Period proliferation in periodic states in cyclically sheared jammed solids

Athermal disordered systems can exhibit a remarkable response to an applied oscillatory shear: After a relatively few shearing cycles, the system falls into a configuration that had already been visited in a previous cycle. After this point the system repeats its dynamics periodically despite undergoing many particle rearrangements during each cycle. We study the behavior of orbits as we approach the jamming point in simulations of jammed particles subject to oscillatory shear at fixed pressure and zero temperature. As the pressure is lowered, we find that it becomes more common for the system to find periodic states where it takes multiple cycles before returning to a previously visited state. Thus, there is a proliferation of longer periods as the jamming point is approached.

Figure 1

(a) The enthalpy $H$ as a function of time. An oscillatory strain with amplitude ${\gamma }_t=0.15$ is applied quasistatically to a system of $N=64$ particles. (b), (c) The probability $P_{\mathrm{active}}$ of a system (with $N=256$) to still be “active” and not in a periodic cycle vs the number of applied shear cycles $t$, for a fixed ${\gamma }_t=0.05$ in (b) and for a fixed pressure $p\approx 0.004$ in (c). The error bars for the ${\gamma }_t=0.09$ data are smaller than the symbol sizes.

Figure 2

Characteristic time ${\tau }_a$ to reach a periodic state, as a function of the training amplitude ${\gamma }_t$ at fixed pressure $p\approx 0.004$ (a) and as a function of $p$ at fixed ${\gamma }_t=0.05$ (b). In (a), we see that ${\tau }_a$ appears to diverge at a critical amplitude ${\gamma }_t^{*}\approx 0.13$ (dotted line). The dashed line shows a fit to $a{({\gamma }_t^{*}-{\gamma }_t)}^{-\nu }$, with $\nu =2.4\pm 0.3$ and $a=0.0043$. In (b), ${\tau }_a$ grows very slowly as $p$ decreases. The dashed line shows a linear fit to the data ${\tau }_a=alogp+b$ with $a=-0.88$ and $b=-0.38$.

Figure 3

Comparison for a system with $N=256$ of the first training cycle (red $\times$'s) with the eventual periodic cycle (purple diamonds and blue squares) into which it settles, for fixed pressure $p\approx 0.004$ and varying training amplitude ${\gamma }_t$ in (a) and (b) and for a fixed ${\gamma }_t=0.05$ (and varying $p$) in (c) and (d). We plot the average enthalpy drops $\mathrm{\Delta }H$ during particle rearrangements in (a) and (c) and the number of rearrangements $N_r$ in (b) and (d). In (b), the dashed line shows a linear fit through the first cycle data, $N_r=480{\gamma }_t$. In (c) and (d), the dashed lines show power-law fits to the first cycle data, with $\mathrm{\Delta }H=0.11p^{1.17}$ and $N_r=3.5p^{-0.33}$, respectively. The errors in the data points are smaller than the symbol sizes.

Figure 4

The fraction of $N=256$ particle systems $P\left(T\right)$ that have settled into a periodic cycle with period $T$, at training amplitude ${\gamma }_t=0.05$, and pressures $p\approx 0.04$ (blue squares), $p\approx 0.004$ (green triangles), and $p\approx 0.0004$ (red circles). Lines are a guide to the eye. We also include results at $p\approx 0.004$ at a higher training amplitude of ${\gamma }_t=0.07$ (open purple pentagons). The fraction of systems with multicycle periods increases with decreasing $p$.