Influence of Third Invariant of Deviatoric Stress and Intermediate Principal Stress on Constitutive Models and Localization Conditions for High Porosity Sandstone

Abstract:

High porosity sandstone (HPS) is observed to fail, in field and laboratory settings, by the formation of localized bands. Bands perpendicular to the direction of the minimum principal stress (σ_III), with pure compactant strain, are defined as compaction bands. Dilation bands form parallel to the σ_III direction, with pure dilatant strain. Shear bands form at an angle to the σ_III direction and have shear strain, with either compactant or dilatant strain normal to the band. Recent experimental evidence indicates that the behavior of HPS depends on the third stress invariant (J₃) since it yields at a lower stress in extension, compared to loading under compression. In this work, existing constitutive models, which depend only on two stress invariants, were modified and new constitutive models, which incorporate the J₃ dependence, were developed. Band orientation predictions were determined using the Rudnicki and Rice (1975) bifurcation theory. While HPS are typically tested under axisymmetric (AS) stress states in the laboratory, in the field non-AS states are more likely. The stress state is characterized by the intermediate principal stress (σ_II). Therefore, to investigate the influence of σ_III on band orientation predictions, localization conditions were determined for non-AS stress states. The AS stress states are specialized cases where the dependence on J₃ is inconsequential. Incorporating the J₃ dependence significantly influenced localization conditions for stress states perturbed from AS compression, while the influence was small for stress states perturbed from AS extension. Under non-AS stress states, when a single yield surface constitutive model is appropriate, including the J₃ dependence typically inhibits band formation, due to strongly negative critical hardening modulus (h_cr). For load paths where a two yield surface constitutive model is applicable, with J₃ dependence, shear bands were predicted. Compaction bands and dilation bands were predicted with a positive h_cr, only under AS stress states. These results could be used to explain why these bands are rarely reported in the field, compared to shear bands.