Self-Similar Structural Systems with No-Unloading and Scale-Invariant Strength Distributions

  • Dmitry A. Onishchenko
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)


The problem of structural strength is tightly connected with the study of multiscale processes of damage accumulation and fracture propagation which are inherent for various materials and for complicated systems and structures of both engineering and natural origin. Among typical examples are quasibrittle fracture in solids, the interaction of faults in earthquakes, breakage of fibre structures, failure of complex structural systems, progressive fire propagation. In the paper, multi-element systems whose elements have random strength are considered, and a general class of multi-element systems with no-unloading is described for which the transfer loads on survival elements do not decrease along any possible failure path. A number of specific properties is established for such systems, the most important of which is that the probability of system collapse does not depend on failure path, provided external loading is treated as quasistatic. Moreover, it is shown that the failure probability can be calculated through an explicit algebraic equation of recurrence type. Starting from this general motivation, an earlier introduced model of hierarchical failure that utilised a multilevel tree-like structure, or fractal tree, composed of statistically homogeneous elements is studied in a more general case and in more detail. Applying the interpretation of the governing equations as a dynamical system with discrete time in a specially introduced Banach space of infinite sequences, it is shown that the asymptotic behaviour of the cumulative probability distribution function (cdf) of system’s strength, when the hierarchy depth of the system becomes large enough, can be essentially different, depending on the cdf of element’s strength. In particular, it is proved that there can exist scale-invariant strength distributions which are stable by Lyapunov fixed points of the dynamical system introduced. This means that, for some specific load redistribution laws, the absence of the size effect can be observed within the framework of multilevel tree-like model structures.


Multi-element structural systems probability strength distribution systems with no-unloading collapse probability calculation hierarchical modelling fractal tree (FT) structure reloading function FT strength asymptotic behaviour scale-invariant strength distributions 


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  1. 1.
    Newman WI, Gabrielov AM. “Failure of hierarchical distributions of fibre bundles. I”, Int. J Fract., 50, pp. 1–14, 1991.Google Scholar
  2. 2.
    Onishchenko DA. “Probabilistic modeling of multiscale fracture”, Mech. Solids, 34(No. 5), pp. 21–38, 1999.Google Scholar
  3. 3.
    Daniels HE. “The statistical theory of strengths of bundles of threads”, Proc. R. Soc. Lond., A183, pp. 405–435, 1945.MathSciNetGoogle Scholar
  4. 4.
    Sutherland LS, Soares CG. “Review of probabilistic models of the strength of composite materials”, Reliab. Eng. Sys. Saf., 56, pp. 183–196, 1997.CrossRefGoogle Scholar
  5. 5.
    Bažant ZP, Pang SD. “Revision of structural reliability concepts for quasibrittle structures and size effect on probability distribution of structural strength.” Proc. 9th Int. Conf. on Struct. Safety and Reliability (ICOSSAR-9), Rome, Italy, June 19–23, 2005, pp. 377–386.Google Scholar
  6. 6.
    Herrmann HJ. “Fractures”, in: Fractals and Disordered Systems (eds. A. Bunde, S. Havlin), Berlin, Springer-Verlag, pp. 201–231, 1996.Google Scholar
  7. 7.
    Madsen HO, Krenk S, Lind NC. Methods of Structural Safety, NY, Prentice Hall, 1986.Google Scholar
  8. 8.
    Onishchenko DA. Some Principles of Construction and Analysis of Quasistatic Models in Probabilistic Fracture Mechanics of Discrete Systems, Preprint No. 572, Institute for Problems in Mechanics of Russian Academy of Sciences, Moscow, 1996 (in Russian).Google Scholar
  9. 9.
    Onishchenko DA. “A probability analysis of the problem of the breakthrough of a coarsely cellular foam in a porous medium”, J. Appl. Maths Mechs., 66 (2), pp. 261–270, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Wagner HD. “Statistical concepts in the study of fracture properties of fibres and composites”, in: Application of Fracture Mechanics to Composite Materials (ed. K. Friedrich), Elsevier, Amsterdam, 1989, pp. 39–77.Google Scholar
  11. 11.
    Onishchenko DA. “Scale-invariant distributions in the strength problem for stochastic systems with hierarchical structure”, Doklady Physics, 44, pp. 645–647, 1999.zbMATHGoogle Scholar
  12. 12.
    Onishchenko DA. “Hierarchical failure modeling and related scale-invariant probability distributions of strength”, Fracture of Nano and Engineering Materials and Structures. Proc. 16th European Conference of Fracture (ECF-16), Alexandroupolis, Greece, July 3–7, 2006, CD-version , Springer.Google Scholar
  13. 13.
    Bažant ZP, Yavari A. “Is the cause of size effect on structural strength fractal or energetic–statistical?”, Eng. Fract. Mech., 72, pp. 1–31, 2005.CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Dmitry A. Onishchenko
    • 1
    • 2
  1. 1.Scientific-Research Institute of Natural Gases and Gas TechnologiesRussia
  2. 2.Institute for Problems in MechanicsRussian Academy of Sciences Prospect VernadskogoRussia

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