parameter or fatigue life under constant-amplitude loading cannot reveal the definite effect of the loading sequence on fatigue life.

Initiated by Kachanov in 1958, continuum damage mechanics has been employed in modeling cumulative-damage models. Shang et al. [22], Hua et al. [23], and Yuan [24] proposed cumulative-damage models based on the damage-evolution model proposed by Lemaitre, which predicts the life expectancy of engineering structures subjected to variable loading. However, many material parameters in these models have to be identified, making them inconvenient for applications.

Here, we propose a new accumulative-damage model based on continuum damage mechanics, to improve the abovementioned shortcomings. First, the equivalent completely reversed stress amplitude, which accounts for the effect of additional hardening behavior related to nonproportional loading paths, mean stress, stress gradients, and loading history on high-cycle fatigue under variable loading, is elaborated using the stress-field intensity concept. The effect of loading history on fatigue life is established by combing damage parameters and fatigue life under constant-amplitude loading. Here, the equivalent completely reversed stress amplitude under single-stage loading is called the damage parameters. By introducing equivalent completely reversed stress amplitudes into the damage-evolution model, a new cumulative-damage model is established. The obtained rule compares very well with experimental data in the literature, and it is consistent with the previously proposed models, including the Palmgren–Miner law [17], corrected Palmgren–Miner law [15], Morrow’s plastic work interaction rule [18], and Wang’s rule [19]. Moreover, only one parameter is evaluated in our model, and it is very simple to obtain the model parameter.

2. Equivalent Completely Reversed Stress Amplitude

High-cycle fatigue failure is a local phenomenon, and the locality, also called the critical domain, is taken as a research object in stress-field intensity concepts. Nevertheless, the application of stress-field intensity concepts in investigating multiaxial high-cycle fatigue, especially under nonproportional loading, has not been extensively studied. Here, we propose damage parameters, based on continuum damage mechanics and stress-field intensity concepts, to model the cumulative-damage model.

Continuum damage mechanics has provided a method for analyzing damage development by introducing a damage variable into the constitutive stress-strain relationship. As one of the outstanding examples, Lemaitre et al. [25] proposed a differential equation for damage development. Distinguishingly, the damage development equation can be expressed as follows for the components subjected to fully reversed tension:

To develop the damage development equation available for multiaxial high-cycle fatigue, equation (1) is modified as follows:where is the damage parameter describing the effect of additional hardening behavior related to nonproportional loading paths and mean stress on high-cycle fatigue under single-stage constant loading. The damage parameter is expressed as equation (3), as proposed by Freitas et al. [26]. The parameter c proposed by Yao et al. [27], which is employed when considering the effect of stress gradients, is also utilized here, and it is expressed as equation (4). Considering the variety of c under situations where the mechanical component is subjected to nonproportional loading, c is calculated when reaches the maximum value. Certainly,the damage evolution equation can also be obtained from the explicit expression of damage variables as follows:where is the fatigue life of a point located at the critical domain and is the predicted life equivalent to that of structural components subjected to fully reversed tensile loading. Both are, respectively, expressed as follows:

Introducing stress-field intensity concepts, the average superior limit of the intrinsic damage dissipation work [28] in the critical domain can be obtained as the left side of equation (8). The hypothetical condition that the same average superior limit of the intrinsic damage dissipation work in the critical domain implies the same fatigue life is captured to formulate the damage parameters in our model. To simplify, the average superior limit of the intrinsic damage dissipation work in the critical domain for smooth structural components subjected to fully reversed tensile loading can be obtained as the right side of the following equation:

Then, expanding equation (8), the following equation is obtained:

Substituting equations (5)–(7) into equation (9) and employing first-order approximation, the damage parameters are finally expressed as follows. Notably, several material constants are contracted to consider the self-consistency of the damage parameters:

Numerous fatigue experiments of variable amplitude have shown that loading history has a remarkable effect on fatigue life. For structural components subjected to uniaxial loading, different loading sequence makes Miner’s cumulative critical value fall in different regions. When structural components are subjected to a high-low loading sequence, Miner’s cumulative critical value falls below one, and conversely, for a low-high loading sequence, the value is greater than one. Here, by combing the damage parameters, the equivalent completely reversed stress amplitude considering the loading sequence is proposed as follows:where is the contraction towards the partial variables in equation (11).

