Continuum Modeling of Secondary Rheology in Dense Granular Materials

Recent dense granular flow experiments have shown that shear deformation in one region of a granular medium fluidizes its entirety, including regions far from the sheared zone, effectively erasing the yield condition everywhere. This enables slow creep deformation to occur when an external force is applied to a probe in the nominally static regions of the material. The apparent change in rheology induced by far-away motion is termed the “secondary rheology,” and a theoretical rationalization of this phenomenon is needed. Recently, a new nonlocal granular rheology was successfully used to predict steady granular flow fields, including grain-size-dependent shear-band widths in a wide variety of flow configurations. We show that the nonlocal fluidity model is also capable of capturing secondary rheology. Specifically, we explore creep of a circular intruder in a two-dimensional annular Couette cell and show that the model captures all salient features observed in experiments, including both the rate-independent nature of creep for sufficiently slow driving rates and the faster-than-linear increase in the creep speed with the force applied to the intruder.

Figure 1

Schematic of the two-dimensional analogue of the experiments of Ref. [9].

Figure 2

Force applied to the intruder as a function of intruder velocity when the inner wall is fixed $\mathrm{\Omega }=0$. The dashed line indicates the rate-independent critical force $F_c/P_{a}D=2.91$. Insets show contour plots of the stress ratio $\mu$ in the region of the intruder; black regions denote where $\mu >{\mu }_s$.

Figure 3

Intruder creep speed as a function of inner wall speed for $F/F_c=0.75$ and $L/d=24$, demonstrating nominal rate independence in a range of sufficiently small inner wall speeds; the dashed line denotes the linear relation $V_c/\mathrm{\Omega }{R}_i=1.3\times 10^{-3}$.

Figure 4

Intruder creep speed as a function of applied force for $L/d=18$, 24, and 34 and $\mathrm{\Omega }{R}_i\sqrt{{\rho }_s/P_a}=1\times 10^{-5}$. Simulated creep data are shown as symbols; the dashed line represents the drag force versus intruder velocity relation of Fig. 2.

Figure 5

(a) Contour plots of the stress ratio $\mu$ in the region of the intruder during steady creep for $F/F_c=0.75$ and 0.95, $L/d=24$, and $\mathrm{\Omega }{R}_i\sqrt{{\rho }_s/P_a}=1\times 10^{-5}$. (b) The granular fluidity along a radial path between the inner wall and the intruder for these two cases.