A unified constitutive framework for transient granular rheology through shear rate-induced dilatancy.

Abstract

A central concept in critical state theory is stress dilatancy, which relates the equivalent bulk friction, or deviatoric-stress-to-pressure ratio, <i>μ</i>, to dilatancy, defined as the ratio of plastic volumetric to plastic shear strain increments. The empirical <i>ϕ</i>(<i>I</i>) rheology, based on steady-state shear flow experiments and discrete element method simulations, suggests an additional form of dilatancy: under constant pressure, the solid volume fraction <i>ϕ</i> decreases as the inertial number <i>I</i> increases, a behavior often associated with dynamic processes, including interparticle collisions and microstructural rearrangements. While <i>ϕ</i>(<i>I</i>) is sufficient for describing steady flows, an additional mechanism must become active when <i>I</i> changes, allowing dilatancy to develop during transients. We attribute this mechanism to volumetric hardening induced by shear-rate variations and develop a unified constitutive framework in which stress- and rate-induced dilatancy govern quasi-static and dynamic stress evolution, respectively. While any critical-state plasticity model may be used for the quasi-static component, we select the classical modified Cam-Clay model to demonstrate how a dynamic extension can be formulated and what the emergent features are. By assuming a constant ratio between the deviatoric and isotropic dynamic stress components and linking rate-induced dilatancy to the critical-state pressure-<i>ϕ</i> relation, we obtain expressions for the dynamic stress increments, which depend on the rate-induced plastic volumetric strain increment and the shear rate increment through a nonlinear hardening modulus and a nonlinear shear viscosity. The resulting model is applied to homogeneous deformations under pressure- and volume-controlled triaxial shear. The results show that the model recovers classical critical-state behavior in the quasi-static regime and predicts a steady-state relationship between the equivalent bulk friction <i>μ</i> and <i>I</i>, resembling the empirical <i>μ</i>(<i>I</i>) rheology, without using it as input. Notably, the model also captures transient responses across a wide range of shear rates and variations thereof, including acceleration-deceleration cycles. Future work will involve characterizing rate-induced dilatancy through laboratory and numerical experiments and relating it to granular temperature.