Plastic response and correlations in athermally sheared amorphous solids

The onset of irreversible deformation in low-temperature amorphous solids is due to the accumulation of elementary events, consisting of spatially and temporally localized atomic rearrangements involving only a few tens of atoms. Recently, numerical and experimental work addressed the issue of spatiotemporal correlations between these plastic events. Here, we provide further insight into these correlations by investigating, via molecular dynamics (MD) simulations, the plastic response of a two-dimensional amorphous solid to artificially triggered local shear transformations. We show that while the plastic response is virtually absent in as-quenched configurations, it becomes apparent if a shear strain was previously imposed on the system. Plastic response has a fourfold symmetry, which is characteristic of the shear stress redistribution following the local transformation. At high shear rate we report evidence for a fluctuation-dissipation relation, connecting plastic response and correlation, which seems to break down if lower shear rates are considered.

Figure 1

Plastic response field $R_2(\mathbf{r},\mathrm{\Delta }t)$, as defined in Eq. (5), induced by a shear transformation (at the center of the cell) in the long time limit $\mathrm{\Delta }t={10}^3$ for as-quenched configurations. Data for two different values of the shear strain $\epsilon$ are shown: 0.05 (left) and 0.1 (right). A region of size $200\times 200$ around the ST is shown and the color corresponds to the amplitude of the plastic response field.

Figure 2

Plastic response field $R_2(\mathbf{r},\mathrm{\Delta }t)$, as defined in Eq. (5), induced by a ST in sheared configurations for different time lags $\mathrm{\Delta }t$. Data are shown for different shear rates $\dot{\gamma }={10}^{-6}$ (top), $\dot{\gamma }={10}^{-5}$ (middle), and $\dot{\gamma }={10}^{-4}$ (bottom) and for different time lags $\mathrm{\Delta }t=4$ (left column), $\mathrm{\Delta }t=20$ (central column) and $\mathrm{\Delta }t=100$ (right column). A region of size $200\times 200$ around the ST is shown.

Figure 3

(a) Angular dependence of the plastic response ${\overline{R}}_2(\theta ,\mathrm{\Delta }t)=\alpha {\int }_{2a}^{L/2}{R}_2(r,\theta ,\mathrm{\Delta }t)dr$ (see text). Data are shown for $\mathrm{\Delta }t=100$ and for different shear rates $\dot{\gamma }={10}^{-4}$ (triangles), $\dot{\gamma }={10}^{-5}$ (squares), and $\dot{\gamma }={10}^{-6}$ (crosses). (b) Plastic response along the shear direction $R_2^{\parallel }$ in lin-lin (main panel) and lin-log (inset) plots. Data are shown for different time lags $\mathrm{\Delta }t=4$ (black), $\mathrm{\Delta }t=8$ (red), $\mathrm{\Delta }t=20$ (green), and $\mathrm{\Delta }t=40$ (blue), and for different shear rates $\dot{\gamma }={10}^{-4}$ (triangles) and $\dot{\gamma }={10}^{-6}$ (crosses).

Figure 4

Top: plastic response field $R_2(\mathbf{r},\mathrm{\Delta }t)$ at different time lags, $\mathrm{\Delta }t=4$ (left), $\mathrm{\Delta }t=4$ (center), and $\mathrm{\Delta }t=100$ (right), induced by a ST whose main axis is rotated by an angle $\varphi ={90}^{\circ }$ with respect to the shear direction ($\dot{\gamma }={10}^{-5})$. Bottom, panel (a), corresponding angular dependence for $\mathrm{\Delta }t=100$ (triangles); for comparison, data for $\varphi =0^{\circ }$ are also shown (squares). Bottom, panel (b), spatial decay along a principal direction ${\theta }^{*}$ at different time lags, $\mathrm{\Delta }t=4$ (black), $\mathrm{\Delta }t=8$ (red), $\mathrm{\Delta }t=20$ (green), and $\mathrm{\Delta }t=40$ (blue), for $\varphi ={90}^{\circ }$ with ${\theta }^{*}={45}^{\circ }$ (triangles) and the reference $\varphi =0^{\circ }$ with ${\theta }^{*}=0^{\circ }$ (squares).

Figure 5

Decay of the plastic response $R_1(r,\mathrm{\Delta }t)$ (symbols) and correlation $C_1(r,\mathrm{\Delta }t)$ (lines) in the direction parallel and perpendicular to the shear direction. Data are shown for different shear rates $\dot{\gamma }={10}^{-4}$ (top panels), $\dot{\gamma }={10}^{-5}$ (bottom panels), and for different time lags $\mathrm{\Delta }t=4$ (black), $\mathrm{\Delta }t=8$ (red), $\mathrm{\Delta }t=20$ (green), and $\mathrm{\Delta }t=40$ (blue). Data are vertically rescaled in order to have $R_1(r,\mathrm{\Delta }t)=C_1(r,\mathrm{\Delta }t)=1$ at $r=4$. The relative distance between $R_1$ and $C_1$ can be defined as ${\mathrm{\Delta }}_1\left(\mathrm{\Delta }t\right)={\int }_{2a}^{L/4}|R_1(r,\mathrm{\Delta }t)-\beta C_1(r,\mathrm{\Delta }t)|dr/{\int }_{2a}^{L/4}\left|R_1(r,\mathrm{\Delta }t)\right|dr$, where the integral is calculated along a given direction (parallel or perpendicular to the flow). We note that ${\mathrm{\Delta }}_1$ is smaller than 0.2 for all directions at large strain rate $\dot{\gamma }={10}^{-4}$ and larger for the small one $\dot{\gamma }={10}^{-5}$.

Figure 6

Time dependence of the decay length $l$ for the plastic response (open squares) and correlation (crosses). Data are shown for the directions parallel and perpendicular to the shear direction and for different shear rates.