Reynolds Pressure and Relaxation in a Sheared Granular System

We describe experiments that probe the evolution of shear jammed states, occurring for packing fractions ${\varphi }_S\le \varphi \le {\varphi }_J$, for frictional granular disks, where above ${\varphi }_J$ there are no stress-free static states. We use a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This fixed $\varphi$ system exhibits coupling between the shear strain, $\gamma$, and the pressure, $P$, which we characterize by the “Reynolds pressure” and a “Reynolds coefficient,” $R(\varphi )=({\partial }^{2}P/\partial {\gamma }^2)/2$. $R$ depends only on $\varphi$ and diverges as $R\sim ({\varphi }_c-\varphi {)}^{\alpha }$, where ${\varphi }_c\simeq {\varphi }_J$ and $\alpha \simeq -3.3$. Under cyclic shear, this system evolves logarithmically slowly towards limit cycle dynamics, which we characterize in terms of pressure relaxation at cycle $n\mathrm{\text{:}}\Delta P\simeq -\beta \ln (n/n_0)$. $\beta$ depends only on the shear cycle amplitude, suggesting an activated process where $\beta$ plays a temperaturelike role.

Figure 1

(a) Setup schematics. (b) The three close-up images that the camera captures at each step: particle positions (upper), force response under polariscope (middle), and particle orientation images under UV light (lower). (c) The $x$ and $y$ displacements of particles vs their horizontal positions in the system. (d) The coarse-grained [12, 13] density profile after 27% linear shear.

Figure 2

(a) Reynolds pressure $P({\gamma }^2)$ observed in forward shear (see the text) tests for $\varphi =0.691–0.816$. (b) Reynolds coefficient $R$ extracted from linear fitting, obtained from up to 54% forward shear (red squares), up to 27% forward shear (blue dots), and cyclic shear tests under limit cycle behavior (black triangles). The inset shows the same data on double logarithmic scales with ${\varphi }_c=0.841\pm 0.004$. The error bar is smaller than the size of the symbols unless marked. The dashed line shows a fit to a power law. A line corresponding to an exponent $-3.3$ is also shown for reference.

Figure 3

(a) $P$ vs $\gamma$ for a symmetric cyclic shear run with $\varphi =0.825$, which started from $\gamma =0$ and sheared between ${\gamma }_{\max }=2.25\%$ and ${\gamma }_{\min }=-2.25\%$. Only cycles 1, 2, 28, and 29 are shown in the plot. (b) $\tau$ vs $\gamma$ for the same run and the same shear cycles. (c) $P$ vs $\gamma$ at cycles 1, 2, 28, and 29 for a nonsymmetric cyclic shear run (${\gamma }_{\max }=4.5\%$, ${\gamma }_{\min }=0$) with the same density.

Figure 4

(a) $\Delta P$ vs number of shear cycles for various runs with ${\gamma }_A=2.7\%$ and $\overline{\gamma }=1.35\%$ (solid line), 5.85% (open circles), and 10.35% (open triangles). Different color or shading indicates different $\varphi$. (b) $\Delta P$ vs $\log (n/n_0)$ for three different ${\gamma }_A$’s: 1.35% (open circles), 2.70% (open triangles), and 6.75% (solid lines). Data come from runs with various $\varphi$ (shown by color or shading) and various $\overline{\gamma }$ (not shown). (c) The decay factor, $\beta$, shows a strong increase with ${\gamma }_A$. (d) The universal decay curve, $\Delta P/\beta ({\gamma }_A)$ vs $\log (n/n_0)$, including all data from (b).