Computational modeling of deformation bands in granular media. I. Geological and mathematical framework Original Research Article

Failure of granular media under natural and laboratory loading conditions involves a variety of micromechanical processes producing several geometrically, kinematically, and texturally distinct types of structures. This paper provides a geological framework for failure processes as well as a mathematical model to analyze these processes. Of particular interest is the formation of tabular deformation bands in granular rocks, which could exhibit distinct localized deformation features including simple shearing, pure compaction/dilation, and various possible combinations thereof. The analysis is carried out using classical bifurcation theory combined with non-linear continuum mechanics and theoretical/computational plasticity. For granular media, yielding and plastic flow are known to be influenced by all three stress invariants, and thus we formulate a family of three-invariant plasticity models with a compression cap to capture the entire spectrum of yielding of geomaterials. We then utilize a return mapping algorithm in principal stress directions to integrate the stresses over discrete load increments, allowing the solution to find the critical bifurcation point for a given loading path. The formulation covers both the infinitesimal and finite deformation regimes, and comparisons are made of the localization criteria in the two regimes. In the accompanying paper, we demonstrate with numerical examples the role that the constitutive model and finite deformation effects play on the prediction of the onset of deformation bands in geomaterials.