Skip to main content
SpringerLink
Log in

Critical Plane Orientation Influence on Multiaxial High-Cycle Fatigue Assessment

  • Published:
Physical Mesomechanics Aims and scope Submit manuscript

Abstract

In the present paper, the multiaxial fatigue lifetime of structural components failing in the high-cycle fatigue regime is evaluated by employing the modified Carpinteri-Spagnoli (C-S) multiaxial fatigue criterion based on the critical plane approach. In the above criterion, the critical plane position is linked to averaged principal stress directions through an off-angle 8. Then, the fatigue damage parameter used is determined by a nonlinear combination of an equivalent normal stress amplitude and the shear stress amplitude acting on the critical plane. In the present paper, some modifications of the original expression for the off-angle 8 are implemented in the modified Carpinteri-Spagnoli criterion. In particular, modified expressions recently proposed by Lagoda et al. are in accordance with the assumption originally developed by Carpinteri and co-workers, that is, the off-angle is a function of the ratio between the fatigue limit under fully reversed shear stress and that under fully reversed normal stress. Such expressions can be employed for metals ranging from mild to very hard fatigue behaviour. Some experimental data available in the literature are compared with the theoretical estimations and, only for materials with hard and very hard fatigue behaviour, the modified 8 relationships are shown to yield fatigue lifetime results slightly better than those determined through the original 8 expression.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Abbreviations

C :

shear stress vector acting on the critical plane

C(t):

modulus of the shear stress vector C

C a :

shear stress amplitude

m :

slope of the S-N curve for fully reversed normal stress (R = -1)

m * :

slope of the S-N curve for fully reversed shear stress (R = -1)

N :

normal stress vector perpendicular to the critical plane

N(t):

modulus of the normal stress vector N

N ʹa, eq :

equivalent normal stress amplitude

t :

time

S w :

stress vector at material point P and related to the critical plane

w :

normal unit vector perpendicular to the critical plane

δ:

angle between the averaged direction î of the maximum principal stress and the normal w to the critical plane (Fig. 1)

σaf,-1 :

fully reversed normal stress fatigue limit

σu :

material ultimate tensile strength

τaf,-1 :

fully reversed shear stress fatigue limit.

References

  1. Socie, D.F. and Marquis, G.B., Multiaxial-Fatigue, Warrendale, USA: Society of Automative Engineers, 1999.

    Google Scholar 

  2. Carpinteri, A., De Freitas, M., and Spagnoli, A., Biaxial/Multiaxial Fatigue and Fracture, Amsterdam: Elsevier Science Ltd., 2003.

    Google Scholar 

  3. Susmel, L., Multiaxial Notch Fatigue: from Nominal to Local Stress/Strain Quantities, Cambridge, UK: CRC Press, 2009.

    Book  Google Scholar 

  4. Shanyavsky, A.A., Scales of Metal Fatigue Cracking, Phys. Mesomech., 2015, vol. 18, no. 2, pp. 163–173.

    Article  Google Scholar 

  5. Marquis, G.B. and Karjalainen-Roikonen, P., Long-Life Multiaxial Fatigue of a Nodular Graphite Cast Iron, in Biaxial/Multiaxial Fatigue and Fracture, Carpinteri, A., de Freitas, M., Spagnoli, A., Eds., Amsterdam: Elsevier Science Ltd., 2003.

    Google Scholar 

  6. Lopez-Crespo, P., Moreno, B., Lopez-Moreno, A., and Zapatero, J., Study of Crack Orientation and Fatigue Life Prediction in Biaxial Fatigue with Critical Plane Models, Eng. Fract. Mech., 2015, vol. 136, pp. 115–130.

    Article  Google Scholar 

  7. Liu, X.Y., Su, T.X., Zhang, Y., and Yuan, M.N., A Multiaxial High-Cycle Fatigue Life Evaluation Model for Notched Structural Components, Int. J. Fatigue, 2015, vol. 80, pp. 443–448.

    Article  Google Scholar 

  8. Li, B.C., Jiang, C., Han, X., and Li, Y., A New Path-Dependent Multiaxial Fatigue Model for Metals under Different Paths, Fatigue Fract. Eng. Mater. Struct., 2014, vol. 37, pp. 206–218.

