Microscopic Origin of Nonlocal Rheology in Dense Granular Materials

We study the microscopic origin of nonlocality in dense granular media. Discrete element simulations reveal that macroscopic shear results from a balance between microscopic elementary rearrangements occurring in opposite directions. The effective macroscopic fluidity of the material is controlled by these velocity fluctuations, which are responsible for nonlocal effects in quasistatic regions. We define a new micromechanically based unified constitutive law describing both quasistatic and inertial regimes, valid for different system configurations.

Figure 1

(a) System geometry and illustration of the velocity profile and inertial (red) and quasistatic (gray) zones. (b),(c) Friction coefficient $\mu$ and normalized velocity fluctuations $\mathrm{\Delta }$ versus inertial number $I$ for different values of inner cylinder velocity $\mathrm{\Omega }$. For a given value of $\mathrm{\Omega }$, the different data points correspond to different radial positions $r$ within the sample.

Figure 2

Probability density distributions of local shear rate ${\dot{\gamma }}_{\mathrm{mic}}$ at two radial positions $r$ (indicated by the dashed circles in the insets). (a),(b) Case $\mathrm{\Omega }=20\mathrm{rad}{\mathrm{s}}^{-1}$: coexistence between an inertial (red) and a quasistatic (gray) zone within the sample. (c),(d) Case $\mathrm{\Omega }=0.05\mathrm{rad}{\mathrm{s}}^{-1}$: the whole sample is in the quasistatic regime.

Figure 3

(a),(b) Macroscopic shear rate $\dot{\gamma }$ versus average shear rate evaluated from the distribution of local shear rate ${\dot{\gamma }}_{\mathrm{mic}}$ (see text). Dashed line: $1:1$ line. (c) Dimensionless local shear rate $⟨{\dot{\gamma }}^{-}⟩d\sqrt{\rho /P}$ versus normalized velocity fluctuations $\mathrm{\Delta }$. Dashed line: Eq. (2). (d) Quantity $(1-\eta )/(1+\eta )$ versus friction coefficient $\mu$. Dashed line: Eq. (3). In all plots, symbols represent different values of inner cylinder velocity $\mathrm{\Omega }$ (see Fig. 1 for legend).

Figure 4

(a) Evolution of quantity $C_f{\mathrm{\Delta }}^{1.2}f(\mu )$ versus inertial number $I$ in our simulations. Symbols represent different values of inner cylinder velocity $\mathrm{\Omega }$. Dashed line: $1:1$ line. Inset: $I/(C_f{\mathrm{\Delta }}^{1.2})$ versus $\mu$. Dashed line: function $f(\mu )$. (b) Same plot as in Fig. [4] using data from [3] (plane shear) and [4] (annular shear with different system sizes: see legend).