Stochastic Diffusion Models for Fatigue Crack Growth and Reliability Estimation

  • B. F. SpencerJr.
Part of the International Centre for Mechanical Sciences book series (CISM, volume 334)


Crack propagation is a major task in the design and life prediction of fatigue-critical structures such as aircraft, offshore platforms, bridges, etc. Experimental data indicate that fatigue crack propagation involves a large amount of statistical variation and is not adequately modeled deterministically. The lectures presented herein discuss the basic analysis and use of fracture mechanics-based random process fatigue crack growth models that can be represented by Markov diffusion processes. For completeness, the random variable models are presented as a special case of the random process models. The use of the models in fatigue reliability estimation is also discussed.


Crack Length Crack Growth Rate Fatigue Crack Growth Fatigue Crack Propagation Crack Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    ASCE Committee on Fatigue and Fracture Reliability, Series of articles on fatigue and fracture reliability, J. of the Struct. Div., ASCE 108 (ST1) (1982), 3–88.Google Scholar
  2. 2.
    Bannantine, J. A., J. J. Corner, and J. L. Handrock, Fundamentals of Metal Fatigue Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1990.Google Scholar
  3. 3.
    Bluham, J. I.: Crack propagation laws, in Fracture Mechanics of Aircraft Structures, AGARD-AG-176, 95–109, 1974.Google Scholar
  4. 4.
    Miller, M. S. and G. P. Gallagher: An analysis of several fatigue crack growth rate descriptions, in Measurements and Data Analysis, ASTM STP 738, 205–251, 1981.Google Scholar
  5. 5.
    Paris, R C. and F. Erdogan: A critical analysis of crack propagation laws, J. Basic Engrg., Trans. ASME, Series D, 85 (1963), 528–534.CrossRefGoogle Scholar
  6. 6.
    Forman, R. G., et al.: Numerical analysis of crack propagation in cyclic load structures, J. Basic Engrg, Trans. ASME, Series D, 89 (1967), 459–465.CrossRefGoogle Scholar
  7. 7.
    Larsen, J. M., B. J. Schwartz and C. G. Annis, Jr.: Cumulative fracture méchanics under engine spectra, Tech Report AFML-TR-794159, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, 1980.Google Scholar
  8. 8.
    Kung, C. J., and K. Ortiz: Objective comparison of fatigue crack growth laws, in Structural Safety and Reliability, Vol. 2 (A. H-S. Ang, M. Shinozuka, and G. I. Schuëller, eds.) ASCE, New York, 1627–1630, 1990.Google Scholar
  9. 9.
    Virkler, D. A., B. M. Hillberry, and P. K. Goel: The statistical nature of fatigue crack propagation, J. of Engrg. Mat. and Tech., ASME, 101 (1979), 148–153.CrossRefGoogle Scholar
  10. 10.
    Ghonem, H., and S. Dore: Experimental study of the constant probability crack growth curves under constant amplitude loading, Engrg. Fract. Mech., 27 (1987), 1–25.CrossRefGoogle Scholar
  11. 11.
    Bolotin, V. V.: On safe crack size under random loading, lzvestiia Akademiia Nauk SSSR, Mekhanika Tverdogo Tela, 1 (1980) (in Russian).Google Scholar
  12. 12.
    Bolotin, V. V.: Lifetime distribution under random loading, Zhurnal Priklandnoi Mekhaniki, Tekhnicheskoi Fiziki, 5 (1980) (in Russian).Google Scholar
  13. 13.
    Ditlevsen, O.: Random Fatigue Crack Growth — A first passage problem, Engrg. Fract. Mech., 23 (1986), 467–477.CrossRefGoogle Scholar
  14. 14.
    Ditlevsen, O., and R. Olesen: Statistical analysis of the Virkler data on fatigue crack growth, Engrg. Fract. Mech., 25 (1986), 177–195.CrossRefGoogle Scholar
  15. 15.
    Dolinski, K.: Stochastic loading and material inhomogeneity in fatigue crack propagation, Engrg. Fract. Mech., 25 (1986), 809–818.CrossRefGoogle Scholar
  16. 16.
    Dolinski, K.: Stochastic modelling and statistical verification of crack growth under constant amplitude loading, (manuscript) (1991).Google Scholar
  17. 17.
    Guers, F., and R. Rackwitz: Time-Variant Reliability of Structural Systems subject to Fatigue, in Proc. of ICASP-5, Vol. 1, Vancouver, Canada, 497–505, 1987.Google Scholar
  18. 18.
    Kozin, F., and J. L. Bodganoff: A Critical Analysis of some Probabilistic Models of Fatigue Crack Growth, Engrg. Fract. Mech., 14 (1981), 59–89.CrossRefGoogle Scholar
  19. 19.
