Shear transformation distribution and activation in glasses at the atomic scale

We characterize shear transformations (STs) at the atomic scale in a model of amorphous silicon using a mapping on Eshelby inclusions. We investigate the effect of pressure, glass relaxation, as well as damage on the ST characteristics. We show that the characteristic ST effective volume, ${\gamma }_0{V}_0$, product of the ST plastic shear strain ${\gamma }_0$ and volume $V_0$, does not depend significantly on an applied pressure but increases with accumulated plastic deformation from about $10{\AA }^3$ in the pseudoelastic regime to about $60{\AA }^3$ once plastic flow sets in. Furthermore, by using nudged elastic band calculations, we measure the energy barrier against ST activation. Analyzing different paths leading to either an isolated ST or an avalanche, we show that the barrier is systematically controlled by the first ST with an activation volume equal to the effective volume of the ST at the activated state, which represents only a fraction of the complete ST volume. The activation volume is also found smaller for avalanches, presumably because of accumulated local damage. This work provides essential information to build reliable mesoscale models of plasticity.

Figure 1

Quasistatic shear deformation simulations under a constant pressure P. (a) Sketch of the loading conditions: positive shear strain increments $\delta {\gamma }_{xy}$ are applied in the $xy$ plane, giving rise to a total strain $\gamma$ and shear stress ${\sigma }_{xy}$, at a constant pressure $P$. (b) Yield stress ${\sigma }_Y$ as a function of the applied pressure $P$. Both ${\sigma }_Y$ and $P$ are normalized by ${\sigma }_{Y,0}$, the yield stress at $P=0$ GPa. (c) Stress-strain curves obtained under quasistatic shear at a constant applied pressure $P$, ranging from $-10$ to 10 GPa.

Figure 2

Comparison between stress-strain curves obtained with atomistic simulations (solid color lines) and Eshelby model (dashed gray lines) for different values of the applied pressure.

Figure 3

Probability density function (Pdf) of the effective Eshelby volume ${\gamma }_0{V}_0$ in different regions of strain: (a) $\gamma <{\gamma }_{\mathrm{I}}$ ($\sim 0.1$), the deformation is mainly elastic; (b) ${\gamma }_{\mathrm{I}}\le \gamma \le {\gamma }_{\mathrm{II}}$, transition region, which contains the yield stress and the first shear band; (c) $\gamma >{\gamma }_{\mathrm{II}}$, plastic flow. ${\gamma }_{\mathrm{II}}$ varies with the applied pressure $P$ and corresponds to the development of the first shear band. Three pressures, from $-10$ to +10 GPa are considered. Inset: characteristic effective volume ${\mathrm{\Omega }}_0$, calculated as the width of the ${\gamma }_0{V}_0$ distributions fitted with a decaying exponential function, as a function of $P$ in the three selected $\gamma$ ranges.

Figure 4

Probability density function (Pdf) of the Eshelby volume variation $\mathrm{\Delta }{V}_0={\varepsilon }_{\text{Tr}}{V}_0$ in the same three strain regimes described in the caption of Fig. 3. ${\varepsilon }_{\text{Tr}}$ is the trace of the Eshelby strain tensor and $V_0$, the Eshelby volume.

Figure 5

Plastic stress drops due to (a) an isolated ST and (b) a cascade of STs are identified on the stress-strain curve at different critical shear strains ${\gamma }_c$. Pairs of equilibrium configurations, one before and the other after the plastic event, are shown as green and red circles, respectively, for different $\gamma -{\gamma }_c$ values.

Figure 6

Effective Eshelby volume ${\gamma }_0{V}_0$ for the two most active STs identified along the NEB path calculated at ${\gamma }_c$ = 0.16. Inset: characteristic size $\lambda$ of the two STs.

Figure 7

Variation of the internal energy $\mathrm{\Delta }E$ (a), shear stress $\mathrm{\Delta }{\sigma }_{xy}$ (b), and pressure $\mathrm{\Delta }P$ (c) along the NEB paths of the isolated ST, that occurred at ${\gamma }_c$ = 0.16 in Fig. 5. The different curves were obtained at increasing values of ${\gamma }_c-\gamma$. Inset: Barrier height $\mathrm{\Delta }{E}_a$ as a function of the distance from the instability. Empty diamonds are used to visualize the position of the activated state on each curve.

Figure 8

Effective Eshelby volume ${\gamma }_0{V}_0$ for the four most active STs identified along the NEB path characteristic of a plastic rearrangement involving several STs.

Figure 9

Variation of the internal energy $\mathrm{\Delta }E$ (a), shear stress $\mathrm{\Delta }{\sigma }_{xy}$ (b), and pressure $\mathrm{\Delta }P$ (c) along the NEB paths of the cascade of STs that occurred at ${\gamma }_c$ = 0.1624 in Fig. 5. The different curves were obtained for increasing values of ${\gamma }_c-\gamma$. Inset: Magnification at the beginning of the energy path in (a), showing the growth of the energy barrier.

Figure 10

ST activation volume calculated in two different ways: as the negative derivative of the energy barrier $\mathrm{\Delta }{E}_a$ with respect to the shear stress ${\sigma }_{xy}$ (red circles) and as the Eshelby effective volume ${\gamma }_0^{*}{V}_0$ of the first plastic event identified along the NEB path computed at the saddle point (blue diamonds). Activation volumes are calculated for an isolated ST (top curve, $\left|\mathrm{\Delta }\sigma \right|\sim |\mathrm{\Delta }{\sigma }_0|$) and for a cascade of STs (bottom curve, $\left|\mathrm{\Delta }\sigma \right|\gg |\mathrm{\Delta }{\sigma }_0|$).

Figure 11

Activation volume as a function of ${\gamma }_c$ (a) and of shear stress drop amplitude produced by the plastic event $|\mathrm{\Delta }{\sigma }_{xy}|$ (b). $\mathrm{\Delta }{V}_a$ is calculated as the negative derivative of the energy barrier $\mathrm{\Delta }{E}_a$ with respect to the shear stress ${\sigma }_{xy}$ at $({\gamma }_c-\gamma )$ = 0.005 for several plastic events involving either isolated or cascades of STs.