Onsager-symmetric constitutive laws for three-dimensional granular flow in the inertial regime

This paper introduces a new mathematical technique for deriving continuum rheological models of granular matter. Specifically, it is shown that, under the hypothesis of Onsager symmetry, three-dimensional dynamic constitutive laws for general strain rates can be derived from a three-dimensional yield condition plus steady-state empirical data of quasi-two-dimensional flow. To illustrate the technique, a new rate-dependent three-dimensional yield condition, suitable for dry granular materials in the inertial regime, is proposed and combined with discrete-element method (DEM) particle simulation data of simple shear flow. In combination with Onsager symmetry, this generates a complete three-dimensional viscoplastic model for such materials. Despite the simplicity of the inputs, the resulting constitutive laws agree very well with the pioneering non-planar DEM simulations of Clemmer et al. Phys. Rev. Lett. 127 (2021). Unlike several previous theories, the novel Onsager-symmetric constitutive relations incorporate a non-zero second normal stress difference in simple shear and are able to distinguish between general triaxial deformations via dependence on the Lode angle.