Elastic properties of fractured rock masses

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Abstract
We derive the relationships that link the general elastic properties of rock masses to the geometrical properties of fracture networks, with a special emphasis to the case of frictional crack surfaces. We extend the well-known elastic solutions for free-slipping cracks to fractures whose plane resistance is defined by an elastic fracture (shear) stiffness k_s and a stick-slip Coulomb threshold. A complete set of analytical solutions have been derived for i) the shear displacement in the fracture plane for stresses below the slip threshold and above, ii) the partitioning between the resistances of the fracture plane in the one hand, and of the elastic matrix in the other hand, and iii) the stress conditions to trigger slip. All the expressions have been checked with numerical simulations. The Young’s modulus and Poisson’s ratio were also derived for a population of fractures. They are controlled both by the total fracture surface for fractures larger than the mechanical length l_M (defined by k_s and the intact matrix elastic properties), and by the percolation parameter of smaller fractures. These results were applied to power-law fracture size distributions, which are likely relevant to geological cases. We show that, if the fracture size exponent is in the range -3 to -4, which corresponds to a wide range of geological fracture networks, the elastic properties of the bulk rock are almost exclusively controlled by k_s and the mechanical length, meaning that the fractures of size l_M play a major role in the definition of the elastic properties.

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