The inverse hall–petch relation in nanocrystalline metals: A discrete dislocation dynamics analysis

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Abstract

When the grain size in polycrystalline materials is reduced to the nanometer length scale (nanocrystallinity), observations from experiments and atomistic simulations suggest that the yield strength decreases (softening) as the grain size is decreased. This is in contrast to the Hall–Petch relation observed in larger sized grains. We incorporated grain boundary (GB) sliding and dislocation emission from GB junctions into the classical DDD framework, and recovered the smaller is weaker relationship observed in nanocrystalline materials. This current model shows that the inverse Hall–Petch behavior can be obtained through a relief of stress buildup at GB junctions from GB sliding by emitting dislocations from the junctions. The yield stress is shown to vary with grain size, d, by a d1/2 relationship when grain sizes are very small. However, pure GB sliding alone without further plastic accomodation by dislocation emission is grain size independent.

Introduction

When a polycrystalline material is deformed and undergoes plastic deformation, yield strength scales with average grain size by d1/2, which implies that the yield strength increases when we decrease the grain size. This is the classical Hall–Petch relationship proposed by Hall (1951) and Petch (1953). The relationship is phenomenological, but one of several ways to understand it is by considering dislocation pile-ups. If the grain boundaries in a polycrystalline material are assumed to obstruct dislocation motion, which is often the case, dislocation multiplication within a grain will cause dislocations to pile up against the grain boundary (GB). Ahead of the leading dislocation in the pile-up, the shear stress induced by the pile-up, στL/x (cf. Leibfried, 1951, Hirth and Lothe, 1992), where τ is the applied stress, L is the length of the pile-up, and x is the distance ahead of the leading dislocation in the pile-up, which is obstructed from moving beyond the GB. This is akin to a crack with the crack tip at the GB. If we define the yield stress of the material as the applied stress when the leading dislocation of the pile-up can penetrate through the GB into the adjacent grain, the stress ahead of the pile-up σ must exceed a critical stress barrier, σ, of the GB. Alternatively, one can think of σ as the stress required to operate a dislocation source on the opposite side of the GB as the dislocation pile-up. Activation of this source due to the stress induced from the pile-up is similar to describing the transmission of the leading dislocation in the pile-up through the GB as discussed by Quek et al. (2014). The yield stress of the material is therefore given byσyσx/LSince the size of the pile-up, L is limited by the size of the grain, d, we can assume that Ld, and Eq. (1) gives the relationship that yield stress, σy scales with d1/2. The above equation simply suggests that as the grain size decreases, a larger applied stress is required for the dislocation to penetrate into the adjacent grain, hence, a higher yield stress.

Nevertheless, one quickly realises that there is a limitation to the pile-up theory as described in the last paragraph. As the grain size becomes very small (typically less than tens of nanometer), the grain interior becomes starved of dislocations and dislocation sources and there would be no dislocation pile-ups. We therefore expect to see a deviation from the Hall–Petch relation when the grain size reaches this regime and this has indeed been observed in experiments (e.g., see Chokshi et al., 1989, Nieman et al., 1989, Fougere et al., 1992). Chokshi et al. (1989) performed hardness tests on nanocrystalline copper (Cu) and palladium (Pd) samples processed by inert gas condensation (IGC). Their results showed a clear decrease in hardness (or yield stress) with a decrease in grain size, i.e., inverse Hall–Petch behavior. Nieman et al. (1989) performed similar experiments and while they did not clearly show a ”smaller-is-softer” trend for their nanocrystalline Cu and Pd samples, their hardness vs. d1/2 plot showed a shallower positive slope as compared with coarse grain samples. Interestingly, Fougere et al. (1992) tested the grain size dependence of hardness from samples that progressively increase in grain size by annealing and found that hardness increases with increasing grain size (inverse Hall–Petch) to a critical size followed by softening with further increase in grain size (Hall–Petch). Lu and Sui (1993), while discussing the experiments of Fougere et al. (1992) and Chokshi et al. (1989), suggested that the annealing process relaxed the grain boundaries and changed their interfacial properties. These observation suggest that when the grain size is very small, the contribution from lattice dislocation plasticity is decreased and a different mechanism – one accommodated by grain boundaries – may dominate the plastic deformation.

Chokshi et al. (1989) argued that the inverse Hall–Petch relation they observed is because of Coble creep (diffusional creep along grain boundaries) at room temperature. At such small grain sizes, diffusional creep along the grain boundaries is enhanced and has a similar effect as GB sliding in coarse-grain polycrystals at high temperature. Masumura et al. (1998) generalized the Hall–Petch relation based on conventional Hall–Petch for large grains and Coble creep for very smaller grains. They considered a log-normal distribution of grain sizes and proposed a model that integrates over the grain size distribution such that for a given grain microstructure, there are contributions from both the Hall–Petch and Coble creep mechanisms. Their model provided a reasonable fit to some experimental observations.

