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
Articles
  • 23 Downloads

Abstract

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. F. Richter,Elementary Seismology (Freeman, San Francisco, 1958); T. Utsu,Seismology, 2nd ed. (Kyoritsu, Tokyo, 1984) [in Japanese].Google Scholar
  2. 2.
    B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982); H. Takayasu,Fractal (Asakura, Tokyo, 1986) [in Japanese].Google Scholar
  3. 3.
    D. J. Andrews,J. Geophys. Res. 85:3867 (1980); K.Yamashina,EOS 63:1157 (1982); M. Matsushita,J. Phys. Soc. Japan 54:857 (1985).Google Scholar
  4. 4.
    B. B. Mandelbrot, D. E. Passoja, and A. J. Paullay,Nature 308:721 (1984).Google Scholar
  5. 5.
    C. J. Allègre, J. L. LeMoulel, and A. Provost,Nature 297:47 (1982); T. R. Madden,J. Geophys. Res. 88:585 (1983); R. F. Smalley and D. L. Turcotte,J. Geophys. Res. 90:1894 (1985).Google Scholar
  6. 6.
    H. Oda, H. Koami, and K. Seya,Zisin 38:331 (1985) [in Japanese].Google Scholar
  7. 7.
    B. Paul, inFracture II, H. Liebowitz, ed. (Academic Press, New York, 1968).Google Scholar
  8. 8.
    K. Itô and H. P. McKean, Jr.,Diffusion Processes and Their Sample Paths, 2nd ed. (Springer, Berlin, 1974).Google Scholar
  9. 9.
    K. Aki, inEarthquake Prediction, D. W. Simpson and D. G. Richerds, eds. (AGU, Washington, D.C., 1981).Google Scholar
  10. 10.
    K. Itô,Stochastic Processes II (Yale University Press, 1963), Chapter 5.Google Scholar
  11. 11.
    Y. Kasahara,Japan J. Math. 1:67 (1975); S. Kotani and S. Watanabe, inFunctional Analysis in Markov Processes, M. Fukushima, ed. (Springer, Berlin, 1982), p. 235.Google Scholar
  12. 12.
    E. Seneta,Regularly Varying Functions, A. Dold and B. Eckham, eds. (Springer, Berlin, 1976).Google Scholar
  13. 13.
    J. Lamperti,Z. Wahrsch. Verw. Geb. 22:205 (1972).Google Scholar
  14. 14.
    Y. B. Belyaev,Theor. Prob. Appl. 4:402 (1959).Google Scholar
  15. 15.
    H. Cramer and M. R. Leadbetter,Stationary and Related Stochastic Processes (Wiley, New York, 1967).Google Scholar
  16. 16.
    T. Tsuboi,Proc. Imp. Acad. 16:449 (1940).Google Scholar
  17. 17.
    H. Resten,J. Math. Mech. 12:391 (1963).Google Scholar
  18. 18.
    C. Stone,Ill. J. Math. 7:631 (1963).Google Scholar
  19. 19.
    K. Itô,Stochastic Processes (Springer, 1984), Section 4.5.Google Scholar
  20. 20.
    R. M. Blumenthal and R. K. Getoor,Ill. J. Math. 6:308 (1962).Google Scholar

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

Personalised recommendations