Nonlinear plastic modes in disordered solids

We propose a theoretical framework within which a robust micromechanical definition of precursors to plastic instabilities, often termed soft spots, naturally emerges. They are shown to be collective displacements (modes) $\widehat{z}$ that correspond to local minima of a barrier function $b\left(\widehat{z}\right)$, which depends solely on inherent structure information. We demonstrate how some heuristic searches for local minima of $b\left(\widehat{z}\right)$ can a priori detect the locus and geometry of imminent plastic instabilities with remarkable accuracy, at strains as large as ${\gamma }_c-\gamma \sim {10}^{-2}$ away from the instability strain ${\gamma }_c$. Our findings suggest that the a priori detection of the entire field of soft spots can be effectively carried out by a systematic investigation of the landscape of $b\left(\widehat{z}\right)$.

Figure 1

Plastic instability in a sheared two-dimensional model glass. (a) Destabilizing eigenfunction ${\widehat{\mathrm{\Psi }}}_c$. (b) Elementary shear transformation: the postinstability displacements that followed the instability of (a). (c) Nonaffine displacements $\vec{v}$ calculated at $\delta \gamma \equiv {\gamma }_c-\gamma =0.007$. This delocalized field is used as the initial conditions ${\widehat{z}}_{\text{ini}}$ for the minimization of $b\left(\widehat{z}\right)$ (see the main text), the result of which is the plastic mode $\widehat{\pi }$ displayed in (f). (d) and (e) Intermediate states along the minimization of $b\left(\widehat{z}\right)$.

Figure 2

(a) Variations $\delta U_{\widehat{\pi }}\left(s\right)$ of the potential energy upon displacing the particles a distance $s$ along plastic modes $\widehat{\pi }$, obtained as described in the text. The curves correspond to $\delta \gamma ={10}^{-5},{10}^{-4},3\times {10}^{-4},8\times {10}^{-4},2\times {10}^{-3},7\times {10}^{-3}$. (b) Stiffnesses ${\kappa }_{\widehat{z}}={\mathcal{M}}_{ij}:{\widehat{z}}_i{\widehat{z}}_j$ associated with the plastic modes $\widehat{\pi }$ used to calculate the variations plotted in (a) (circles) and with the destabilizing eigenfunction ${\widehat{\mathrm{\Psi }}}_c$ vs $\delta \gamma$ (squares). Inset: overlaps ${\widehat{\mathrm{\Psi }}}_c{{|}_{{\gamma }_c}·\widehat{\pi }|}_{\gamma }$ vs $\delta \gamma$.

Figure 3

(a) Scatter plot of the barrier function (2) evaluated for eigenfunctions ${\widehat{\mathrm{\Psi }}}_{\omega }$ of $\mathcal{M}$ vs their eigenvalues ${\omega }^2$. The eigenfunction ${\widehat{\mathrm{\Psi }}}_{\text{min}}$ represented by the circled data point is plotted in (d) and is used as the initial conditions ${\widehat{z}}_{\text{ini}}$ for the minimization of the barrier function $b\left(\widehat{z}\right)$; the resulting plastic mode $\widehat{\pi }$ is displayed in (e). (b) Variations $\delta U_{\widehat{z}}\left(s\right)$, calculated by displacing the particles according to $\delta \vec{x}=s{\widehat{\mathrm{\Psi }}}_{\text{min}}$ (solid curve) and by $\delta \vec{x}=s\widehat{\pi }$ (dashed curve). (c) Products $N|{\tau }_{\widehat{\mathrm{\Psi }}}|$, averaged over bins of the participation ratio; see the text for definitions.

Figure 4

(a) Spatial decay of plastic modes; see the text for definitions. All modes analyzed decay as $r^{1-d}$, as indicated by the solid lines, where $r$ is the distance to the core center. (b) Plastic mode found by choosing a random ${\widehat{z}}_{\text{ini}}$. (c) Plastic mode calculated in a disordered network of relaxed Hookean springs. (d) Plastic mode found in a Lennard-Jones glass under isotropic tension.