Sinking in a bed of grains activated by shearing

We show how a weak force $f$ enables intruder motion through dense granular materials subject to external mechanical excitations, in the present case, stepwise shearing. A force acts on a Teflon disk in a two-dimensional system of photoelastic disks. This force is much smaller than the smallest force needed to move the disk without any external excitation. In a cycle, the material plus intruder are sheared quasistatically from $\gamma =0$ to ${\gamma }_{\text{max}}$, and then backwards to $\gamma =0$. During various cycle phases, fragile and jammed states form. Net intruder motion $\delta$ occurs during fragile periods generated by shear reversals. $\delta$ per cycle, e.g., the quasistatic rate $c$, is constant, linearly dependent on ${\gamma }_{\text{max}}$ and $f$. It vanishes as $c\propto {({\varphi }_c-\varphi )}^a$, with $a\simeq 3$ and ${\varphi }_c\simeq {\varphi }_J$, reflecting the stiffening of granular systems under shear [J. Ren, J. A. Dijksman, and R. P. Behringer, Phys. Rev. Lett. 110, 018302 (2013)] as $\varphi \to {\varphi }_J$. The intruder motion induces large-scale grain circulation. In the intruder frame, this motion is a granular analog to fluid flow past a cylinder, where $f$ is the drag force exerted by the flow.

Figure 1

(a) Sketch of experimental setup: A $76.2\text{−}\mathrm{mm}$-diam intruder immersed in a bidisperse layer of photoelastic disks was pulled with a constant force $f$. The intruder started at low $x$ values (i.e., to the left), centered along the $y$ axis; under shear, it moved along the $x$ axis. The system was imaged from above by a high-definition camera following each shear step: in (c) white light, (d) cross-polarized light, and (e) UV. Each particle was marked with a small UV-sensitive bar. (b) Shear protocol: One cycle of shear strain was applied to the whole system stepwise to ${\gamma }_{\text{max}}$ by deforming the initially square box to a parallelogram, at constant area. The shear was then reversed, also stepwise, back to the original boundary configuration. This protocol was repeated $N_c$ times.

Figure 2

(a) Cumulative net intruder motion $\delta$ vs $N$, for different constants, for different packing fractions $\varphi$. Here, $f=1.09\mathrm{N}$ and ${\gamma }_{\text{max}}=10\%$. The data are consistent with a linear dependence of $\delta$ on $N$: $\delta =cN$. (b) Variation of the quasistatic rate $c$ as a function of the distance to the jamming packing fraction. This varies as a power law: $c=0.054{({\varphi }_J-\varphi )}^{2.95}$ ($\varphi$ in percent). (c) Constant drag force $f$ vs $c$. Here, $\varphi =80.83\%$, and ${\gamma }_{\text{max}}=10\%$. The intruder quasistatic rate increases linearly with $f$: $c=4.28f-1.46$ mm. (d) Shear amplitude ${\gamma }_{\text{max}}$ vs $c$ with $f=1.09\mathrm{N}$ and $\varphi =80.83\%$. $c$ increases linearly with ${\gamma }_{\text{max}}$: $c=2.56{\gamma }_{\text{max}}+1.11$ mm.

Figure 3

Net intruder displacement (blue) and average contact number for nonrattler particles (red) of the first four cycles and 20th–24th cycles ($f=1.09\mathrm{N}$, $\varphi =80.83\%$, and ${\gamma }_{\text{max}}=20\%$).

Figure 4

(a) Net intruder displacement (blue) and mean particle displacement (red) during a typical cycle ($f=1.09\mathrm{N}$, $\varphi =80.83\%$, and ${\gamma }_{\text{max}}=10\%$). (b) Evolution of $P$ to the right of the intruder, $\langle \overline{G_R^2}\rangle$ (black), to the left of the intruder, $\langle \overline{G_L^2}\rangle$ (blue), and the difference, $\langle \overline{G_R^2}\rangle -\langle \overline{G_L^2}\rangle$ (red). The last measures the net granular drag force on the intruder. Inset: Sketch of the regions to the left (light blue) and right (dark blue) of the intruder center, used to average $G^2$. (c)–(f) $G^2$ for each particle for different steps of the shear cycle presented in (a) and (b). In (c), the intruder compresses grains to the right. In (d), the shear jammed state is nucleating. The intruder moves in the negative $x$ direction because of high differential granular pressure. In (e), following shear reversal, the particles rearrange and the strong granular force network almost vanishes. In (f), the force network has reformed but it is weaker than and has a different orientation than the network in (d).

Figure 5

Large-scale quasistatic convective flow. Images (a) and (b) show the non-affine flow of particles in the frame of the intruder averaged over 50 cycles corresponding to positions (c) and (e) of the shear cycle in Figs. 4 and 4, respectively. There is large-scale collective motion extending to the boundaries of the box including major vortices that partially change direction between (a) and (b). The average nonaffine displacement of the particles is given in Fig. S2 of the SM [33]. These data pertain to a typical experiment: $f=1.09\mathrm{N}$, $\varphi =80.83\%$, and ${\gamma }_{\text{max}}=15\%$. See SM for a video of the quasistatic flow [33].