Athermal shear-transformation-zone theory of amorphous plastic deformation. II. Analysis of simulated amorphous silicon

In the preceding paper, we developed an athermal shear-transformation-zone (STZ) theory of amorphous plasticity. Here we use this theory in an analysis of numerical simulations of plasticity in amorphous silicon by Demkowicz and Argon (DA). In addition to bulk mechanical properties, those authors observed internal features of their deforming system that challenge our theory in important ways. We propose a quasithermodynamic interpretation of their observations in which the effective disorder temperature, generated by mechanical deformation well below the glass temperature, governs the behavior of other state variables that fall in and out of equilibrium with it. Our analysis points to a limitation of either the step-strain procedure used by DA in their simulations, or the STZ theory in its ability to describe rapid transients in stress-strain curves, or perhaps to both. Once we allow for this limitation, we are able to bring our theoretical predictions into accurate agreement with the simulations.

Figure 1

(Color online) Theoretical stress-strain curves (solid lines) compared to the DA numerical simulation data [3]. The parameters used are ${ϵ}_0=1$, $s_f=1.06\mathrm{GPa}$, $\mu =46\mathrm{GPa}$, $s_y=0.35\mathrm{GPa}$, $c_0=0.18$, and ${\chi }_{\infty }=0.065$. The initial effective temperatures are ${\chi }_0=0.0559,0.0580,0.0612,0.0680$ (from top to bottom). The curves are shifted by $0.5\mathrm{GPa}$ for clarity.

Figure 2

(Color online) Upper panel: The equation of state $\varphi ={\varphi }_0\left(\chi \right)$ obtained by fitting the four data points (filled circles) plus ${\varphi }_{\infty }={\varphi }_0\left({\chi }_{\infty }\right)=0.458$ (filled square). The ${\varphi }_0$’s are the DA numerical simulation data [3]; the adjusted ${\chi }_0$’s are 0.0559, 0.0589, 0.0663, and 0.0749, with ${\chi }_{\infty }=0.065$. The solid line is our fit to these points in Eq. (5.1). The open circles are the observed ${\varphi }_0$’s plotted against the unadjusted, best-fit, ${\chi }_0$’s. The dashed line is intended only as a guide for the eye. Lower panel: The equation of state $\rho ={\rho }_0\left(\chi \right)$ obtained by a procedure similar to the one described above except that the point ${\rho }_{\infty }=1.0335$ (filled square) is not included in the fit. We attribute its displacement from the equation of state to a stress-induced dilation. The density is normalized by ${\rho }_c$, the density of crystalline silicon (diamond cubic). The solid line is our smooth fit to the data. Again, the open circles and and dashed line indicate the unadjusted equation of state.

Figure 3

(Color online) Modified stress-strain curves using ${\chi }_0=0.0559$, 0.0589, 0.0663, and 0.0749 (from top to bottom). All other parameters are the same as those used in Fig. 1.

Figure 4

(Color online) Upper panel: The DA numerical simulation data for $\varphi$ versus the theoretical $\chi$ for the four cases considered in Ref. 3. The discrete markers sets, from top to bottom, correspond to the largest initial $\varphi$ (fastest quench) down to the smallest initial $\varphi$ (slowest quench), respectively. The equation of state $\varphi ={\varphi }_0\left(\chi \right)$ (solid line) was added to stress the nonequilibrium nature of the deformation-induced dynamics of $\varphi$. Lower panel: The corresponding plot for $\rho ∕{\rho }_c$. The discrete markers sets, from top to bottom, correspond to the largest initial $\rho$ (fastest quench) down to the smallest initial $\rho$ (slowest quench), respectively. The points $({\chi }_{\infty },{\varphi }_{\infty })$ and $({\chi }_{\infty },{\rho }_{\infty })$ are marked by solid squares.

Figure 5

(Color online) Upper panel: $\varphi$ as a function of $\chi$ using the nonequilibrium Eq. (6.1) with $c_1=0.42$. The equilibrium equation of state was added for illustration. The curves, from top to bottom, correspond to the largest initial $\varphi$ (fastest quench) down to the smallest initial $\varphi$ (slowest quench), respectively. These curves should be compared to the DA data shown in the upper panel of Fig. 4. Lower panel: $\rho$ as a function of $\chi$ using the nonequilibrium Eq. (6.2) with $c_1=0.42$. The curves, from top to bottom, correspond to the largest initial $\rho$ (fastest quench) down to the smallest initial $\rho$ (slowest quench), respectively. These curves should be compared to the DA data shown in the lower panel of Fig. 4. The points $({\chi }_{\infty },{\varphi }_{\infty })$ and $({\chi }_{\infty },{\rho }_{\infty })$ are marked by solid squares.

Figure 6

(Color online) Theoretical predictions of $\varphi \left(\gamma \right)$ for all four quenches, based on Eq. (6.1) with $c_1=0.42$, compared with the DA simulation data. The discrete markers sets and the solid lines, from top to bottom, correspond to the largest initial $\varphi$ (fastest quench) down to the smallest initial $\varphi$ (slowest quench), respectively.

Figure 7

(Color online) Theoretical predictions of $\rho \left(\gamma \right)$ for all four quenches, based on Eq. (6.2) with $c_1=0.42$, compared with the DA simulation data. The discrete markers sets and the solid lines, from top to bottom, correspond to the largest initial $\rho$ (fastest quench) down to the smallest initial $\rho$ (slowest quench), respectively.