Critical transport and failure in continuum crack percolation

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Abstract
Critical transport and failure properties of a new class of continuum percolation systems (blue cheese model), where the transport medium is the space between randomly placed clefts, are discussed. The critical behaviour of electrical conductivity, fluid permeability, elastic constants and failure thresholds are, for most of them, distinct from their counterparts in both the discrete-lattice and Swiss-cheese continuum percolation models. Furthermore, it is argued that the asymmetric elastic response of a crack submitted to compression or extension leads, for the macroscopic mechanical properties, to a new percolation threshold at a crack concentration higher than that of usual geometric percolation. This is due to the presence of hooks or overhang crack configurations which locally transform the macroscopic applied extensional stress into a compressional stress. In two dimensions, an analogy with directed percolation is suggested. When the cracks have a non-vanishing width, one recovers the usual geometric percolation threshold and the analysis of the transport properties is similar to that of the Swiss cheese model developed by Halperin et al. For elastic sheets which are not constrained to lie in a flat plane, the two edges of a crack may overlap : this leads to the existence of two additional elastic deformation modes (crack opening and buckling) which exhibit distinct critical behaviours. The brittle Griffith failure scenario which corresponds to the growth of a crack is analysed and introduces new critical failure exponents