    Article  Google Scholar 

  9. Xia, T., Yao, W., Zou, J., and Gao, D., A Novel Accumulative Fatigue Damage Model for Multiaxial Step Spectrum Considering the Variations of Loading Amplitude and Loading Path, Fatigue Fract. Eng. Mater. Struct., 2015, doi 10.1111/ffe.12349.

    Google Scholar 

  10. Kadhim, N.A., Mustafa, M., and Varvani-Farahani, A., Fatigue Life Prediction of Low-Alloy Steel Samples Undergoing Uniaxial Random Block Loading Histories Based on Different Energy-Based Damage Descriptions, Fatigue Fract. Eng. Mater. Struct., 2014, vol. 38, pp. 69–79.

    Article  Google Scholar 

  11. Wang, C., Shang, D.G., Wang, X.W., Chen, H., and Liu, J.Z., Comparison of HCF Life Prediction Methods Based on Different Critical Planes under Multiaxial Variable Amplitude Loading, Fatigue Fract. Eng. Mater. Struct., 2015, vol. 38, pp. 392–401.

    Article  Google Scholar 

  12. Wu, Z.R., Hu, X.T., Li, Z.X., and Song, Y.D., Evaluation of Fatigue Life for Titanium Alloy TC4 under Variable Amplitude Multiaxial Loading, Fatigue Fract. Eng. Mater. Struct., 2015, vol. 38, pp. 402–409.

    Article  Google Scholar 

  13. Susmel, L. and Lazzarin, P., A Stress-Based Method to Predict Lifetime under Multiaxial Fatigue Loadings, Fatigue Fract. Eng. Mater. Struct., 2013, vol. 26, pp. 1171–1187.

    Google Scholar 

  14. Łagoda, T. and Ogonowski, P., Criteria of Multiaxial Random Fatigue Based on Stress, Strain and Energy Parameters of Damage in the Critical Plane, Mat. Wiss. U. Werkstofftech., 2005, vol. 36, pp. 429–437.

    Article  Google Scholar 

  15. Susmel, L., Tovo, R., and Socie, D.F., Estimating the Orientation of Stage I Crack Paths through the Direction of Maximum Variance of the Resolved Shear Stress, Int. J. Fatigue, 2014, vol. 58, pp. 94–101.

    Article  Google Scholar 

  16. Anes, V., Reis, L., Li, B., and de Freitas, M., Crack Path Evaluation on HC and BCC Microstructures under Multiaxial Cyclic Loading, Int. J. Fatigue, 2014, vol. 58, pp. 102–113.

    Article  Google Scholar 

  17. Wang, C., Shang, D.G., and Wang, X.W., A New Multiaxial High-Cycle Fatigue Criterion Based on the Critical Plane for Ductile and Brittle Materials, J. Mater. Eng. Perform., 2015, vol. 24, pp. 816–824.

    Article  Google Scholar 

  18. Macha, E., Simulation Investigations of the Position of Fatigue Fracture Plane in Materials with Biaxial Loads, Mat. Wiss. U. Werkstofftech., 1989, vol. 20, pp. 132–136, 153–163.

    Article  Google Scholar 

  19. Carpinteri, A., Brighenti, R., and Spagnoli, A., A Fracture Plane Approach in Multiaxial High-Cycle Fatigue of Metals, Fatigue Fract. Eng. Mater. Struct., 2000, vol. 23, pp. 355–364.

    Article  Google Scholar 

  20. Carpinteri, A., Karolczuk, A., Macha, E., and Vantadori, S., Expected Position of the Fatigue Fracture Plane by Using the Weighted Mean Principal Euler Angles, Int. J. Fracture, 2002, vol. 115, pp. 87–99.

    Article  Google Scholar 

  21. Karolczuk, A. and Macha, E., A Review of Critical Plane Orientations in Multiaxial Fatigue Failure Criteria of Metallic Materials, Int. J. Fracture, 2005, vol. 134, pp. 267–304.

