Dynamically strained ferroelastics: Statistical behavior in elastic and plastic regimes

The dynamic evolution in ferroelastic crystals under external shear is explored by computer simulation of a two-dimensional model. The characteristic geometrical patterns obtained during shear deformation include dynamic tweed in the elastic regime as well as interpenetrating needle domains in the plastic regime. As a result, the statistics of jerk energy differ in the elastic and plastic regimes. In the elastic regime the distributions of jerk energy are sensitive to temperature and initial configurations. However, in the plastic regime the jerk distributions are rather robust and do not depend much on the details of the configurations, although the geometrical pattern formed after yield is strongly influenced by the elastic constants of the materials and the configurations we used. Specifically, for all geometrical configurations we studied, the energy distribution of jerks shows a power-law noise pattern $P\left(E\right)\sim E^{-\left(\gamma -1\right)}(\gamma -1=1.3-2)$ at low temperatures and a Vogel-Fulcher distribution $P$($E$) ∼ exp-($E/E_0$) at high temperatures. More complex behavior occurs at the crossover between these two regimes where our simulated jerk distributions are very well described by a generalized Poisson distributions $P\left(E\right)\sim E^{-(\gamma -1)}$ exp-($E/E_0$)${}^n$ with $n$ $=$ 0.4–0.5 and γ−1 ≈ 0 (Kohlrausch law). The geometrical mechanisms for the evolution of the ferroelastic microstructure under strain deformation remain similar in all thermal regimes, whereas their thermodynamic behavior differs dramatically: on heating, from power-law statistics via the Kohlrausch law to a Vogel-Fulcher law. There is hence no simple way to predict the local evolution of the twin microstructure from just the observed statistical behavior of a ferroelastic crystal. It is shown that the Poisson distribution is a convenient way to describe the crossover behavior contained in all the experimental data without recourse to specific scaling functions or temperature-dependent cutoff lengths.

Figure 1

The model with nearest-neighbor (black springs), next-nearest-neighbor (red springs), and third-nearest-neighbor (dashed green lines) interactions.

Figure 2

The initial configurations used in our calculation. (a) Hard material with simple twin configuration, (b) hard material with sandwich twin configuration, and (c) soft material with sandwich twin configuration.

Figure 3

Potential energy ($Pe$) versus applied strain ($\varepsilon$) curve for the hard material with sandwich twin configuration under applied shear strain at high temperature 1.25 $T_{\mathrm{VF}}$ (here and elsewhere, $T_{\mathrm{VF}}$ refers to the Vogel-Fulcher temperature for the hard material with a simple twin configuration).

Figure 4

The microstructural evolution in the elastic regime (before point A in Fig. 3). (a) The dynamic tweed formed in the hard and soft materials with sandwich twin configuration [see Figs. 2b and 2c]. (b) Nucleation and propagation of needle twins in the hard material with simple twin configuration. The behavior in the elastic regime is independent of the temperature of the system. The color scheme represents the local shear angle from the underlying bulk structure ($|{\Theta }_{\mathrm{vertical}}|-4^{\circ }+{\Theta }_{\mathrm{horizontal}}$); for details, see Ref. 8.

Figure 5

Twin patterns at low yield point (e.g., point B in Fig. 3). The twin density decreases with temperature for the hard material but not significantly for the soft material. Shown are domains formed at high temperature (a), (c), and (e) and low temperature (b), (d), and (f). Here, (a) and (b) are for hard materials with simple twin configuration; (c) and (d) show those for a hard material with the sandwich twin configuration, and (e) and (f) are for a soft material with the sandwich configuration. The color scheme represents the local shear angle from the underlying bulk structure ($|{\Theta }_{\mathrm{vertical}}|-4^{\circ }+{\Theta }_{\mathrm{horizontal}}$); for details, see Ref. 8.

Figure 6

Microstructure at and after point C in Fig. 3. At point C, most of the region has been switched to the new orientation (a); further loading will lead to a perfect single domain at point D. The above behavior is independent of the configurations we used (in Fig. 2). The color scheme represents the local shear angle from the underlying bulk structure ($|{\Theta }_{\mathrm{vertical}}|-4^{\circ }+{\Theta }_{\mathrm{horizontal}}$); for details, see Ref. 8.

Figure 7

Energy distributions of jerks $P$($E$) in the plastic regime at high (1.25 $T_{\mathrm{VF}}$) and low temperatures (0.125 $T_{\mathrm{VF}}$) that show that the energy distributions of the jerks depend on temperature, whereas they are not sensitive to the configurations we used. The energy distributions of the jerks at high temperature (1.25 $T_{\mathrm{VF}}$) for the three configurations (a), (c), and (e) follow simple exponential distributions, whereas those (b), (d), and (f) at lower temperatures (0.125 $T_{\mathrm{VF}}$) show Poisson distributions $P$($E)\sim E^{-(\gamma -1)}$exp-($E/E_0$)${}^n$ with stretching exponent $n$ $=$ 0.4.

Figure 8

At very low temperature, the energy distributions of the jerks $P$($E$) in the plastic regime show power-law behavior $P$($E)\sim E^{-(\gamma -1)}$, where (a) the exponent γ−1 $=$ 1.81 for the hard material with simple twin configuration for $8.75\times {10}^{-2}$ $T_{\mathrm{VF}}$; (b) γ−1 $=$ 2.07 for the hard material with the sandwich twin configuration for 1.25 × 10${}^{-2}$ $T_{\mathrm{VF}}$ and (c) γ−1 $=$ 1.77 for soft material with the sandwich twin configuration for $1.25\times {10}^{-2}$ $T_{\mathrm{VF}}$.

Figure 9

At high temperature (1.25 $T_{\mathrm{VF}}$), the waiting time in the plastic regime for all considered different materials follows the stretched exponential $P$($\tau$) ∼ exp-($\tau /{\tau }_0$)${}^n$, where (a) $n$ $=$ 0.43 for a hard material with double-twin configuration, (b) $n$ $=$ 1 for the hard material with a sandwich configuration, and (c) $n$ $=$ 1 for the soft material with sandwich configuration.

Figure 10

Energy distributions of jerks $P$($E$) in the elastic regime at high (1.25 $T_{\mathrm{VF}}$) and low temperature (0.125 $T_{\mathrm{VF}}$), which show the energy distributions of jerks in the elastic regime is very sensitive to both the temperature and the configurations we used. The left column shows results at high temperature (1.25 $T_{\mathrm{VF}}$), The energy distribution of the jerks for (a) the hard material with the simple twin configuration shows power-law distribution $P$($E)\sim E^{-(\gamma -1)}$ with an exponent γ−1 of 2.1, whereas (c) hard and (e) soft materials with sandwich configuration only show Poisson distributions $P$($E)\sim E^{-(\gamma -1)}$exp-($E/E_0$)${}^n$, in which (γ−1) $=$ −0.07, $n$ $=$ 0.39 and (γ−1) $=$ 0.13, $n$ $=$ 0.53, respectively. The right column shows results at low temperature (0.125 $T_{\mathrm{VF}}$). The energy distributions of jerks for (b) a hard material with simple twin configuration is erratic (local front propagation), whereas (d) hard and (f) soft materials with sandwich twin configuration show power-law distributions $P$($E)\sim E^{-(\gamma -1)}$ with the exponent γ−1 of 1.77 and 1.53, respectively.