3. Proposed Cumulative-Damage Model

For smooth structural components subjected to single-stage fully reversed tensile loading, the damage-evolution equation can be obtained from the explicit expression of the damage variable:

Substituting equation (11) into (12), the expression of the damage-evolution associated with ith loading step can be obtained as follows:

Then, the expression of damage-evolution associated with the first loading step (equation (13)) should be rewritten as follows:

For structural components subjected to two-step loading, the expression of damage-evolution associated with the second loading step can be expressed as follows:

Based on the damage-equivalent concepts [19], if the fatigue life and applied cycles under th constant-amplitude loading are and , then the damage caused by cycle loading can be equivalent to that caused by cycle loading at the ith level, i.e., . Then, the equivalent damage of the first loading step can be obtained through the second loading step by combining equations (14) and (15). The equivalent relation can be expressed as follows:

Applying the superposable fatigue-life condition to a single-loading step, , and the cumulative-damage model for two steps loading can be established as follows:

Distinguishingly, the cumulative-damage rule for structural components is subjected to two-step uniaxial loading (fully reversed tensile loading), and we conclude that different loading sequence makes Miner’s accumulative critical value fall in different regions. The principle associated with the conclusion is expressed in equation (18). For structural components subjected to high-low two-step uniaxial loading (fully reversed tensile loading), Miner’s cumulative critical value falls below one, and conversely, the sum is greater than one:

Furthermore, for structural components subjected to three-step loading, the expression of damage-evolution associated with the third loading step can be expressed as equation (19), and it is obtained from equation (13). The damage-evolution associated with the second loading step can be obtained from equation (15), and it is explicitly expressed in equation (20):

Then, one can obtain the cumulative-damage model for three-step loading based on the damage-equivalent concepts. The resulting cumulative-damage model for three-step loading is expressed as follows:

Similarly, the recurrence formula of the ith level loading can be expressed as follows:

The application of the new cumulative-damage model for predicting the fatigue life of structural components under variable loading involves identifying the material parameter p. Material parameter p can be obtained by analyzing the experimental data associated with a specimen subjected to fully reversed tensile loading employing the least square method. Certainly, the resulting identification model (equation (23)) is derived by combing equation (12) and the least-square method:

4. Evaluation by the Experimental Data

4.1. Uniaxial Loading Condition

The uniaxial two-level step-loading test data for C35, SAE4130, and 7050-T7451 (Table 1) were used to evaluate the proposed damage-cumulative model. For better comparison, existing popular cumulative-damage models, including Palmgren–Miner law, corrected Palmgren–Miner law, and Wang’s rule, were employed to predict the fatigue life under the loading conditions. All experimental data and material parameters required in these models are listed in Table 2.


MaterialsMin.C. MinerWan.Pro.Exp.

C353532750.1689200461200645282181625353280
C353532750.25583000355000522964124743226560
C353532750.540600017800035007884092108840
C353532750.7522900010001955786367280040
C353342750.1695000467000665427282970425960
C353342750.25597500369500555925205401209140
C353342750.5435000207000395480149872203200
C353342750.7527250044500248607123416141780
C353532940.1365200245200347626159006218400
C353532940.25313000193000288484114354200600
C353532940.522600010600020282680373129200
C353532940.75139000190001250346253580600
C353342940.1371000251000360633233743245800
C353342940.25327500207500312617182448214300
C353342940.5255000135000240633141918173000
C353342940.7518250062500173737121174137700
C352753530.1122800107200125055128000134240
C352753530.25229000213400232769242000243560
C352753530.75583000567400585675617765614200
C352943340.1139000106000141485149837156600
C352943340.25182500149500186429207822205600
C352943340.75327500294500330098361257385800
SAE41306485520.2522487514027521066511042992617
SAE41306485520.5167750831501541928037657488
SAE41306485520.75110625260251024106376255353
SAE41306075520.2523900015440023127416843494898
SAE41306075520.5196000111400188474137549109990
SAE41306075520.7515300068400148386120257104496
SAE41306485650.25181375114175171584101796118431
SAE41306485650.5138750715501293537648184542
SAE41306485650.759612528925904126228963421
SAE41306075650.25195500128300190834153281132108
SAE41306075650.516700099800162428130673121080
SAE41306075650.7513850071300135687117725113412
70501761330.0745885640436578855592954852
70501761330.1485631237892549245231158708
70501761330.54421325793425883986830449
70501761330.7413592917509349033324937627
7050176850.074211090143350201638176399274446
7050176850.148196381128641183362153578204383
7050176850.7417850910769696465540192428
7050176850.3951472847954413181410330180475
7050133850.082212319144579206720191883200205
7050133850.164198838131098191046172476244318
7050133850.82190827230878692579630216409
7050133850.51436007586013502311775768700

Note: Min.: Palmgren–Miner damage cumulative law, C. Miner: corrected Palmgren–Miner damage cumulative law, Wan.: Wang YY’s damage cumulative law, Pro.: our proposal, Exp.: median fatigue experimental life, 7050-7050-T7451 aluminum alloy.

