Spectral Classification of Non-Coaxiality for Two-Dimensional Incremental Stress-Strain Response

The present study examines the non-coaxial aspects of incremental material behavior, and attempts to classify the incremental non-coaxiality that relates stress and strain increments. In the solid mechanics literature, non-coaxiality (NC) refers usually to incremental strains and stress states having different principal directions. Departing from conventional non-coaxiality, the analysis investigates the incremental non-coaxiality (INC) of linearized rate-type solids. This study uses the concept of deviatoric second-order work for examining the relations between stability and incremental non-coaxiality. Based on a spectral analysis of the constitutive compliance matrix, it proposes three classifications for distinguishing various degrees of incremental non-coaxiality and stability. These classifications determine the conditions for the existence of incremental coaxiality (i.e., colinearity of stress and strain increments), stability, instability, and stable-instable transition (i.e., positive, negative, or zero second-order deviatoric work). The study illustrates these classifications in the cases of generic elastic and elastoplastic constitutive models. The analysis pertains to two-dimensional cases. Additional research is required to extend the analysis from two to three dimensions. 1. Introduction The anisotropic and non-coaxial behaviors of geomaterials are challenging to model using the conventional flow theory of plasticity, which assumes that the strain increments and principal stress have identical direction, that is, are coaxial [1–3]. For instance, the associative flow rule of plasticity, which assumes that strain increments are normal to the yield surface, disagrees with many experimental evidences [4, 5] and micromechanical observations [6, 7], which show non-coaxiality, that is, different principal directions for stress states and strain increments. As early as Hill [8], several theories have been proposed to introduce noncoaxiality. For instance, one approach added tangent plasticity to classical coaxial models [1, 9]. Another approach defined strain increment in terms of stress states and material anisotropy, which may lead to an anisotropic hardening law [10–12]. Others have described noncoaxial behaviors using double-shear models [13–15]. Micromechanical studies have related non-coaxiality to the anisotropic fabric resulting from the arrangement of material particles and associated voids [16, 17]. Most studies of non-coaxiality focused on the case of non-proportional loading, and assumed coaxiality under proportional loading [18].