Journal of Materials Science

, Volume 28, Issue 4, pp 931–941 | Cite as

Combined weakest link and random defect model for describing strength variability in fibres

  • W. F. Knoff


A mathematical model which describes the strength variability along the length of a fibre was developed. The model is a combination of the modified weakest link and random defect models. This combined model describes very well the strength variability data of aramid fibres.


Polymer Mathematical Model Material Processing Combine Model Defect Model 
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Specimen length


Cumulative frequency distribution of link strengths

1 — F(s)

Survival function of a link


Cumulative frequency distribution of strengths of a specimen of length L

1 — FL(s)

Survival function of a specimen of length L


Strength variable


Fibre defect-free strength for a random defect or combined model

s1, s2...

Fibre strength at the point of a defect

s1′, s2′ ...

Strength a fibre must have at the location of the defect to have a strength of s at the location of the defect


Length of a hypothetical link in a weakest link model

ϱ2, ϱ2 ...

Defect frequencies (mean number per unit length)

v1, v2 ...

Defect severities, 0 ≦ v ≦ 1


Defect frequency distribution function defined in terms of the strength at the defect


Defect frequency distribution function defined in terms of the defect severity

α, β

Defect frequency distribution parameters (Equation 14)

a, b

Weibull distribution parameters (Equation 4)


Probability that m defects will occur in a given specimen length


Number of defects occurring


Mean strength


Coefficient of variation of strength


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  1. 1.
    H. E. Daniels, Proc. R. Soc. 83A (1945) 405.Google Scholar
  2. 2.
    B. D. Coleman, J. Mech. Phys. Solids 7 (1958) 60.CrossRefGoogle Scholar
  3. 3.
    W. B. Rosen, American Institute of Aeronautics and Astronautics Journal 2 (1964) 1985.CrossRefGoogle Scholar
  4. 4.
    C. Zweben, ibid. 6 (1968) 2325.CrossRefGoogle Scholar
  5. 5.
    F. T. S. Peirce, J. Text. Inst. 17 (1926) 355.CrossRefGoogle Scholar
  6. 6.
    W. F. Knoff, J. Mater. Sci. 22 (1987) 1024.CrossRefGoogle Scholar
  7. 7.
    W. Weibull, “Handlingar”, No. 151 (Royal Swedish Academy of Energy Sciences, 1939).Google Scholar
  8. 8.
    S. L. Pheonix, in “Composite Materials: Testing and Design (Third Conference)”, ASTM STP 546 (American Society for Testing and Materials, Philadelphia, 1974) p. 130.CrossRefGoogle Scholar
  9. 9.
    K. K. Phani, Compos. Sci. Technol. 30 (1987) 59.CrossRefGoogle Scholar
  10. 10.
    C. A. Bennett and N. L. Franklin, “Statistical Analysis in Chemistry and Chemical Industry” (Wiley, New York, 1954) pp. 319–321.Google Scholar
  11. 11.
    M. B. Wilk and R. Ganandesikann, Biometrica 55(1968) 1.Google Scholar
  12. 12.
    A. S. Watson and R. L. Smith, J. Mater. Sci. 20 (1985) 3260.CrossRefGoogle Scholar

Copyright information

© Chapman & Hall 1993

Authors and Affiliations

  • W. F. Knoff
    • 1
  1. 1.Fibers Research DivisionE. I. DuPont De Nemours & Co. Inc.RichmondUSA

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