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 lawCl −pasl↓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.
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Ito, H.M., Kotani, S. & Yokota, T. Size distribution of fractured areas in one-dimensional systems. J Stat Phys 51, 569–584 (1988). https://doi.org/10.1007/BF01028473
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DOI: https://doi.org/10.1007/BF01028473