Non-linear elastic behaviour of damaged rocks

The pervasive damage of rocks by microcracks and voids strongly affects their macroscopic elastic properties. To evaluate the damage effects, we derive here the macroscopic stress-strain relations for a 3-D elastic solid with non-interacting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky O'Connell (1976). The solution describes the stress-strain relations in terms of λ^d and μ^d, which are the modified Lame constants, and an additional parameter γ. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ε<0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension. We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.