The Fractal-Statistical Nature of Size-Scale Effects on Material Strength and Toughness

  • Alberto Carpinteri
  • Simone Puzzi
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)


The size-scale effects on the mechanical properties of materials are a very important topic in engineering design. Three different modeling approaches have been proposed and analyzed at least, i.e. the statistical, the energetical and the fractal one. Aim of this paper is to revisit the fractal approach and to reject the most recurrent criticisms against it. Moreover, we will show that it is wrong to set the fractal approach to size-scale effects against the statistical one, since they are deeply connected. More in detail, by analyzing a fractal distribution of micro-cracks in the framework of Extreme Value theory, we will obtain a scaling law for tensile strength characterized, in the bi-logarithmic plot, by the slope -1/2. Conversely, by considering a fractal grain size distribution in a grained material, we will obtain a scaling law or fracture energy characterized, in the bi-logarithmic plot, by the positive slope 1/2. These slopes are the natural consequence of perfect self-similarity of the flaw (or grain) size distribution. Eventually, the theoretical results regarding the link between fractals and statistics will be confirmed by numerical simulations.


Size-scale effects self-similarity fractals 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Alberto Carpinteri
    • 1
  • Simone Puzzi
  1. 1.Department of Structural Engineering and GeotechnicsPolitecnico di Torino Corso Duca degli Abruzzi 24Italy

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