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Multiaxial notch fatigue probability modeling based on three-parameter Weibull distribution and effective strain energy density

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Abstract

Multiaxial fatigue life analysis of notched specimens is a critical issue for structural integrity design. However, there is still a lack of multiaxial fatigue life prediction models with coupled notch effect and probability distributions, which are vital for the fatigue design of key components. Therefore, in this study, a new probabilistic fatigue life prediction model is proposed for notched specimens in structural integrity design, which combines the three-parameter Weibull distribution (TPWD) with the effective strain energy density (ESED) to estimate the fatigue life of notched specimens with different survival probabilities. Meanwhile, the study involves the numerical calculation of the critical plane, extraction of the stress–strain distribution, calculation of strain energy density (SED), determination of critical damage region (CDR), and introduction of a weight function to quantify the damage weights of stress–strain at different locations around the notch on the effective damage parameters of the notch. The proposed model and three other energy-based models are validated using experimental data from notched specimens of Al7050, GH4196 alloys and medium steel En8, and the comparison results exhibit that the proposed model yields higher accuracy than three other models within the ± 3 life factor. Meanwhile, P-Weff-Nf curves of notched specimens with different survival probabilities are presented and analyzed under multiaxial loading. Overall, the proposed approach provides a new way to estimate the fatigue life of notched specimens, which can be useful for the fatigue design of key components.

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Abbreviations

TPWD:

Three-parameter Weibull distribution

ESED:

Effective strain energy density

SED:

Strain energy density

CDR:

Critical damage region

HSV:

High stress volume

TCD:

Theory of critical distances

HCF:

High cycle fatigue

LCF:

Low cycle fatigue

\(\varepsilon^{\prime}_{xy}, \varepsilon^{\prime}_{xz}\) :

Shear strains after coordinate transformation

\(\varepsilon^{\prime}_{xx}\) :

Normal strain after conversion

n%:

Experience parameter to determine the HSV range

\(\sigma_{n}\) :

Lower stress limit of HSV

\(A^{\prime } ,r^{\prime }\) :

Material constants

W c :

Plastic work per cycle per unit volume

\(\varepsilon_{p}\) :

Plastic normal strain

\(\gamma_{p}\) :

Plastic shear strain

\(\sigma\) :

Normal stress

\(\tau\) :

Shear stress

\(\Delta W_{t}\) :

Total cyclic SED

\(k_{0}, \alpha, C\) :

Material constants

\(W^{*}\) :

SED at any point

W max :

Maximum local SED

W n :

Lower limit of SED in the effective region

L LM :

Critical distance

W(r):

Energy density function

\(\gamma\) :

Relative stress gradient

r :

Radius from the notch root

\(\sigma (r)\) :

Stress function

Y w :

Strain energy gradient of the notched component

\(S_{\sigma ,0.5} ,S_{\tau ,0.5}\) :

Area surrounded by normal stress and shear stress normalization curves

\(\omega (r)\) :

Normalized curve expression

p, q, z :

Material constants

\(\varphi (r)\) :

Weight function

\(\varphi^{\prime}(r)\) :

Modified weight function

S :

Maximum stress

N :

Number of fatigue cycles

S 0 :

Maximum stress correction parameter

N 0 :

Life correction parameter

D:

Constant

x :

Independent variable

\(\alpha\) :

Position parameter

\(\beta\) :

Scale parameter

\(\gamma\) :

Shape parameter

A, B :

Calibration parameters

\(\mu^{\prime}\) :

Number of fatigue cycles

W eff,i :

Effective strain energy density

\(n^{\prime}\) :

Sample size

N f,i :

Number of fatigue failure cycles corresponding to each effective strain energy density Weff,i

N e :

Experimental life

N p :

Predicted life

E N :

Prediction error

μ :

Mean value

δ :

Standard deviation of EN

References

  1. Hu Y, Wu S, Xie C et al (2021) Fatigue life evaluation of Ti–6Al–4V welded joints manufactured by electron beam melting [J]. Fatigue Fract Eng Mater Struct 44(8):2210–2221

    Article  Google Scholar 

  2. Chen G, Zhang X, Xu DK et al (2017) Multiaxial ratcheting behavior of zirconium alloy tubes under combined cyclic axial load and internal pressure [J]. J Nucl Mater 489:99–108

    Article  Google Scholar 

  3. Amjadi M, Fatemi A (2020) Multiaxial fatigue behavior of thermoplastics including mean stress and notch effects: experiments and modeling [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2020.105571

    Article  Google Scholar 

  4. Zhu SP, Liu Q, Zhou J et al (2018) Fatigue reliability assessment of turbine discs under multi-source uncertainties [J]. Fatigue Fract Eng Mater Struct 41(6):1291–1305

    Article  Google Scholar 

  5. Carpinteri A, Spagnoli A (2001) Multiaxial high-cycle fatigue criterion for hard metals [J]. Int J Fatigue 23(2):135–145

    Article  Google Scholar 

  6. Song W, Liu X, Xu J et al (2021) Multiaxial low cycle fatigue of notched 10CrNi3MoV steel and its undermatched welds [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2021.106309

