Comparison of critical plane models for multiaxial fatigue life prediction




multiaxial fatigue, critical plane criteria, fatigue life prediction, proportional and non-proportional loading


Background. The operation of numerous machines and units takes place under conditions of multi-axial cyclic loading, which, as a rule, is non-proportional. Evaluating the fatigue durability of metal alloys under conditions of multi-axial non-proportional loading is a relevant task in modern engineering. Solving this problem requires fatigue calculation methods that would consider operating conditions and properties of structural materials, including factors such as the type of stress state, loading trajectory, material sensitivity to non-proportional loading, and so on.
Objective. To conduct a comparative analysis of a range of fatigue life models based on the concept of the critical plane, including the Fatemi-Socie, Wang-Brown, Smith-Watson-Topper, Liu I, and Liu II approaches, and to identify the limits and peculiarities of their application.
Methods. The fatigue lives calculated using the selected models were compared with experimental results obtained for various metal alloys subjected to uniaxial tension-compression, alternating torsion, and proportional and non-proportional multiaxial loading.
Results. The applicability limits of fatigue life models based on the critical plane concept were analyzed for different metal alloys under conditions of proportional and non-proportional multiaxial loading.
Conclusions. The research results demonstrated that models requiring the use of material constants obtained from tests in both tension-compression and alternating torsion provide reliable fatigue life estimates for various types of metal alloys. Calculations based solely on fatigue curves from alternating torsion better correlate with the results of tests on ductile materials, while calculations based on criteria utilizing fatigue curves from tension-compression align more closely with results from tests on brittle materials.


  1. A. Fatemi and N. Shamsaei, “Multiaxial fatigue: An overview and some approximation models for life estimation,” Inter-national Journal of Fatigue, 33(8), pp. 948–958, 2011.
  2. B.R. You and S.B. Lee, “A critical review on multiaxial fatigue assessments of metals,” International Journal of Fatigue, 18(4), pp. 235–244, 1996.
  3. ASME Code Case N-47-23 (1988) Case of ASME Boiler and Pressure Vessel Code, American Society of Mechanical Engineers.
  4. K. Kanazawa, K.J. Miller and M.B. Brown, “Low-cycle fatigue under out-of-phase loading conditions,” ASME J. Eng. Mater. Techn., Vol. 99, pp. 222–228, 1977.
  5. N.S. Mozharovskyj and S.N. Shukaev, “Dolgovechnost konstrukcyonnuh materyalov pry neproporcyonalnuh putjah malocyklovogo nagruzhenyja,” Problemu prochnosty, No. 10. pp. 47–54, 1988.
  6. C. Lu and J. M. Martínez-Esnaola, “Multiaxial fatigue space: A three-dimensional space constituted of fatigue basic units,” International Journal of Fatigue, Vol. 143, 105995, 2021.
  7. V.T. Troshchenko, “Rasseyannoe ustalostnoe povrezhdenie metallov i splavov. Soobshchenie 3. Deformatsionnye i en-ergeticheskie kriterii,” Problemy prochnosti, No. 1, pp. 5–31, 2006.
  8. A. Karolczuk and E. Macha, “A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materi-als,” International Journal of Fracture, 134, pp. 267–304, 2005.
  9. Z. Y. Yu, S. P. Zhu, Q. Liu and Y. Liu, “Multiaxial fatigue damage parameter and life prediction without any additional material constants,” Materials, 10(8), 923, 2017.
  10. G. He, Y. Zhao and C. Yan, “Multiaxial fatigue life prediction using physics-informed neural networks with sensitive fea-tures”, Engineering Fracture Mechanics, Vol. 289, 109456, 2023.
  11. M.W. Brown and K.J. Miller, “A Theory For Fatigue Failure Under Multiaxial Stress-Strain Conditions,” Proc. Inst. Mech. Engrs., Vol.187. pp. 745–755, 1973.
  12. C.H. Wang and M.W. Brown, “A path‐independent parameter for fatigue under proportional and non‐proportional load-ing,” Fatigue & fracture of engineering materials & structures, 16(12), pp. 1285–1297, 1993.
  13. D. Socie and G.B. Marquis, Multiaxial fatigue. Society of Automotive Engineers Warrendale, PA, 2000.
  14. A. Fatemi and D.F. Socie, “A critical plane approach to multiaxial fatigue damage including out-of-phase loading,” Fa-tigue Fract Eng Mater Struct, Vol. 11(3), pp. 149–165, 1988.
  15. K.N. Smith, P. Watson and T.H. Topper, “A stress-strain function for the fatigue of metals,” Journal Material 5 (1970), pp. 767–778.
  16. K.C. Liu, “A method based on virtual strain-energy parameters for multiaxial fatigue life prediction,” ASTM special tech-nical publication, 1191, pp. 67–67, 1993.
  17. K.C. Liu and J.A. Wang, “An energy method for predicting fatigue life, crack orientation, and crack growth under multiax-ial loading conditions,” International Journal of Fatigue, Vol. 23, pp. 129–134, 2001.
  18. D. Skibicki and Ł. Pejkowski, “Low-cycle multiaxial fatigue behaviour and fatigue life prediction for CuZn37 brass using the stress-strain models,” International Journal of Fatigue, Vol. 102, pp. 18–36, 2017.
  19. Z.-R. Wu, X.-T. Hu and Y.-D. Song, “Multiaxial fatigue life prediction for titanium alloy TC4 under proportional and non-proportional loading,” International Journal of Fatigue, Vol. 59, pp. 170–175, 2014.
  20. D.J. Jones and P.Kurath (1988). Cyclic Fatigue Damage Characteristics observed for Simple loadings extended to multi-axial fatigue life prediction. NASA contractor report NAS 1.26:182126.
  21. T. Zhao and Y. Jiang, “Fatigue of 7075-T651 aluminum alloy,” International journal of fatigue, Vol. 30(5), pp. 834–849, 2008.
  22. S. Ma, B. Markert and H. Yuan, “Multiaxial fatigue life assessment of sintered porous iron under proportional and non-proportional loadings,” International Journal of Fatigue, Vol. 97, pp. 214–226, 2017.



How to Cite

Y. Savchuk and S. Shukayev, “Comparison of critical plane models for multiaxial fatigue life prediction”, Mech. Adv. Technol., vol. 7, no. 3 (99), pp. 279–293, Dec. 2023.