Journal of Statistical Physics

, Volume 51, Issue 3–4, pp 569–584 | Cite as

Size distribution of fractured areas in one-dimensional systems

  • H. M. Ito
  • S. Kotani
  • T. Yokota


We study a one-dimensional model for fracture, identifying fractured areas with intervals on which a stress fieldξ exceeds a threshold valueΔ. Whenξ is a diffusion process, the cumulative numberN(l) of fractured areas whose length is greater thanl obeys a power lawClpasl↓0 with probability one. The exponentp and the constantC are determined. The exponentp agrees with the Hausdorff dimension of the end points of fractured areas, i.e.,ξ−1(Δ). Even ifξ is self-similar with parameterH>0, i.e.,ξ(cx)−Δ is equivalent toc H {ξ(x)−Δ} for anyc>0, the exponentp does not depend solely onH;p=λH, whereλɛ(0, 1/H) is another parameter characterizingξ. Non-diffusion processes are given whereN(l) does not follow a power law.

Key words

Fracture size distribution power law diffusion process Hausdorff dimension self-similar 


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Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • H. M. Ito
    • 1
  • S. Kotani
    • 2
  • T. Yokota
    • 1
  1. 1.Seismology and Volcanology DivisionMeteorological Research InstituteIbarakiJapan
  2. 2.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

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