Statistics of Critical Load in Arrays of Nanopillars on Nonrigid Substrates

  • Tomasz Derda
  • Zbigniew DomańskiEmail author
Conference paper


Multicomponent systems are commonly used in nano-scale technology. Specifically, arrays of nanopillars are encountered in electro-mechanical sense devices. Under a growing load weak pillars crush. When the load exceeds a certain critical value the system fails completely. In this work we explore distributions of such a critical load in overloaded arrays of nanopillars with identically distributed random strength-thresholds (\(\sigma _{th}\)). Applying a Fibre Bundle Model with so-called local load transfer we analyse how statistics of critical load are related to statistics of pillar-strength-thresholds. Based on extensive numerical experiments we show that when the \(\sigma _{th}\) are distributed according to the Weibull distribution, with shape and scale parameters k, and \(\lambda = 1\), respectively, then the critical load can be approximated by the same probability distribution. The corresponding, shape and scale, parameters K and \(\varLambda \) are functions of k.


Array of pillars Fracture Load transfer Scaling Statistics Weibull probablity distribution 


  1. 1.
    A. Hansen, P.C. Hemmer, S. Pradhan, The Fiber Bundle Model: Modeling Failure in Materials (Wiley, 2015)Google Scholar
  2. 2.
    S. Pradhan, A. Hansen, B.K. Chakrabarti, Failure processes in elastic fiber bundles. Rev. Mod. Phys. 82, 499–555 (2010)Google Scholar
  3. 3.
    M.J. Alava, P.K.V.V. Nukala, S. Zapperi, Statistical models of fracture. Adv. Phys. 55, 349–476 (2006)CrossRefGoogle Scholar
  4. 4.
    F. Kun, F. Raischel, R.C. Hidalgo, H.J. Herrmann, Extensions of fibre bundle models, in Modelling Critical and Catastrophic Phenomena in Geoscience, Lecture Notes in Physics, vol. 705 (2006) pp. 57–92Google Scholar
  5. 5.
    Z. Domański, T. Derda, and N. Sczygiol, Critical avalanches in fiber bundle models of arrays of nanopillars, in Lecture Notes in Engineering and Computer Science: Proceedings of The International MultiConference of Engineers and Computer Scientists 2013, IMECS 2013, 13–15 Mar 2013, Hong Kong, pp. 765-768Google Scholar
  6. 6.
    Z. Domański, T. Derda, Distributions of critical load in arrays of nanopillars, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering 2017, 5–7 July 2017, London, UK, pp. 797–801Google Scholar
  7. 7.
    W. Weibull, A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297 (1951)zbMATHGoogle Scholar
  8. 8.
    N.M. Pugno, R.S. Ruoff, Nanoscale Weibull statistics. J. Appl. Phys.99 (2006) (id. 024301)Google Scholar
  9. 9.
    A. Azzalini, A.R. Massih, A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178 (1985)MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Azzalini The Skew-normal And Related Families (Cambridge University Press, 2013)Google Scholar
  11. 11.
    H.E. Daniels, The statistical theory of the strength of bundles of threads I. P. Roy. Soc. Lond. A Mat. 183, 405–435 (1945)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R.L. Smith, The asymptotic distribution of the strength of a series-parallel system with equal load sharing. Ann. Probab. 10, 137171 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    L.N. McCartney, R.L. Smith, Statistical theory of the strength of fiber bundles. ASME J. Appl. Mech. 105, 601608 (1983)Google Scholar
  14. 14.
    D.G. Harlow, S.L. Phoenix, The chain-of-bundles probability model for the strength of fibrous materials I: analysis and conjectures. J. Compos. Mater. 12, 195–214 (1978)Google Scholar
  15. 15.
    D.G. Harlow, S.L. Phoenix, The chain-of-bundles probability model for the strength of fibrous materials II: a numerical study of convergence. J. Compos. Mater. 12, 314–334 (1978)Google Scholar
  16. 16.
    S.L. Phoenix, R.L. Smith, A comparison of probabilistic techniques for the strength of fibrous materials under local load-sharing among fibers. Int. J. Sol. Struct. 19, 479–496 (1983)CrossRefGoogle Scholar
  17. 17.
    L.S. Sutherland, C. Guedes, Soares, Review of probabilistic models of the strength of composite materials. Reliab. Eng. Syst. Saf. 56, 183–196 (1997)Google Scholar
  18. 18.
    M. Ibnabdeljalil, W.A. Curtin, Strength and reliability of fiber-reinforced composites: Localized load-sharing and associated size effects. Int. J. Sol. Struct. 34, 2649–2668 (1997)CrossRefGoogle Scholar
  19. 19.
    S. Mahesh, S.L. Phoenix, I.J. Beyerlein, Strength distributions and size effects for 2D and 3D composites with Weibull fibers in an elastic matrix. In. J. of Fracture 115, 41–85 (2002)CrossRefGoogle Scholar
  20. 20.
    P.K. Porwal, I.J. Beyerlein, S.L. Phoenix, Statistical strength of a twisted fiber bundle: an extension of Daniels equal-load-sharing parallel bundle theory. J. Mech. Mater. Struct. 1, 1425–1447 (2007)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsCzestochowa University of TechnologyCzestochowaPoland

Personalised recommendations