Resolving Structural Avalanches in Amorphous Carbon with Arclength Continuation

Abstract

Plastic deformation in amorphous solids is carried by localized shear transformations that self-organize into avalanches. In amorphous carbon modeled with a machine-learned interatomic potential, we find that the energetics and organization of these avalanches can be resolved by systematically following the underlying energy landscape. With a pseudo-arclength numerical continuation framework, we decompose avalanches into constituent shear transformations and determine their strain-dependent energetics. Our analysis shows that, prior to onset, avalanches have a latent structure that consists of well-separated local minima. We further demonstrate that arclength continuation yields an event driven framework for following avalanche dynamics, eliminating time-step effects on statistical avalanche properties such as distributions of stress drops.

Figures (4)

• Figure 1: Structure and plastic events in amorphous carbon. (a) A 4096-atom structure of amorphous carbon with a density of 2.7 $\text{gcm}^{-3}$. (b) The shear-stress shear-strain curve of the amorphous carbon structure. Note that $\gamma$ is the simple shear strain. (c) An enlargement of the stress-strain curve corresponding to the position of the red box in (b). The stress drop encircled corresponds to a three-bond structural avalanche plastic event. (d) The 3-bond structural avalanche highlighted in (c). The three bonds involved are colored blue, orange and green.
• Figure 2: The strain-dependent energy landscape of single-bond plastic events is explored using AC. (a) For a reversible event, the continuation curve follows the minimum associated with the open bond (lower branch, solid black) until the critical strain $\gamma_c$, where the barrier vanishes. AC then passes through the bifurcation to the central branch, corresponding to the index-1 saddle traced as strain decreases (dashed black), before reaching the upper branch associated with the formed bond (solid black). (b) Nudged elastic band (NEB) energy profiles sampled across this landscape, with color-coded strains indicated in (a), show excellent agreement between NEB and continuation saddle energies (triangles and squares denote minima and saddles). (c)–(d) The same analysis for an irreversible event shows no continuous path from the central to the upper branch, consistent with the strongly asymmetric barrier revealed by the NEB profiles.
• Figure 3: The strain-dependent energy landscape of a six-bond avalanche. (a) A set of continuation curves reveals the latent structure of the avalanche. The AQS trajectory (thick black line) triggers the full avalanche at the critical strain $\gamma_c$. In contrast, AC can traverse the bifurcation at $\gamma_c$ onto the index-1 saddle associated with the first bond-breaking event (dashed blue). Relaxation from this saddle at lower strain leads to an intermediate basin in which a single bond has broken (orange), and repeating this procedure (green, red, purple, and brown) resolves the avalanche into successive events. All intermediate basins disappear below $\gamma_c$, implying that even an infinitesimal-step AQS protocol would still trigger the full avalanche. (b) A nudged elastic band (NEB) energy landscape at $\gamma_{xy}-\gamma_c=-0.01$, assembled from sequential NEBs between adjacent intermediate basins, with continuation minima and saddles marked by triangles and squares. The NEB and continuation saddle energies agree closely. Insets in (a) and (b) indicate the ordering of events and the full avalanche.
• Figure 4: A comparison between AQS and continuation simulations is shown. (a) Overlaid stress–strain curves from AQS simulations with step sizes $\Delta\gamma=10^{-3}$ (blue) and $10^{-4}$ (orange), together with continuation results (green). The curves diverge upon entering the plastic flow regime ($\gamma \gtrsim 0.17$). (b–d) Log–log plots of the binned probability density of stress drops for $\gamma>0.17$ over 5 repeats. AQS results are shown for $\Delta\gamma=10^{-3}$ in (b) and $\Delta\gamma=10^{-4}$ in (c); in both cases, finite step size leads to an underestimation of the fitted power-law exponent (dashed black). In contrast, the continuation algorithm in (d) eliminates step-size dependence, yielding a steeper and more clearly defined power-law scaling.