Figure 1

Particle tracks (solid lines) in a small window of a $N=128$ system of bidisperse disks undergoing (a) point-reversible, (b) loop-reversible, and (c) irreversible behavior during cyclic simple shear. Panels (d), (e), and (f) show the disks’ tracks in panels (a), (b), and (c), respectively, after subtracting off the affine motion. The systems in (a) and (d) correspond to $\varphi =0.64$ and ${\gamma }_{\max }=0.8$, (b) and (e) to $\varphi =0.8$ and ${\gamma }_{\max }=0.8$, (c) and (f) to $\varphi =0.82$ and ${\gamma }_{\max }=0.8$. Pluses, circles, and crosses mark the beginning, middle, and end of the particle tracks, and particle outlines correspond to the beginning of the cycle.Reuse & Permissions

Figure 2

(a) “Dynamical phase diagram” for $N=128$ bidisperse disks in 2D after $n={10}^4$ cycles showing point-reversible (circles), period-one (pluses), or multiperiod (crosses) loop-reversible, and transient (triangles) or steady-state (squares) irreversible dynamics versus ${\gamma }_{\max }$ and $\varphi$ for 16 independent initial conditions. The solid and dashed lines indicate ${\varphi }_L\left({\gamma }_{\max }\right)$ and ${\varphi }_{\mathrm{I}}\left({\gamma }_{\max }\right)$, the boundaries between point- and loop-reversible dynamics and between loop-reversible and irreversible dynamics, respectively. The vertical dotted and dot-dashed lines define ${\varphi }_R=\pi /4\simeq 0.785$ and ${\varphi }_J\approx 0.84$. (b) Arc-length $L$ vs. intracycle mean-square displacement $\Delta r_1$ after $n=10$ (blue), $3\times {10}^2$ (red), and ${10}^4$ (black) cycles. The regions labeled $\mathcal{L}$, $\mathcal{T}$, and $\mathcal{I}$ define loop-reversible, transient irreversible, and steady-state irreversible dynamics, respectively. The point-reversible ($\mathcal{P}$) region extends from $L=\Delta r_1={10}^{-6}$ to ${10}^{-16}$ but is not shown. The dashed boundaries indicate $\Delta r_1={\tau }_r$, $\Delta r_1=0.3$, $L={\tau }_L$, $L=30\Delta r_1$, and $L=\Delta r_1$ discussed in the main text. The inset shows the fraction $F_t$ of systems in (a) categorized as point-reversible (solid), loop-reversible (dashed), transient irreversible (dotted), and irreversible (dot-dashed) dynamics versus $n$. Panels (c) and (d) show the same data as (a) and (b) but for 3D systems.Reuse & Permissions

Figure 3

(a) Single-cycle mean-square displacement $\Delta r_1$ versus $n$ for a point-reversible system at $\varphi =0.64$ and ${\gamma }_{\max }=0.5$ (circles) and loop-reversible system at $\varphi =0.8$ and ${\gamma }_{\max }=0.5$ (pluses) with best fits to $\Delta r_1^f$ [Eq. (7)] indicated by solid and dashed lines. (b) Comparison of $\Delta r_1\left(n\right)$ (averaged over 16 initial conditions for each ${\gamma }_{\max }$ and $\varphi$) to $\Delta r_1^f\left(n\right)$ (black dots) for point-reversible systems in Fig. 2 with $\Delta <0.18$. $\Delta r_1^f\left(n\right)=\Delta r_1\left(n\right)$ is indicated by the dashed line. The top left inset shows ${\log }_{10}\Delta r_1\left(n\right)$ versus $\alpha {\log }_{10}n/n_c$ (black dots). The dashed line indicates $\Delta r_1\left(n\right)={(n/n_c)}^{-\alpha }$. The bottom right inset shows ${\log }_{10}\Delta r_1\left(n\right)$ versus $\beta (n-n_c)$ (black dots). $\Delta r_1\left(n\right)=e^{-\beta (n-n_c)}$ is indicated by the dashed line. (c) ${\log }_{10}\Delta r_1\left(n\right)$ versus $\alpha {\log }_{10}n$ (black dots) for systems in Fig. 2 that evolve to loop-reversible states with $\Delta <0.04$. $\Delta r_1\left(n\right)=n^{-\alpha }$ is indicated by the dashed line. The inset shows $\Delta r_1$ versus $n$ for three independent initial conditions at $\varphi =0.76$ and ${\gamma }_{\max }=0.8$. Exponential fits to the large-$n$ regime are shown as solid, dashed, and dotted lines with slopes $\beta =0.029$, $0.026$, and $0.016$, respectively.Reuse & Permissions

