This paper describes a model that predicts the moduli for brittle rock loaded by compressive principal stresses. All inelastic deformation is assumed due to microcracks that open under local tensile stresses caused by small scale heterogeneities. For axisymmetric loading and an isotropic initial distribution of cracks, the model predicts that crack growth begins in the axial direction and expands to a cone of angle 12-15 as loading continues to peak stress. The macroscopic response evolves from isotropic to transversely isotropic and the elastic moduli decrease with ongoing deformation. For combined axisymmetric compression ) and torsion axial loading causes the damage surface in vs. space, separating stress states causing unloading from those that cause continued damage, to evolve from an ellipse to a surface with a sharp vertex at the current stress point.
Understanding the localization of shear deformation into narrow zones is essential to predicting macroscopic rock fracture in compression, progressive reduction in load bearing ability (strain softening behavior) and the evolution of faults and certain other geologic structures, e.g., kink bands. Understanding highly localized deformation is also essential for predicting complex failures around underground excavations and boreholes where the mode and extent of rock decrepitation cannot be described by means of conventional yield or fracture criteria. The onset of localization typically involves an abrupt change in the deformation pattern or, in other words, in the orientation of the principal axes of the rate of deformation. Consequently, attempts to predict the onset of localization in terms of the constitutive description of homogeneous deformation (Rudnicki and Rice, 1975) are strongly dependent on the moduli governing such abrupt changes in the pattern of deformation. These moduli are, however, difficult to measure in experiments. The goal of this work has been to predict these moduli using a macroscopic constitutive relation based on the inelastic deformation caused by the growth of tensile microcracks.
Our approach has its origin in a model of Costin (1983a, 1983b, 1985). He proposed an ad hoc expression for the local tensile stress that arises due to microscale inhomogeneities. These inhomogeneities may include neighboring grains with different elastic constants, grain boundaries, or sliding frictional cracks with opening wing cracks. Regardless of the details of the microstructure, these inhomogeneities have the effect of creating local tensile stresses that drive crack growth even when all the applied principal stresses are compressive. Costin (1985) introduced this function as an approximation to numerical results of Horii and Nemat-Nasser (1983) for the dependence of the stress intensity factors for a row of colinear cracks. More generally, the function g (a) describes the decrease of local tensile stress with distance away from the inhomogeneity and the increase due to crack interaction. The first effect causes initially stable growth and the second is the source of a peak in the stress strain curve. Different forms of this function may be needed to describe data on different rock types. A few other forms have been explored by Rudnicki (1994).