    Article  MATH  Google Scholar 

  22. Marciniak, Z., Rozumek, D., and Macha, E., Verification of Fatigue Critical Plane Position According to Variance and Damage Accumulation Methods under Multiaxial Loading, Int. J. Fatigue, 2014, vol. 58, pp. 84–93.

    Article  Google Scholar 

  23. Carpinteri, A., Spagnoli, A., and Vantadori, S., Multiaxial Fatigue Assessment Using a Simplified Critical Plane-Based Criterion, Int. J. Fatigue, 2011, vol. 33, pp. 969976.

  24. Carpinteri, A. and Spagnoli, A., Multiaxial High-Cycle Fatigue Criterion for Hard Metals, Int. J. Fatigue, 2001, vol. 23, pp. 135–145.

    Article  Google Scholar 

  25. Carpinteri, A., Spagnoli, A., Vantadori, S., and Bagni, C., Structural Integrity Assessment of Metallic Components under Multiaxial Fatigue: the C-S Criterion and Its Evolution, Fatigue-Fract. Eng. Mater. Struct., 2013, vol. 36, pp. 870–883.

    Article  Google Scholar 

  26. Carpinteri, A., Spagnoli, A., and Vantadori, S., Reformulation in the Frequency Domain of a Critical Plane-Based Multiaxial Fatigue Criterion, Int. J. Fatigue, 2014, vol. 67, pp. 55–61.

    Article  Google Scholar 

  27. Carpinteri, A., Spagnoli, A., Ronchei, C., Scorza, D., and Vantadori, S., Critical plane criterion for fatigue life calculation: time and frequency domain formulations, Proc. Eng., 2015, vol. 101, pp. 518–523.

    Article  Google Scholar 

  28. Carpinteri, A., Spagnoli, A., Vantadori, V., Viappiani, D., A Multiaxial Criterion for Notch High-Cycle Fatigue Using a Critical-Point Method, Eng. Fract. Mech., 2008, vol. 75, pp. 1864–1874.

    Article  Google Scholar 

  29. Ellyin, F. and Kujawski, D., Generalization of Notch Analysis and Its Extension to Cyclic Loading, Eng. Fract. Mech, 1989, vol. 32, pp. 819–826.

    Article  Google Scholar 

  30. Ye, D., Hertel, O., and Vormwald, M., A Unified Expression of Elastic-Plastic Notch Stress-Strain Calculation in Bodies Subjected to Multiaxial Cyclic Loading, Int. J. Solids Struct., 2008, vol. 45, pp. 6177–6189.

    Article  MATH  Google Scholar 

  31. Salavati, H., Alizadeh, Y., and Berto, F., Effect of Notch Depth and Radius on the Critical Fracture Load of Bainitic Functionally Graded Steels under Mixed Mode I + II Loading, Phys. Mesomech., 2014, vol. 17, no. 3, pp. 178–189.

    Article  Google Scholar 

  32. Berto, F., A Criterion Based on the Local Strain Energy Density for the Fracture Assessment of Cracked and V-Notched Components Made of Incompressible Hyperelastic Materials, J. Theor. Appl. Mech., 2015, vol. 76, pp. 17–26.

    Article  Google Scholar 

  33. Berto, F., Campagnolo, A., and Lazzarin, P., Fatigue Strength of Severely Notched Specimens Made of Ti-6Al-4V under Multiaxial Loading, Fatigue Fract. Eng. Mater. Struct., 2015, vol. 38, pp. 503–517.

    Article  Google Scholar 

  34. Torabi, A.R., Campagnolo, A., and Berto, F., Tensile Fracture Analysis of V-Notches with End Holes by Means of the Local Energy, Phys. Mesomech., 2015, vol. 18, no. 3, pp. 194–202.

    Article  Google Scholar 

  35. Łagoda, T. and Wala, K., Lifetime of Semi-Ductile Materials through the Critical Plane Approach, Int. J. Fatigue, 2014, vol. 67, pp. 73–77.

    Article  Google Scholar 

  36. Lee, S.B., A Criterion for Fully Reversed Out-of-Phase Torsion and Bending, in Multiaxial Fatigue, Miller, K.J. and Brown, M.W., Eds., Philadelphia: ASTM STP 853, 1985, pp. 553–568.

