Advertisement

Statistical Length Scale in Weibull Strength Theory and Its Interaction with Other Scaling Lengths in Quasibrittle Failure

  • Miroslav Vořechovský
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)

Abstract

The main result of the paper is the introduction of a statistical length scale into the Weibull theory. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line in double logarithmic plot). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, we show that such behavior is a direct implication of the assumption that the structural elements have independent random strengths. We show that by introduction of statistical dependence in a form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such model, the autocorrelation length plays a role of a statistical length scale. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and stochastic finite element method in the context of quasibrittle failure.

Keywords

Autocorrelation length strength random field chain of bundles extreme value theory extremes of random fields statistical length scale 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Daniels. H. E., “The statistical theory of the strength of bundles of threads”, Proceedings of the Royal Society. (London), 183A, pp. 405–435, 1945.Google Scholar
  2. 2.
    Coleman, B. D., “On the strength of classical fibres and fibre bundles”, Journal of the Mechanics and Physics of Solids, 7, pp. 60–70, 1958.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gücer, D. E., Gurland, J. “Comparison of the statistics of two fracture modes”, Journal of Mechanics and Physics Solids 10, pp. 365–373, 1962.CrossRefGoogle Scholar
  4. 4.
    Phoenix, S. L., Taylor, H. M. “The asymptotic strength distribution of a general fiber bundle”, Advances in Applied Probability, 5, pp. 200–216, 1973.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Phoenix, S. L. “The random strength of series-parallel structures with load sharing among members”, Probabilistic Mechanics, pp. 91–95, 1978.Google Scholar
  6. 6.
    Pan, N., Hua, T., Qiu, Y. “Relationship between fiber and yarn strength”, Textile Research Journal, 71(11), pp. 960–964. 2001.Google Scholar
  7. 7.
    Vořechovský, M., Chudoba, R. “Stochastic modeling of multi-filament yarns: II. Random properties over the length and size effect”, International Journal of Solids and Structures, 43 (3–4), pp. 435–458, ISSN 0020-7683, 2006.MATHCrossRefGoogle Scholar
  8. 8.
    Bažant, Z.P., Pang, S.-D. “Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture”, Journal of the Mechanics and Physics of Solids, 55 (1), pp. 91–134, 2007.CrossRefMATHGoogle Scholar
  9. 9.
    Weibull, W. “A statistical theory of the strength of materials”, Royal Swedish Institute of Engineering Research (Ingenioersvetenskaps Akad. Handl.), Stockholm, Vol. 151, Stockholm, 1939.Google Scholar
  10. 10.
    Fisher, R. A., Tippett, L. H. C. “Limiting forms of the frequency distribution of the largest and smallest member of a sample”, Proceedings of the Cambridge Philosophical Society, 24, pp. 180–190, 1928.MATHCrossRefGoogle Scholar
  11. 11.
    Gnedenko, B. V. “Sur la distribution limite du terme maximum d’une série aléatorie”, Annals of Mathematics, 2nd Ser, 44(3), pp. 423–453, 1943.MathSciNetGoogle Scholar
  12. 12.
    Bažant, Z. P., Vořechovský, M., Novák, D. “Asymptotic prediction of energetic-statistical size effect from deterministic finite element solutions”, Journal of Engineering Mechanics (ASCE), 133 (2), pp. 153–162, ISSN 0733-9399, 2007.Google Scholar
  13. 13.
    Bažant, Z. P., Pang. S. D., Vořechovský, M., Novák, D. “Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model”, International Journal of Numerical Methods in Engineering, 71(11), 1297–1320, 2007, DOI: 10.1002/nme. 1986.CrossRefGoogle Scholar
  14. 14.
    Vořechovské, M. “Statistical length scale in the Weibull strength theory and its interaction with other scaling lengths in quasibrittle failure”, International Journal of Solids and Structures, in preparation, 2007.Google Scholar
  15. 15.
    Vořechovské, M. “Stochastic fracture mechanics and size effect”, Brno University of Technology, Brno, Czech Republic, ISBN 80-214-2695-0, 2004.Google Scholar
  16. 16.
    Vořechovské, M. “Statistical alternatives of combined size effect on nominal strength for structures failing at crack initiation”, In: Stibor (Ed.) Problémy lomové mechaniky IV (Problems of Fracture Mechanics IV), Brno Univ. of Techn., invited lecture at the Academy of Sciences, Inst. of Physics of Materials, ISBN 80-214-2585-7 pp. 99–106, 2004.Google Scholar
  17. 17.
    Leadbetter, M. R., Lindgren, G., Rootzen, H. “Extremes and related properties of random sequences and processes”, Springer-Verlag, New York, ISBN-10: 0387907319, 1983.Google Scholar
  18. 18.
    Harlow, D. G., Smith, R. L., Taylor, H. M. “Lower tail analysis of the distribution of the strength of load-sharing systems”, Journal of Applied Probability, 20(2), pp. 358–367, doi:10.2307/3213808, 1983.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Smith, R. L. “The asymptotic distribution of the strength of a series-parallel system with equal load-sharing”, The Annals of Probability, 10(1), pp. 137–171, 1982.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Smith, R. L. “Limit theorems for the reliability of series-parallel load-sharing systems”, Ph.D. Thesis, Cornell University, Ithaca, New York, 1979.Google Scholar
  21. 21.
    Smith, R. L., Phoenix, S.L. “Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load-sharing”, Journal of Applied Mechanics, 48, pp. 75–82, 1981.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Cramér, H. Mathematical methods of statistics, Princeton University Press, ISBN-10: 0691080046, 1946.Google Scholar
  23. 23.
    Phoenix, S.L. “Statistical theory for the strength of twisted fiber bundles with applications to yarns and cables”, Textile Research Journal 49, pp. 407–423, 1979.CrossRefGoogle Scholar
  24. 24.
    Bažant, Z.P., Planas, J. “Fracture and size effect in concrete and other quasibrittle materials”, CRC Press, Boca Raton Florida and London, 1998.Google Scholar
  25. 25.
    Vořechovský, M. “Interplay of size effects in concrete specimens under tension studied via computational stochastic fracture mechanics”, International Journal of Solids and Structures, 44(9), pp. 2715–2731, ISSN 0020-7683, 2007.MATHCrossRefGoogle Scholar
  26. 26.
    Vořechovský, M. Simulation of simply cross correlated random fields by series expansion methods, Structural safety, 30(4), pp. 337–363, 2008. ISSN 0167-4730, available on-line, 2007, DOI 0.1016/j.strusafe.2007.05.002.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Miroslav Vořechovský
    • 1
  1. 1.Faculty of Civil Engineering Institute of Structural MechanicsBrno University of TechnologyBrnoCzech Republic

Personalised recommendations