Micromechanics of nonlinear plastic modes

Nonlinear plastic modes (NPMs) are collective displacements that are indicative of imminent plastic instabilities in elastic solids. In this work we formulate the atomistic theory that describes the reversible evolution of NPMs and their associated stiffnesses under external deformations. The deformation dynamics of NPMs is compared to those of the analogous observables derived from atomistic linear elastic theory, namely, destabilizing eigenmodes of the dynamical matrix and their associated eigenvalues. The key result we present and explain is that the dynamics of NPMs and of destabilizing eigenmodes under external deformations follow different scaling laws with respect to the proximity to imminent instabilities. In particular, destabilizing modes vary with a singular rate, whereas NPMs exhibit no such singularity. As a result, NPMs converge much earlier than destabilizing eigenmodes to their common final form at plastic instabilities. This dynamical difference between NPMs and linear destabilizing eigenmodes underlines the usefulness of NPMs for predicting the locus and geometry of plastic instabilities, compared to their linear-elastic counterparts.

Figure 1

Review of the micromechanics of a plastic instability. (a) An illustration of the basic setup considered in this work: an athermal glass under quasistatic simple shear deformation. (b) Cartoon of a typical stress $\sigma$ vs strain $\gamma$ signal in our setup; at some instability strain ${\gamma }_c$ a plastic instability occurs. The dashed frame shows that close to the instability the stress follows $\sigma -{\sigma }_c\sim \sqrt{{\gamma }_c-\gamma }$, as shown, e.g., in Ref. [7]. (c) Lowest eigenvalue ${\lambda }_p=\mathcal{M}:{\widehat{\mathrm{\Psi }}}_p{\widehat{\mathrm{\Psi }}}_p$ of the dynamical matrix $\mathcal{M}$, vs the distance in strain ${\gamma }_c-\gamma$ to the instability. Away from the instability the eigenmode ${\widehat{\mathrm{\Psi }}}_p$ associated with ${\lambda }_p$ is delocalized, as demonstrated in panel (f), and ${\lambda }_p$ is largely insensitive to the deformation. As the solid is further deformed ${\widehat{\mathrm{\Psi }}}_p$ destabilizes and localizes, as demonstrated in panel (e). ${\lambda }_p$ then vanishes as $\sqrt{{\gamma }_c-\gamma }$. (d) Energy variations $\delta U_{\widehat{\mathrm{\Psi }}}\left(s\right)$ upon displacing the particles about the mechanical equilibrium state according to $\delta \vec{x}=s{\widehat{\mathrm{\Psi }}}_p$, measured at distances ${\gamma }_c-\gamma ={10}^{-14},{10}^{-41/3},{10}^{-40/3}$, and ${10}^{-13}$ away from the instability strain. These curves demonstrate the well-understood saddle-node bifurcation which characterizes plastic instabilities, in which a saddle point and minimum on the potential energy landscape coalesce and mutually annihilate, as shown in Refs. [6, 7, 8]. The continuous lines are obtained by a cubic Taylor expansion of the energy variation, for which the expansion coefficients were calculated using inherent state information.

Figure 2

(a) The eigenvalues ${\lambda }_p$ associated with destabilizing modes ${\widehat{\mathrm{\Psi }}}_p$ vs distance in strain ${\gamma }_c-\gamma$ to a plastic instability strain ${\gamma }_c$, for various system sizes as shown in the legend. The instabilities were the first encountered upon shearing a randomly selected freshly quenched glass. (b) Rescaled ${\lambda }_p$ vs the rescaled strain interval reveals that Eq. (9) holds on intervals below the scale $\delta \gamma \sim 1/\left({\tau }_p{\nu }_p{L}^4\right)$. ${\tau }_p\equiv U^{\prime \prime \prime }\stackrel{\mathbf{\text{.}}}{:}{\widehat{\mathrm{\Psi }}}_p{\widehat{\mathrm{\Psi }}}_p{\widehat{\mathrm{\Psi }}}_p$ and ${\nu }_p\equiv \frac{{\partial }^{2}U}{\partial \gamma \partial \vec{x}}·{\widehat{\mathrm{\Psi }}}_p$ were calculated at ${\gamma }_c-\gamma \lesssim {10}^{-13}$.

