Abstract
We reanalyzed a model introduced by T. R. Madden for the evaluation of the state of connectivity among the microcrack population existing inside crystalline rocks. The model assumes that cracks are distributed randomly in a cubic lattice with a basic occupation probability, p.Depending on the value of p, the stale of connectivity can give rise to macroscopic paths, which allow electrical conduction of the sample, or to extended crack surfaces, which would be responsible for rock failure. The position of the phase boundaries, that is, the threshold values of pfor the onset of conductivity or macroscopic fracture, are estimated by a real-space renormalizalion-group (RG) technique. By identifying all the relevant configurations of the lattice model, we have been able to provide explicit analytic formulae for the critical lines. The criterion used by Madden to “accept” the existence of microscopic linear connectivity is modified and the new consequences discussed. We analyze the limitations of simple versions of the RG technique, in particular when concerned with anisotropic spatial distributions of cracks. Finally, we emphasize the interest of acquiring experimental data, especially to test the position of the conduction thresholds.
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Gómez, J.B., Pacheco, A.F. & Seguí-Santonja, A.J. A model for crack connectivity in rocks, a discussion. Math Geol 27, 23–39 (1995). https://doi.org/10.1007/BF02083566
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DOI: https://doi.org/10.1007/BF02083566