Materials

C35353520003341100002944000002757600002164.30
SAE4130648535006071100005652240005522820003914.17
7050-T7451176270271336140085225800----230.46

For better comparison, the total fatigue life under uniaxial two-level step loading and that predicted using the Palmgren–Miner law, corrected Palmgren–Miner law, Wang’s rule, and the proposed model are plotted on the same coordinate plane (Figure 1). 97.8% of the predicted data fall within an error factor of 2, and the proportion for the data predicted by the other models is 78.2%, 80.4%, and 82.6% for Palmgren–Miner law, corrected Palmgren–Miner law, and Wang’s rule, respectively. To predict fatigue life for uniaxial two-level loading, the proposed model yielded better results than other tested models.

4.2. Multiaxial Loading Condition

To evaluate the proposed model for predicting structural components under multiaxial variable loading, multiaxial two-level and two-stage block-loading test data for LY12CZ (Table 3) were used. The cylindrical specimens were subjected to combined torsion and tension. First, the equivalent completely reversed stress amplitude in the specimen under the stress condition was evaluated. Then, the proposed cumulative-damage model was employed to predict the fatigue life of the cylindrical specimen under multiaxial variable loading.


No.Loading pathsMin.Mor.Wan.Pro.Exp.

10.54837642383370751973819149
20.54837647476500235312851304
30.252972129271311243243328888
40.51972631707891627356458274635
50.5197263195890198935204827196934
60.51795641551621415834952863348
70.5179564178732180794184378184835
80.51394913908174211408411811
90.513949138879718137479357
100.18113941138111667118155236
110.568212677766883372565132923
120.56821267602673366241475311
130.568726865837972526104
140.579917969468471664673
150.43605632048154061513524138
160.577703749357784668579111423
170.57770371814775508855298732
180.42787327674284773046639078
190.43735435565335922132316875
200.42558225039272252724029967
210.52031819910207292090120619
220.52031818721189121835622650
23130421343012112120008746
24131061313013916800011986
25130251256514052900018052

Note: : single program sequence including loading paths; : each life score for single program sequence.

Considering the cylindrical specimen subjected to combined torsion and tension, at each point located in the critical domain of the specimen, we define the coordinate system (Figure 2). In this frame, the stress state under combined torsion and tension is described by the following components and aimed at each point located in the critical domain:

Then, von Mises equivalent stress is expressed as follows (equation (24)):

Considering , the following approximate equation for equation (25) is established:

Substituting equation (26) into (4), the coefficient c will be zero, which expresses the effect of the stress gradient on fatigue life. Based on the same approximate principle, the following approximate equations are established:

Ultimately, the equivalent completely reversed stress amplitude for the cylindrical specimen subjected to combined torsion and tension can be obtained by substituting equations (26)–(28) into equation (10):

Note that equation (29) cannot be applied in the case of pure torsion since a large error is obtained by making c = 0. However, one can immediately establish the equivalent relationship between the endurance limits under fully reversed tension and torsion from equation (10):

Analogously, the following equation for expressing the equivalent completely reversed stress amplitude under pure torsion can be obtained:

A concise form of equation (31) can be obtained by introducing equation (30) into equation (31):

Therefore, the proposed cumulative-damage model for predicting the fatigue life of structural components under multiaxial two-level variable loading and combined torsion and tension can be explicitly expressed as follows:

Engineering structures are usually subjected to block program sequence loading. It is important to study certain fatigue issues, such as cumulative-damage under program sequence loading. Hence, we investigated cumulative-damage under multiaxial two-stage block loading and combined torsion and tension using the proposed model. Based on damage-equivalent concepts, the damage caused by cycle loading is equivalent to that caused by cycle loading at the ith level. The variable expresses the ratio of the absorbed fatigue with the total fatigue life under single-stage loading, and its expression is derived from equations (22) and (33) as follows:

For the fatigue-life prediction under program sequence loading, we may be ignorant of the level loading causing component failure, and consequently, cannot be calculated. However, the sequence of , i.e., , can be calculated from equation (34). According to the sequence of , component failure will result at the th level due to fatigue once . There may be two cases of cumulative-damage at th level, even under at . The predicted fatigue life for two cases can be, respectively, expressed as follows:

