The loss of ellipticity of the governing equations is the criteria used to determine the emergence of localization bands, i.e. when the characteristic surfaces of the governing equations become real. This is equivalent to the traditional Rice- Rudnicki approach. However, it becomes essential for the investigation of the post bifurcation behavior of deformation bands. To determine the orientation of the bands Rice-Rudnicki theory requires a minimization of the hardening. In the current paper this condition is not used. An important result of the study is that preconsolidation of the reservoir must be present and is crucial to the appearance of deformation bands. Within the framework of a classical Modified Cam-Clay model the thickness of the bands cannot be calculated. The spacing of the resulting bands will be discussed in a further publication.
We start with the earlier work where the Eshelby approach has been used to successfully model the stress, strain and displacement fields associated with a depleting reservoir. A modified Cam-Clay material model was implemented for the constitutive equations for the reservoir material . The exterior of the reservoir was modeled as a linear elastic material. This enabled solutions to be calculated far more quickly than using a purely numerical approach to the problem. We have extended the calculation to investigate the occurrence of deformation bands in the reservoir subject to pressure depletion. This is accomplished by examining the stability of the equilibrium stress and strain fields to determine the conditions at which a bifurcation of the equilibrium state occurs. These conditions are equivalent to the appearance of deformation bands. We have used an approach similar to Rice-Rudnicki . The loss of ellipticity of the governing equations of equilibria is used as the criteria for the emergence of localization bands, i.e. when the characteristic surfaces of the governing equations become real. This is equivalent to the traditional Rice-Rudnicki approach. It allows, however, the investigation of the post bifurcation behavior of deformation bands. This is beyond the scope of this paper.
We investigate the stability of the homogeneous stress field in the ellipsoidal reservoir embedded in the elastic host rock . To do so, we first linearize the equations of equilibrium in the vicinity of this homogeneous stress field. For simplicity, we show the equation for normal stress conditions, with the two horizontal stresses equal. We also note that the formation of deformation bands is different from the occurrence of negative rate of work of plastic strains (NRW), which implies a violation of the second law of thermodynamics. We have interpreted this as meaning that the constraints in the model have been violated, since elastic unloading is impossible and therefore delamination from the matrix has occurred .
As discussed above, in order to form deformation bands, the yield surface has to be encountered to the left of the critical state line.