, Volume 23, Issue 4, pp 2701–2713 | Cite as

Influence of the sample number for the prediction of the tensile strength of high tenacity viscose fibres using a two parameters Weibull distribution

Original Paper


The objective of this work is to determine an adequate number of samples for an accurate prediction of tensile strength of fibres using a Weibull distribution. Theory will be compared to experimental results in order to know the effect of experimental errors on the theoretical expectations. The diameter and strength distribution of high tenacity viscose were evaluated. The chosen Weibull distribution was with two parameters. First, the best probability estimator for Weibull was determined using a random selection of experimental datas. Then, in order to be able to predict the number of sample knowing the variation of the Weibull modulus m, different relationship between the coefficient of variation of m and the number of sample n were tested. The relationship CV = 0.78/\(\sqrt n\) presented good agreement with experimental datas. The influence of the variation of m on the predicted value of strength was studied in order to determine an adequate number of sample to obtain limited variation of the predicted strength. The last part focus on experimental verification of the points previously developed. It was shown that it is possible to determine the influence of the variation of the Weibull modulus on the predicted strength. However, no correlation was found between the variations of the Weibull modulus and the error on the predicted strength.


Regenerated cellulose High tenacity viscose Weibull distribution Tensile strength prediction 



The authors acknowledge Carnot Institute “MICA” for financial funding. The authors would like to thank the French National Association of Research and Technology (ANRT) for funding the research through the Ph.D. Grant awarded to B.P. Revol. The authors would also like to thank the reviewers for their useful comments which helped in improving this work.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Andersons J, Joffe R, Hojo M, Ochiai S (2002) Glass fibre strength distribution determined by common experimental methods. Compos Sci Technol 62:131–145. doi: 10.1016/S0266-3538(01)00182-8 CrossRefGoogle Scholar
  2. Andersons J, Spārniņš E, Joffe R, Wallström L (2005) Strength distribution of elementary flax fibres. Compos Sci Technol 65:693–702. doi: 10.1016/j.compscitech.2004.10.001 CrossRefGoogle Scholar
  3. Asloun EM, Donnet JB, Guilpain G et al (1989) On the estimation of the tensile strength of carbon fibres at short lengths. J Mater Sci 24:3504–3510CrossRefGoogle Scholar
  4. Bergman B (1984) On the estimation of the Weibull modulus. J Mater Sci Lett 3:689–692CrossRefGoogle Scholar
  5. Ganster J, Fink H-P, Uihlein K, Zimmerer B (2008) Cellulose man-made fibre reinforced polypropylene—correlations between fibre and composite properties. Cellulose 15:561–569. doi: 10.1007/s10570-008-9204-x CrossRefGoogle Scholar
  6. Gindl W, Reifferscheid M, Adusumalli RB et al (2008) Anisotropy of the modulus of elasticity in regenerated cellulose fibres related to molecular orientation. Polymer 49:792–799. doi: 10.1016/j.polymer.2007.12.016 CrossRefGoogle Scholar
  7. Glandus JC, Boch P (1984) Uncertainty on the mean strength and Weibull’s modulus of an alumina batch as a function of the number of samples. J Mater Sci Lett 3:74–76CrossRefGoogle Scholar
  8. Graupner N, Herrmann AS, Müssig J (2009) Natural and man-made cellulose fibre-reinforced poly(lactic acid) (PLA) composites: an overview about mechanical characteristics and application areas. Compos Part Appl Sci Manuf 40:810–821. doi: 10.1016/j.compositesa.2009.04.003 CrossRefGoogle Scholar
  9. Gupta PK (1987) Combined effect of flaw distribution and diameter variation on the statistics of glass fiber strength. J Am Ceram Soc 70:486–492CrossRefGoogle Scholar