    Un, Y. K., and J. N. Yang: On Statistical Moments of Fatigue Crack Propagation, Engrg. Fract. Mech., 18 (1983), 243–256.CrossRefGoogle Scholar
  20. 20.
    Un, Y. K., and J. N. Yang: A Stochastic Theory of Fatigue Crack Propagation, J. of the AIAA, 23 (1985), 117–124.CrossRefGoogle Scholar
  21. 21.
    Madsen, H. O.: Deterministic and probabilistic models for damage accumulation due to time varying loading, DIALOG 5–82, Danish Engrg. Academy, Lyngby, Denmark, 1983.Google Scholar
  22. 22.
    Madsen, H. O.: Random fatigue crack growth and inspection, in Structural Safety and Relia bility, Vol. 1, Proc. of ICOSSAR ‘85, Kobe, Japan (l. Konishi, A. H-S. Ang, and M. Shinozuka, eds.), Elsevier, Amsterdam, The Netherlands, 475–484, 1985.Google Scholar
  23. 23.
    Madsen, H. O., S. Krenk, and N. C. Lind, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.Google Scholar
  24. 24.
    Ortiz, K., and A. S. Kiremidjian: Time series analysis of fatigue crack growth data, Engrg. Fract. Mech., 24 (1986), 657–676.CrossRefGoogle Scholar
  25. 25.
    Ortiz, K., and A. S. Kiremidjian: Stochastic modeling of fatigue crack growth, Engrg. Fract. Mech., 29 (1988), 657–676.Google Scholar
  26. 26.
    Sobczyk, K.: On the Markovian models for fatigue accumulation, J. de Mécanique Theorique et Appliqué, ( Numor Special ) (1982), 147–160.Google Scholar
  27. 27.
    Sobczyk, K.: Stochastic modeling of fatigue crack growth, in Proc. of the IUTAM Symposium on ‘Probabilistic Methods in Mechanics of Solids and Structures,’ Stockholm, Sweden, Springer-Verlag, Berlin, 111–119, 1984.Google Scholar
  28. 28.
    Sobczyk, K.: Modelling of random fatigue crack growth, Engrg. Fract. Mech., 24 (1986), 609–623.CrossRefGoogle Scholar
  29. 29.
    Solomos, G. P.: First-passage solutions in fatigue crack propagation, Prob. Engrg. Mech., 4 (1989), 32–39.CrossRefGoogle Scholar
  30. 30.
    Spencer, B. F., Jr., and J. Tang: A Markov process model for fatigue crack growth, J. of Engrg. Mech., ASCE, 114 (1988), 2134–2157.CrossRefGoogle Scholar
  31. 31.
    Spencer, B. F., Jr., J. Tang, and M. E. Artley: A stochastic approach to modeling fatigue crack growth, J. of the AIAA, 27 (1989), 1628–1635.CrossRefGoogle Scholar
  32. 32.
    Tang, J., and B. F. Spencer, Jr.: Reliability solution for the stochastic fatigue crack growth problem, Engrg. Fract. Mech., 34 (1989), 419–433.CrossRefGoogle Scholar
  33. 33.
    Tanaka, H., and A. Tsurui: Random propagation of a semi-elliptical surface crack as a bivariate stochastic process, Engrg. Fract. Mech., 33 (1989), 787–800.CrossRefGoogle Scholar
  34. 34.
    Tsurui, A., and H. Ishikawa: Application of Fokker-Planck equation to a stochastic fatigue crack growth model, Struct. Safety, 4 (1986), 15–29.Google Scholar
  35. 35.
    Tsurui, A., J. Nienstedt, G.i. Schuëller, and H. Tanaka: Time variant structural reliability using diffusive crack growth models, Engrg. Fract. Mech., 34 (1989), 153–167.CrossRefGoogle Scholar
  36. 36.
    Veers, P. J.: Fatigue crack growth due to random loading, Ph.D. Dissertation, Department of Mechanical Engrg., Stanford University, Stanford, California, 1987.Google Scholar
  37. 37.
    Veers, P. J., S. R. Winterstein, D. V. Nelson, and C. A. Cornell: Variable amplitude load models for fatigue damage and crack growth, in Development of Fatigue Loading Spectra, STP-1006, ASTM, Philadelphia, 172–197, 1989.Google Scholar
  38. 38.
    Winterstein, S. R., and P. S. Veers: Diffusion models of fatigue crack growth with sequence effects due to stationary random loads, Structural Safety and Reliability, Vol. 2 (A. H-S. Ang, M. Shinozuka, and G. I. Schuëller, eds.), ASCE, New York, 1523–1530, 1990.Google Scholar
  39. 39.
    Yang, J. N., G. C. Salivar, and C. G. Annis: Statistical modeling of fatigue-crack growth in a nickel-based superalloy, Engrg. Fract. Mech., 18 (1983), 257–270.CrossRefGoogle Scholar
  40. 40.
