Physical scaling laws in dislocation microstructures and avalanches from dislocation dynamics simulations

TL;DR

This work tackles the variability of avalanche statistics in crystal plasticity by performing large-scale 3D Dislocation Dynamics simulations of Cu across wide ranges of dislocation density and crystal orientation under a fixed strain rate. The study demonstrates a robust power-law regime with $α \approx -1.6$ to $-1.7$ that is invariant to density and loading direction, and identifies a density- and orientation-dependent upper cutoff for avalanche sizes, roughly obeying $\Delta \gamma_{max} \approx D_{hkl} - C_{hkl} b \sqrt{\rho_{obs}}$; it also shows that triggering stresses $τ_c$, when rescaled by the local obstacle density and averaged stress, collapse to a Frechet distribution, and that per-system avalanche dynamics exhibit clear correlations among slip systems. These findings reconcile disparate experimental and simulation reports by providing precise scaling laws linking mesoscopic dislocation interactions to macroscopic plasticity, enabling improved upscaling to mesoscale and continuum models. The results also emphasize that dislocation-density evolution modulates the statistics, underscoring the need for controlling strain rate and microstructure in predictive constitutive laws.

Abstract

Avalanche-like plastic bursts in crystalline materials follow power law statistics, but the scaling exponents and cutoff parameters vary widely in the literature ($α$ ranging from -1 to -2.2), hindering predictive modeling. Since distributions do not follow Gaussian behavior, the average of plastic kinetics is not correctly defined. Larger-scale models that rely on average behavior are therefore fundamentally flawed. We performed extensive 3D Dislocation Dynamics simulations of FCC Cu deformation across three orders of magnitude in dislocation density ($ρ= 5 \ 10^{10} \ \text{à} \ 2 \ 10^{12} \ \text{m}^{-2}$) under constant strain rates. Our results demonstrate that the power law exponent ($α\approx $ -1.6 to -1.7 ) is invariant to both dislocation density and loading direction, resolving previous inconsistencies. However, dislocation density strongly controls the power law truncation scaling ($Δγ_{max} \propto \ b/\sqrtρ$) and the distribution of avalanche triggering stresses. We quantify correlations between slip system activities and show how individual system contributions evolve with avalanche size. These findings reconcile experimental scatter in avalanche statistics and provide quantitative scaling laws for mesoscale-to-continuum plasticity models.

Figures (13)

• Figure 1: Typical mesoscopic plastic behavior simulated by DDD for Cu [001] single crystals under constant strain rate. (a) Deformation curves with the shear stress $\tau$ as function of the total shear strain $\gamma$ for different initial density $\rho_0$. Inset: zoom in on one of the serrations seen on the deformation curve. We define the strain-bursts $\Delta\gamma$ and the stress-drop $\Delta\tau$ for every plastic events. (b) Corresponding evolutions of the dislocation density $\rho$ as function of the total shear strain $\gamma$. (c) Amplitude of individual strain-bursts $\Delta\gamma)$. (d) Correlation between the strain-bursts $\Delta\gamma$ and the stress-drop $\Delta\tau$, for $\Delta\gamma > 1.10^{-7}$, well into the power law regime. The legend also shows the values for the correlation exponent $1/\beta$ (see main text).
• Figure 2: Impact of the dislocation density upon the strain resolved avalanche statistics during [001] deformation of Cu single crystals. (a) and (c) Probability density function (pdf) of strain-bursts $\Delta\gamma$ ($=x$) . The avalanche extension $L$ is also given in units of $\mu$m to give a sense of the size of simulated plastic events. Complementary stress resolved data are provide in Supp. Mat. (b) and (d) Complementary cumulative probability function of strain-bursts $\Delta\gamma$. For a) and b), the box dimensions are fixed and dislocation density is changed. For c) and d), box dimensions are resized changing the dislocation density while preserving the numerical efficiency of DDD simulations at large densities. The figures also show the values for the parameters $\alpha_i$ and $\lambda_i$ appearing in the truncated power law modeling strain burst statistics (see main text for more details).
• Figure 3: Impact of the loading conditions on the avalanche statistics. (a) Probability density function of strain-bursts $\Delta\gamma$. (b) CCDF of strain-bursts $\Delta\gamma$. The largest events may differ by about an order of magnitude depending on the loading condition, diminishing by as much the extend of the power law domain, while the power law exponent is mostly unchanged.
• Figure 4: Histograms of avalanche contributions $c_{sa}$ for the heigh active slip systems in a [001] simulation at $\rho_0 = 10^{12}$ m$^{-2}$. Histograms are further separated into three bins depending upon the size of the avalanches shown on the top left: (a) small plastic bursts, (b) intermediate avalanches well in the power law regime and (c) largest avalanches.
• Figure 5: Correlations among slip system contributions to avalanches. Correlation (blue) between the primary system $1/2 [\bar{1}01](1\bar{1} 1)$ and its collinear system $1/2[\bar{1}01](111)$; correlation (red) between the primary system $1/2 [\bar{1}01](1\bar{1} 1)$ and all other systems except its collinear system $1/2[\bar{1}01](111)$.