Strain rate dependency of dislocation plasticity

What is it about?

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.

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Why is it important?

Metals are mostly used for their excellent load-bearing capacity, enabled by their mechanical strength and damage tolerance. Serving in practically all engineering fields such as transportation, energy, health, construction, and safety, they create an annual global market above 3000 billion Euros1. In many safety-relevant loading scenarios, the in-service mechanical response of metals depends significantly on loading rate, for instance, during vehicle crash, metal forming, medical implants or bird strike impact on jet engines. A strain rate hardening response is generic for metallic materials deforming by dislocation slip2, with exception of a limited regime of deformation conditions in solution-hardened alloys where dislocation-solute interactions may lead to strain rate softening3,4. Nevertheless, the relationship between the strain rate and microscale deformation mechanisms is still poorly understood, and most dynamic constitutive models (e.g., Johnson-Cook, Zerilli-Armstrong) were formulated in a phenomenological or semi-phenomenological manner with several empirical parameters that do not reflect microscale deformation mechanisms and need to be fitted to specific experiments with loss of generality5. Therefore, it is essential to develop a general understanding of the microscopic mechanisms that control strain rate effects, in order to develop physics-based models that are able to reflect and predict the strain rate dependence of the mechanical properties of metals. In BCC (body-centered cubic) metals, such as many steels, strain rate effects are often related to dislocation core properties (the relatively high atomic-scale Peierls barriers and the associated kink-pair mechanism), which control screw dislocation motion. The resulting temperature and stress dependent mobility of screw dislocations has been incorporated into numerous physics-based plasticity models (see refs. 6,7). In FCC (face-centered cubic) metals, such as Al and Cu, where dislocation motion is controlled by phonon drag, the situation becomes more complicated because dislocation motion is strongly affected by various collective phenomena related to the mutual elastic interactions among the dislocations. Investigating these phenomena and establishing their strain rate dependence is the aim of the present study. Experimental studies on single-crystalline Cu8, Al9, and LiF10 as well as on polycrystalline Cu11,12, Al5 spanning many orders of magnitude in strain rate showed that the flow stress exhibits a weakly strain rate-dependent response at low strain rates followed by a strain rate hardening response at high strain rates. It has been argued that the strain rate-independent regime is dominated by dislocation forest interactions and/or dislocation interactions with grain boundaries or precipitates. On the other hand, the strain rate hardening regime was attributed to viscous drag forces acting on dislocations5. In this case, the stress acting on dislocations was related to the dislocation velocity through the dislocation drag coefficient, and the dislocation velocity to the strain rate through the Orowan relationship. Accordingly, the direct relationship between stress and strain rate depends on the ratio between the drag coefficient and the density of ‘mobile’ dislocations. This poses serious problems: drag coefficients predicted from strain rate-dependent stress–strain curves, under the assumption that all dislocations are mobile, are always significantly higher than theoretical estimates, and also higher than drag coefficients deduced from direct velocity measurements13,14. Such discrepancy persists even if additional scattering mechanisms beyond viscous phonon drag are considered15,16,17. Kumar et al. conversely used measurements of strain rate-dependent stress–strain curves in conjunction with directly measured drag coefficients to determine mobile dislocation densities, leading to a very low fraction of mobile dislocations, ~10−5 18. The problem in all these studies resides in the fact that the mobile dislocation density is not a directly observable quantity. Also, it may be argued that the attribute ‘mobile’ is somewhat ill-defined since, depending on the loading conditions, any dislocation (including those were temporarily rendered immobile) can become mobile again. This is particularly important when load path or strain rate changes are imposed. As a consequence of the conceptual difficulties engendered by introducing the distinct categories of ‘mobile’ and ‘immobile’ dislocations, many fundamental questions regarding the relationship between the externally imposed strain rate and the internal collective dynamics of dislocations have never been properly answered. These questions concern not only the relationship between strain rate and average dislocation velocity and its dependence on dislocation density, but also the relationship between individual and collective dislocation behaviors. To settle these questions, a systematic investigation is required that focuses on the problem: how dislocations move. Discrete dislocation dynamics (DDD) simulations allow in situ observations of collective dislocation behavior during plastic flow and can therefore provide fundamental insights into the mechanisms controlling strain rate effects of dislocation-mediated plasticity without the need of relying on ad hoc assumptions. In DDD simulations19,20,21,22,23, dislocations are coarse-grained as discrete elastic lines and most relevant dislocation mechanisms are accounted for in a physics-based fashion (dislocation glide, cross-slip, multiplication, annihilation, long-range interaction, junction formation, etc.). Over the past two decades, DDD has been extensively employed to investigate various aspects of dislocation-mediated plasticity. The two dimensional (2D) DDD approach was previously employed to study dislocation mobility at high strain rates24, and showed that dislocation inertia effects may be important for the accurate prediction of the dynamical properties of dislocations at high strain rates above 105 s−1 25. Using three-dimensional (3D) DDD simulations combined with finite element method, Liu et al. observed that dislocation patterns change from nonuniform to uniform under high strain rates26. Wang et al. performed 3D-DDD simulations and found that while almost all dislocations are mobile at high strain rates27, a very small percentage of the dislocations move at a speed approaching the shear wave velocity28. Under shock loading at super high strain rates, dislocation homogeneous nucleation plays an important role in dynamical plasticity29,30. 3D-DDD simulations were also employed to study shock deformation in silicon crystals under laser shock peening, and the dislocation density and dislocation multiplication rate are strongly dependent on the laser processing conditions31. While DDD simulations were applied to a wide range of problems in dislocation plasticity, the aforementioned fundamental questions pertaining to strain rate dependency have not been systematically investigated. Especially, essential quantities such as the mean dislocation velocity and distribution of dislocation velocity, which are difficult to be determined experimentally, were rarely studied, although they can be naturally obtained from 3D-DDD simulations. In this work, we perform a total of 194 simulations using 3D-DDD and MD (molecular dynamics) methods to analyze the strain rate dependence of collective dislocation plasticity. In these simulations, the effects of dislocation density (varied over 9 orders of magnitude) and strain rate (varied over 10 orders of magnitude) on the plastic deformation behavior of bulk copper and aluminum single crystals are studied. The mean dislocation velocity and velocity distribution are analyzed in detail and universal characteristics of collective dislocation behavior are revealed. Based on this comprehensive database, we derive a universal analytical relationship between dislocation density, strain rate, material strength, and dislocation mobility, which predicts strain rate and dislocation density effects on the plastic properties of metals in terms of a single parameter that combines dislocation density and strain rate.