Aging in the shear-transformation-zone theory of plastic deformation

Aging phenomena in the plastic response of amorphous solids are studied within the theory of shear transformation zones (STZs), which describes the kinetic rearrangement of localized defects in response to external stress. To account for the slow, nonequilibrium dynamics after a quench below the glass transition temperature, two possible models are considered. In the first model, transition rates between the internal states of STZs decrease with time, while in the second model aging occurs due to the relaxation of an effective temperature that determines the number density of STZs and other out-of-equilibium degrees of freedom. It is shown that for reasonable choices of parameters, both models capture qualitatively typical aging features seen in computer simulations and experiments: (i) compliance curves measured for different waiting times $t_{\mathrm{w}}$ after the quench can be superimposed, when the observation times are rescaled with relaxation times $\propto t_{\mathrm{w}}^{\mu }$, $0<\mu ⩽1$, and (ii) stress-strain curves show a stationary plateau stress independent of $t_{\mathrm{w}}$ and a peak stress that increases logarithmically with both $t_{\mathrm{w}}$ and the strain rate. Trends of the aging behavior with the quench temperature are also discussed.

Figure 1

(Color online) Compliance curves (red, dot dashed) for ${\sigma }_{\mathrm{ext}}=0.1$ and two different “temperatures”: (a) $t_{\mathrm{w}}={10}^3–{10}^6$, $\mu =1$, ${\tau }_0=0.1$, and ${\tau }_0^{\infty }=10$; (b) $t_{\mathrm{w}}={10}^8–{10}^{11}$, $\mu =0.5$, ${\tau }_0=1000$, and ${\tau }_0^{\infty }=100$. The compliance curves collapse onto a common curve (blue, solid) when time is rescaled with $t_{\mathrm{w}}^{\mu }$.

Figure 2

(Color online) Stress-strain curves for several waiting times $t_{\mathrm{w}}={10}^4–{10}^{10}$ (bottom to top), fixed strain rate ${\dot{ϵ}}_{\mathrm{ext}}=0.01$ and otherwise the same parameters as in Figs. 1a, 1b, respectively.

Figure 3

(Color online) Stress-strain curves for varying strain rates ${\dot{ϵ}}_{\mathrm{ext}}={10}^{-5}–{10}^1$ (bottom to top), fixed $t_{\mathrm{w}}={10}^4$ and otherwise the same parameters as is Fig. 1a, 1b, respectively.

Figure 4

(Color online) Peak stress ${\sigma }_{\max }$ (a) as a function of waiting time at fixed ${\dot{ϵ}}_{\mathrm{ext}}=0.1$, and (b) as a function of strain rate at fixed $t_{\mathrm{w}}={10}^4$, for four different “temperatures” represented by ($\mu =1$, ${\tau }_0=0.1$, ${\tau }_0^{\infty }=10$), ($\mu =0.75$, ${\tau }_0=10$, ${\tau }_0^{\infty }=30$), ($\mu =0.5$, ${\tau }_0=1000$, ${\tau }_0^{\infty }=100$), ($\mu =0.25$, ${\tau }_0=100000$, ${\tau }_0^{\infty }=300$) (from bottom to top). Solid lines in panel (a) are logarithmic fits with slope ${\sigma }_{\max }^{\left(1\right)}$, and ${\sigma }_{\max }^{\left(1\right)}$ is plotted against $\mu$ in the inset. Solid lines in panel (b) have slope ${\sigma }_{\max }^{\left(2\right)}=0.9$ independent of $\mu$. (c) Steady-state plateau stress ${\sigma }_{\mathrm{ss}}$ as a function of strain rate for the same four temperatures.

Figure 5

(Color online) Scaling plot of the peak stress as predicted by Eq. (4) for the four different “temperatures” analyzed in Figs. 4a, 4b.

Figure 6

(Color online) Numerical solution of Eq. (25) for $\Gamma =0$ and the parameters ${\alpha }_2=1$, $\rho =1$, and ${\chi }_T=0.025$ and three values of $\beta =1$, 1.5, and 2 (top to bottom). Also shown is the asymptotic result Eq. (27) in the aging regime (dashed lines).

Figure 7

(Color online) Compliance curves (red, dot-dashed) for several wait times $t_w={10}^5–{10}^8$ and ${\sigma }_{\mathrm{ext}}=0.05$ in the effective time formulation of the STZ theory. The other parameters are $\tau =1∕\rho =1$, ${\chi }_T=0.01$, and $\beta =1$. The compliance curves collapse (blue, solid) when time is rescaled with $t_w^{\mu }$; here we achieve the best collapse with $\mu =0.97$.

Figure 8

(Color online) Stress-strain curves for several waiting times $t_{\mathrm{w}}={10}^2–{10}^8$ (bottom to top) at fixed strain rate ${\dot{ϵ}}_{\mathrm{ext}}=0.1$. All other parameters are as in Fig. 7.

Figure 9

(Color online) Stress-strain curves for varying strain rates ${10}^{-7}–{10}^1$ (bottom to top) and fixed $t_{\mathrm{w}}={10}^3$. All other parameters are as in Fig. 7.

Figure 10

(Color online) (a) Peak stress against wait time at ${\dot{ϵ}}_{\mathrm{ext}}=0.1$ for three different “temperatures” represented by the parameter sets ($\beta =1$, $\tau =1$, ${\chi }_T=0.01$), ($\beta =1.5$, $\tau =1000$, ${\chi }_T=0.005$), ($\beta =2$, $\tau =20000$, ${\chi }_T=0.001$) (from bottom to top). Solid lines show logarithmic fits to the data and have slopes ${\sigma }_{\max }^{\left(1\right)}\propto 1∕\beta$. (b) Peak stress against rate for the same three temperatures at $t_{\mathrm{w}}={10}^6$. The solid lines have slope ${\sigma }_{\max }^{\left(2\right)}=0.91$. (c) Steady-state plateau stress against strain rate for the same three temperatures.

Figure 11

(Color online) Scaling plot of the peak stress as predicted by Eq. (4) for the three different “temperatures” analyzed in Figs. 10a, 10b and several waiting times.