Locality and nonlocality in elastoplastic responses of amorphous solids

A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elastoplastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is subextensive. We offer a number of measures of the magnitude of the plastic deformation, all of which display subextensive scaling characterized by nontrivial exponents. We provide some evidence that the scaling exponents governing the subextensive scaling laws are nonuniversal, depending on the degree of disorder and on the parameters of the systems. Nevertheless, understanding what determines these exponents should shed considerable light on the physics of amorphous solids.

Figure 1

(Color online) The potential (1) used in the present simulations in comparison to the more standard $\sigma /r^{12}$ potential and to the Lennard-Jones potential.

Figure 2

(Color online) The configuration of the pulling experiment. The center particle (in dark green) is pulled in a random direction and the force exerted on it is measured, in addition to the total energy of the system. The outer particles (in red) are stationary and are not allowed to move. We are interested in the influence of the radius $R$ on the measured quantities.

Figure 3

(Color online) The force exerted on the center particle and the potential energy of the system as a function of time (left panels) and as a function of displacement (right panels) for a typical run with $R=18.0$. Note the linearity of the force vs displacement before the plastic failure; this is the elastic branch which is linear up to the maximum. Sometime nonaffine elasticity destroys the linearity seen here. At zero time, the velocity increases smoothly from zero (up to second derivative) in order to minimize shock waves.

Figure 4

(Color online) Upper panel: the distributions of maximal force before plastic event in the pulling experiment, for different radii of system sizes. The radius increases from right to left. Lower panel: the scaling of the mean maximal force as a function of the system size, demonstrating the scaling law (2) in linear coordinates and in the inset as a log-log plot.

Figure 5

(Color online) Typical stress-strain and potential-energy-strain curves for a system with 4096 particles. The blowup at the right panel demonstrates the procedure of reducing the increment steps to increase numerical precision in determining the stress and energy drops.

Figure 6

(Color online) The raw distributions of energy drops (upper left) and stress drops (upper right). The lower panels exhibit the system-size dependence of the mean energy drop (lower left) and mean stress drop (lower right). This system-size dependence appears very accurately to conform with power laws $⟨\Delta U⟩\sim N^{\alpha }$ and $⟨\Delta \sigma ⟩\sim N^{\beta }$ with $\alpha \approx 0.37$ and $\beta \approx -0.63$. These scaling laws are used to rescale the distributions as shown in Fig. 7

Figure 7

(Color online) The distributions of energy drops (left panel) and stress drops (right panel) shown in the upper panels of Fig. 5 after rescaling by energy drops by $N^{\alpha }$ and the stress drops by $N^{\beta }$. Note that the data collapse is quite perfect for probabilities larger than about 1% and then begin to meander systematically; we cannot determine at this point whether this meandering is due to paucity of data or due to multiscaling of the higher moments.

Figure 8

(Color online) Left panel: an example for which the initial and final energy and stress values are the same for the energy minimization algorithm and the real-time dynamics. In dotted blue line is the latter trajectory and in continuous green line is the former. For the dynamics, the green diamond represents the quenched configuration after the system has reached a steady state without flow (with only thermal fluctuations around a minimum of the energy landscape). Right panel: an example for which the true real-time dynamics results in larger energy and stress drops compared to the energy minimization procedure. Both panels are for systems with $N=2500$ in the steady-state plastic flow.

Figure 9

(Color online) The distributions of $n$ for different size systems. Inset: the scaling of the average of $n$ with the system size. A power law is detected (see text).

Figure 10

(Color online) The potential chosen for the model of Sec. 4.

Figure 11

(Color online) The distribution of the number of particles participating in a purely plastic event as a function of the system size. The number $n_b$ is defined such that elastic processes cannot contribute to this measure by construction. In the inset, we show a log-log plot of the average $⟨n_b⟩$ as a function of $N$; the scaling law is Eq. (11).

Figure 12

(Color online) The measured (red dots) energy difference $\delta U\equiv U_{\text{aff}}-U_0$ as a function of the strain increment $\delta ϵ$ and the fit (blue continuous line) to $\delta U=A{\left(\delta ϵ\right)}^2$, with $A=9285.0$.

Figure 13

(Color online) The scaled energy difference $\frac{\delta U}{\kappa N}$ (symbols) compared to $\frac{\delta U}{\kappa N}={\left(\delta ϵ\right)}^2$ (black continuous line). Double precision numerics allows us to rely on this quantity only up to strain increments of $\delta ϵ\sim {10}^{-7}$, below which large errors are accumulated.

Figure 14

(Color online) Demonstration of the qualitative behavior of $\kappa$ (lower panel, see text), in relation with the plastic stress drops (upper panel), for a typical $N=1024$ run approaching a steady-state flow.

Figure 15

(Color online) Top panel: comparison of $\delta U$ calculated by energy differences (blue circles) to the same quantity calculated via Eq. (A1) (green diamonds). The black line corresponds to $\frac{\delta U}{\kappa N}={\left(\delta ϵ\right)}^2$. Using the Hessian allows us to go to $\delta ϵ$ values that are orders of magnitude smaller than those enabled by the energy difference. Lower panels: analysis of the onset of a typical flow event for a $N=625$ system; the lower left panels show the stress-strain and the $\kappa$-strain curves. The lower right panel demonstrates the $\frac{1}{\sqrt{{ϵ}_0-ϵ}}$ divergence, in qualitative agreement with [12]. The red continuous line has a slope of $-1/2$.