Fast crack bounds method applied to crack propagation models under constant amplitude loading

  • Rodrigo Villaca SantosEmail author
  • Waldir Mariano MachadoJr.
  • Cláudio R. Ávila da SilvaJr.
Technical Paper


In linear elastic fracture mechanics (LEFM), several models under constant amplitude loading describe the propagation of a crack, and these models are formulated by an initial value problem (IVP). However, for most applications, it is not possible to obtain exact numerical solutions for IVP due to the mathematical formulation of the stress intensity factor. From this, approximate numerical solutions are used for the IVP solution, which may reflect in aspects such as time and high computational cost. Thus, this work presents a new method called Fast Crack Bounds (FCB), to improve the way to obtain the IVP solution. This method was applied to the Paris–Erdogan, Forman, Walker, McEvily and Priddle models by establishing two functions, the upper and lower bounds, for the crack size function. Also, this work presents, for the same models, two new possible solutions through the arithmetic and geometric means of the bounds. For both, bounds and bounds means, the results were compared with the numerical solution obtained by the fourth-order Runge–Kutta method, applied to two numerical examples. As a result, the study presented, for all the models analyzed, an efficient and accurate way to obtain the propagation of an initial crack, reflecting in a considerable computational improvement.


Linear elastic fracture mechanics Crack propagation Initial value problem Bounds Bounds means 


  1. 1.
    Short JS, Hoeppner DW (1989) A global/local theory of fatigue crack propagation. Eng Fract Mech 33:175–184CrossRefGoogle Scholar
  2. 2.
    Anderson TL (2005) Fracture mechanics: fundamentals and applications. CRC Press, FloridaCrossRefGoogle Scholar
  3. 3.
    Wang K, Wang F, Cui W, Hayat T, Ahmad B (2014) Prediction of short fatigue crack growth of Ti-6Al-4V. Fatigue Fract Eng Mater Struct 37:1075–1086CrossRefGoogle Scholar
  4. 4.
    Bang D, Ince A, Tang L (2018) A modification of UniGrow 2-parameters driving force model for short fatigue crack growth. Fatigue Fract Eng Mater Struct. CrossRefGoogle Scholar
  5. 5.
    Toor PM (1973) A review of some damage tolerance design approaches for aircraft structures. Eng Fract Mech 5:837–880CrossRefGoogle Scholar
  6. 6.
    Hoeppner DW, Krupp WE (1974) Prediction of component life by application of fatigue crack growth knowledge. Eng Fract Mech 6:47–70CrossRefGoogle Scholar
  7. 7.
    Nelson DV (1977) Review of fatigue-crack-growth prediction methods. Exp Mech 17:41–49CrossRefGoogle Scholar
  8. 8.
    Beden SM, Abdullah S, Ariffin AK (2009) Review of fatigue crack propagation models for metallic components. Eur J Sci Res 28:364–397Google Scholar
  9. 9.
    Machniewicz T (2012) Fatigue crack growth prediction models for metallic materials. Part I: overview of prediction concepts. Fatigue Fract Eng Mater Struct 36:293–307CrossRefGoogle Scholar
  10. 10.
    Ávila CRS, Santos RV (2015) Bounds for the propagation model of crack Forman. Int J Sci Basic Appl Res 22:219–231Google Scholar
  11. 11.
    Ávila CRS, Santos RV, Beck AT (2016) Analytical bounds for efficient crack growth computation. Appl Math Model 40:2312–2321. MathSciNetCrossRefGoogle Scholar
  12. 12.
    Paris PC, Gomez MP, Anderson WE (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14Google Scholar
  13. 13.
    Paris P, Erdogan F (1963) A critical analysis of crack propagation laws. J Fluids Eng 85:528–533Google Scholar
  14. 14.
    Forman RG, Kearney VE, Engle RM (1967) Numerical analysis of crack propagation in cyclic-loaded structures. J Fluids Eng 89:459–463Google Scholar
  15. 15.
    Forman RG (1972) Study of fatigue crack initiation from flaws using fracture mechanics theory. Eng Fract Mech 4:333–345CrossRefGoogle Scholar
  16. 16.
    Walker K (1970) The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. In: Effects of environment and complex load history on fatigue life, vol 462. ASTM International, pp 1–14Google Scholar
  17. 17.
    McEvily AJ, Groeger J (1977) On the threshold for fatigue crack growth. In: ICF4, Waterloo (Canada)Google Scholar
  18. 18.
    Priddle EK (1976) High cycle fatigue crack propagation under random and constant amplitude loadings. Int J Press Vessels Pip 4:89–117CrossRefGoogle Scholar
  19. 19.
    Tada H, Paris PC, Irwin GR (2000) The stress analysis of crack handbook. ASME Press, New YorkCrossRefGoogle Scholar
  20. 20.
    Barsom JM, Rolfe ST (1999) Fracture and fatigue control in structures: applications of fracture mechanics. ASTM, PhiladelphiaCrossRefGoogle Scholar
  21. 21.
    Castro JTP, Meggiolaro MA (2009) Fadiga – Técnicas e práticas de dimensionamento estrutural sob cargas reais de serviço. Create SpaceGoogle Scholar
  22. 22.
    Machado WM (2015) Application of numerical method “Fast Bounds Crack” for a estimate efficient evolution of crack size. Dissertation, Universidade Tecnológica Federal do ParanáGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Rodrigo Villaca Santos
    • 1
    Email author
  • Waldir Mariano MachadoJr.
    • 1
  • Cláudio R. Ávila da SilvaJr.
    • 1
  1. 1.NuMAT/PPGEM, Federal University of Technology of ParanaCuritibaBrazil

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