Determination of effective elastic properties of microcracked rocks based on asymptotic approximation

In this paper, we present a study on the effective elastic properties of finely fractured rock based on the concept of energy equivalence. The effective elastic properties a rock body weakened by many penny-shaped microcracks, such as the apparent Young's modulus, shear modulus and Poisson's ratio, are particularly important in many applications. To determine these macroscopic parameters, we first adopt the dilute solution approach outlined in Kachanov (1992), where a crack compliance tensor is defined and used extensively to express the energy perturbation caused by the presence of the cracks. In parallel, a self-consistent method is employed where an asymptotic form of the Eshelby's tensor (Eshelby, 1957) is developed to treat a penny-shaped microcrack as an extreme case of a spheroid-shaped inhomogeneity. The asymptotic approximation permits the local Eshelby tensor as well as its global expression to be derived analytically, and to be further used to construct an equivalent eigenstrain problem based on energy equivalence. The effective values for the parameters of interest can then be evaluated. The formulation and results obtained through the asymptotic self-consistent approach are compared to those predicted using a non-interacting scheme.