Consequences of Anomalous Diffusion in Disordered Systems under Cyclic Forcing

We study the particle scale response of a 2D frictionless disk system to bulk forcing via cyclic shear with reversal amplitude ${\gamma }_r$. We find a subdiffusive ${\gamma }_r$-dependent regime, which is consistent with models of anomalous diffusion with scale-invariant cage dynamics, and a crossover to diffusive grain motion at high ${\gamma }_r$. Analysis of local displacements of a particle relative to its cage of neighbors reveals a key distinction from thermal systems. Particles are moved by fluctuations of their cage of neighbors rather than rattling in their cage, indicating a distinct cage-breaking mechanism.

Figure 1

A schematic of how shear is applied to the 2D model granular system. The initial state has a repeat cell which is square with linear size $L$. The strain is applied by shifting the lower and upper adjacent repeat cells horizontally by up to $\mathrm{\Delta }{x}_r$ as per Lees-Edwards boundary conditions. Then, the shear strain $\mathrm{\Delta }{x}_r$ is reversed back to zero.

Figure 2

Crossover grain and bulk behaviors at ${\gamma }_r=0.15$ (arrows). (a) MSD measured relative to the initial isotropic state. Different colors correspond to different reversal amplitudes, with blue (bottom curve) being the lowest and black (top curve) being the highest. A solid black line with a slope of 1 is added for reference. (b) (inset) Time- (or cycle-)averaged MSDs. The best linear fits are shown for each reversal amplitude. (b) The slopes from the linear fits ($\beta$) are shown as a function of reversal amplitude. Error bars stemming from the linear fits are shown. (c) (inset) Packing fraction and (d) (inset) shear stress ${\sigma }_{xy}$ averaged over many cycles as a function of strain $\gamma$, for varying ${\gamma }_r$. (c) ${\gamma }_r$ dependence for strain at peak $\varphi$ and (d) slope of ${\sigma }_{xy}$ at reversal.

Figure 3

(Top) The unscaled displacement distributions for cycle intervals $\mathrm{\Delta }t=10,50,100$ (black asterisks, blue $+$, and red $\times$). (Bottom) The displacement distributions scale according to Eq. (1), using the slope from the time-averaged MSD for ${\gamma }_r=0.075$. The markers with solid curves represent Gaussian fits about the mode.

Figure 4

(Left) (inset) The mean waiting time $⟨\tau ⟩$ as a function of candidate cage size $\epsilon$ for ${\gamma }_r=0.075$. (Left) The logarithmic derivative $\mathrm{\Delta }$, with the location of ${\epsilon }_c$ marked with a solid line in both plots. (Right) The waiting time distributions for ${\gamma }_r=0.05$, 0.075, and 0.1. They fall on a coinciding power law of slope $-1.6$.

Figure 5

(Left) An illustration of the modification (inset) to the standard MSD approach (main). Open circles correspond to the initial configuration and filled circles represent the new configuration some time later. The reference grain (blue) is marked with arrows and the green star is the centroid of neighboring grains (green). (Right) The mMSD, relative to the packing configuration at $t=0$, is shown. Solid lines have a slope of 1. (Right) (inset) MSD (top curve) and mMSD (bottom curve) for ${\gamma }_r=0.225$.