Dislocations Jam at Any Density

Crystalline materials deform in an intermittent way via dislocation-slip avalanches. Below a critical stress, the dislocations are jammed within their glide plane due to long-range elastic interactions and the material exhibits plastic response, while above this critical stress the dislocations are mobile (the unjammed phase) and the material flows. We use dislocation dynamics and scaling arguments in two dimensions to show that the critical stress grows with the square root of the dislocation density. Consequently, dislocations jam at any density, in contrast with granular materials, which only jam below a critical density.

Figure 1

Proposed phase diagram for dislocation systems. Notice the absence of a jamming point.

Figure 2

(Top) Time dependence of the collective speed of all $N=64$ dislocations in a square box of side $L=100$. (Bottom) The stress plotted against the total dislocation displacement for the same run. Displacement at time $t$ is the total distance all the dislocations traveled from the beginning of the simulation ($t=0$) till time $t$: ${\int }_0^{t}d{t}^{\prime }\sum _{i=1}^N{b}_{i}d{x}_i(t^{\prime })$. The arrows indicate the start and finish of the last large avalanche. (Note that in the stress vs displacement bottom figure the equilibration occurs at zero external stress.)

Figure 3

(Top) The cumulative distribution of critical stresses ${\tau }_c$ for 5 different numerical densities. Each curve is extracted from 288 runs with $N=64$, 32, 16 dislocations and 96 runs with $N=48$, 24 dislocations in a square box of side $L=100$. The smaller the density, the narrower the distribution and smaller the mean ${\tau }_c$ (see next Fig. 4). (Bottom) We obtain a good collapse using the curves with the three larger densities based on the expression $p({\tau }_c,\rho )\sim {\rho }^{\lambda }f[{\tau }_c{\rho }^{{\lambda }^{\prime }}]$. The collapse quantifies the fact that the distributions get steeper and have a smaller mean for lower density $\rho$. $\lambda =0$ since the cumulative probability is restricted in $[0,1]$. ${\lambda }^{\prime }=-0.5\pm 0.02$. The rescaling of the horizontal axis indicates that ${\tau }_c\sim {\rho }^{0.5}$.

Figure 4

The mean critical stress $⟨{\tau }_c⟩$ plotted against the inverse numerical density $\rho$. The system is jammed for $\tau <{\tau }_c$ and unjammed above it. Each point is extracted from 288 runs with $N=64$, 32, 16, 8 dislocations and 96 runs with $N=48$, 24, 12 dislocations in a square box of side $L=100$. The critical stress has a similar qualitative dependence on the density as in the proposed jamming phase diagram by Liu and Nagel [17]. However for dislocations ${\tau }_c(\rho )>0$ for all nonzero densities $\rho$.