The proposed model was assessed for predicting the fatigue life of structural components under multiaxial two-level and two-stage block loading and combined torsion and tension using relevant data for LY12CZ (Table 3). The proposed model for predicting fatigue life under two kinds of loading is expressed in equations (33) and (35). For better comparison with other cumulative-damage models, including the Palmgren–Miner law, Morrow’s plastic work interaction rule, and Wang’s rule, the models were synchronously used to predict the fatigue life of LY12CZ, and all text data under single-stage loadings and material parameter required in these models are listed in Tables 4 and 5. Notably, multiaxial equivalent stress parameters are employed in Morrow’s plastic work interaction rule. However, in the proposed model, the equivalent stress parameters are replaced by Matake’s critical plane stress parameters [32], and the interaction exponent is set to −0.45 [33].


Loading paths

A1247.520142.910016831333242
A2176.810102.0800377694238173
A3250000056316250179
A400144.30075299204146
A500202.08008911285202
A6350000031725350250
B1247.520142.9104510229326246
B2176.810102.08045348899234176
B3158012504557004246188
C1247.520142.9109011067308246
C2176.810102.0809085684220176
C32500144.340904634312248


Materials

LY12CZ350317253001406702502721231692.15

The predicted fatigue life of LY12CZ under multiaxial two-level step and two-stage block loading based on the test cumulative-damage models is shown in Figure 3. The percentage of the predicted data falling within the factor of 2.05 scatter band is 80%, 80%, 88%, and 96% for the Palmgren–Miner law, Morrow’s plastic work interaction rule, Wang’s rule, and our model, respectively. For predicting the fatigue life of LY12CZ under multiaxial variable loading, including two-level step loading and two-stage block loading, our model yielded results comparable to those of the test cumulative-damage models.

5. Conclusions

Here, we propose a cumulative-damage model based on continuum damage mechanics to evaluate the fatigue life of smooth structural components under axial and multiaxial variable loading. The conclusions can be obtained as follows:(1)The model is very competitive with the existing test cumulative-damage models. It adopts a kind of equivalent completely reversed stress amplitude that expresses the effect of mean stress, stress gradients, loading history, and additional hardening behavior related to nonproportional loading paths on high-cycle fatigue under variable loading.(2)Only one parameter is evaluated for the application of our model, and it is very simple to obtain the model parameter which is a simple function on the slope of S-N curve of materials.(3)The concept of equivalent completely reversed stress amplitude provides a new conception to investigate the cumulative damage for structural components under variable loading. Cumulative damage during fatigue progress and fatigue life can be precisely predicted by combining the equivalent completely reversed stress amplitude and damage development equation.

Abbreviations

:Volume of the critical domain
, , and :Stress amplitude under fully reversed loading in tension, damage parameters, and equivalent completely reversed stress amplitude, respectively
:Internal damage variable
:Maximum von Mises equivalent stress in the critical domain
:Polar diameter within polar coordinates and polar angle within polar coordinates, respectively
:Material parameters dependent on the slope of S-N curve
and :Damage parameters for the ith level loading and equivalent completely reversed stress amplitude for the ith level loading, respectively
:Fatigue life and applied cycles, respectively, under the ith level constant-amplitude loading
:Endurance limits in reversed tension
:Applied cycles under fully reversed loading in tension
:Internal damage variable of the ith step loading
:Equivalent internal damage variable and applied cycles, respectively, based on damage-equivalent concepts, respectively
:ith level stress amplitude under fully reversed loading in tension
:Stress amplitude and fatigue life plotted in S-N curve
:Damage parameters proposed by Matake et al.
:Elastic modulus
:von Mises equivalent stress
:Three axis factor
:Maximum hydrostatic stress in one cycle
:Maximum value in symbol
:Maximum generalized force of damage driving in one cycle
:Poisson ratio
:Hydrostatic stress
:Stress distribution function in the critical domain
:Predicted life of smooth specimen
:The critical value of damage
:Normal stress amplitude, normal mean stress, shear stress amplitude, and shear mean stress, respectively
:Phase difference between normal loading path and shear-loading path
:Radius of smooth specimen
:Polar diameter of the critical domain.

Data Availability

The data are from previously reported studies, and these prior studies are cited at relevant places within the text as references [21, 2931].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the research support for this work provided by the Scientific Research Start-Up Project of Zhejiang Sci-Tech University (11133132612019) and National Natural Science Foundation of China (52075471).

