International Journal of Fracture

, Volume 103, Issue 1, pp 71–94 | Cite as

The fractal growth of fatigue defects in materials

  • M. Rybaczuk
  • P. Stoppel


Material defects growing during fatigue or damage process are described in terms of fractals. The assumed, uniform energy distribution over fractal defects corresponds to generalized energy density, treated as material characteristics. It has been shown that the way of evolution as well as the main features of an irreversible process are determined by characteristic (for a given material) fractal measures. The macroscopic range of length scales has been introduced via additional energy dependence upon macroscopic volume limiting defects evolution. Under certain constrains imposed upon defects growth, the effect similar to phase transition can be observed. The transition point coincides with the singularity of characteristic measures. In turn, the singularity comes from macroscopic limitations of defects growth. Theoretical results are compared with numerical simulations of the simplified stochastic fibre break process in composites. The simplified model has been generated in a way allowing to exclude heat outflow from the simulated system. This makes possible to examine defects growth over full range of scales beginning with the microscopic level. The calculated singularity appears at percolation point when observed correlated defects approach macroscopic size in accordance with the proposed theoretical model.

Key words:  Fatigue process, fractal defects theory, numerical simulations.


Fatigue Defect Growth Fibre Break Fatigue Process Break Process 
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. Bažant, Z. P. (1997a). Scaling of Quasibrittle Fracture: Asymptotic Analysis, International Journal of Fracture, 83, No. 1, 19–40.Google Scholar
  2. Bažant, Z. P. (1997b). Scaling of Quasibrittle Fracture: Hypothesis of Invasive and Lacunar Fractality, Their Critique and Weibull Connection, International Journal of Fracture, 83, No. 1, 41–65.Google Scholar
  3. Bažant, Z. and Planas, J. (1998). Fracture and size effect, CRC Press, Boca Raton.Google Scholar
  4. Carpinteri, A. and Chiaia, B. (1996). Power Scaling Laws and Dimensional Transitions in Solid Mechanics, Chaos, Solitons and Fractals, 7, 1343–1364.Google Scholar
  5. Datta, D., Munshi, P. and Kishore, N. N. (1996). Automated Utrasonic NDE of Composite Specimens by Cluster and Fractal Analysis, Nondestr. Test. Eval. 13, 15–30.Google Scholar
  6. Falconer, K. (1990). Fractal geometry, Chichester, New York, John Wiley & Sons.Google Scholar
  7. Gandmaher, F. R. (1988). Matrix theory, Nauka, Moscow (in Russian).Google Scholar
  8. Goldsztein, R. W. and Mosolov, A. B. (1991). Cracks with fractal surfaces, WAN SSSR, 319 Nr 4 (in Russian).Google Scholar
  9. Goldsztein, R. W. and Mosolov, A. B. (1992). Fractal cracks, PMM. 56 (in Russian).Google Scholar
  10. Kasprzak, W., Lysik, B. and Rybaczuk, M. (1990). Dimensional Analysis in the Identification of Mathematical Models, World Scientific, Singapore New Jersey.Google Scholar
  11. Kohomoto, M. (1988). Entropy Function for Multifractals, Phys. Rev. A37, 1345–1350.Google Scholar
  12. Nottale, L. (1996). Scale, Relativity and Fractal Space- Time: Applications to Quantum Physics, Cosmology and Chaotic Systems, Chaos, Solitons & Fractals, 7, 877–938.Google Scholar
  13. Rybaczuk, M. (1992). The Fatigue Evolution of Fractal Defects in Metals (Edited by K. T. Rie), proceedings of Third International Conference on Low Cycle Fatigue and Elasto-Plastic Behavior of Materials, Elsevier London and New York.Google Scholar
  14. Rybaczuk, M. (1997). The Fractal Model of Defects Growth in Solids (Edited by R. C. Tennyson, A. E. Kiv), Computer Modelling of Electronic and Atomic Processes in Solids, Kluwer Academic Publishers, 309–320.Google Scholar
  15. Stoppel, P. (1999). Simulations of Fibres Breaking in the Stochastic Model of Composite, unpublished thesis (in Polish).Google Scholar
  16. Stoppel, P. and Rybaczuk, M. (1997). Simulations of Random Fractal Fiber Breaking in Composites (Edited by Karihaloo B. L., Mai Y.-W., Ripley M. I. and Richie R. O.), Advances in Fracture Research (ICF9), Amsterdam, Oxford, Pergamon, 849–856.Google Scholar
  17. Volterra, V. and Hostinsky, B. (1938). Opérations Infinitésimales linéares, Herman, Paris.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • M. Rybaczuk
    • 1
  • P. Stoppel
    • 2
  1. 1.Institute of Materials Science and Applied MechanicsWrocław University of TechnologyWrocławPoland
  2. 2.Institute of Materials Science and Applied MechanicsWrocław University of TechnologyWrocławPoland

Personalised recommendations