Localized deformation such as shear bands, compaction bands, dilation bands, combined shear/compaction or shear/dilation bands, fractures, and joint slippage are commonly found in rocks. Thus, modeling their inception, development and propagation, and effect on stress response is important. This paper will focus on modeling the inception of these localized deformations-the onset of bifurcation to a localized material deformation response-for a three-invariant, isotropic/kinematic hardening cap plasticity model. Bifurcation analysis is the first step in developing a constitutive model for representing the transition of continuous rock-like material to fragmented rock material. Developing a post-bifurcation constitutive model and numerical implementation, whether via the finite element method or a meshfree method, is the next step and will not be discussed in this paper (but is part of our ongoing research). Applications of a constitutive model for modeling localized deformation in geomaterials include assessing the long term performance of nuclear waste repositories, designing tunneling construction, oil and natural gas production, and depleted reservoirs used for subsurface sequestration of greenhouse gases.
1. INTRODUCTION
Localized deformation such as shear bands, compaction bands, dilation bands, combined shearcompaction or shear-dilation bands, fractures, and joint slippage are commonly found in rocks. These localized deformations can be triggered by either material inhomogeneities such as joint sets in rocks, inhomogeneous stress resulting from boundary conditions such as friction at end platens in a con- fined compression test, or by some microstructurally driven material instability. We can account for material inhomogeneities by constitutivemodeling in conjunction with a numerical simulation method such as the finite element method. Significant material inhomogeneities such as strata and joint sets can be meshed discretely, assigning different material properties for each spatial region of the finite element mesh, or they can be incorporated in an average sense into a continuum constitutive model via directional structure/anisotropy tensors or the like. Either way, depending on boundary and loading conditions, the material deformation response predicted by the constitutive model could become mathematically unstable. This mathematical instability could be made to coincide with the natural material instability observed in the field or laboratory. The most straightforward way to do this is to endow the constitutive model with as much material characterization and representative deformation response that is deemed significant for the problem of interest. For example, if joint sets are plentiful and dominate the material deformation response, they must be represented in the constitutive model. Depending on the boundary and loading conditions, the model must predict the onset of gross localized deformation resulting from activity of certain critical joint sets. In essence, the ability of a continuum constitutive model to predict material instability in the form of localized deformation is only as good as the model's sophistication in terms of representing material behavior. Some questions we should ask when choosing and developing constitutive models for geomaterials are: Is thematerial isotropic or anisotropic elastically and/or plastically? Is the material temperature and ratesensitive?Are joint sets or other in-situ material inhomogeneities prominent?
Given a relatively sophisticated continuum constitutive model for geomaterials, this paper focuses on determining stress states at which the constitutive model predicts mathematical instabilities.