Figure 4

(a) Contour plot of the power-law exponent $\alpha$ [Eq. (7)] versus $\varphi$ and ${\gamma }_{\max }$ for systems that evolve to either point- (circles) or loop-reversible (crosses) states. (b) Contour plot of $n_c$ [Eq. (7)] that controls the crossover from power-law to exponential decay versus $\varphi$ and ${\gamma }_{\max }$ for point-reversible systems in (a).Reuse & Permissions

Figure 5

(a) Global ${\psi }_6^g$ and (b) local ${\psi }_6^l$ bond-orientational order parameters and layer order parameters (c) $l_{\perp }$ and (d) $l_{\left|\right|}$ for $N=128$ systems after $n={10}^4$ cycles with no applied shear (solid black) and maximum simple shear strain ${\gamma }_{\max }=0.4$ (red dashed), $0.8$ (green dash-dash-dotted), $1.6$ (blue dash-dotted), and $2.4$ (purple dotted) as a function of packing fraction $\varphi$.Reuse & Permissions

Figure 6

Dynamical phase diagram for (a) $N=64$ and (b) 256 bidisperse disks subjected to cyclic simple shear after $n={10}^3$ cycles, showing point-reversible (circles), period-one (pluses), or multiperiod (crosses) loop-reversible, and transient (triangles) or steady-state (squares) irreversible dynamics versus ${\gamma }_{\max }$ and $\varphi$ (for 16 independent initial conditions). The solid and dashed lines indicate ${\varphi }_L\left({\gamma }_{\max }\right)$ and ${\varphi }_{\mathrm{I}}\left({\gamma }_{\max }\right)$, the boundaries between point- and loop-reversible dynamics and between loop-reversible and irreversible dynamics, respectively. The vertical dotted and dot-dashed lines define ${\varphi }_R=0.785$ and ${\varphi }_J\approx 0.84$, respectively.Reuse & Permissions

Figure 7

The arc length $L$ versus the single-cycle mean-square displacement $\Delta r_1$ after $n={10}^3$ cycles for 2D systems with $N=64$ (red points) and 256 (black points). The regions labeled $\mathcal{L}$, $\mathcal{T}$, and $\mathcal{I}$ define loop-reversible, transient-irreversible, and steady-state irreversible dynamics, respectively, using the same boundary lines as those in Fig. 2b. In the inset, we show the fraction of systems that are point-reversible (solid), loop-reversible (dashed), transient (dotted), and irreversible (dot-dashed) as a function of cycle number $n$ for $N=64$ (thin) and 256 (thick). (b) The maximum shear strain amplitude ${\gamma }_{\max }$ versus the packing fraction ${\varphi }_L\left({\gamma }_{\max }\right)$, above which the system transitions from point- to loop-reversible dynamics after $n={10}^3$ cycles for $N=64$ (dashed) and 256 (solid). The best fit to Eq. (A1) is shown as a thin dotted line.Reuse & Permissions

Figure 8

(a) Deviation ${\Delta }_r$ between the particle coordinates of the original and perturbed trajectories after $n={10}^3$ cycles following the perturbation as a function of the perturbation amplitude $\delta$ for a point-reversible system at $\varphi =0.62$ and ${\gamma }_{\max }=2.0$. Data is shown for 40 independent random perturbations at each $\delta$. (b) ${\Delta }_r$ versus $\delta$ for stable loop-reversible systems at $\varphi =0.8$ and ${\gamma }_{\max }=0.2$ (circles), $0.83$ and $0.2$ (squares), $0.74$ and $0.8$ (diamonds), $0.8$ and $0.8$ (pluses), and $0.74$ and $1.4$ (crosses). (c) ${\Delta }_r$ versus $\delta$ for a loop-reversible system with 1 dynamical floater at $\varphi =0.83$ and ${\gamma }_{\max }=0.1$ (circles), and with the the dynamical floater removed from the calculation of ${\Delta }_r$ (crosses).Reuse & Permissions

Figure 9

Arc-length $\langle L\rangle$ averaged over 16 independent trajectories versus cycle number $n$ for systems of size $N=128$ at maximum shear strain amplitude ${\gamma }_{\max }=0.3$, packing fraction $\varphi =0.32$ (black), $0.56$ (red), $0.72$ (green), $0.76$ (blue), $0.78$ (yellow), $0.81$ (purple), and $0.83$ (cyan). (a) Effect of shear strain increment $\Delta \gamma ={10}^{-2}$ (solid), ${10}^{-3}$ (dotted), and ${10}^{-4}$ (dashed) for energy minimization tolerance $V_{\mathrm{tol}}={10}^{-16}$. (b) Effect of energy minimization tolerance $V_{\mathrm{tol}}={10}^{-16}$ (solid), ${10}^{-8}$ (dotted), and ${10}^{-6}$ (dashed) for shear strain increment $\Delta \gamma ={10}^{-3}$.Reuse & Permissions