Scaling of slip avalanches in sheared amorphous materials based on large-scale atomistic simulations

Atomistic simulations of binary amorphous systems with over 4 million atoms are performed. Systems of two interatomic potentials of the Lennard-Jones type, LJ12-6 and LJ9-6, are simulated. The athermal quasistatic shearing protocol is adopted, where the shear strain is applied in a stepwise fashion with each step followed by energy minimization. For each avalanche event, the shear stress drop $\left(\mathrm{\Delta }\sigma \right)$, the hydrostatic pressure drop $\left(\mathrm{\Delta }{\sigma }_h\right)$, and the potential energy drop $\left(\mathrm{\Delta }E\right)$ are computed. It is found that, with the avalanche size increasing, the three become proportional to each other asymptotically. The probability distributions of avalanche sizes are obtained and values of scaling exponents fitted. In particular, the distributions follow a power law, $P\left(\mathrm{\Delta }U\right)\sim \mathrm{\Delta }{U}^{-\tau }$, where $\mathrm{\Delta }U$ is a measure of avalanche sizes defined based on shear stress drops. The exponent $\tau$ is $1.25\pm 0.1$ for the LJ12-6 systems, and $1.15\pm 0.1$ for the LJ9-6 systems. The value of $\tau$ for the LJ12-6 systems is consistent with that from an earlier atomistic simulation study by Robbins et al. [Phys. Rev. Lett. 109, 105703 (2012)], but the fitted values of other scaling exponents differ, which may be because the shearing protocol used here differs from that in their study.

Figure 1

A sample with 8000 atoms using potential I (large: atom A; small: atom B).

Figure 2

Stress vs. strain and potential energy vs. strain of [(a) and (b)] potential I and [(c) and (d)] potential II. The number of atoms is 256,000.

Figure 3

Relationship between hydrostatic pressure drop ($\mathrm{\Delta }{\sigma }_h$), shear stress drop ($\mathrm{\Delta }\sigma$), and energy drop ($\mathrm{\Delta }E$) for potential I. The number of atoms is 64,000. The strain increments are labeled by the subcaptions. Each blue cross represents a single event. The red curves are obtained by first logarithmically binning the events according to their stress drops and then averaging the hydrostatic pressure drops (a) or the energy drops [(b) and (c)] within each bin. The black dashed lines have a slope of 1. $\mathrm{\Delta }{\sigma }_h$ is defined by Eq. (3). In (b), $\mathrm{\Delta }\sigma$ and $\mathrm{\Delta }E$ are calculated using Eqs. (4) and (5), respectively. In (c), $\mathrm{\Delta }\sigma$ and $\mathrm{\Delta }E$ are calculated using the corrected formulas Eqs. (6) and (7), respectively.

Figure 4

(a) Average hydrostatic pressure drop ($〈\mathrm{\Delta }{\sigma }_h〉$), (b) average energy drop ($〈\mathrm{\Delta }E〉$), and (c) scaled average energy drop ($〈\mathrm{\Delta }E〉/L^2$) for potential I and potential II with different system sizes. $L$ is the side length of the system. The strain increment is $\mathrm{\Delta }\epsilon =5\times {10}^{-6}$. In each subfigure, the curves are obtained by first logarithmically binning the events according to their stress drops and then averaging the hydrostatic pressure drops (a) or the energy drops [(b) and (c)] within each bin. $\mathrm{\Delta }{\sigma }_h$ is defined by Eq. (3). In (b) and (c), $\mathrm{\Delta }\sigma$ and $\mathrm{\Delta }E$ are calculated using the corrected formulas Eqs. (6) and (7), respectively. The black dashed lines have a slope of 1.

Figure 5

Original $P\left(\mathrm{\Delta }U\right)$ [probability distribution of $\mathrm{\Delta }U$, with $\mathrm{\Delta }U$ defined in Eq. (9)] and $P\left(\mathrm{\Delta }E\right)$ [probability distribution of energy drop $\mathrm{\Delta }E$, with $\mathrm{\Delta }E$ defined in Eq. (7)] at different strain increments, which are indicated by the legends. The number of atoms in each system is 64,000.

Figure 6

Original $P\left(\mathrm{\Delta }U\right)$ [probability distribution of $\mathrm{\Delta }U$, with $\mathrm{\Delta }U$ defined in Eq. (9)] and $P\left(\mathrm{\Delta }E\right)$ [probability distribution of energy drop $\mathrm{\Delta }E$, with $\mathrm{\Delta }E$ defined in Eq. (7)]. The system sizes, given in the numbers of atoms, are indicated by the legends. The strain increment is $5\times {10}^{-6}$. The slopes of the straight reference lines are $-1.25$ in (a) and (b), $-1.15$ in (c), and $-1.0$ in (d). Plotting the reference lines in (b) and (d) is just to show the rough trend.

Figure 7

Finite-size scaled $P\left(\mathrm{\Delta }U\right)$ [probability distribution of $\mathrm{\Delta }U,\mathrm{\Delta }U$ defined in Eq. (9)] and $P\left(\mathrm{\Delta }E\right)$ [probability distribution of energy drop $\mathrm{\Delta }E,\mathrm{\Delta }E$ defined in Eq. (7)]. $L$ is the side length of the system. The definition of the finite-size scaling is given in Eq. (11). The system sizes, given in the numbers of atoms, are indicated by the legends. The strain increment is $5.0\times {10}^{-6}$. The slopes of the straight reference lines are $-1.25$ in (a) and (b), $-1.15$ in (c), and $-1.0$ in (d). Plotting the reference lines in (b) and (d) is just to show the rough trend, since the shapes of the curves cannot be fit with a straight line. In (a), $\alpha =1.25$ and $\beta =-0.5$. In (b), $\alpha =1$ and $\beta =0$. In (c), $\alpha =1.2$ and $\beta =-0.3$. In (d), $\alpha =1.2$ and $\beta =-0.05$. A better collapse in (b) and (d) cannot be achieved by choosing different values for $\alpha$ and $\beta$.

Figure 8

$P\left(\mathrm{\Delta }U\right)$ [probability distribution of $\mathrm{\Delta }U,\mathrm{\Delta }U$ defined in Eq. (9)] and $P\left(\mathrm{\Delta }E\right)$ [probability distribution of energy drop $\mathrm{\Delta }E,\mathrm{\Delta }E$ defined in Eq. (7)] scaled by $L^{\gamma }$. $L$ is the side length of the system. The system sizes, given in the numbers of atoms, are indicated by the legends. The strain increment is $5.0\times {10}^{-6}$. $\gamma$ is 1.05 for (a), 1.00 for (b), and 1.15 for (c) and (d). The slopes of the straight reference lines are $-1.25$ in (a) and (b), $-1.15$ in (c), and $-1.0$ in (d). Plotting the reference lines in (b) and (d) is just to show the rough trend.

Figure 9

$P\left(w\right)$ (probability distribution of the “waiting time” $w$) for different system sizes. The subfigures are (a) $P\left(w\right)$ for potential I; (b) $P\left(w\right)$ for potential II; (c) scaled $P\left(w\right)$ for potential I, where $\eta =0.74$ and $\xi =1.5$; and (d) scaled $P\left(w\right)$ for potential II, where $\eta =0.97$ and $\xi =1.8$. $L$ is the side length of the system. $\xi$ and $\eta$ are scaling exponents shown in Eq. (A1) in this document. The strain increment is $5\times {10}^{-5}$. The way of calculating $P\left(w\right)$ is such that $P\left(w\right)dw$ is the number of events that fall in the range $[w,w+dw]$ per unit strain interval.