Figure 3

(a) Spatial decay profiles $\mathcal{C}\left(r\right)$ (see text for definition) of a NPM $\widehat{\pi }$ and of a destabilizing eigenmode ${\widehat{\mathrm{\Psi }}}_p$ calculated in a system of $N=102400$ at a distance ${\gamma }_c-\gamma \sim {10}^{-14}$ away from a plastic instability. Also plotted is the decay profile of the response $\delta \vec{R}={\mathcal{M}}^{-1}\vec{d}$ to a local dipolar force $\vec{d}$. All these modes are found to decay as $r^{-1}$ (notice that $\mathcal{C}$ scales as the magnitude squared of a mode's components). (b) The nonlinear force responses of the same modes analyzed in panel (a) decay as $r^{-4}$. We verify that the double contraction of a spatially decaying mode with $U^{\prime \prime \prime }$, e.g., $U^{\prime \prime \prime }:{\widehat{\mathrm{\Psi }}}_p{\widehat{\mathrm{\Psi }}}_p$, picks up the square of the spatial gradient of that mode, in consistency with Eq. (20). Notice that the linear and nonlinear force responses of the NPM $\widehat{\pi }$ are parallel, therefore $|\mathcal{M}·\widehat{\pi }|\left(r\right)\sim r^{-4}$ as well, while $|\mathcal{M}·\widehat{\mathrm{\Psi }}|\left(r\right)\sim r^{-1}$.

Figure 4

(a) Stiffnesses ${\kappa }_{\widehat{\pi }}$ vs strain interval ${\gamma }_c-\gamma$. The pale symbols represent the eigenvalues ${\lambda }_p$, which perfectly coincide with the ${\kappa }_{\widehat{\pi }}$ as $\gamma \to {\gamma }_c$. (b) Rescaling the stiffnesses by their asymptotic form ${\kappa }_{\widehat{\pi }}\simeq \sqrt{2{\tau }_{\widehat{\pi }}{\nu }_{\widehat{\pi }}}\sqrt{{\gamma }_c-\gamma }$ verifies Eq. (24) and shows that this scaling breaks down at a strain scale with no clear system-size dependence, in stark contrast with the eigenvalues ${\lambda }_p$ shown in Fig. 2. Nevertheless, up to strain intervals ${\gamma }_c-\gamma \lesssim {10}^{-3}$, we find that the deviations from the asymptotic form remain less than roughly 50%.

Figure 5

(a) The NPM $\widehat{\pi }$ and the destabilizing mode ${\widehat{\mathrm{\Psi }}}_p$ converge to a common final form at the instability strain ${\gamma }_c$, as indicated by the vanishing of $1-\widehat{\pi }·{\widehat{\mathrm{\Psi }}}_p$. (b) We find that $1-\widehat{\pi }·{\widehat{\mathrm{\Psi }}}_p\sim {\gamma }_c-\gamma$; see Sec. 5 for a theoretical explanation of this scaling. We also find that the same strain scale $\delta \gamma \sim 1/\left({\tau }_p{\nu }_p{L}^4\right)$ controls the convergence of both modes to their common final form.

Figure 6

(a) The total derivative squared with respect to strain of destabilizing eigenmodes, $|\frac{d{\widehat{\mathrm{\Psi }}}_p}{d\gamma }{|}^2$ vs the distance to the imminent plastic instability strain ${\gamma }_c-\gamma$. (b) An appropriate rescaling (see text) reveals that the same strain scale $\delta \gamma \sim 1/\left({\tau }_p{\nu }_p{L}^4\right)$ controls the deformation dynamics of destabilizing modes, as well as their associated eigenvalues.

Figure 7

Norm squared of the total derivatives of NPMs vs ${\gamma }_c-\gamma$. Although NPM stiffnesses and eigenvalues associated with destabilizing modes follow the same scaling ${\kappa }_{\widehat{\pi }}\sim {\lambda }_p\sim \sqrt{{\gamma }_c-\gamma }$, the two modes' deformation dynamics follow different scaling laws, namely, $|d{\widehat{\mathrm{\Psi }}}_p/d\gamma {|}^2\sim {({\gamma }_c-\gamma )}^{-1}$, while $|d\widehat{\pi }/d\gamma {|}^2\sim {({\gamma }_c-\gamma )}^0$.

Figure 8

The field $\frac{d{\widehat{\mathrm{\Psi }}}_p}{d\gamma }$ calculated at ${\gamma }_c-\gamma \sim {10}^{-14}$ away from a plastic instability in a system of $N=6400$ particles.

Figure 9

The field $\frac{d\widehat{\pi }}{d\gamma }$ calculated at ${\gamma }_c-\gamma \sim {10}^{-14}$ away from a plastic instability in a system of $N=6400$ particles.

Figure 10

Norm $\mathrm{\Delta }\equiv |\widehat{\pi }-{\widehat{\mathrm{\Psi }}}_p|$ of the difference vector between $\widehat{\pi }$ and ${\widehat{\mathrm{\Psi }}}_p$ vs the product ${\lambda }_p{L}^2$.

Figure 11

$1-\widehat{\pi }\left(\gamma \right)·\widehat{\pi }\left({\gamma }_c\right)$ (outlined symbols) and $1-{\widehat{\mathrm{\Psi }}}_p\left(\gamma \right)·{\widehat{\mathrm{\Psi }}}_p\left({\gamma }_c\right)$ (solid symbols) vs ${\gamma }_c-\gamma$. NPMs converge must faster scaling-wise to their final form at instabilities compared to destabilizing modes and are therefore better predictors of imminent plastic instabilities.