Strain rate dependency of dislocation plasticity

Dislocation glide is a general deformation mode, governing the strength of metals. Via discrete dislocation dynamics and molecular dynamics simulations, we investigate the strain rate and dislocation density dependence of the strength of bulk copper and aluminum single crystals. An analytical relationship between material strength, dislocation density, strain rate and dislocation mobility is proposed, which agrees well with current simulations and published experiments. Results show that material strength displays a decreasing regime (strain rate hardening) and then increasing regime (classical forest hardening) as the dislocation density increases. Accordingly, the strength displays universally, as the strain rate increases, a strain rate-independent regime followed by a strain rate hardening regime. All results are captured by a single scaling function, which relates the scaled strength to a coupling parameter between dislocation density and strain rate. Such coupling parameter also controls the localization of plasticity, fluctuations of dislocation flow and distribution of dislocation velocity.


Supplementary Note 1: Simulation method
Since we conducted serial simulations over a very wide range of dislocation densities and strain rates, it is mandatory to keep the computational cost per simulation as low as possible.
This indicates to use a small simulation cell size L, which, however, must be at the same time large enough for the dislocations to sample configurations representative of bulk behavior both during initial relaxation and during loading. Because of scaling relations governing dislocation simulations 1 , simulation cell size must, in this context, be measured in units of In engineering, the yield stress is defined as the stress at the point on the stress-strain curve where a prescribed plastic strain value (e.g., 0.2% plastic strain) is reached. While this offset is normally chosen by convention, we aim to make sure that it correctly reflects basic aspects of dislocation behavior. To this end we consider the two extreme regimes (forest hardening regime vs exhaustion regime) separately.
In the forest hardening regime, yielding can, in dislocation terms, be defined as the stress driven transition of the dislocation system from an initial, stable configuration of the dislocation network (energy minimum) to a flowing state. For an initially random dislocation system as considered in our simulations, the mean glide distance of dislocations between two statistically independent energy minima simply equals the mean dislocation spacing -1/ 2 0  . Accordingly, the saddle point, which marks the transition out of the original minimum energy configuration, is reached, if all dislocations on the active slip systems move by half a dislocation spacing on average, . The corresponding axial strain serves as an estimated lower bound to the plastic strain that should be accomplished at yield: To obtain an appropriate definition of the yield plastic strain p y  , we note that according to At very low strain rates, however, a practical problem arises, which stems from the computational cost of DDD simulations again. As is well-known, the total computational cost of DDD simulations increases significantly as one moves to low strain rates. This makes it very challenging to conduct dislocation simulations at strain rates as low as 0.1s -1 . Such a low strain rate is easy to run in a simulation with only one dislocation 3

Supplementary Note 2: A fluctuation-dissipation theorem for dislocation plasticity
It is a standard assumption in continuum plasticity theory, which is motivated by thermodynamic arguments, that the work expended in creating plastic deformation is entirely dissipated into heat. Hence, the dissipated work per unit volume is equal to the plastic work For dislocation plasticity, this statement needs to be qualified: strain hardening is associated with the change of an internal variable (the dislocation density). Since dislocations carry elastic energy in form of stress and strain fields and the much smaller contribution from the dislocation core energy, this leads to a stored internal energy contribution (a stored defect energy 4 ). We estimate the defect energy storage rate as where L E is the dislocation line energy.
where II  is the initial hardening slope in the limit of athermal strain rate independent deformation (in FCC single crystals: hardening stage II). The numerical factor in the bracket on the right-hand side of Supplementary Eq. (7)  On the dislocation level, dissipation occurs because the work that is expended in moving dislocations is dissipated into the phonon system. The situation is particularly simple for non-relativistic over-damped dislocation motion, because there, due to absence of inertia, all work is instantaneously dissipated. Using where the integral runs over the set V  of all dislocation lines in the volume V considered.
Introducing the dislocation density and the mean square dislocation velocity as we thus find that the mean velocity square is related to the microscopically dissipated work by It is clear that the microscopically dissipated work and the macroscopic dissipated work must be identical. We therefore obtain a relationship between the 'macroscopic' quantities in Supplementary Eq. (5) and the 'microscopic' quantities in Supplementary Eq. (10): We write the plastic strain rate now in terms of microscopic quantities (segment velocities) where we note that the motion of dislocations on inactive slip systems (Schmidt factor: We can envisage Supplementary Eq. (13) as an expression for the magnitude of fluctuations.
We define the weighted coefficient of variation of dislocation velocities as 1/2 0.5 2 2 3/4 3 This quantity measures the magnitude of dislocation velocity fluctuations. As we move to very small strain rates, this quantity diverges, which indicates a critical behavior. We note that the general idea of the above derivation goes back to Hähner 6 and the case of a linear drag law has, in embryonic form, been previously considered by one of the present authors 7 .

Supplementary Note 3: Data analysis
Comparison of our theoretical predictions and simulation data with experimental data it is easy to determine dislocation densities pertaining to a given stress and plastic strain in the loaded state, in experiments this is exceedingly difficult. As a consequence, most experiments report the initial dislocation density before loading or the dislocation density after unloading.
It is expected that such data somewhat underestimate the actual dislocation density under load, since bowed-out dislocations relax into stable configurations, see our discussion of ρ0 vs ρy in the main paper.
These difficulties imply that, in selecting experimental data to substantiate our theoretical conjectures, certain compromises are inevitable. For instance, experimental studies published in the literature may provide the initial dislocation densities of single crystal specimens and the flow stresses at a finite strain level. We have included such data if stresses were measured at strains less than 5%, because, in FCC single crystals, the strain hardening rate after the onset of deformation is low (hardening stage I, easy glide) 8 and the dislocation density increases at most by a factor of 2 below a strain of 5% in Cu 9 and Al 10 . In Table 1 Table 2 of Supplementary Data 2.
A complete compilation of our own yield stress data used in Fig. 3 of the main paper is provided in Table 3 Supplementary Figures 5 and 6, respectively.

Supplementary Note 4: Statistical analysis of local plastic strain distribution
For the simulation depicted in Fig. 4i of the main paper, we conducted a statistical analysis of the plastic strain distribution. The simulation cell is divided into 64000 sub-elements. Then, in Supplementary Fig. 3, the probability density is plotted as a function of normalized plastic strain at different total plastic strain levels. In Supplementary Fig. 3, below the total plastic strain of 0.2%, the distribution does not change much. For total plastic strain > 0.2%, we can see two humps appear on the curves. The second hump corresponds to the slip plane with the largest plastic strain, while the first hump corresponds to other slip planes. Both humps increase in height and move rightwards as the total plastic strain increases, indicating that the plastic strain localizes more significantly. Therefore, the influence of plastic strain localization on dislocation plasticity becomes stronger as the total plastic strain increases. However, at the total plastic yield strain of 0.2%, which is considered in the current work, the influence of plastic strain localization on the material strength is still slight. Note that the simulation cell size changes as the initial dislocation density increases. DDD is abbreviation of discrete dislocation dynamics.