    Yang, J. N., and R. C. Donath: Statistical crack propagation in fastener holes under spectrum loading, J. of Aircraft, AIAA, 20 (1983), 1028–1032.CrossRefGoogle Scholar
  41. 41.
    Yang, J. N., W. H. Hsi, and S. D. Manning: Stochastic crack propagation with application to durability and damage tolerance analyses, in Probabilistic Fracture Mechanics and Reliability ( J. Provan, ed.) Martinus Nijhoff Publishers, The Netherlands, 1987.Google Scholar
  42. 42.
    Sobczyk, K. and B.F. Spencer, Jr.: Random Fatigue: From Data to Theory, Academic Press, 1992.MATHGoogle Scholar
  43. 43.
    Manning, S. D., and J. N. Yang: USAF Durability Design Handbook: Guidelines for the Analysis and Design of Durable Aircraft Structures, Technical Report AFFDL-TR-84–3027, Wright-Patterson Air Force Base, Ohio, February, 1984.Google Scholar
  44. 44.
    Yang, J. N., and S. D. Manning: Distribution of equivalent initial flaw size, in Proc. of the Reliability and Maintainability Conference, 112–120, 1980.Google Scholar
  45. 45.
    PROBAN-2, A.S. Veritas Research, Det norske Veritas, Oslo, Norway, 1989.Google Scholar
  46. 46.
    Sobczyk, K., Stochastic Differential Equations with Application to Physics and Engrg., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.Google Scholar
  47. 47.
    Cox, D. R., and H. D. Miller: The Theory of Stochastic Processes, Chapman and Hall, London, 1977.MATHGoogle Scholar
  48. 48.
    Oh, K. P.: “A diffusion model for fatigue crack growth,” Proc. of Royal Soc. of London, A367 (1979), 47–58.CrossRefGoogle Scholar
  49. 49.
    Fichera, G.: On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in Boundary Problems in Differential Equations (R. E. Langer, ed.), University of Wisconsin Press, Madison, Wisconsin, 97–102, 1960.Google Scholar
  50. 50.
    Enneking, T. J., B. F. Spencer, Jr., and I. P. E. Kinnmark, “Stationary two-state variable problems in stochastic mechanics,” J. of Engrg. Mech., ASCE 116 (1990), 334–358.Google Scholar
  51. 51.
    Ostergaard, D. F., and B. M. Hiliberry: Characterization of variability in fatigue crack propagation data, in Probabilistic Methods for Design and Maintenance of Structures, in STP-798, ASTM, Philadelphia, 97–115, 1983.Google Scholar
  52. 52.
    Ferguson, R. I.: River loads underestimated by rating curves, Water Resources Res., 22 (1986), 74–76.CrossRefGoogle Scholar
  53. 53.
    ASTM: Appendix I, E647–86a, Recommended data reduction techniques, in Annual Book of ASTM Standards. Vol. 3.01: Metals Test Methods and Analytical Procedures, ASTM, Philadelphia, 919–920, 1987.Google Scholar
  54. 54.
    IMSL, The International Math and Statistics Subroutine Library, IMSL, Inc., Houston, Texas, 1984.Google Scholar
  55. 55.
    Johnson, W. S.: Multi-parameter yield zone model for predicting spectrum crack growth, in Methods and Models for Predicting Fatigue Crack Growth under Random Loading (J. B. Chang, and C. M. Hudson, eds.), STP-748, ASTM, Philadelphia, 85–102, 1981.CrossRefGoogle Scholar
  56. 56.
    Enneking,T. J.: On the stochastic fatigue crack growth problem, Ph.D. Dissertation, Department of Civil Engineering, University of Notre Dame, Notre Dame, Indiana, 1991.Google Scholar
  57. 57.
    MIL-A-87221: General Specifications for Aircraft Structures, U. S. Air Force Aeronautical Systems Division, Wright-Patterson Air Force Base, Ohio, 1985.Google Scholar
  58. 58.
    Madsen, H. O., R. Skjong, A. G. Tallin, and F. Kirkemo: Probabilistic fatigue crack growth analysis of offshore structures with reliability updating through inspection, in Proc. of the Marine Structural Reliability Symposium, Arlington, Virginia, 45–55, 1987.Google Scholar
  59. 59.
    Yang, J. N., and S. Chen: Fatigue reliability of structural components under scheduled inspection and repair maintenance, Probabilistic Methods in Mechanics of Solids and Structures, Proc. of the IUTAM Symposium, Stockholm, 1984, Springer-Verlag, Berlin, 559–568, 1985.Google Scholar
  60. 60.
    Berens, A. R, and R W. Hovey: Evaluation of nde reliability characterization, Report No. AFWAL-TR-81–4160, Volume I, University of Dayton Research Institute, Dayton, Ohio, 1981.Google Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • B. F. SpencerJr.
    • 1
  1. 1.University of Notre DameNotre DameUSA

Personalised recommendations