INTRODUCTION
A continuum damage mechanics model simulates the stress and damage fields and the corresponding paths of pressurized en-echelon cracks. The model includes gradual strength degradation and subcritical crack growth, together with development of process zones, strain localization and brittle failure. Prior to onset of damage, the calculated stress field around the pressurized cracks is the same as that produced by linear-elastic models. After the onset of damage, localized damage zonespread out around the crack planes, and their shape is sensitive to the state of stress. The model reproduces shapes of damage lobes and geometries of crack connections that are commonly observed around en-echelon dyke segments in sandstone. Thus, the distributed damage and the crack paths may help to better estimate the state of stress acting during growth of pressurized cracks (dykes). The close kinship between dykes and artificial hydrofracturing provides an economic incentive to incorporate our results in hydraulic fracturing analyses.
Distributed damage in the form of microcrack arrays profoundly affects rock strength and rock elastic coefficents (e.g. Reches & Lockner 1994, Weinberger et al. 1994, Pestman & Munster 1996, Lyakhovsky et al. 1997b) and leads to vanishing elastic moduli at large stresses immediately before failure (Lockher et al. 1992). Rock-mechanics experiments in which damage evolve indicate that fracturing cannot be described in terms of single- crack propagation and that the inelastic process zone at the crack tip has a significant size (Yukutake 1989, Reches & Locknet, 1994). The linear elastic fracture mechanics (LEFM) approach often fails to account for the distributed damage as it assumes that the size of the process zone is negligibly small. The finite-size effect of the crack process zone is often modeled by specifying a cohesive zone near the crack tip along the crack plane (Dugdale 1960, Barenblatt 1962, 1996). This approach removes the physical unrealisticrack- tip singularity prevailing in LEFM, and is useful when the crack geometry is well defined. However, experiments in most engineering and rock-like materials indicate that a slowly propagating crack is preceded by distributedamage off the crack plane (e.g. Bazant & Cedolin 1991, Lockher et al. 1991), which possibly controls the macro-crack trajectory and the growth rate (Huang et al. 1991, Chai, 1993, Zietlow & Labuz, 1998). Thus it is desirable to account explicitly for the off-plane distribution of damage in studies of crack evolution.
In accordance with laboratory experiments, rheological models of fracturing should include subcritical crack growth, material degradation due to increasing crack concentration, macroscopic brittle failure, post-failure deformation, and healing. Continuum damage mechanics (CDM) uses irreversible thermodynamics to quantitatively account for the above deformational aspects (Kachanov 1994). Our model uses the balance equations of energy and entropy to establish a thermodynamical foundation for a rheological model under CDM (Lyakhovsky & Myasnikov 1985, Lyakhovsky et al. 1993, 1997a). The present CDM model describes the variations of elastic moduli and Poisson's ratio by a scalar damage intensity, tz, that is scaled properly with the ratio of strain invariants. Lyakhovsky et al. (1997a,b) develop the model further, and constrain the final model parameters by comparing theoretical predictions with various laboratory results.
The present study applies the damage model to investigate the propagation paths, connections, and damage distribution associated with en-echelon pressurized cracks. These cracks are abundant in a variety of engineering and geological environments,
