User: unknown  ||   Login  ||  NEWS: MMP method published, WCFA2017-DFW, FKM_LFSA


Title PragTic


PragTic SW





Fatigue Lounge



User Profile

Conference List

Photo Gallery





Reports & References

Paper for ULM 2006 Conference

This is a revised version of the paper presented on the conference Fatigue and Fracture Mechanics 2006. It is shown here with a kind permission of the conference organizers. The paper is in Czech language, but at least figures are comprehensible. The work on experimental validation of PI and PC criteria has not been finished yet, so I do not want to translate it into English.


PhD Thesis of Jan Papuga


The project described here is focused on multiaxial fatigue calculation in high cycle fatigue. The phenomenon of multiaxial fatigue, some basic assumptions and formulas are given first. Then the set of contemporary commercial fatigue postprocessors utilizing FE-data is reviewed in short. Since the postprocessors are aimed at commercial use, a possibility to use them for research is very limited. Therefore a PragTic software package, which is presented here, was developed. A set of experimental data (129 experiments altogether) was compiled from references. Thanks to the PragTic software, they could be evaluated using 12 existing high cycle fatigue criteria. Fruitfulness of the individual criteria is commented and also the effects of phase shift and mean stress can be separately examined. These results serve as a basis for design of two new criteria. First of them is of integral nature, the second one is based on critical plane evaluation. The integral version demands great computational effort. The critical plane criterion is thus emphasized too, though it leads to slightly worse prediction and its mathematical basis embodies some problems commented here. Nevertheless, the both proposed criteria give results, which largely overcome results of any other criterion gathered here. The both criteria were generated from a set of approximately twenty other versions. Here the advantage of PragTic was taken, because it can evaluate more criteria and load regimes together simultaneously.

Keywords: multiaxial fatigue, critical plane criteria, integral criteria, fatigue postprocessors, mean stress effect, phase shift effect



  • zipped *.doc version (1.02 MB)
  • zipped *.pdf version (1.46 MB) - converted from the *.doc file via Create Adobe PDF Online service. This is the reason why there are minor problems concerning the graphics of the thesis; the content is nevertheless O.K.