    Chapter  Google Scholar 

  37. Sanetra, C., Untersuchungen zum Fetigkeitsvwrhalten bei mehrachsiger Randombeanspruchung unter Biegung und Torsion: Dissertation, Clausthal-Zellerfeld: Technische Universität Clausthal, 1991.

    Google Scholar 

  38. Łagoda, T. and Macha, E., Estimated and Experimental Fatigue Lives of 30CrNiMo8 Steel under In-and Outof-Phase Combined Bending and Torsion with Variable Amplitudes, Fatigue Fract. Eng. Mater. Struct., 1994, vol. 17, pp. 1307–1318.

    Google Scholar 

  39. Niesłony, A., Łagoda, T., Walat, K., and Kurek, M., Multiaxial Fatigue Behaviour of Selected Aluminium Alloys under Bending with Torsion Loading Condition, Mat. Wiss. U. Werkstofftech., 2014, vol. 45, pp. 947–952.

    Article  Google Scholar 

  40. Kurek, M. and Łagoda, T., Including of Ratio of Fatigue Limits from Bending and Torsion for Estimation Fatigue Life under Cyclic Loading, in 6th New Methods of Damage and Failure Analysis of Structural Parts (MDFA), Ostrava, Czech Republic, 2014.

    Google Scholar 

  41. Krzysztof, K. and Łagoda, T., New Energy Model for Fatigue Life Determination under Multiaxial Loading with Different Mean Values, Int. J. Fatigue, 2014, vol. 66, pp. 229–245.

    Article  Google Scholar 

  42. Brown, M.W. and Miller, K.J., A Theory of Fatigue under Multiaxial Stress-Strain Conditions, Proc. Inst. Mech. Eng., 1973, vol. 187, pp. 745–755.

    Google Scholar 

  43. Araójo, J.A., Dantas, A.P., Castro, F.C., Mamiya, E.N., and Ferreira, J.L.A., On the Characterization of the Critical Plane with a Simple and Fast Alternative Measure of the Shear Stress Amplitude in Multiaxial Fatigue, Int. J. Fatigue, 2011, vol. 33, pp. 1092–1100.

    Article  Google Scholar 

  44. Araójo, J.A., Carpinteri, A., Ronchei, C., Spagnoli, A., and Vantadori, S., An Alternative Definition of the Shear Stress Amplitude Based on the Maximum Rectangular Hull Method and Application to the C-S (Carpinteri-Spagnoli) Criterion, Fatigue Fract. Eng. Mater. Struct., 2014, vol. 37, pp. 764–771.

    Article  Google Scholar 

  45. Carpinteri, A., Ronchei, C., Spagnoli, A., and Vantadori, S., On the Use of the Prismatic Hull Method in a Critical Plane-Based Multiaxial Fatigue Criterion, Int. J. Fatigue, 2014, vol. 68, pp. 159–167.

    Article  Google Scholar 

  46. Papadopoulos, I.V., Critical Plane Approaches in High-Cycle Fatigue: on the Definition of the Amplitude and Mean Value of the Shear Stress Acting on the Critical Plane, Fatigue Fract. Eng. Mater. Struct., 1998, vol. 21, pp. 269–285.

    Article  Google Scholar 

  47. Goodman, J., Mechanics Applied to Engineering, London: Longmans Green, 1899.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Parma, Parma, 43124, Italy

    A. Carpinteri, C. Ronchei, D. Scorza & S. Vantadori

Authors
  1. A. Carpinteri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Ronchei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Scorza
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Vantadori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Vantadori.

Additional information

Original Text © A. Carpinteri, C. Ronchei, D. Scorza, S. Vantadori, 2015, published in Fizicheskaya Mezomekhanika, 2015, Vol. 18, No. 5, pp. 74-79.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carpinteri, A., Ronchei, C., Scorza, D. et al. Critical Plane Orientation Influence on Multiaxial High-Cycle Fatigue Assessment. Phys Mesomech 18, 348–354 (2015). https://doi.org/10.1134/S1029959915040074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1029959915040074

Keywords

Search

Navigation