Atomic nonaffinity as a predictor of plasticity in amorphous solids

Structural heterogeneity of amorphous solids presents difficult challenges that stymie the prediction of plastic events, which are intimately connected to their mechanical behavior. Based on a perturbation analysis of the potential energy landscape, we derive the atomic nonaffinity as an indicator with intrinsic orientation, which quantifies the contribution of an individual atom to the total nonaffine modulus of the system. We find that the atomic nonaffinity can efficiently characterize the locations of the shear transformation zones, with a predicative capacity comparable to the best indicators. More importantly, the atomic nonaffinity, combining the sign of the third-order derivative of energy with respect to coordinates, reveals an intrinsic softest shear orientation. By analyzing the angle between this orientation and the shear loading direction, it is possible to predict the protocol-dependent response of one shear transformation zone. Employing this method, the distribution of orientations of shear transformation zones in model two-dimensional amorphous solids can be measured. The resulting plastic events can possibly be understood from a simple model of independent plastic events occurring at variously oriented shear transformation zones. These results shed light on the characterization and prediction of the mechanical response of amorphous solids.

Figure 1

Analysis of configuration that was sheared to be close to the triggering of a plastic event. (a) The spatial distribution of the normal mode with lowest eigenvalue. (b) The $D_{\min }^2$ field [26] after shear with strain of $\mathrm{\Delta }\gamma =6\times {10}^{-5}$ in the orientation of ${\theta }_{\mathrm{L}}=-\pi /4$. The inset shows the stress-strain curve. (c) The predicted triggering strain (line) and triggering strain from simulation (circles) as a function of shear angle ${\theta }_{\mathrm{L}}$. (d) The magnitude of nonaffine modulus contribution from the lowest mode at different ${\theta }_{\mathrm{L}}$. The blue line represents the range of ${\theta }_{\mathrm{L}}$, where the plastic event can be triggered, while the red line represents the range of ${\theta }_{\mathrm{L}}$, where the plastic event cannot be triggered. (e) The spatial distribution of $\widehat{G}\left({\theta }_{\mathrm{L}}\right)\mathrm{sign}\left(\mathrm{\Delta }{\gamma }_c\left({\theta }_{\mathrm{L}}\right)\right)$ for different orientations. (f) The atomic shear nonaffinity in different orientations for the atom that has the maximum magnitude of atomic shear nonaffinity in (e).

Figure 2

Predicting plastic events by analyzing initial configurations. (a) The correlation field of atomic shear nonaffinity $\widehat{G}({\theta }_{\mathrm{L}}=0)$ and the locations of the first ten plastic events (black circles) triggered in shear protocols with ${\theta }_{\mathrm{L}}=0$. (b) Correlation between the indicators including local yield stress (LYS), mean-square vibrational amplitude (MSVA), atomic shear nonaffinity (ASN), participation fraction (PF), and nonaffine rate (NR) with the locations of plastic events as a function of the index of the events. Averages are taken over windows of five events. The error bar at each window represents the standard deviation of the mean. (c) Orange circles represent the atoms with ${\mathrm{RC}}_{\widehat{G}}>0$ and $|{\theta }_s|<\frac{\pi }{4}$. Black circles mark the locations of the first ten plastic events with ${\theta }_{\mathrm{L}}=0$. (d) Green circles represent the atoms with ${\mathrm{RC}}_{\widehat{G}}>0$ and $|{\theta }_s|>\frac{\pi }{4}$. Triangles mark the locations of the first ten plastic events with ${\theta }_{\mathrm{L}}=\frac{\pi }{2}$.

Figure 3

Distribution of the softest shear orientations for plastic regions. (a) A schematic diagram introducing the notations used in this figure. (b) The distribution of the softest shear orientations for all the plastic events that are triggered before shear strain 0.12 in ten samples. The softest shear orientations are calculated based on the configuration just before each event. The red line follows the function $\frac{4}{\pi }{cos}^2\left(2\mathrm{\Delta }\theta \right)$. (c) Similar to (b), but the softest shear orientations for each event are calculated based on the configurations just after the previous event.