  1. BALDA, M.; RŮŽIČKA, M.; SVOBODA, J.; PAPUGA, J.; FRÖHLICH, V.: Vliv náhodného víceosého neproporcionálního namáhání na únavovou životnost strojních konstrukcí. [Research Report Z-1353/04]. Czech Academy of Science, Institute of Thermomechanics 2004.
  2. BANNANTINE, J. A.; SOCIE, D. F.: A multiaxial fatigue life estimation technique. In: Advances in Fatigue Lifetime Predictive Techniques, ASTM STP 1122. Red M. R. Mitchell a R. W. Landgraf. Philadelphia, American Society for Testing and Materials 1992, pp. 249-275.
  3. BANVILLET, A.; LAGODA, T.; MACHA, E.; NIESLONY, A.; PALIN-LUC, T.; VITTORI, J.-F.: Fatigue life under non-Gaussian random loading from various models. Int. J. Fatigue, Vol. 26, 2004, pp. 349-363.
  4. BANVILLET, A.; PALIN-LUC, T.; LASSERRE, S.: A volumetric energy based high cycle multiaxial fatigue criterion. Int. Journal of Fatigue 25, 2003, pp. 755-769.
  5. BENNEBACH, M.: Fatigue d'une fonte GS. Influence de l'entaille et d'un traitement de surface. [PhD thesis]. ENSAM CER Bordeaux 1993, p. 157.
  6. BERNASCONI, A.: Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications. Int J Fatigue 24, 2002, pp. 649-657.
  7. BOLLER, C.: Der Einfluss von Probengrösse und Oberflächenrauhigkeit auf Lebensdaruerabschätzungen bei Betrachtung der örtlichen Beanspruchungen. Veröffentlichungen des Instituts für Stahlbau and Werkstoffmechanik der Technischen Hochschule Darmstadt, Heft 46. Darmstadt 1988. p. 160.
  8. BROWN, M. W.; MILLER, K. J.: A theory for fatigue under multiaxial stress-strain conditions. Proc. Inst. Mech. Engrs 187, 1973, pp. 745-755.
  9. BROWN, M. W.; SUKER, D. K.; WANG, C. H.: An analysis of mean stress in multiaxial random fatigue. Fatigue Fract. Engng. Mater. Struct. 19, 1996, No. 2/3, pp. 323-333.
  10. CARPINTERI, A.; BRIGHENTI, R.; MACHA, E.; SPAGNOLI, A.: Expected principal stress directions under multiaxial random loading. Part II: numerical simulation and experimental assessment through the weight function method. Int J Fatigue 21, 1999, pp. 89-96.
  11. CARPINTERI, A.; MACHA, E.; BRIGHENTI, R.; SPAGNOLI, A.: Expected principal stress directions under multiaxial random loading. Part I: theoretical aspects of the weight function method. Int J Fatigue 21, 1999, pp. 83-88.
  12. CARPINTERI, A.; SPAGNOLI, A.: Multiaxial high-cycle fatigue criterion for hard metals. Int. Journal of Fatigue 23, 2001, pp. 135-145.
  13. CROSSLAND, B.: Effect of large hydrostatic pressure on the torsional fatigue strength of an alloy steel. In: Proc. Int. Conf. on Fatigue of Metals, Institution of Mechanical Engineers, London, 1956, pp. 138-149.
  14. DANG VAN, K.: Sur la résistance a la fatigue des métaux. These de Doctorat es Sciences, Sci. Techniq. l´Armement, 47, 1973, p. 647.
  15. DANG VAN, K., CAILLETAUD, G.; FLAVENOT, J. F.; DOUARON, L.; LIEURADE, H. P.: Criterion for high-cycle failure under multiaxial loading. In M. Brown and K. Miller, editors, Biaxial and Multiaxial Fatigue, pp. 459-478, Sheffield, 1989. ESIS.
  16. DAVOLI, P.; BERNASCONI, A.; FILIPPINI, M.; FOLETTI, S.; PAPADOPOULOS, I. V.: Independance of the torsional fatigue limit upon a mean shear stress. Int J Fatigue 23, 2003, pp. 471-480.
  17. DELAHAY, T.: Développemenent d'une méthode probabiliste de calcul en fatigue multiaxiale prenant en compte la répartition volumique des contraintes. [PhD thesis], ENSAM CER de Bordeaux 2004.
  18. DOWLING, N. E.: Mean Stress Effects in Stress-Life and Strain-Life Fatigue. SAE Paper No. 2004-01-2227, Fatigue 2004: Second SAE Brasil International Conference on Fatigue, Sao Paulo, Brasil, June 2004, 14 pages.
