Statistical physics of elastoplastic steady states in amorphous solids: Finite temperatures and strain rates

The effect of finite temperature $T$ and finite strain rate $\dot{\gamma }$ on the statistical physics of plastic deformations in amorphous solids made of $N$ particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, $T<T_{\text{cross}}\left(N\right)$, and the strain is quasistatic. In this regime the elastoplastic steady state exhibits highly correlated plastic events whose statistics are characterized by anomalous exponents. In the second regime $T_{\text{cross}}\left(N\right)<T<T_{\text{max}}\left(\dot{\gamma }\right)$ the system-size dependence of the stress fluctuations becomes normal, but the variance depends on the strain rate. The physical mechanism of the crossover is different for increasing temperature and increasing strain rate, since the plastic events are still dominated by the mechanical instabilities (seen as an eigenvalue of the Hessian matrix going to zero), and the effect of temperature is only to facilitate the transition. A third regime occurs above the second crossover temperature $T_{\text{max}}\left(\dot{\gamma }\right)$ where stress fluctuations become dominated by thermal noise. Throughout the paper we demonstrate that scaling concepts are highly relevant for the problem at hand, and finally we present a scaling theory that is able to collapse the data for all the values of temperatures and strain rates, providing us with a high degree of predictability.

Figure 1

(Color online) A typical stress vs strain curve in a system of 4096 particles under simple shear deformation in two dimensions obtained in the athermal quasistatic limit. Every elastic (reversible) increase in stress is followed by a sudden plastic (irreversible) drop in stress.

Figure 2

(Color online) Mean energy drop $⟨\Delta U⟩$ (left panels) and mean strain interval $⟨\Delta \gamma ⟩$ (right panels) as functions of system size, measured in AQS simulations of steady plastic flow of the system described in Sec. 2. Upper panels: two dimensions. Lower panels: three dimensions. The continuous lines represent the scaling laws [Eqs. (26, 27)]. The scaling exponents are the same in 2D and 3D.

Figure 3

(Color online) Stress fluctuations $\widetilde{⟨\delta {\sigma }^2⟩}$ of the steady flow state of the studied model in two dimensions. The continuous line represents the scaling law [Eq. (29)].

Figure 4

(Color online) The mean energy drop (see text) in plastic events as a function of system size, measured in the two dimensional system for different temperatures, see figure legend. The strain rate is $\dot{\gamma }=2\times {10}^{-5}$. The straight line represents the AQS scaling of mean energy drops.

Figure 5

(Color online) Left panel: the variance of the stress fluctuations (multiplied by $N$) as a function of the system size $N$ for a 2D system and for various temperatures, at a strain rate $\dot{\gamma }=2\times {10}^{-6}$. Right panel: the scaling function $g_2\left(x\right)$, cf. Eq. (36) for the data in the left panel. Note the crossover for $x$ of the order of unity as predicted by Eq. (34). The power law decrease at low values of $x$ are in agreement with the prediction of $\zeta \approx 0.33$. The two black lines represent the theoretical prediction for the scaling function $g_2\left(x\right)$ for $x⪡1$ and for $x⪢1$. Note that the full scaling function depends on $\dot{\gamma }$, and the present one is an approximate version for $\dot{\gamma }\to 0$.

Figure 6

(Color online) Illustration of the procedure of finding the potential barriers. Since the exact direction of the barrier $\widehat{\mathit{z}}$ is unknown, the initial guess brings the system to a state in which $U>U_{\text{saddle}}$. The point $s_{\star }$ is the displacement along the estimated $\widehat{\mathit{z}}$ at which the system leaves the original basin of attraction. At $s_{\star }$ the saddle point ${\mathit{x}}^s$ can be detected by minimizing the square of the gradient function ${\left|{\nabla }_{i}U\right|}^2$.

Figure 7

(Color online) Scaling of energy barriers for various system sizes. The slope of the continuous lines is 3/2.

Figure 8

(Color online) The probability for thermally induced plastic event as a function of $\gamma$ for different temperatures increasing from right to left. The system is of size $N=484$ and it has a mechanical instability at ${\gamma }_P=0.0346$. Inset: The mean distance ${\gamma }_P-⟨\gamma ⟩$ of the distributions for different temperatures. Here $\dot{\gamma }=2\times {10}^{-6}$.

Figure 9

(Color online) The mean flow stress $⟨{\sigma }_{\infty }⟩$ and the mean configurational stress $⟨{\sigma }_c⟩$ as a function of strain rate for a system with $N=1024$. Inset: a typical drop in the stress when the system is minimized to the nearest local minimum.

Figure 10

(Color online) Left panel: the average difference between the steady state stress ${\sigma }_{\infty }$ and the corresponding stress at the local minimum ${\sigma }_c$ for different system sizes as a function of $\dot{\gamma }$ in two dimensions. System sizes are $N=10000$ (△), $N=4096$ (○), $N=2116$ $(★)$, $N=1024$ (◇), $N=484$ (◻). Strain rates are $\dot{\gamma }={10}^{-5}$ (full symbols), $\dot{\gamma }={10}^{-4}$ (dotted symbols) and $\dot{\gamma }={10}^{-3}$ (empty symbols). Right panel: data collapse by replotting the same data as shown. The continuous line has a slope of unity.

Figure 11

(Color online) Left panel: the average difference between the steady state stress ${\sigma }_{\infty }$ and the corresponding stress at the local minimum ${\sigma }_c$ for different system sizes as a function of $\dot{\gamma }$ in three dimensions. System sizes are $N=16000$, (▽), $N=8000$ (△), $N=4000$ (○), $N=2000$ $(★)$, $N=1000$ (◇), $N=512$ (◻). Right panel: data collapse by replotting the same data as shown. The continuous line has a slope of unity.

Figure 12

(Color online) Left panel: the stress variance in athermal simulations (multiplied by $N$) for different strain rates as a function of the system size, see inset in the right panel. The change in scaling upon increasing the strain rate is obvious. Right panel: the data collapse in agreement with the scaling function ${\tilde{g}}_1$ of Eq. (54).

Figure 13

(Color online) Left panel: the measured stress fluctuations as a function of the system size for different temperatures and strain rates. Here the temperature ranges from $T=0$ to $T=0.15$ and the strain rate ranges from $\dot{\gamma }=1\times {10}^{-6}$ to $\dot{\gamma }=5\times {10}^{-4}$. Right panel: data collapse as predicted by the scaling function Eq. (57).