References

  1. J. F. Barbosa, J. A. Correia, R. Freire Júnior, S.-P. Zhu, and A. M. De Jesus, “Probabilistic S-N fields based on statistical distributions applied to metallic and composite materials: state of the art,” Advances in Mechanical Engineering, vol. 11, no. 8, p. 168781401987039, 2019. View at: Publisher Site | Google Scholar
  2. N. Ottosen, R. Stenström, and M. Ristinmaa, “Continuum approach to high-cycle fatigue modeling,” International Journal of Fatigue, vol. 30, no. 6, pp. 996–1006, 2008. View at: Publisher Site | Google Scholar
  3. M. V. Borodii, “Determination of the non-proportional cyclic hardening coefficient sensitive to the loading amplitude,” Strength of Materials, vol. 52, no. 6, pp. 919–929, 2021. View at: Publisher Site | Google Scholar
  4. A. Nourian-Avval and A. Fatemi, “Variable amplitude fatigue behavior and modeling of cast aluminum,” Fatigue & Fracture of Engineering Materials & Structures, vol. 44, no. 6, 2021. View at: Publisher Site | Google Scholar
  5. C. Ronchei, A. Carpinteri, G. Fortese et al., “Fretting high-cycle fatigue assessment through a multiaxial critical plane-based criterion in conjunction with the Taylor’s point method,” Solid State Phenomena, vol. 258, pp. 217–220, 2017. View at: Google Scholar
  6. I. Milošević, G. Winter, F. Grün et al., “Influence of size effect and stress gradient on the high-cycle fatigue strength of a 1.4542 steel,” Procedia Engineering, vol. 160, pp. 61–68, 2016. View at: Google Scholar
  7. H.-R. Li, Y. Peng, Y. Liu, and M. Zhang, “Corrected stress field intensity approach based on averaging superior limit of intrinsic damage dissipation work,” Journal of Iron and Steel Research International, vol. 25, no. 10, pp. 1094–1103, 2018. View at: Publisher Site | Google Scholar
  8. C. Wang, D.-G. Shang, and X.-W. Wang, “A new multiaxial high-cycle fatigue criterion based on the critical plane for ductile and brittle materials,” Journal of Materials Engineering and Performance, vol. 24, no. 2, pp. 816–824, 2015. View at: Publisher Site | Google Scholar
  9. X.-W. Wang and D.-G. Shang, “Determination of the critical plane by a weight-function method based on the maximum shear stress plane under multiaxial high-cycle loading,” International Journal of Fatigue, vol. 90, pp. 36–46, 2016. View at: Publisher Site | Google Scholar
  10. Y.-L. Wu, S.-P. Zhu, J.-C. He, D. Liao, and Q. Wang, “Assessment of notch fatigue and size effect using stress field intensity approach,” International Journal of Fatigue, Article ID 106279, 2021. View at: Publisher Site | Google Scholar
  11. G. Qilafku, N. Kadi, J. Dobranski et al., “Fatigue of specimens subjected to combined loading. Role of hydrostatic pressure,” International Journal of Fatigue, vol. 23, no. 8, pp. 689–701, 2001. View at: Publisher Site | Google Scholar
  12. D. Taylor, “Geometrical effects in fatigue: a unifying theoretical model,” International Journal of Fatigue, vol. 21, no. 5, pp. 413–420, 1999. View at: Publisher Site | Google Scholar
  13. Y. Zeng, M. Li, Y. Zhou, and N. Li, “Development of a new method for estimating the fatigue life of notched specimens based on stress field intensity,” Theoretical and Applied Fracture Mechanics, vol. 104, p. 102339, 2019. View at: Publisher Site | Google Scholar
  14. D. S. Paolino and M. P. Cavatorta, “On the application of the stochastic approach in predicting fatigue reliability using Miner’s damage rule,” Fatigue & Fracture of Engineering Materials & Structures, vol. 37, no. 1, pp. 107–117, 2014. View at: Publisher Site | Google Scholar
  15. S. B. Zhao, “Study on the accuracy of fatigue life predictions by the generally used damage accumulation theory,” Journal of Mechanical Strength, vol. 22, no. 3, pp. 206–209, 2000. View at: Google Scholar
