Emergence of grain-size effects in nanocrystalline metals from statistical activation of discrete dislocation sources

Abstract

Nanocrystalline (NC) metals are exceptionally strong because they contain an unusually high density of grain boundaries (GBs) that act as sources and sinks for dislocations and significantly modify dislocation motion. In this study, we seek to understand the relationship between discrete dislocation emissions from GBs and grain size effects seen in the strength of NC metals. We propose a statistical GB dislocation source model responsible for discrete dislocation slip events. We find that the grain size limitation on dislocation source sizes gives rise to grain size effects on the statistical distribution for the critical resolved shear stress (CRSS) for discrete slip events. To establish its impact on mechanical behavior, this GB source model is integrated into a 3D crystal plasticity finite element model for NC Ni. We observe that a Hall–Petch scaling in yield strength emerges from the calculations. Further the predictions achieve quantitative agreement with experimental data from several studies across a wide range of average nanograin sizes. It is revealed that statistical dispersion in the CRSS for discrete slip events causes (a) strain hardening in the macroscale flow stress–strain response and (b) the fraction of grains that accommodate the applied strain and the fraction of active grains that undergo multi-slip to strongly depend on grain size. It is also found to lead to an unusual texture effect on slip activity that is significant in the finer nanograins but weak in larger grains (>100 nm). In addition, it causes increases in heterogeneity in strain concentrations with decreasing grain size, suggesting that plastic instabilities are more likely to form as nano-scale grain sizes decrease.

Introduction

Nanocrystalline (NC) metals with grain sizes smaller than 100 nm have attracted a lot of attention in the materials science field for decades because of their ultra-high strength and hardness [1], [2]. Many experiments have shown that the yield strength of NC metals increases with decreasing grain size D for D larger than a critical size, e.g., ∼10 nm [3]. Most often this size effect closely follows an empirical Hall–Petch (H–P) relationship, in which the yield strength is inversely proportional to the square root of D [4]. This H–P relationship, however, originates from studies of coarse-grained (CG) metals and for several decades has been remarkably successful in describing grain size effects in an impressively broad range of CG metals [3], [4]. Most attempts to explain H–P effects are based on deformation mechanisms characteristic of CG metals, such as dislocation pile ups [3], [4]. It is, therefore, interesting that this grain size scaling relationship persists in NC metals as well, although many of the basic dislocation processes and mechanisms operating in NC and CG metals are believed not to be the same [3], [4]. Although much knowledge on the deformation mechanisms particular to NC metals has been obtained over the past few decades, it has yet to be clarified why the H–P relationship also fits the yield strength of NC metals.

In some respects, NC and CG metals accommodate mechanical strains in a similar way. Both experimental and computational studies have indicated, for instance, that dislocation-mediated plasticity prevails inside the grains of NC metals, as they do in CG metals, even for grain sizes as small as ∼10 nm (the actual grain size varies with material) [5], [6]. However, unlike in CG metals, it is generally believed that dislocations, whether perfect or partial, originate predominantly from the GBs and not from the grain interiors [5], [7]. Also, in great contrast to CG metals, dislocations seldom accumulate inside NC grains [8]. These two differences lead to the following physical picture of how strain is distinctly accommodated in NC metals: dislocations are emitted from grain boundaries due to local stress concentration, then continue to bow out, slip across the relatively clean grain interior, and eventually are absorbed by the GBs at the other end of the grain, which has been seen in numerical simulations [2], [9], [10], [11]. However, how this discrete action can lead to grain size effects, especially one nearly described by a traditional H–P relationship, is still unclear.

In addition to predicting this mechanism, molecular dynamics (MD) simulations have added much insight. Through a series of statistical analyses of atomistic simulations on the plastic deformation of NC metals, Van Swygenhoven and coworkers found that dislocations preferred to nucleate at GB triple junctions or stress concentrators at GBs and the resolved shear stress (RSS) required for the nucleation of dislocations was lower than that for propagation [5], [12], [13]. Specifically, dislocation sources have already nucleated from and are readily available at GB triple junctions or stress concentration points well before the RSS in a slip system reaches or exceeds the CRSS for dislocation slip across the grain. Physically these sources are dislocation segments that have emerged from the triple junction and are pinned at each end of the adjacent grain boundaries. Propagating this source requires unpinning the dislocation segment from the GBs. Although depinning can be thermally assisted, it appears to be largely mechanically driven. Taken together, these results suggest that dislocation-mediated plasticity in NC fcc metals is predominantly determined by dislocation propagation rather than nucleation [5].

