Mechanical excitation and marginal triggering during avalanches in sheared amorphous solids

We study plastic strain during individual avalanches in overdamped particle-scale molecular dynamics (MD) and mesoscale elastoplastic models (EPM) for amorphous solids sheared in the athermal quasistatic limit. We show that the spatial correlations in plastic activity exhibit a short length scale that grows as $t^{3/4}$ in MD and ballistically in EPM, which is generated by mechanical excitation of nearby sites not necessarily close to their stability thresholds, and a longer lengthscale that grows diffusively for both models and is associated with remote marginally stable sites. These similarities in spatial correlations explain why simple EPMs accurately capture the size distribution of avalanches observed in MD, though the temporal profiles and dynamical critical exponents are quite different.

Figure 1

Persistent homology clustering. (a) Space-time map of the plastic activity monitored by the $D_{\min }^2$ field for a typical avalanche in a glass with $N=160\times 160$. (b) Birth-death phase diagram of individual local maxima of our $D_{\min }^2$ grid. The persistent $p$ of a given event is given as the distance to the line y(death) = x(birth). (c) Probability distribution of the persistent averaged over an ensemble of avalanches. The red area indicates our threshold $p>0.1$ for persistent events. (d) Superposition of the actual dissipation $-dU/dt$ and birth time of individual events with $p>0.1$.

Figure 2

Spatiotemporal plastic evolution. Typical spatiotemporal map of an avalanche in molecular dynamics (MD) with gradient descent dynamics (left) and in the elastoplastic model (EPM) with synchronous dynamics (right). Each plastic event is colored according to its normalized birth time $t/T$, with $T$ the avalanche duration. Insets show the particle resolved plastic field (left) and EPM local stress redistribution (right). System sizes are $L=320$ and $L=256$ for MD and EPM, respectively. For visibility only a subset of the system is shown for the EPM.

Figure 3

Plastic activity profile. (a) Typical dissipation rate $-dU/dt$ in MD plotted as a function of the reduced time $t/T$ for duration $1500<T<3000$ and $L=320$. (b) Scatter plot of avalanche size vs duration. The blue and black empty symbols are running means $\langle S\rangle \sim T$ and $\langle T\rangle \sim S$, respectively. Solid and dashed lines indicate the scaling $\langle S\rangle \sim T^{\delta }$, with exponent $\delta =0.65$ and $\delta =4/3$, respectively. (c) Average normalized activity for $400<T<600$ [orange region in (b)]. (d) Average duration $\langle T\rangle$ at a fixed avalanche size $S$ for different system sizes. Insets show the same data collapsed using $\langle T\rangle L^{-\nu }$, with $\nu \simeq 0.8$. Panels (e), (f), (g), and (h) show the same results as in (a), (b), (c), and (d) but for the EPM where $-dU/dt$ is measured as the energy dissipation per sweep and $L=256$.

Figure 4

Spatiotemporal plastic propagation. (a) Particle-based (MD) two-points two-times normalized correlation function $C(x,y)/C_{\max }$ in ${log}_{10}$ for $\mathrm{\Delta }t=4$ and 40. White regions correspond to a negative correlation. The system size is $L=320$. (b) Correlation function along the main stress redistribution for different delay times $\mathrm{\Delta }t=4$, 10, 20, 40, 60, and 100. Inset shows the correlation in semilog scale. Solid lines are fits of the form $C\left(x\right)/C_{\max }\sim e^{-x/{\xi }_{\mathrm{excite}}}$, with ${\xi }_{\mathrm{excite}}$ a short-length-scale decay. (c) Change of $C\left(x\right)/C_{\max }$ for different system size at a fixed $\mathrm{\Delta }t=6$. Inset shows the negative crossing in linear scale. (d) Transverse normalized correlation function $C\left(y\right)/C_{\max }$ at a fixed $x_{\mathrm{MD}}=50$ for different $\mathrm{\Delta }t$. Solid lines are fits with $C\left(y\right)\sim e^{-{(y/w)}^2/2}$, where $w$ is the width of the plastic localization. (e), (f), (g), and (h) The same results as in (a), (b), (c), and (d) but for our mesoscale model (EPM) with $\mathrm{\Delta }t=1$, 2, 4, 6, 8, and 12 and $L=256$. $C\left(x\right)/C_{\max }$ in (g) is for $\mathrm{\Delta }t=3$. $C\left(y\right)/C_{\max }$ in (h) is evaluated at $x_{\mathrm{EPM}}=40$. The meanings of ${\xi }_{\mathrm{excite}}$, ${\xi }_{\mathrm{marginal}}$, and $w$ are illustrated in (a) and (e).

Figure 5

Time translational invariance. (a) Comparison of the MD incremental correlation $C=\langle \overline{\mathrm{\Delta }f({\vec{r}}_0,t_0)\mathrm{\Delta }f({\vec{r}}_0+\vec{r},t_0+\mathrm{\Delta }t)}\rangle$ for $t_0=0$ (green) and $t_0\ge 0$ (black). (b) Same data as in (a) for our EPM but with $t_0=0$ (green) and $t_0/T=1/5$ (black). System sizes are $L=320$ and $L=256$ for MD and EPM, respectively.

Figure 6

Plastic length scales. (a) MD data for ${\xi }_{\mathrm{excite}}$ (orange) and ${\xi }_{\mathrm{marginal}}$ (purple) plotted against $\mathrm{\Delta }t$ with $L=320$. The purple line corresponds to the diffusive response associated with the stress field generated by a single event at $\mathrm{\Delta }t=0$. The orange line indicates a superdiffusive regime with $\sim t^{3/4}$. Gray data are the plastic width $w$ extracted from fitting $C\left(y\right)$. The gray line indicates a subdiffusive regime with $\sim t^{1/10}$. (b) ${\xi }_{\mathrm{excite}}$ and ${\xi }_{\mathrm{marginal}}$ for $\mathrm{\Delta }t=t_{\mathrm{STZ}}=6$ plotted as a function of the system size $L$. ${\xi }_{\mathrm{marginal}}\left(L\right)$ scales as $\sim L^{1/3}$. (c) and (d) are the same results as in (a), (b), but for our mesoscale model (EPM) with $L=256$. The solid orange line in (g) indicates $v_{0}t$, with $v_0=3$. Results in (d) are for $\mathrm{\Delta }t=1$, where ${\xi }_{\mathrm{marginal}}\left(L\right)$ scales as $\sim L^{2/3}$. ${\xi }_{\mathrm{excite}}\left(L\right)$ is fairly constant for both MD and EPM. The mean and error bar for ${\xi }_{\mathrm{marginal}}$ are estimated as the arithmetic mean and the spread between the first negative crossing and the median over all negative crossings (if any), respectively.