  19. ELLYIN, F.; XIA, Z.: A general fatigue theory and its applications to out-of-phase cyclic loading. Journal of Engng. Mater. Technol., Transactions of the ASME, 115, 1993, pp. 411-416.
  20. ELLYIN, F.: Fatigue Damage, Crack Growth and Life Prediction. Chapman & Hall 1997.
  21. FINDLEY, W. N.: Fatigue of Metals Under Combinations of Stresses, Transactions of ASME (vol. 79), 1957.
  22. FROUSTEY, C.: Fatigue multiaxiale en endurance de l'acier 30NCD16. [PhD thesis]. ENSAM CER de Bordeaux 1987.
  23. FROUSTEY, C.; LASSERRE, S.: Multiaxial fatigue endurance of 30NCD16 steel. Int J Fatigue 11 (1989), No. 3, pp.169-175.
  24. FROUSTEY, C.; LASSERRE, S.; DUBAR, L.: Validité des critéres de fatigue multiaxiale a l'endurance en flexion-torsion. In: Fatigue des structures industrielles. 40 promenade Marx-Dormoy, F-93460 Gournay-sur-Marne, France, IITT International 1989, pp. 126-138.
  25. FATEMI, A.; SOCIE, D. F.: A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract. Engng. Mater. Struct., 11, 1988, No. 3, pp. 149-165.
  26. GAIER, C.; LUKACS, A.; HOFWIMMER, K.: Investigation on a statistical measure of the non-proportionality of stresses. Int. Journal of Fatigue 26, 2004, pp. 331-337.
  27. GALTIER, A.: Contribution a l'étude de l'endommagement des aciers sous solicitations uni ou multiaxials. [PhD thesis]. ENSAM CER de Bordeaux 1993, p. 193.
  28. GONÇALVES, C. A.; ARAÚJO, J. A.; MAMIYA, E. N.: Multiaxial fatigue: a stress based criterion for hard metals. Int. Journal of Fatigue 27, 2005, pp. 177-187.
  29. GOUGH, H. J.: Engineering Steels Under Combined Cyclic and Static Stresses. Journal of Applied Mechanics, June 1950. pp. 113-125.
  30. GOUGH, H. J.; POLLARD, H. V.; CLENSHAW, W. J.: Some Experiments on the Resistance of Metals to Fatigue under Combined Stresses. London, His Majesty's Stationery Office 1951.
  31. HAN, C.; CHEN, X.; KIM, K.: Evaluation of multiaxial fatigue criteria under irregular loading. Int. J. Fatigue, Vol. 24, 2002, pp. 913-922.
  32. HEIDENREICH, R.; ZENNER, H.: Schubspannungsintensitätshypothese – Erweiterung und experimentelle Abschützung einer neuen Festigkeitshypothese für schwingende Beanspruchung. Forschungshefte FKM, 1979, Heft 77.
  33. HOFFMANN, M.; SEEGER T.: A Generalized Method for Estimating Multiaxial Elastic-Plastic Notch Stresses and Strains, Part I & II. . J. of Engng. Mater. & Technology, Vol. 107, 1985, pp. 250-260.
  34. ILYUSHIN; A. A.: Plasticity Foundations of General Mathematical Theory, Akad. Nauk, Moscow, 1963, p. 16.
  35. KENMEUGNE, J. L.; VIDAL-SALLE, E.; ROBERT, J. L.; BAHUAUD, R. J.: On a new multiaxial fatigue criterion based on a selective integration approach. In: Fatigue '96, Proc. of the Sixth Int. Fatigue Congress, Vol. II. Red. G. Lütjering, Berlin, Pergamon 1996, pp. 1013-1018.
  36. KIM, K.; PARK, J.: Shear strain based multiaxial fatigue parameters applied to variable amplitude loading. Int. J. of Fatigue, Vol. 21, 1999, pp. 475-483.
  37. KIM, K.; PARK, J.; LEE, J.: Multiaxial fatigue under variable amplitude loads. J. of Engng. Mater. Tech., Transactions of the ASME, 121, 1999, pp. 286-293.
  38. LASSERRE, S.; FROUSTEY, C.: Multiaxial fatigue of steel - testing out of phase and in blocks: validity and applicability of some criteria. Int. Journal of Fatigue 14, No. 2, 1992. pp. 113-120.
  39. LIU, J.; ZENNER, H.: Berechnung der Dauerschwingfestigkeit bei mehrachsiger Beanspruchung. Mat. Wiss und Werkstofftech 24(S), 1993, pp. 240-249.