Predicting how the foregoing effects would impact NC strength and give rise to grain size effects calls for larger scale material models. At present, it is not feasible to use atomistic models exclusively to span a broad range of nanograin sizes (up to 100 nm and higher) and applied strain rates (∼10⁻⁴–10³/s) realized experimentally. Toward this end, several mesoscopic models based on the motion of discrete or homogenized dislocations have been proposed to understand experimental observations of grain size effects on the strength and ductility of NC metals. Asaro and Suresh [14] adopt the concept of dislocation nucleation at GBs and calculate an athermal critical nucleation stress for full as well as partial lattice dislocations. The critical stress scaled with 1/D, thus introducing a length scale. Zhu et al. [15], [16] extends this work by including the grain size distributions for predicting the overall mechanical response of NC aggregates. They assumed that the total plastic rate of deformation is given as the sum of the contribution by stacking fault emission and by the grain boundary sliding. The grain size distributions were represented by log-normal distributions. Their simulation results showed that the mechanical response of NC Ni is sensitive to grain size distribution, that is, not simply to the mean grain size, but also to the spread (variance) about the mean grain size. Fu et al. [17] propose a two-phase (core and mantle) model to investigate how yield stress depends on grain size in the NC regime. The core represents material that work hardens in the vicinity of GBs, where multiple slip systems are activated to satisfy compatibility. They predicted that the thickness of the work-hardened layer is equal to one half the grain diameter and the H–P slope decreases with grain size in NC domains. Inspired by MD simulation results on the discrete motion of dislocation loops in nanograins, Li et al. [18] developed a quantized crystal plasticity finite element model to characterize the distinctive features of NC Ni, one that included a quantized jump in shear strain rate when a slip system from a GB was activated. They applied the model to study the effect of statistically varying the critical resolved shear stress (CRSS) among the grains on the stress–strain responses of NC Ni. They found that a smaller fraction of grains plastically deforms to carry the global plastic strain in NC metals than coarser grained metals. While prior modeling efforts have predicted size effects in mechanical response, the link between discrete dislocation slip from GBs and the grain size effects seen in the macroscopic deformation response of NC metals is still unclear.

To make this link, a question that needs to be addressed is how would the emission of dislocations from discrete GB sources depend on grain size? It is often conjectured that the size of the dislocation source scales with D [6], [19]. Thus, larger length sources accompany larger grains and require less stress to propagate. At the same time, it has been shown that when source lengths are made constant, the stress to nucleate is insensitive to grain size [10]. We envision that grain boundaries may nucleate many dislocation sources, which are doubly pinned dislocation segments, but only few experience a sufficiently high stress needed to activate them, enabling them to bow out and propagate across the grain. The required characteristic stress, the CRSS, will depend on the initial length of the dislocation source, the ‘source length’. This source length is not expected to be the same for all dislocations produced by the grain boundary, even when originating from the same boundary and from the same location on that boundary. Consequently, the formation of dislocation sources varies statistically in time and position due to fluctuations in the defect state and local stresses at the boundary. Dispersion in source lengths will lead to dispersion in the CRSS to activate them. Grain size D presents a geometric upper bound on the source length. In this way, D can also impact the distribution of the CRSS values to activate them.

In this study, we aim to understand the effect of variations in GB source lengths on grain size effects in nanocrystalline strength. We develop a statistical model for random source lengths and show that it gives rise to an extreme value distribution in the characteristic stress ${\tau }_{\text{CRSS}}$ to activate the source. Decreasing grain size is found to not only increase the mean but also the statistical dispersion in the distribution of ${\tau }_{\text{CRSS}}$ By incorporating this model into a 3D CPFE model for the deformation of NC Ni, we show that the statistical variability is responsible for strain hardening and the Hall–Petch scaling seen experimentally. The model also reveals that the statistical dispersion can give rise to a pronounced texture effect, higher propensity for multi-slip, and increased heterogeneity in strain concentrations as D decreases further in nanoscale dimensions. These findings help to explain the reduced ductility observed in NC metals as nanograin sizes decrease.

The rest of the paper is organized as follows. Section 2 introduces the statistical model for ${\tau }_{\text{CRSS}}$ and briefly describes the crystal plasticity finite element model. In Section 3, the results of grain size dependent behavior are discussed and compared with experimental data on NC Ni. This is followed by conclusions and implications for plasticity in NC metals in Section 4.