    Article  Google Scholar 

  7. Liao D, Zhu SP (2019) Energy field intensity approach for notch fatigue analysis [J]. Int J Fatigue 127:190–202

    Article  Google Scholar 

  8. Qi X, Liu T, Shi X et al (2021) A sectional critical plane model for multiaxial high-cycle fatigue life prediction [J]. Fatigue Fract Eng Mater Struct 44(3):689–704

    Article  Google Scholar 

  9. Shamsaei N, Gladskyi M, Panasovskyi K et al (2010) Multiaxial fatigue of titanium including step loading and load path alteration and sequence effects [J]. Int J Fatigue 32(11):1862–1874

    Article  Google Scholar 

  10. Glinka G, Shen G, Plumtree A (1995) A multiaxial fatigue strain energy density parameter related to the critical fracture plane [J]. Fatigue Fract Eng Mater Struct 18(1):37–46

    Article  Google Scholar 

  11. Neuber H (1961) Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law[J]. J Appl Mech 28:544–549

    Article  MathSciNet  MATH  Google Scholar 

  12. Sharpe WN, Wang KC (1991) Evaluation of a modified monotonic neuber relation[J]. J Eng Mater Technol 113(1):1–8

    Article  Google Scholar 

  13. Molski K, Glinka G (1981) A method of elastic-plastic stress and strain calculation at a notch root[J]. Mater Sci Eng 50(1):93–100

    Article  Google Scholar 

  14. Moftakhar A (1995) Calculation of time-independent and time-dependent strains and stresses in notches [D]. University of Waterloo (Canada)

  15. Ince A, Glinka G, Buczynski A (2014) Computational modeling of multiaxial elasto-plastic stress–strain response for notched components under non-proportional loading[J]. Int J Fatigue 62:42–52

    Article  Google Scholar 

  16. Taylor D (1999) Geometrical effects in fatigue: a unifying theoreticalmodel[J]. Int J Fatigue 21(5):413–420

    Article  Google Scholar 

  17. Susmel L (2008) The theory of critical distances: a review of its appli cations in fatigue [J]. Eng Fract Mech 75(7):1706–1724

    Article  Google Scholar 

  18. Liao D, Zhu SP, Correia JAFO et al (2020) Recent advances on notch effects in metal fatigue: a review [J]. Fatigue Fract Eng Mater Struct 43(4):637–659

    Article  Google Scholar 

  19. Wu YL, Zhu SP, He JC et al (2021) Assessment of notch fatigue and size effect using stress field intensity approach [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2021.106279

    Article  Google Scholar 

  20. Ye WL, Zhu SP, Niu XP et al (2021) Fatigue life prediction of notched components under size effect using stress gradient- based approach [J]. Int J Fract. https://doi.org/10.1007/s10704-021-00580-5

    Article  Google Scholar 

  21. Garrido A, Caro-Carretero R, Jimenez-Octavio JR et al (2019) A new approach to fitting the three-parameter Weibull distribution: An application to glass ceramics [J]. Commun Stat Theory Methods 50(14):3403–3420

    Article  MathSciNet  MATH  Google Scholar 

  22. Khalil A, Ijaz M, Ali K et al (2020) A novel flexible additive Weibull distribution with real-life applications [J]. Commun Stat Theory and Methods 50(7):1557–1572

    Article  MathSciNet  MATH  Google Scholar 

  23. Strzelecki P (2021) Determination of fatigue life for low probability of failure for different stress levels using 3-parameter Weibull distribution [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2020.106080

    Article  Google Scholar 

  24. Park JP, Park C, Cho J et al (2017) Effects of cracking test conditions on estimation uncertainty for Weibull parameters considering time-dependent censoring interval [J]. Materials 10(1):1–26

    Google Scholar 

  25. Shang DG, Sun GQ, Jing D et al (2007) Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach [J]. Int J Fatigue 29(12):2200–2207

    Article  Google Scholar 

  26. Liu JH, Hua FL, Lang SS et al (2022) Evaluation of fatigue strength on multiaxial notched specimens considering failure probability [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2021.106649

    Article  Google Scholar 

  27. He JC, Zhu SP, Liao D et al (2020) Probabilistic fatigue assessment of notched components under size effect using critical distance theory [J]. Eng Fracture Mech. https://doi.org/10.1016/j.engfracmech.2020.107150

    Article  Google Scholar 

  28. Zhu SP, Ai Y, Liao D, et al (2021) Recent advances on size effect in metal fatigue under defects: a review [J]. Int J Fracture pp 1–23

  29. Kuguel R (1961) A relation between theoretical stress concentration factor and fatigue notch factor deduced from the concept of highly stressed volume [J]. Soc Mater Sci 61:732–748

    Google Scholar 

  30. Sonsino CM, Moosbrugger E (2008) Fatigue design of highly loaded short-glass-fibre reinforced polyamide parts in engine compartments[J]. Int J Fatigue 30(7):1279–1288

    Article  Google Scholar 

  31. Wang Y, Du W, Luo Y (2017) A mean plastic strain fatigue–creep life prediction and reliability analysis of AISI H13 based on energy method[J]. J Mater Res 32(22):4254–4262

