Size of Plastic Events in Strained Amorphous Solids at Finite Temperatures

We address the system-size dependence of plastic flow events when an amorphous solid is put under a fixed external strain rate at a finite temperature. For small system sizes at low strain rates and low temperatures the magnitude of plastic events grows with the system size. We explain, however, that this must be a finite-size effect; for larger systems there exist two crossover length scales ${\xi }_1$ and ${\xi }_2$, the first determined by the elastic time scale and the second by the thermal energy scale. For systems of size $L\gg \xi$ there must exist $(L/\xi {)}^d$ uncorrelated plastic events which occur simultaneously. We present a scaling theory that culminates with the dependence of the crossover scales on temperature and strain rate. Finally, we relate these findings to the temperature and size dependence of the stress fluctuations. We comment on the importance of these considerations for theories of elastoplasticity.

Figure 1

A typical equilibrium configuration with 65 356 particles. The particles are all point objects, and the ball around each particle is of radius ${\lambda }_i$.

Figure 2

The variance of the stress fluctuations as a function of the system size $N$ for a 2D system and for various temperatures. The first power law (data in squares) is obtained under athermal quasistatic conditions where we determine for the present model $\beta =-0.61$, $\theta =-0.40$. The other plots go up in temperature as indicated. The plots are displaced by a fixed amount for clarity. Note that the slope becomes more negative as the temperature increases.

Figure 3

The variance of the stress fluctuations as a function of the system size $N$ for a 3D system and for various temperatures. The plots are displaced by a fixed amount for clarity. Note that the slope becomes more negative as the temperature increases.

Figure 4

The scaling function $g(x)$, cf. Eq. (15), for the 3D data (left panel) and the 2D data (right panel). Note the crossover for $x$ of the order of unity as predicted by Eq. (8). The power-law decrease at low values of $x$ is in agreement with the prediction of $\zeta \approx 0.33$ in both cases. The two black lines represent the theoretical prediction for the scaling function $g(x)$ for $x\ll 1$ and for $x\gg 1$. Note that the full scaling function depends on $\dot{\gamma }$, and the present one is an approximate version for $\dot{\gamma }\to 0$. This is the reason for the curved data sets of the low temperature runs.