Yielding in amorphous solids reveals an age-dependent intrinsic lengthscale

TL;DR

This work tackles the challenge of predicting local yielding in amorphous solids by introducing the soft matrix method, which isolates a local sub-region to yield within a minimally constrained elastic background. It reveals an intrinsic length scale $\zeta$ that governs local failure and demonstrates that $\zeta$ grows with the material's age, linking aging to the spatial extent of independent plastic events. The study also shows that local yield-stress statistics follow a Weibull form with a pseudogap exponent $\theta$ that increases with age, indicating enhanced marginal stability in aged samples. Collectively, these insights provide a robust framework for age-aware mesoscale elastoplastic modeling and offer connections to other static/dynamic length scales in disordered solids.

Abstract

Understanding how amorphous solids yield under shear is central to predicting material failure, yet prescribing reliable local yielding criteria remains a fundamental challenge. Here, through a mesoscale analysis of localized yielding, we reveal an intrinsic length scale (ζ) that governs local failure, and demonstrate that ζgrows with the age of the system. The age dependence shows up not only in the features of the distribution of local yield stress but also in the pseudogap exponent θ, which provides a measure of marginal stability of the amorphous solids. These insights are made possible by a new method, termed the soft matrix approach, that allows local regions of an amorphous solid to yield within a minimally constrained, elastically coupled environment. By overcoming key limitations of earlier techniques, our approach provides a robust platform for probing failure mechanisms, particularly in soft disordered materials and paves the way for improved elastoplastic modeling of disordered solids.

Figures (15)

• Figure 1: Soft Matrix Methodology. (a) Schematic representing the soft matrix method. The sub-system (centre blue box) is subjected to unconstrained relaxation, whereas the background (surrounding pink region) relaxes with a constraint imposed by the spring tether (see text for details). (b.) Snapshot from 2D BMLJ simulation shows a plastic event confined to the subsystem, even as the background is allowed to relax. Here, x is the shearing or flow direction, and y is the gradient direction. The coloring scheme is based on particle displacement, where white indicates no displacement and black corresponds to the largest (of magnitude 0.0001). (bottom) Variation of stress ($\sigma_{xy} \equiv \sigma$) as a function of strain ($\gamma$) for (c) fixed subsystem size ($L_s=10 a$) and varying $k$ value and (d) fixed $k$ value ($k=10 \epsilon/a^2$) and varying subsystem size.
• Figure 2: The k dependence. (a) Displacement maps obtained from 2D BMLJ simulations for varying k values (0.1, 1, 10 and 100). The coloring scheme - heatmap is based on the particle displacements (white indicates no displacement, and black corresponds to the displacement of magnitude 0.0001). The region where the soft matrix is imposed is made translucent to emphasize the sub-system (with size $L_s = 20 a$). The top two panels (k = 0.1 and 1 $\epsilon/a^2$) are displayed with higher translucency than the lower ones (k = 10 and 100 $\epsilon/a^2$). The reduced translucency in the lower panels is intentional, as it highlights the displacements near the sub-system boundaries that are otherwise less visible. With the increasing k value, the displacement outside the probe region is suppressed in a gradual manner; notably, near the boundaries, the displacements are softened rather than sharply cut off. (b) Relative yield strain $\Delta \gamma_y$ as a function of sub-system size $L_s$ for varying k values, computed for a poorly aged sample. Inset of (b) shows the same plot in lin-log scale to highlight the exponential behavior. The dashed line corresponds to the exponential fit $\propto exp(-L_s/\zeta)$.
• Figure 3: Characteristic Length Scale. Relative yield strain $\Delta \gamma_y$ as a function of sub-system size $L_s$ for four different systems, which are (a) 2D BMLJ (b) 3D soft-Rep (c) 3D SiO2 and (d) 3D metallic glass. The open symbols are data obtained from the soft matrix method. The filled symbols are from the frozen matrix method. Different symbols correspond to different system sizes. The thick dashed line is the exponential fit, and the thin dashed line is just a guide to the eye.
• Figure 4: Age Dependence. (a) Relative yield strain as a function of sub-system size for different ages of the initial samples (2D BMLJ). (inset) The same plot in log-linear scale for a selected few ages. Dashed lines represent fits to exponential functions using which characteristic lengths $\zeta$ are extracted. (b) Characteristic length $\zeta$ as a function of energy $U_{init}$ (energy of samples at zero strain) showing the change $\zeta$ with an increase in the samples' age. Note that $U_{init}$ decreases with an increase in age. The solid line is just a guide to the eye and the dashed line is drawn for $\zeta = 5a$.
• Figure 5: Local mechanical properties: comparison of estimates via soft matrix and frozen matrix. (a.) Average yield strain $\gamma_y$, obtained from soft Matrix (left panel) and frozen Matrix (right panel) methods, as a function of sub-system size $L_s$ for two different ages ($U_{init}=-3.6 \epsilon$ corresponding to poorly-aged states and $U_{init}=-3.78 \epsilon$ corresponding to the well-aged samples). Fit lines (dashed dark green line - exponential fit and solid lines - power law fit) show a cross-over from an exponential regime at small $L_s$ to a power law regime at large $L_s$. The cross-over length (indicated in (a.) by dotted vertical lines) varies from $\sim 12 a$ in the the poorly-aged samples to $\sim 18 a$ in the well-aged samples, as obtained via the soft matrix method, unlike in frozen matrix (see main text) (b.) (inset) Distribution of local yield stress threshold $P(X)$ where $X = \sigma_y - \sigma_0$ and (main panel) the corresponding cumulative distribution $F(X)$ for the two different ages discussed in (a), with left and right panels corresponding to measurements from soft Matrix and frozen Matrix respectively. The fit lines are from the Weibull distribution (see main text). (c.) Comparison of the distribution of local storage modulus ($\mu$), obtained via the two methods, as marked, measured using the 3D soft-rep model (poorly-aged); the lines correspond to Gaussian fits. The bulk storage modulus is marked with an orange circle.