    Article  Google Scholar 

  32. Bandyopadhyay R, Prithivirajan V, Peralta AD et al (2020) Microstructure-sensitive critical plastic strain energy density criterion for fatigue life prediction across various loading regimes [J]. Proc Royal Soc A. https://doi.org/10.1098/rspa.2019.0766

    Article  MATH  Google Scholar 

  33. Morrow J D (1964) Cyclic plastic starin energy and fatigue of metals[J]. Astm Stp

  34. Garud Y (1981) A new approach to the evaluation of fatigue under multiaxial loadings[J]. J Eng Mater Technol 103(2):118–125

    Article  Google Scholar 

  35. Ellyin F, Kujawski D (1984) Plastic strain energy in fatigue failure[J]. J Pressure Vessel Technol 106(4):342–347

    Article  Google Scholar 

  36. Li XK, Zhu SP, Liao D et al (2022) Probabilistic fatigue modelling of metallic materials under notch and size effect using the weakest link theory [J]. Int J Fatigue. https://doi.org/10.1016/j.ijfatigue.2022.106788

    Article  Google Scholar 

  37. Lin CK, Lee WJ (1998) Effects of highly stressed volume on fatigue strength of austempered ductile irons [J]. Int J Fatigue 20(4):301–307

    Article  Google Scholar 

  38. Qylafku G, Azari Z, Kadi N (1999) Application of a new model proposal for fatigue life prediction on notches and key-seats[J]. Int J Fatigue 21(8):753–760

    Article  Google Scholar 

  39. Bo Z, Wang Y, Wei D et al (2017) Multiaxial fatigue life prediction for powder metallurgy superalloy FGH96 based on stress gradient effect[J]. Int J Fatigue 109:26–36

    Google Scholar 

  40. Caiza PDT, Ummenhofer T (2018) A probabilistic Stüssi function for modelling the SN curves and its application on specimens made of steel S355J2+ N [J]. Int J Fatigue 117:121–134

    Article  Google Scholar 

  41. Cai W, Li W, Xu J (2022) Study on the PSN curve of sucker rod based on three-parameter Weibull distribution [J]. Materials. https://doi.org/10.3390/ma15020560

    Article  Google Scholar 

  42. Liao D, Zhu SP, Qian G (2019) Multiaxial fatigue analysis of notched components using combined critical plane and critical distance approach [J]. Int J Mech Sci 160:38–50

    Article  Google Scholar 

  43. Sun GQ, Shang DG (2010) Prediction of fatigue lifetime under multiaxial cyclic loading using finite element analysis [J]. Mater Des 31(1):126–133

    Article  Google Scholar 

  44. Faruq NZ, Susmel L (2019) Proportional/nonproportional constant/variable amplitude multiaxial notch fatigue: cyclic plasticity, non-zero mean stresses, and critical distance/plane [J]. Fatigue Fract Eng Mater Struct 42(9):1849–1873

    Article  Google Scholar 

  45. Chen X, Xu S, Huang D (2010) A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading. Fatigue Fract Eng Mater Struct 22(8):679–686

    Article  Google Scholar 

  46. Chu CC, Conle FA, Bonnen J (1993) Multiaxial stress-strain modeling and fatigue life prediction of SAE axle shafts In: McDowell D, Ellis J. Editors. Advances in Multiaxial Fatigue. West Conshohocken, PA: ASTM International, STP, 1191: 37–54.

  47. Liu KC (1993) Method based on virtual strain-energy parameters for multiaxial fatigue life prediction. In: McDowell DL, Ellis R, editor. American Society for Testing and Materials STP 1191, Philadelphia, 1191: 67–84.

  48. Gan L, Wu H, Zhong Z (2019) Use of an energy-based/critical plane model to assess fatigue life under low-cycle multiaxial cycles [J]. Fatigue Fract Eng Mater Struct 42(12):2694–2708

    Article  Google Scholar 

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Acknowledgements

This research was supported by the National Natural Science Foundation of China, Grant/Award Number: 51605212; Natural Science Foundation of Gansu Province, Grant/Award Number: 20JR10RA161; Project of Hongliu Excellent Youth Program of Lanzhou University of Technology Grant/Award Number: 2020062001. Innovation Star Project for Outstanding Graduate Students of Gansu Province, Grant/Award Number: 2022CXZX-413; Innovation Star Project for Outstanding Graduate Students of Gansu Province, Grant/Award Number: 2022CXZX-417.

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Authors and Affiliations

  1. School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou, 730050, China

    Jianhui Liu, Jie Wang, Feilong Hua, Yingbao He & He Zhao

  2. Henan Sida Electric Power Equipment Co. LTD, Xuchang, China

    Kongyong Wu

  3. Gansu Precision Machining Technology and Equipment Engineering Research Center, Lanzhou, China

    Xiangbin Yi

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  1. Jianhui Liu
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Liu, J., Wang, J., Wu, K. et al. Multiaxial notch fatigue probability modeling based on three-parameter Weibull distribution and effective strain energy density. J Braz. Soc. Mech. Sci. Eng. 45, 404 (2023). https://doi.org/10.1007/s40430-023-04312-9

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