Yielding and memory in a driven mean-field model of glasses

TL;DR

This work shows that a Hamiltonian mean-field model of glasses, previously matched to the non-phononic vibrational density of states, naturally reproduces oscillatory yielding, absorbing-to-diffusive transitions, and mechanical memory observed in driven glasses without adding free parameters. By driving the model in the athermal quasi-static limit, the authors map yielding to an elasto-plastic transition and demonstrate dynamic slowing-down near the yield, analogous to particle-based simulations. They further reveal a non-equilibrium ensemble equivalence: the post-yielding dynamics of a single driven realization increasingly mirrors quenched-disorder averages of the non-driven system, as seen in the buildup of the $D_G(\omega) \sim \omega^4$ tail. Varying quenched disorder $J$ and performing thermal annealing elucidate how landscape properties control yielding and memory, with brittle-like behavior at small $J$ and enhanced annealability at larger $J$. Overall, the model provides a unifying, predictive framework linking static energy landscapes to driven glassy dynamics and memory formation.

Abstract

Glassy systems reveal a wide variety of generic behaviors, which lack a unified theoretical description. Here, we study a mean-field model, recently shown to reproduce the universal non-phononic vibrational spectra of glasses, under oscillatory driving forces. The driven mean-field model, featuring a disordered Hamiltonian structure, naturally predicts the salient dynamical phenomena in cyclically deformed glasses. Specifically, it features an oscillatory yielding transition, characterized by an absorbing-to-diffusive transition in the system's microscopic trajectories and large-scale hysteresis. The model also reveals dynamic slowing-down from both sides of the transition, as well as mechanical and thermal annealing effects that mirror their glass counterparts. Finally, we demonstrate a non-equilibrium ensemble equivalence between the driven post-yielding dynamics at fixed quenched disorder and quenched disorder averages of the non-driven system, along with memory formation.

Figures (10)

• Figure 1: (a) The energy per oscillator $U(t)/N$ vs. the periodic driving $f(t)$ in steady-state at various amplitudes $f_0$ (see legend), see text for discussion. Here, we used $N\!=\!8192$ and $J\!=\!0.5$, and the results are averaged over more than $M\!=\!20$ realizations of quenched disorder and more than $10$ steady-state cycles. The curves are displaced vertically for visual clarity, but maintain their order (see Appendix). (b) The same as (a), but for a computer glass driven by an oscillatory shear $\gamma(t)$ (see Appendix for details). (c) Representative stroboscopic trajectories of the dynamics of the mean-field model at various amplitudes $f_0$ (see legend). Shown is the energy at the end of each cycle, $U_{f(t)=0}$, vs. the number of cycles $n$, see text for discussion. (d) The corresponding results for the computer glass. Note that at the intermediate $f_0$ value, the model's trajectory reveals multiperiodicity, cf. panel (c), not observed in the corresponding trajectory of the computer glass, cf. panel (d).
• Figure 2: (a) The steady-state stroboscopic energy $U_{\rm ss}(f_0)$, evaluated at the end of each cycle, for the calculations reported on in Fig. \ref{['fig:fig1']}a. (b) The fraction of $\mathcal{R}$ and $\mathcal{I}$ steady-states vs. $f_0$ ($\mathcal{R}$ denotes reversible/periodic steady-states, while $\mathcal{I}$ denotes irreversible/non-periodic ones). (c) The steady-state cycle-to-cycle mean square displacement $\text{MSD}_{\rm ss}$, defined in Eq. \ref{['eq_sr:MSD_definition']}, vs. $f_0$. (d) The characteristic accumulated driving force $f_{\rm acc}^*$ needed to reach a steady-state as a function of $f_0$, obtained by fitting a stretched exponential function $U_{\rm ss}\!+\!\Delta U\exp\!{[-(f_{\rm acc}/f_{\rm acc}^*)^{1/2}]}$ (where $f_{\rm acc}\!\equiv\!4f_0n$) to storoboscopic trajectories (see Appendix for details). The dashed lines are guides to the eye.
• Figure 3: (a) The non-phononic VDoS ${\cal D}_{\rm G}(\omega)\!\sim\!\omega^4$ (see line and power-law triangle) obtained by averaging over $M\!=\!1000$ realizations (with $N\!=\!8192$) of the quenched disorder with $J\!=\!0.5$ (large gray squares) and by periodically driving a single realization with $f_0\!=\!0.28\!>\!f_{\rm y}(J\!=\!0.5)$ over sufficiently large number of cycles $n$ (see legend). It is observed that ${\cal D}_{\rm G}(\omega;n)$ entirely lacks the $\omega^4$ tail initially ($n\!=\!0$) and that the latter progressively builds up with increasing $n$, see text for additional discussion. The arrows mark the edge of the vibrational spectrum per $n$. (b) The number density of the lowest 100 eigenfrequencies, denoted as $\mathcal{D}_{\rm G}(\omega_{\rm min};n)$, with increasing $n$ (see color bar on the right, using the same color code of panel (a)). The progressive build up of the $\omega^4$ tail is highlighted by the power-law triangle.
• Figure 4: (a) $\Delta{U}_{\rm ss}(f_0)$, similarly to Fig. \ref{['fig:fig2']}a but for various levels of quenched disorder quantified by $J$ (see legend in panel (b)). Here, $\Delta{U}_{\rm ss}(f_0)$ is measured relative to the minimum of $U_{\rm ss}(f_0)$ per $J$, with $N\!=\!4096$ and $M\!=\!16$. (inset) The energy $U_0$ of the non-driven system vs. $J$. (b) The same as panel (a), but vs. $f_0/J$, see text for discussion. (c) $\Delta{U}_{\rm ss}(f_0)$ for thermally annealed states quantified by the "parent temperature" $T_{\rm p}$, see text for discussion and Appendix for technical details.
• Figure A1: (a) One-periodic, (b) multiperiodic, and (c) non-periodic steady-states observed in $J\!=\!0.5$ (and $N\!=\!8192$) systems driven at $f_0\!=\!0.18$, presented in terms of $\text{MSD}(n)$. (d) The steady-state mean-squared-displacement ${\rm MSD}_{\rm ss}$ as a function of the driving amplitude $f_0$, plotted here for the unscreened (green diamonds) and multiperiodicity screened (brown-red squares) values. See text for definitions and discussion.