  16. S. P. Zhu, Y. Z. Hao, J. A. F. Oliveira Correia, G. Lesiuk, and A. M. P. Jesus, “Nonlinear fatigue damage accumulation and life prediction of metals: a comparative study,” Fatigue & Fracture of Engineering Materials & Structures, vol. 42, no. 6, pp. 1271–1282, 2019. View at: Publisher Site | Google Scholar
  17. Ö. G. Bilir, “Experimental investigation of fatigue damage accumulation in 1100 Al alloy,” International Journal of Fatigue, vol. 13, no. 1, pp. 3–6, 1991. View at: Publisher Site | Google Scholar
  18. P. Kurath, H. Sehitoglu, J. D. Morrow et al., “Effect of selected sub-cycle sequences in fatigue loading histories,” American Society of Mechanical Engineers, Pressure Vessels and Piping Division, vol. 72, pp. 43–60, 1983. View at: Google Scholar
  19. Y. Wang, D. Zhang, and W. Yao, “Fatigue damage rule of LY12CZ aluminium alloy under sequential biaxial loading,” Science China Physics, Mechanics and Astronomy, vol. 57, no. 1, pp. 98–103, 2014. View at: Publisher Site | Google Scholar
  20. S.-P. Zhu, D. Liao, Q. Liu, J. A. F. O. Correia, and A. M. P. De Jesus, “Nonlinear fatigue damage accumulation: isodamage curve-based model and life prediction aspects,” International Journal of Fatigue, vol. 128, p. 105185, 2019. View at: Publisher Site | Google Scholar
  21. T. X. Xia, W. X. Yao, and L. P. Xu, “Comparative research on accumulative damage models under multiaxial 2-stage step loading spectra for LY12CZ aluminium alloy,” Journal of Aeronautical Materials, vol. 34, no. 3, pp. 86–92, 2014. View at: Google Scholar
  22. D. Shang and W. X. Yao, “Study on nonlinear continuous damage cumulative model for multiaxial fatigue,” Acta Aeronautica Et Astronautica Sinica, vol. 19, no. 6, pp. 647–656, 1998. View at: Google Scholar
  23. C. T. Hua, Fatigue Damage and Small Crack Growth during Biaxial Loading, University of Illinois, Urbana, IL, USA, 1984.
  24. R. Yuan, H. Li, Z. H. Hong et al., “A new non-linear continuum damage mechanics model for fatigue life prediction under variable loading,” Mechanika, vol. 19, no. 5, pp. 506–511, 2013. View at: Publisher Site | Google Scholar
  25. G. Cheng and A. Plumtree, “A fatigue damage accumulation model based on continuum damage mechanics and ductility exhaustion,” International Journal of Fatigue, vol. 20, no. 7, pp. 495–501, 1998. View at: Publisher Site | Google Scholar
  26. C. Gonçalves, J. A. Araújo, and E. N. Mamiya, “Multiaxial fatigue: a stress based criterion for hard metals,” International Journal of Fatigue, vol. 27, no. 2, pp. 177–187, 2005. View at: Publisher Site | Google Scholar
  27. W. X. Yao, “The prediction of fatigue behaviours by stress field intensity approach,” Acta Mechanica Solida Sinica, vol. 9, no. 4, pp. 337–349, 1996. View at: Google Scholar
  28. A. Öchsner, Continuum Damage Mechanics, Springer, Singapore, Asia, 2016.
  29. L. M. C. André, P. M. Juliana, and J. C. V. Herman, “Fatigue damage accumulation in aluminum 7050-T7451 alloy subjected to block programs loading under step-down sequence,” Procedia Engineering, vol. 2, pp. 2037–2043, 2010. View at: Google Scholar
  30. Y. L. Wang, “Equivalent-conversion method for estimation of various amplitude fatigue life,” Chinese Journal of Applied Mechanics, vol. 22, no. 1, pp. 90–94, 2005. View at: Google Scholar
  31. Y. Y. Wang, Multiaxial Fatigue Behavior and Life Estimation of Metal Materials, Nanjing University of Aeronautics and Astronautics, Nanjing, China, 2005.
  32. T. Matake, “An explanation on fatigue limit under combined stress,” Bulletin of JSME, vol. 20, no. 141, pp. 257–263, 1977. View at: Publisher Site | Google Scholar
  33. T.-X. Xia and W.-X. Yao, “Comparative research on the accumulative damage rules under multiaxial block loading spectrum for 2024-T4 aluminum alloy,” International Journal of Fatigue, vol. 48, no. 1, pp. 257–265, 2013. View at: Publisher Site | Google Scholar

Copyright © 2021 Haoran Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views520
Downloads377
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.