  40. MAMIYA, E. N.; ARAÚJO, J. A.: Fatigue limit under multiaxial loadings: on the definition of the equivalent shear stress. Mechanics Research Communications 29 (2002), pp. 141-151.
  41. MCDIARMID, D. L.: Fatigue under out-of-phase bending and torsion. Fatigue Fract. Engng. Mater. Struct., 9, 1987, pp. 457-475.
  42. MCDIARMID, D. L.: A general criterion for high cycle multiaxial fatigue failure. Fatigue Fract. Engng. Mater. Struct., 14, 1991, No. 4, pp. 429-453.
  43. MCDIARMID, D. L.: A shear stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction. Fatigue Fract Engng Mater Struct, 1994, 17(12), pp. 1475-1484.
  44. MOFTAKHAR, A.; BUCZYNSKI, A.; GLINKA, G.: Calculation of elasto-plastic strains and stresses in notches under multiaxial loading. Int. J. of Fracture, Vol. 70, 1995, pp. 357-373.
  45. MOREL, F.: A fatigue life prediction method based on a mesoscopic approach in constant amplitude multiaxial loading. Fatigue Fract. Engng. Mater. Struct., 21, 1998, No. 3, p. 241-256.
  46. MOREL, F.: A critical plane approach for life prediction of high cycle fatigue under multiaxial variable amplitude loading. Int. J. Fat., 22, 2000, pp. 101-119.
  47. MOREL, F.; PALIN-LUC, T.: A non-local theory applied to high cycle multiaxial fatigue. Fatigue Fract. Engng. Mater. Struct. 25, 2002, pp. 649-665.
  48. NINIC, D.: A stress-based multiaxial high-cycle fatigue damage criterion. Int J Fatigue 28 (2006). pp. 103-113.
  49. NISHIHARA, T.; KAWAMOTO, M.: The strength of metals under combined alternating bending and torsion with phase difference. Mem. College Engng., Kyoto Imperial University 11, 1945, pp. 85-112.
  50. PALIN-LUC, T.: Fatigue multiaxiale d'une fonte GS sous solicitations combinées d'amplitude variable. [PhD thesis]. ENSAM CER de Bordeaux 1996, p. 261.
  51. PAPADOPOULOS, I. V.: Fatigue polycyclique des métaux: Une nouvelle approche. [PhD thesis], Paris, École Nationale des Ponts et Chaussées 1987.
  52. PAPADOPOULOS, I. V. Fatigue limit of metals under multiaxial stress conditions: the microscopic approach. Technical note no. I.93.101. Commission of the European Communities, Joint Research Centre, ISEI/IE 2495/93, 1993.
  53. PAPADOPOULOS, I. V.: A new criterion of fatigue strength for out-of-phase bending and torsion of hard metals. Int. Journal of Fatigue, 16, 1994, pp. 377-384.
  54. 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. Engng. Mater. Struct., 21, 1998, No. 3, pp. 269-285.
  55. PAPADOPOULOS, I. V.; DAVOLI, P.; GORLA, C.; FILIPPINI, M.; BERNASCONI, A.: A comparative study of multiaxial high-cycle fatigue criteria for metals. International Journal of Fatigue, 19, 1997, No. 3, pp. 219-235.
  56. PARK, J.; NELSON, D.: Evaluation of energy-based approach and a critical plane approach for predicting constant amplitude multiaxial fatigue life. Int. J. Fatigue, Vol. 22, 2000, pp. 23-39.
  57. PAPUGA, J.; RŮŽIČKA, M.: Constitutive Equations in Elasto-Plastic Solution. [Research Report 2051/00/26]. CTU in Prague 2000. (in Czech)
  58. PAPUGA, J.; RŮŽIČKA, M.; BALDA, M.; SVOBODA, J.: Several high-cycle fatigue multiaxial criteria compared with experimental results. In: Danubia-Adria 2004. Zagreb 2004.
  59. PAPUGA, J.; RŮŽIČKA, M.; ŠIMEK, D.: Analýza únavového poškození na virtuálním prototypu - přechod k multiaxiálnímu řešení. In: Aplikovaná mechanika 2002. Ostrava: VŠB-TUO, 2002, pp. 273-280. (in Czech)
  60. PAPUGA, J.; RŮŽIČKA, M.; ŠPANIEL, M.: Software solution of fatigue damaging based on the FE-analysis results utilization. In: Numerical Methods in Continuum Mechanics, University of Žilina. 2000.
