Shear-induced mixing of granular materials featuring broad granule size distributions

Granular flows during a shear-induced mixing process are studied using discrete element methods. The aim is to understand the underlying elementary mechanisms of transition from unmixed to mixed phases for a granular material featuring a broad distribution of particles, which we investigate systematically by varying the strain rate and system size. Here the strain rate varies over four orders of magnitude and the system size varies from ten thousand to more than a million granules. A strain rate-dependent transition from quasistatic to purely inertial flow is observed. At the macroscopic scale, the contact stresses drop due to the formation of shear-induced instabilities that serves as an onset of granular flows and initiates mixing between the granules. The stress-drop displays a profound system size dependence. At the granular scale, mixing dynamics are correlated with the formation of shear bands, which result in significantly different timescales of mixing, especially for those regions that are close to the system walls and the bulk. Overall, our results reveal that although the transient dynamics display a generic behavior, these have a significant finite-size effect. In contrast, macroscopic behaviors at steady states have negligible system size dependence.

Figure 1

A schematic of a cubic simulation box of length $L$. The top and bottom planes are covered by two smooth flat walls. The top wall moves with a constant velocity and the bottom wall is static, which generates a shear flow inside the simulation box.

Figure 2

Granular flows at different stages of shear $\gamma =\dot{\gamma }\times \text{time}$ (top). Here the simulation box has length $L=114.94$, contains $131441$ particles, and it is sheared at a constant strain rate $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$. $v_x$, the average velocity of particles along $x$, as a function of $z$ is plotted for the above configurations (bottom).

Figure 3

(a) The average local strain rate $\langle {\dot{\gamma }}_{\text{loc}}\rangle t_{\text{col}}$ of test cubes lying on the same $z$ plane is shown as a function over $\gamma$ for $N=131441$. The dashed line indicates the imposed strain rate $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$. (b) Similarly, the average coordination number $\langle n_c\rangle$ is plotted over $\gamma$.

Figure 4

Pressure $P$ (a), kinetic component of $P$ (b), and shear stress ${\sigma }_{xz}$ (c) rescaled by $k_n$ and plotted over $\gamma$ for $N=131441$ and $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$.

Figure 5

$\langle \delta \mathrm{\Theta }\rangle$, the density fluctuation parameter averaged over cubes locating at the same $z$ plane, versus $\gamma$ for $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$ and $N=131441$.

Figure 6

$P/k_n$ (a) and ${\sigma }_{xz}/k_n$ (b) over $\gamma$ at $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$ for several system sizes $N$.

Figure 7

$v_x$ rescaled by $v_{\text{aff}}$ is plotted against $z/L$ at steady states for $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$ for several system sizes $N$. Error bars are smaller than the symbols.

Figure 8

$\langle \delta \mathrm{\Theta }\rangle$ versus $\gamma$ (a). The bottom plane (left), and the top plane (right) of the final configuration (b). Here $\dot{\gamma }{t}_{\text{col}}=2.4\times {10}^{-5}$ and $N=1314378$.

Figure 9

$P/k_n$ versus $\gamma$ for several strain rates and $N=13147$ (a), respective final configurations as shown in roman numerals (b). $\langle \delta \mathrm{\Theta }\rangle$ versus $\gamma$ at $z=0$ (c, bottom panel) and $z=27$ (c, top panel) for the same strain rates. Here the box length $L=54.241$.

Figure 10

$v_x/v_{\text{aff}}$ versus $z/L$ at steady states for several strain rates and $N=13147$.

Figure 11

Steady-state values of $P/k_n$ and ${\sigma }_{xz}/k_n$ as a function of $\dot{\gamma }{t}_{\text{col}}$.