Mechanics of Composite Materials

, Volume 44, Issue 5, pp 479–486 | Cite as

Analysis of the fiber length dependence of its strength by using the weakest-link approach 1. A family of weakest-link distribution functions

  • Yu. Paramonov
  • J. Andersons

This paper presents a summary and further development of the ideas proposed in the previous papers of the authors, which were dedicated to investigating the fiber scale effect (strength-length relation). In the first part of the paper, some theoretical aspects of the problem are considered; in the second one, an application to the processing of test datasets is discussed. As distinct from our previous publications, two types of defects (“technological,” i.e., existing before loading, and load-dependent) and two types of influence of the number of defects on the fiber strength are considered; the probability of absence of defects is also taken into account. We consider a specimen as a sequence of n elements of the same length. It is supposed that there are defects in K of them, 0 ≤ K ≤ n. Two cases are considered: K is a random variable or a random process K (t). In the second case, the increase in K and the failure of a specimen is described as a Markov chain whose matrix of transition probabilities depends on the current value of the loading process, described as some (increasing to infinity) sequence {x1, x2, …, xt,…}. Three versions of relationships between the specimen strength and the number of defective elements are considered for both the cases. Thus, six probability structures are introduced, and different versions of distribution functions and the corresponding models are considered. The methods for estimating the model parameters, the results obtained in processing glass, flax, carbon fiber, and carbon bundle datasets, as well as a comparison of different models, are presented in the second part of the paper.


composite fiber strength weakest-link model distribution function 


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  1. 1.
    Yu. Paramonov and J. Andersons, “A family of weakest link models for fiber strength distribution,” Composites, Pt. A, 38, 1227–1233 (2007).CrossRefGoogle Scholar
  2. 2.
    Yu. Paramonov and J. Andersons, “A new model family for the strength distribution of fibers in relation to their length,” Mech. Compos. Mater., 42, No. 2, 119–128 (2006).CrossRefGoogle Scholar
  3. 3.
    Yu. Paramonov and J. Andersons, “Analysis of fiber strength dependence on length using an extended weakest link distribution family,” Comp. Model. New Technol., 11, No. 1, 8–20 (2007).Google Scholar
  4. 4.
    Yu. Paramonov and J. Andersons, “New weakest link distribution family,” in: Proc. Int. Conf. Statistical Methods for Biomechanical and Technical Systems, Limassol, Cyprus (2006), pp. 415–419.Google Scholar
  5. 5.
    Yu. Paramonov and J. Andersons, “New widened weakest link distribution family,” in: Proc. 12th Int. Conf. Appl. Stoch. Models Data Analysis (ASMDA-2007), May 29–June 1, Chania, Crete, Greece (2007).Google Scholar
  6. 6.
    Yu. Paramonov and J. Andersons, “Modified weakest link family for tensile strength distribution,” in: Proc. 5th Int. Conf. Mathem. Meth. Reliab. Methodol. Pract. (MMR 2007), July 1–4, Glasgow, UK (2007).Google Scholar
  7. 7.
    F. T. S. Pierce, J. Text. Inst., 17, p. 355 (1926).Google Scholar
  8. 8.
    W. Weibull, “A statistical theory of the strength of materials,” Proc. Roy. Swed. Inst. Eng. Res., 151 (1939).Google Scholar
  9. 9.
    H. E. Daniels, “The statistical theory of the strength of bundles of threads,” Proc. Roy. Soc. London, A183, 405–435 (1945).ADSMathSciNetGoogle Scholar
  10. 10.
    C. Zweben, “Tensile failure of composites,” AIAA J., 12, 2325–2331 (1962).Google Scholar
  11. 11.
    C. Zweben and B. Rosen, “A statistical theory of material strength with application to composite materials,” J. Mech. Phys. Solids, 18, No. 3, 189–206 (1970).CrossRefADSGoogle Scholar
  12. 12.
    W. F. Knoff, “Combined weakest link and random defect model for describing strength variability in fibres,” J. Mater. Sci., 28, 931–941 (1993).CrossRefADSGoogle Scholar
  13. 13.
    Yu. Gutans and V. P. Tamuzh, “To the scale effect of Weibull distribution of the strength of fibers,” Mekh. Kompoz. Mater., No. 6, 1107–1109 (1984).Google Scholar
  14. 14.
    A. S. Watson and R. L. Smith, “An examination of statistical theories for fibrous materials in the light of experimental data,” J. Mater. Sci., 20, 3260–3270 (1985).CrossRefADSGoogle Scholar
  15. 15.
    R. L. Smith, “Weibull regression models for reliability data,” Reliabil. Eng. Syst. Safety, 34, 55–77 (1991).CrossRefGoogle Scholar
  16. 16.
    W. J. Padgett, S. D. Durham, and A. M. Mason, “Weibull analysis of the strength of carbon fibers using linear and power law models for the length effect,” J. Comp. Mater., 29, No. 14, 1873–1884 (1995).Google Scholar
  17. 17.
    W. A. Curtin, “Tensile strength of fiber-reinforced composites: III. Beyond the traditional Weibull model for fiber strengths,” J. Comp. Mater., 34, No. 15, 1301–1332 (2000).CrossRefGoogle Scholar
  18. 18.
    J. Andersons, R. Joffe, M. Hojo, and S. Ochiai, “Glass fibre strength distribution determined by common experimental methods,” Comp. Sci. Technol., 62, 131–145 (2002).CrossRefGoogle Scholar
  19. 19.
    J. Andersons, E. Spārniņš, R. Joffe, and L. Wallstrom, “Strength distribution of elementary flax fibers,” Comp. Sci. Technol., 65, 693–702 (2005).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • Yu. Paramonov
    • 1
  • J. Andersons
    • 2
  1. 1.Aviation InstituteRiga Technical UniversityRigaLatvia
  2. 2.Institute of Polymer MechanicsUniversity of LatviaRigaLatvia

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