  61. PAPUGA, J.; ŠPANIEL, M.; RŮŽIČKA, M.: Datová struktura pro následnou analýzu MKP řešení. In: Výpočtová mechanika 2002, Part 1. Plzeň, ZČU 2002, pp. 333-340. (in Czech)
  62. REIS, L.; LI, B.; DE FREITAS, M.: Biaxial fatigue for proportional and non-proportional loading paths. Fatigue Fract Engng Mater Struct 27, 2004, pp. 775-784.
  63. RŮŽIČKA, M.; CABRNOCH, B.; PAPUGA, J.: Metody a programy pro hodnocení únavové životnosti. [Research Report 211-98-10]. Prague, CTU in Prague 1998. (in Czech)
  64. RŮŽIČKA, M.; PAPUGA, J.: Fatigue Life Evaluation of Components under Multiaxial Loading. [Research Report 205/99/25] Prague, CTU in Prague 1999. (in Czech).
  65. RŮŽIČKA, M.; PAPUGA, J.; BALDA, M.; SVOBODA, J.: Data Processing Line for Multiaxial Fatigue Computation and its Experimental Verification. In: AED 2004 [CD-ROM]. Glasgow 2004.
  66. SINES, G.: Failure of materials under combined repeated stresses with superimposed static stresses. [NACA-TN-3495]. Washington, NACA 1955.
  67. SINES, G.: Behavior of metals under complex static and alternating stresses. In: Metal Fatigue. Red. G. Sines a J.L. Waisman, New York, McGraw Hill 1959, pp. 145-469.
  68. SINGH, M. N. K.; G. GLINKA; R. N. DUBEY: Elastic-plastic stress-strain calculation in notched bodies subjected to non-proportional loading. Vol. 76, 1996, pp. 39-60.
  69. SOCIE, D. F.: Critical plane approaches for multiaxial fatigue damage assessment. In: Advances in Multiaxial Fatigue, ASTM STP 1191. Red. D. L. Dowell a R. Ellis, Philadelphia, American Society for Testing and Materials 1993, pp. 7-36.
  70. SPAGNOLI, A.: A new high-cycle fatigue criterion applied to out-of-phase biaxial stress state. International Journal of Mechanical Sciences 43, 2001, pp. 2581-2595.
  71. STEFANOV, S. H.: The curvilinear integral method: Testing 2 (under non-proportional pulsating axial force and internal pressure). Int. Journal of Fatigue, 18, 1996, No. 1, pp. 41-48.
  72. VIDAL-SALLÉ, E.; KENMEUGNE, B.; ROBERT, J. L.; BAHUAUD, J.: On a new multiaxial fatigue criterion based on a selective integration approach. In: Fatigue '96, Proc. of the Sixth Int. Fatigue Congress, Vol. II. Red. G. Lütjering, Berlin, Pergamon 1996, pp. 983-988.
  73. WANG, C. H.; BROWN, M. W.: A path-independent parametr for fatigue under proportional and nonproportional loading. Fatigue Fract. Engng. Mater. Struct., 16, 1993, No. 12, pp. 1285-1298.
  74. WANG, C. H.; BROWN, M. W.: Life prediction techniques for variable amplitude multiaxial fatigue-Part 1: Theories, Part 2: Comparison with experimental results. J. of Engng. Mater. Tech., Transactions of the ASME, 118, 1996, pp. 367-374.
  75. WEBER, B.: Fatigue multiaxiale des structures industrielles sous chargement quelconque. [PhD thesis]. Lyon, INSA Lyon 1999.
  76. XIA, Z.; KUJAWSKI, D.; ELLYIN, F.: Effect of mean stress and ratchetting strain on fatigue life of steel. Int. Journal of Fatigue, 18, 1996, No. 5, pp. 335-341.
  77. ZENNER, H.; HEIDENREICH, R.; RICHTER, I. Z.: Fatigue strength under nonsynchronous multiaxial stresses, Mat. Wiss und Werkstofftech 16, 1985, p. 101-112.
  78. ZENNER, H.; SIMBÜRGER, A.; LIU, J.: On the fatigue limit of ductile metals under complex multiaxial loading. Int. Journal of Fatigue 22, 2000, pp. 137-145.

Title FL_DB


FADOFF Project


Workshop on Design and Fatigue of Weldments


Workshop on Vibration Fatigue Analysis


D&DT for Aircraft Engineers



Development in 2011-2014 supported by:
Alfa programme