The stress–strain response of nanocrystalline metals: A quantized crystal plasticity approach

Abstract

This paper develops a finite element-based model with quantized crystal plasticity (QCP) to study distinctive features of nanocrystalline (nc) metal behavior, including an enhanced flow stress, extended plastic transition strain and propensity for strain localization. The QCP feature is motivated by molecular dynamics simulations of dislocation loop propagation across nc grains, showing that the grain-averaged plastic strain jumps by discrete amounts. Further, a simple geometric analysis suggests that the magnitude of the jumps is ∼1/grain size, thereby incorporating a grain size effect. QCP simulations of 1000-grain polycrystals can reproduce the unique experimental stress–strain features of nc metals, but only if the probability density distribution for a slip event increases abruptly at a threshold stress ∼1/grain size. Possible explanations for such a unique signature are discussed in terms of dislocation loop expansion conditions that become important in the nc limit.

Introduction

Nanocrystalline (nc) metals are polycrystals with an average grain size less than 100 nm. This dramatic decrease in intrinsic length scale produces several distinct features; for example, the uniaxial stress–plastic strain data for electrodeposited nc Ni in Fig. 1 shows that the flow stress of nc metals can be a factor of two or more times that of conventional (>1 μm) grain size material. Further, an extended plastic transition strain (ε_trans) is observed, over which the flow stress increases from an initial value σ₀ to approximately 90% of the plateau stress (σ_plateau). This apparent hardening is associated with a micro-plastic regime where not all grains are plastically deforming, even when ε^p ∼0.2%, a value typically associated with macroplasticity in conventional grain size polycrystals [1], [2], [3]. Although the strain rate sensitivity is often larger, failure strains (ε_f) are often smaller, consistent with localized deformation [4], [5], [6], [7], [8], [9]. Admittedly, such generalizations can be difficult since various synthesis techniques produce different microstructures with unique processing-induced defects [4], [5].

At present, the large increases in flow stress and ε_trans with decreasing grain size are not fully understood. Experimental observations [6], [7], [8], [9] suggest dislocation activity is present, yet no permanent root mean square (RMS) strain is accumulated during plasticity, indicating little dislocation build-up [10], [11]. This is consistent with molecular dynamics (MD) simulations documenting the nucleation of dislocations at local stress concentrations in grain boundaries, their propagation through the grain and their subsequent absorption into surrounding grain boundaries [12], [13], [14]. Classical stress-reduction experiments on nc Ni show that the long-range (athermal) and short-range (thermal) stresses for dislocation propagation are large [15] and support a hypothesis that dislocation propagation is hindered by grain boundary pinning. Yet, the lack of a permanent RMS strain cited earlier suggests that these pinning sites do not prevent reverse dislocation motion upon unloading.

Several continuum models of nc plasticity have been developed, postulating 1/grain size strengthening due to the energetic contribution of depositing dislocation content at grain boundaries [16], [17], [18] or 1/(grain size)^1/2 strengthening due to the nucleation of loops from grain boundary stress concentrations [17]. A novel finite element-based extension to study discrete slip events in grains predicts the extraordinary hardening observed in the micro-plastic regime, but it is based on the a priori assumption that successive slip events in grains require increasing stress [19]. Such an assumption is compatible with the stress-drop experiments cited earlier in which the effective stress is observed to increase with deformation [15]. Other models propose the accumulation of geometrically necessary dislocations between the grain boundary and interior [20], [21] and employ a two-phase approach with a distinct volume fraction for a “grain boundary phase” [22], [23] that can become dominant in the nc regime. However, high-resolution transmission electron microscopy shows that grain boundaries in electrodeposited nc Ni and vapor-phase condensed and consolidated nc Cu are sharp [4], indicating no separate disordered grain boundary phase.

The present work develops a finite element model based on a quantized crystal plasticity (QCP) constitutive relation to study nc metals [24]. This modeling approach employs a number of features that have been inspired by both atomistic modeling and experiment. In particular, MD work reveals that at the grain-averaged scale, dislocation propagation results in a change in both the average strain and stress for the active grain [25]. The QCP model captures these quantized jumps – slip events – via a crystal plasticity constitutive law. Since the magnitude of the largest strain jumps is ∼1/grain size, a size effect is introduced. Three key findings result from this scale-dependent QCP approach. First, an asymmetric distribution of critical resolved shear stress is needed to reproduce the experimental shape of the stress–strain curve at small grain size. Second, this distribution results in a substantial fraction of grains that have not slipped, even at the end of the transition regime, when plastic strains are typically much larger than 0.2%. Third, an Eshelby-type analysis suggests that the minimum critical resolved shear stress for complete loop expansion across a grain should increase with decreasing grain size. This last finding, coupled with strengthening due to grain boundary pinning sites, offers a rationale for the asymmetric distribution in critical stress mentioned above.