Asaro et al. (2003) proposed a model that assumes that for grain sizes of about hundreds of nanometers, dislocations/stacking faults emission from GBs dominates deformation, and as grain sizes are further reduced to tens of nanometers, the deformation mechanism changes to one dominated by GB sliding. A GB-bulk composite model was proposed to account for the transition as grain size changes, and GB sliding was modeled with a flow law. In such a GB-bulk composite model, deformation mechanisms in different size regimes are treated independently of each other and which grain size regime they dominate depends on the volume fraction of the GBs (and bulk). Using this model together with the averaging approach by Asaro and Needleman (1985) over an aggregate of grains, Zhu et al. (2005) proposed a constitutive model for nanocrsytalline metals and studied the effect of grain size distribution on the deformation response. While an important advance, the deformation mechanisms in the grain interior and in the GBs are not coupled in these models. In reality, we would expect a strong coupling between the deformation mechanisms. Wei et al. (2008a) extended the model to include GB diffusion using the finite element method, in which enforcement of compatibility and equilibrium conditions ensure that the deformation mechanisms are coupled. They also showed that the transition of deformation mechanisms as one reduces the grain size recovers the increase in strain rate sensitivity of nanocrystalline metals. While these models predicted the transition of deformation mechanism from bulk plasticity to GB-based mechanisms as grain size is reduced, they did not explicitly recover the inverse Hall–Petch relation first observed by Chokshi et al. (1989).

Argon and Yip (2006) considered the GB as an amorphous region of which a viscous flow process describes GB shear. They also considered dislocation plasticity as thermally assisted release of dislocations from forest dislocation intersections within the grain interior. The relative contribution to the plastic strain from the two processes is obtained from the volume fraction of the GB region in the material, hence with grain size. Once again, the two deformation mechanisms are assumed to be independent of one another. Nevertheless, they are able to obtain an optimal grain size that gives the highest flow stress. However, the incompatibility arising from their GB shear is not accounted for in their model where they have assumed the polycrystal simply as a set of parallel shearing grain boundaries. Several other models have been proposed to describe the inverse Hall–Petch phenomenon. These models generally describe mechanisms involving lattice dislocations, grain boundaries, and/or their interactions. To avoid including an incomplete list, we refer interested readers to the excellent review papers by Meyers et al. (2006) and Pande and Cooper (2009).

The reality of what happens during the deformation of nanocrystalline materials may also be a synergistic combination of various mechanisms. Molecular dynamics (MD) simulations provide an appealing way to directly observe the deformation in nanocrystalline materials with no a-priori assumption of what mechanisms actually occur during the deformation. Despite the limitation of drastically high strain rates in MD simulations, they can still provide insights on the deformation behavior of nanocrystalline materials. Early MD simulations of the deformation of nanocrystalline metals include studies by Schiøtz et al. (1998), Van Swygenhoven and Caro (1998), Van Swygenhoven and Derlet (2001) and Schiøtz and Jacobsen (2003). MD simulations revealed that nanoscale grains readily exhibit GB sliding at room temperature. The time scale for MD simulations are too short for these observed GB sliding to be a result of long-range diffusion; hence, the observed GB sliding events are the result of stress-induced atomic shuffling. Li et al. (2009) showed that GB sliding and dislocation emission from GBs are common occurrences in MD simulations of nanocrystalline aluminum, and these mechanisms contribute competitively to strain recovery during annealing. Wu et al. (2013) recently performed large scale MD simulations of nanocrystalline Ni nanowires and bulk materials, and showed that GB sliding precedes plastic deformation by lattice dislocations in these nanocrystalline materials regardless of sample size. In order to characterize the early states of deformation, they unloaded their structure after a strain of 0.025 and observed a permanent residual strain attributed to GB sliding, despite the complete absence of dislocation activity. At larger strains, GB sliding was accompanied by dislocation plasticity, where partial and/or full dislocations were emitted from GBs and particularly GB junctions (with few or no dislocations nucleated within the grain interior).

