Metal Crystal and Polycrystal Plastic Strain Hardening

1. Introduction

Crystalline metallic solids are key components of engineering devices and structures whose manufacture and performance often crucially involve, or seek to minimize, plastic deformation phenomena. Inelastic behavior originates at the nanometer scale for individual dislocations, whereas the response of a polycrystalline structure is more complex, spanning scales many orders of magnitude larger, and reflects the complexity of physics, including dislocation and defect interactions, grain and phase boundaries, texture effects, and so on. Vastly different plastic and fracture behaviors emerge among metals and alloys when time scales or loading rates, temperatures, and stress-deformation histories differ [1]. Owing to this complexity, many questions remain unanswered, and physics-based models are neither universal nor satisfactory for all contemporary applications, despite decades of research.

The study of the inelastic mechanical responses of single-crystal and polycrystalline metals has a long history. Pioneering works by G.I. Taylor [2] and R. Hill [3] considered the plastic yield and hardening response of crystals and polycrystals. Size effects (e.g., grain diameter) in plasticity and fracture were addressed in seminal studies by Hall [4], Petch [5], and Armstrong et al. [6] in the mid-twentieth century. Classic texts on dislocations by Nabarro [7] and Hirthe and Lothe [8] provide insight at lower length scales, while many books [9,10,11,12,13,14] and review articles [15,16,17] describe constitutive models of the inelastic responses of crystalline metals. In the late twentieth century [18,19,20,21,22], several prominent studies were published regarding strain hardening or polycrystal response from the perspectives of crystal interactions, dislocation mechanics, or micromechanics. Keen interest in these subjects still persists, as evidenced by numerous scholarly publications (for which a literature review is outside the scope of this brief article) in the fields of solid mechanics, materials science, and materials physics.

The aforementioned topics related to the inelastic behavior of metallic crystals and polycrystals are explored in this Editorial, showcasing their historical and ongoing interest. Research contributions toward these topics, specifically those appearing in a recent Special Issue of the journal, are collectively summarized and thereby placed in a broader context.

2. Discussion

A Special Issue of Metals, entitled “Metal Crystal/Polycrystal Plastic Strain Hardening”, was initiated in 2022 by the late Professor Ronald W. Armstrong to advance the current state of the art. Bibliographic details are listed for each contribution following the Section 3 of this Editorial. In its final form (2025), this issue comprises ten published works: an initial Editorial article (contribution 1), six regular research papers (contributions 2 through 7), one research communication (contribution 8), one review article (contribution 9), and the present closing Editorial (contribution 10, unlisted). The key outcomes from each of the first nine contributions are discussed below.

Armstrong (contribution 1) concisely described advances in our understanding of strength and strain hardening from the historic and recent literature on dislocation mechanics that emerged from G.I. Taylor’s pioneering work. The specific topics explored were partial dislocations and stacking faults, cross-slip, dislocation reactions, and grain size effects. Armstrong also reported on the plastic instability experienced during the tensile loading of FCC and BCC polycrystals.

Saffarini and Voyiadjis (contribution 2) modeled plastic flow in gold metallic foams at very high compressive strain rates. Large-scale atomistic simulations of foam samples with different domain sizes were used to provide numerical data for upscaling. A continuum plasticity framework accounting for dislocation evolution, strain hardening, and strain rate and thermal effects was parameterized to successfully capture the stress–strain behaviors witnessed in the simulations. The outcomes were compared with those obtained with other continuum models.

Vinogradov and Estrin (contribution 3) analyzed grain-size effects in the context of tensile plastic deformation instability. The Kocks–Mecking–Estrin hardening model, based on dislocation evolution, and in which the grain size enters indirectly, was used to evaluate necking stress. The Hall–Petch effect, in which strength increases with decreasing grain size, was shown to be better described by the necking stress from the Considère criterion than the usual 0.2% offset yield strength. The results, encompassing data for nickel and titanium, have ramifications for pragmatic parameter determination and materials design.

Zubelewicz and Clayton (contribution 4) modeled the low-temperature behavior of BCC metals, focusing on yield surfaces, plastic potentials, and plastic volume changes. Their new description included terms in the yield stress and plastic potential that were associated with dislocation core spreading and atomic friction, leading to non-Schmid effects. The model’s accuracy was demonstrated by comparison with experimental data on molybdenum. In the absence of repulsive core pressure, the analysis further predicted a null correlation between plastic dilatation due to anharmonic effects and dislocation core spreading affecting lattice friction.

Kunda et al. (contribution 5) studied the thermo-mechanical responses of copper single crystals and polycrystals by measuring their stress, strain, and temperature evolution for both static and dynamic compression simultaneously at strain rates spanning ${10}^{-3}$/s to 4800/s. A novel finite-strain crystal plasticity constitutive model was implemented in finite element (FE) simulations, with rigorous thermodynamics based on dislocation theory that enabled a non-constant Taylor–Quinney factor. The FE simulations showed promising agreement with experimental stress and temperature data and provided physical insight into dislocation mechanisms and thermodynamic processes at the slip-system scale.

In a series of two papers, You and Hasebe (contributions 6 and 7) incorporated Hasebe’s field theory of multiscale plasticity (FTMP) into FE crystal plasticity simulations of the slip bands responsible for metal fatigue crack initiation. The FTMP accounts for differential-geometric incompatibility, physically associated with strain gradients and dislocation structures. The first paper (contribution 6) showed predictions of laddered band morphologies typically obtained in experiments. The second paper (contribution 7) demonstrated an extended field theoretic model augmented with vacancy sources, showing that vacancy diffusion is affected by dislocation flux and the incompatibility rate. The theory was implemented in multi-physics crystal plasticity simulations with diffusion kinetics, using both indirectly and directly coupled schemes. The results were in agreement with those obtained by transmission electron microscopy and provided new insight into the microstructure-based origins of surface roughening followed by grooving. The latter eventually transitioned to cracks under cyclic straining.

Yan and Fu (contribution 8) studied the ductility of BCC refractory alloys, including high-entropy alloys, using Density Functional Theory (DFT). Their calculations produced a new measure of average bond stiffness for solid (alloy) solutions. Decreasing this stiffness, or alternatively increasing the average static displacement of atoms, was shown to correlate with increasing ductility (e.g., fracture strain measured in experiments). This information provides guidance on composition to optimize the properties of refractory alloys.

Clayton et al. (contribution 9) reviewed classical and contemporary experiments and models, both analytical and numerical, for the spherical indentation of polycrystalline metals. A new dynamic dimensional analysis framework was implemented to extract uniaxial-equivalent stress–strain curves and material properties (e.g., yield and strain hardening) using experimental force–depth data from dynamic spherical indentation tests. The uility and limitations of the coupled experimental–analytical methodology were assessed in the context of dynamic indentation data for the aluminum alloy Al 6061-T6.

3. Conclusions

The research published in this recent Special Issue, primarily focused on the constitutive modeling of crystalline metals’ plastic response, has been summarized above. These works collectively enhance our current understanding of the effects of dislocations, microstructures, chemistry, and loading conditions on the mechanics and thermodynamics of metals across a range of length and time scales, which we encourage future studies to further expand upon.