Universality Class of Nanocrystal Plasticity: Localization and Self-Organization in Discrete Dislocation Dynamics

The universality class of the avalanche behavior in plastically deforming crystalline and amorphous systems has been commonly discussed, despite the fact that the microscopic defect character in each of these systems is different. In contrast to amorphous systems, crystalline flow stress increases dramatically at high strains and/or loading rates. We perform simulations of a two-dimensional discrete dislocation dynamics model that minimally captures the phenomenology of nanocrystalline deformation. In the context of this model, we demonstrate that a classic rate dependence of dislocation plasticity at large rates ($>{10}^3/\mathrm{s}$) fundamentally controls the system’s statistical character as it competes with dislocation nucleation: At large rates, the behavior is statistically dominated by long-range correlations of “dragged” mobile dislocations. At small rates, plasticity localization dominates in small volumes and a spatial integration of avalanche behavior takes place.

Figure 1

The model. (a) The pillar has width $w$ and aspect ratio $h/w=4$. A single slip system is used, oriented 30° relative to the $y$ axis. Distance between planes is $10b$ where $b=0.25\mathrm{nm}$ is the magnitude of the Burgers vector. Red dots stand for dislocation sources while blue dots represent dislocation obstacles. (b) Sample stress strain curves of compression at high ($10^5/\mathrm{s}$) and low ($10^2/\mathrm{s}$) stress rates $\dot{\sigma }$. (c) Strain pattern after deformation at low $\dot{\sigma }$, (d) strain pattern after deformation at high $\dot{\sigma }$.

Figure 2

Effect of loading protocol: Stress-controlled (SC) vs displacement-controlled (DC). (a) Stress-strain curves of different $w$ using two different loading protocols. The strain rate $\dot{\epsilon }$ is $10^4/\mathrm{s}$ in DC and stress rate $\dot{\sigma }$ is $E^{*}\times {10}^4/\mathrm{s}$. A particular strain burst is shown; (b) Size effect of flow stress at 2% strain (blue stands for DC and red stands for SC, results are based on 50 realizations). The inset shows the dependence of dislocation density on $w$ at 2% strain for different loading protocols. (c) Dependence of flow stress (for $w=1\mu \mathrm{m}$) on rate. Strain rate $\dot{\epsilon }$ is used in DC (blue curve) while the elastic corresponding stress rate $\dot{\sigma }=E^{*}\dot{\epsilon }$ is used in SC (red curve). (d) Event (strain jumps) statistics for different loading protocols, different point size represents different $w$. blue: DC, red: SC. Strain jump in DC mode is calculated according to the method in Ref. [44].

Figure 3

SC rate effect crossover. (a) Event statistics for different $\dot{\sigma }$ using SC. The effective $\tau$ changes from $\sim 3.5$ for $\dot{\sigma }=E^{*}\times 10/\mathrm{s}$ to $\sim -1.5$ for $\dot{\sigma }=E^{*}\times {10}^4/\mathrm{s}$. (b) Effect of dislocation source density ${\rho }_{\mathrm{nuc}}$ and mobility $B$ on power law exponent: when $\dot{\sigma }=E^{*}\times {10}^2/\mathrm{s}$, changing ${\rho }_{\mathrm{nuc}}$ from $60\mu {\mathrm{m}}^2$ (purple curve) to $15\mu {\mathrm{m}}^2$ (blue curve) leads to the exponent changing from 2.5 to 2.1. Increasing $B$ from ${10}^{-4}$ Pa s to ${10}^{-3}$ Pa s results in the change of exponent from 2.5 to 2.2.

Figure 4

Spatial and temporal event distribution in SC. Event distribution on all slip planes during the loading up to 10% strain for $\dot{\sigma }=E^{*}\times {10}^2$ (a) and for $\dot{\sigma }=E^{*}\times {10}^4$ (b) $n$ is the total number of slip planes in the model. The color changes from dark purple to yellow with increasing loading strain. (c) Average avalanche size for $\dot{\sigma }=E^{*}\times {10}^2$ in a single sample. (d) Average avalanche size for $\dot{\sigma }=E^{*}\times {10}^4$ in a single sample.