MD simulations suggest that GB sliding and other GB mediated plasticity mechanisms cause the deviation from the Hall–Petch behavior in nanocrystalline materials. The fact that dislocations are nucleated from GBs and GB junctions also suggests that dislocation plasticity is source limited in the nanocrystalline regime rather than limited by dislocation pileups, as in large grain samples. While MD simulations provided us with insights on possible deformation mechanisms, they often do not provide quantitative data on the plastic strain contributions of individual deformation mechanism, and it is therefore not clear how these mechanisms would result in the inverse Hall–Petch trend. Nevertheless, we could garner insights from MD to propose new models for the deformation of such materials. In the same spirit, the discrete dislocation dynamics (DDD) methodology generally imposes constitutive relationships that describe dislocation interaction mechanisms in addition to the dynamics driven by long range stress fields. However, conventional DDD, when applied to polycrsytalline materials, often only treat GBs as obstacles to dislocation motion (see Shishvan and Van der Giessen, 2010, Balint et al., 2005, Balint et al., 2008, Nicola et al., 2005, Kumar et al., 2009, Ahmed and Hartmaier, 2010). This generally resulted in dislocation pileup against the GBs and for large grain sizes, reproduces the Hall–Petch relationship as it rightfully should. Nevertheless, when applied to nanocrystalline grains, two major issues exist: (1) typical dislocation or Frank-Read source densities imply that the average spacing between dislocations or sources is much larger than the average grain size (i.e., there are few if any dislocations/sources), which means that the nanocrystalline material will remain largely elastic; and (2) there is no alternative mechanism available to contribute to plastic slip.

In a previous paper by Quek et al. (2014), a DDD framework was proposed that models GB sliding by having GB dislocations gliding along the GB plane. These GB dislocations are consistent with the displacement shift complete (DSC) lattice of GB structures (see Balluffi et al., 1982) and their Burger's vectors are typically smaller than lattice dislocations. As a result, GB sliding appears to be more continuous than lattice dislocation slip. Furthermore, the implementation allows for these GB dislocation sources to operate at relatively low stress resulting in GB sliding, even in the absence of any lattice dislocation activity. Taking a cue from MD simulations, it would be of interest to study if we can reproduce the deformation behavior of nanocrystalline materials using a DDD model that includes the above mentioned GB sliding mechanism as well as including dislocation emission from GB junctions. As discussed by Wu et al. (2013) and Quek et al. (2014), dislocations are emitted from GB junctions as a result of stress buildup from GB sliding. Such a model would enable us to study how GB sliding and dislocation emission from GB junctions contributes to the observed grain size effects on the yield behavior of nanocrystalline materials.

In Section 2, we review the DDD simulation methodology and describe how we implement GB sliding and dislocation emission from GB junctions in the present research. We also highlight the parameters used for the simulations performed here. Section 3 shows the simulation results using the proposed model for different grain sizes. In Section 4, we discuss the results obtained and propose a theoretical analysis of the mechanisms observed and how they contribute to the observed grain size effect on yield strength. We then generalize the model to describe the transition from inverse Hall–Petch to Hall–Petch behavior with increasing grain size.

Section snippets

Discrete dislocation dynamics framework

The two-dimensional discrete dislocation dynamics (2D DDD) approach is a popular method of explicitly tracking the motion of dislocations and hence, the accumulation of plastic strain during deformation. Gulluoglu et al. (1989) used the approach to track dislocation distributions in materials, and Van der Giessen and Needleman (1995) utilised the framework to incorporate dislocation plasticity during mechanical deformation, which takes into account the long range elastic effects between

Effect of grain size on the yield stress

With the model described in the previous section, we study the effect of grain size on the yield stress. The applied uniaxial stress σyya is increased quasi-statically in increments of Δσyya=0.005μ. In other words, we increment the stress and integrate the dynamical equations for dislocation nucleation and dislocation migration forward in time until all dislocation activity ceases. At this point, the applied stress is incremented again and the same procedure is repeated until the desired

GB sliding

In the model presented above, GB sliding is represented by the nucleation and glide of GB dislocations along the GB plane. This results in dislocation pile ups at GB junctions, which produce a long range stress field that both shuts down the GB dislocation sources and leads to a significant stress concentration in the vicinity of the junctions. To analyze this situation, we consider the simple case shown schematically in Fig. 2 in which a single GB of length l is terminated at its ends by a

Conclusion

We have described a discrete dislocation dynamics model to study the grain size effects of nanocrystalline metals. GB sliding and dislocation emission from GB junctions are incorporated into the DDD framework, and by doing so, reproduced the inverse Hall–Petch relation observed for very small nanocrystalline grains. It was found using the model that for small applied load, the yield stress is dominated by GB sliding and is grain size independent. As the loading increases, stress build-up as a

Acknowledgments

This work was supported by the A*STAR Computational Resource Centre through the use of its high performance computing facilities. ZHC would like to acknowledge the support given by the Industry-sponsored FYP programme from the School of Mechanical and Aerospace Engineering, Nanyang Technological University.

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