  10. Joffe R, Andersons J, Spārniņš E (2009) Applicability of Weibull strength distribution for cellulose fibers with highly non-linear behaviour. In: ICCM 17, Edinburgh: 17th international conference on composite material. Edinburgh, UK. IOM communications, LondonGoogle Scholar
  11. Karlsson JO, Gatenholm P, Blachot J-F, Peguy A (1996) Improvement of adhesion between polyethylene and regenerated cellulose fibers by surface fibrillation. Polym Compos 17:300–304CrossRefGoogle Scholar
  12. Kelly A, Macmillan NH (1986) Strong solids. Clarendon, OxfordGoogle Scholar
  13. Kelly A, Tyson WR (1965) Tensile properties of fibre-reinforced metals: copper/tungsten and copper/molybdenum. J Mech Phys Solids 13:329–350CrossRefGoogle Scholar
  14. Kulkarni AG, Satyanarayana KG, Rohatgi PK (1983) Weibull analysis of strengths of coir fibres. Fibre Sci Technol 19:59–76. doi: 10.1016/0015-0568(83)90032-5 CrossRefGoogle Scholar
  15. Lara-Curzio E, Russ CM (1999) On the relationship between the parameters of the distributions of fiber diameters, breaking loads, and fiber strengths. J Mater Sci Lett 18:2041–2044CrossRefGoogle Scholar
  16. Lorbach C, Hirn U, Kritzinger J, Bauer W (2012) Automated 3D measurement of fiber cross section morphology in handsheets. Nord Pulp Pap Res J 27:264CrossRefGoogle Scholar
  17. Northolt MG, Boerstoel H, Maatman H et al (2001) The structure and properties of cellulose fibres spun from an anisotropic phosphoric acid solution. Polymer 42:8249–8264CrossRefGoogle Scholar
  18. Petry MD, Mah T-I, Kerans RJ (1997) Validity of using average diameter for determination of tensile strength and Weibull modulus of ceramic filaments. J Am Ceram Soc 80:2741–2744CrossRefGoogle Scholar
  19. Pickering KL, Murray TL (1999) Weak link scaling analysis of high-strength carbon fibre. Compos Part Appl Sci Manuf 30:1017–1021CrossRefGoogle Scholar
  20. Ritter J, Bandyopadhyay N, Jakus K (1981) Statistical reproducibility of the dynamic and static fatigue experiments. Bull Am Ceram Soc 60:798–806Google Scholar
  21. Sullivan JD, Lauzon PH (1986) Experimental probability estimators for Weibull plots. J Mater Sci Lett 5:1245–1247CrossRefGoogle Scholar
  22. Thomason JL (2013) On the application of Weibull analysis to experimentally determined single fibre strength distributions. Compos Sci Technol 77:74–80. doi: 10.1016/j.compscitech.2013.01.009 CrossRefGoogle Scholar
  23. Thoman DR, Bain LJ, Antle CE (1969) Inferences on the parameters of the Weibull distribution. Technometrics 9:445–460CrossRefGoogle Scholar
  24. Trustrum K, Jayatilaka ADS (1979) On estimating the Weibull modulus for a brittle material. J Mater Sci 14:1080–1084CrossRefGoogle Scholar
  25. Van der Zwaag S (1989) The concept of filament strength and th weibull modulus. J Test Eval 17:292–298CrossRefGoogle Scholar
  26. Watson AS, Smith RL (1985) An examination of statistical theories for fibrous materials in the light of experimental data. J Mater Sci 20:3260–3270CrossRefGoogle Scholar
  27. Weibull W (1939) A statistical theory of the strength of materials. Ingeniörsvetenskapsakademiens Handligar 151:1–45Google Scholar
  28. Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297Google Scholar
  29. Xia ZP, Yu JY, Cheng LD et al (2009) Study on the breaking strength of jute fibres using modified Weibull distribution. Compos Part Appl Sci Manuf 40:54–59. doi: 10.1016/j.compositesa.2008.10.001 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • B. P. Revol
    • 1
    • 2
  • M. Thomassey
    • 1
  • F. Ruch
    • 1
  • M. Nardin
    • 2
  1. 1.Pôle Ingénierie des Polymères et CompositesCetim-CermatMulhouseFrance
  2. 2.UMR-CNRS 7361Institut de Science des Matériaux de Mulhouse (IS2M)Mulhouse CedexFrance

Personalised recommendations