Nanomechanics of Hall-Petch relationship in nanocrystalline materials

Classical Hall-Petch relation for large grained polycrystals is usually derived using the model of dislocation pile-up first investigated mathematically by Nabarro and coworkers. In this paper the mechanical properties of nanocrystalline materials are reviewed, with emphasis on the fundamental physical mechanisms involved in determining yield stress. Special attention is paid to the abnormal or ‘inverse’ Hall-Petch relationship, which manifests itself as the softening of nanocrystalline materials of very small (less than 12 nm) mean grain sizes. It is emphasized that modeling the strength of nanocrystalline materials needs consideration of both dislocation interactions and grain-boundary sliding (presumably due to Coble creep) acting simultaneously. Such a model appears to be successful in explaining experimental results provided a realistic grain size distribution is incorporated into the analysis. Masumura et al. [Masumura RA, Hazzledine PM, Pande CS. Acta Mater 1998;46:4527] were the first to show that the Hall-Petch plot for a wide range of materials and mean grain sizes could be divided into three distinct regimes and also the first to provide a detailed mathematical model of Hall-Petch relation of plastic deformation processes for any material including fine-grained nanocrystalline materials. Later developments of this and related models are briefly reviewed.Prof. Frank Nabarro was a physicist by training, a metallurgist by profession and a genius by nature, blessed with a unique ability to treat everyone as his equal. During his later years he was very much interested in the mechanical properties of nanocrystalline materials. This review on that topic is our contribution to the special issue of Progress in Materials Science honoring him.     

Mechanical properties of nanocrystalline materials

The mechanical properties of nanocrystalline materials are reviewed, with emphasis on their constitutive response and on the fundamental physical mechanisms. In a brief introduction, the most important synthesis methods are presented. A number of aspects of mechanical behavior are discussed, including the deviation from the Hall-Petch slope and possible negative slope, the effect of porosity, the difference between tensile and compressive strength, the limited ductility, the tendency for shear localization, the fatigue and creep responses. The strain-rate sensitivity of FCC metals is increased due to the decrease in activation volume in the nanocrystalline regime; for BCC metals this trend is not observed, since the activation volume is already low in the conventional polycrystalline regime. In fatigue, it seems that the S-N curves show improvement due to the increase in strength, whereas the da/dN curve shows increased growth velocity (possibly due to the smoother fracture requiring less energy to propagate). The creep results are conflicting: while some results indicate a decreased creep resistance consistent with the small grain size, other experimental results show that the creep resistance is not negatively affected. Several mechanisms that quantitatively predict the strength of nanocrystalline metals in terms of basic defects (dislocations, stacking faults, etc.) are discussed: break-up of dislocation pile-ups, core-and-mantle, grain-boundary sliding, grain-boundary dislocation emission and annihilation, grain coalescence, and gradient approach. Although this classification is broad, it incorporates the major mechanisms proposed to this date. The increased tendency for twinning, a direct consequence of the increased separation between partial dislocations, is discussed. The fracture of nanocrystalline metals consists of a mixture of ductile dimples and shear regions; the dimple size, while much smaller than that of conventional polycrystalline metals, is several times larger than the grain size. The shear regions are a direct consequence of the increased tendency of the nanocrystalline metals to undergo shear localization.The major computational approaches to the modeling of the mechanical processes in nanocrystalline metals are reviewed with emphasis on molecular dynamics simulations, which are revealing the emission of partial dislocations at grain boundaries and their annihilation after crossing them.     

纳米材料微阵列超塑微成形机理与尺度效应

微成形技术是未来批量制造高精密微小零件的关键技术,但是,微小尺度下材料的塑性变形行为不仅表现出明显的尺度效应,而且零件尺度已经接近常规材料的晶粒尺寸,每个晶粒的形状、取向、变形特征对整体变形产生复杂的影响,难以保证微成形的工艺稳定性。本项目采用纳米材料进行微成形,制造微阵列,零件内部包含大量的晶粒,可以排除晶粒复杂性的影响,而且纳米材料具有超塑性,在超塑状态下,变形抗力和摩擦力都明显降低,从而显著降低微成形工艺对模具性能的苛刻要求,提高工艺稳定性和成形精度。目前,纳米材料超塑性微成形技术方面的研究极少,变形时纳米材料的力学行为、变形机理、尺度效应、位错演化、力学模型等关键问题还有待研究。采用电沉积技术制备晶粒尺寸可控的纳米材料,将工艺实验研究、性能测试、组织分析、力学性能表征、数值模拟相结合,深入探究了纳米材料微阵列超塑性微成形机理和成形规律,以促进该技术的广泛应用。     

Grain boundary sliding in nanomaterials at elevated temperatures

The unique deformation behavior of nanocrystalline materials is considered to be caused by suppression of conventional lattice dislocation slip (which dominates in coarse-grained materials) and effective action of alternative deformation mechanisms occurring through motion of grain boundary defects. A significant role of grain boundary sliding in deformation processes in nanocrystalline materials was shown in models and was revealed experimentally.     

Deforming Nanocrystalline Metals: New Insights,New Puzzles

Recent experiments have brought new insights into the mechanisms which govern the plasticity of nanocrystalline metals. In particular, new opportunities have arisen from the finding that bulk nanocrystalline samples with extremely small grain size, prepared by the inert gas condensation technique, can be deformed to large true strain. The findings elucidate the roles of creep, partial dislocation activity along with its consequences, faulting and twinning, as well as grain boundary sliding and grain rotation. However, they also rise intriguing new questions, specifically with respect to the mechanisms of dislocation nucleation at grain boundaries, and with respect to slip system selection and alignment in twinned grains. An emerging insight is that there is not ‘the’ deformation mechanism at small grain size; instead, deformation mechanism maps in, for instance, the parameter space spanned by the strain rate and the grain size, are more appropriate representations of the various processes that control the materials behavior.     

Stress-dependent deformation behaviour in bulk nanocrystalline titanium–nickel alloys

In this study, the deformation mechanisms operating with stress in bulk nanocrystalline (NC) titanium–nickel with an average grain size below a critical size of 10–20?nm have been investigated. We demonstrate a sequential variation of the deformation mechanism from grain boundary (GB) sliding and grain rotation to grain growth and dislocation activity with the increase of the deformation stress. These deformation mechanisms are different from the previous understanding that below a critical grain size of 10–20?nm, GB sliding and grain rotation govern plastic deformation of NC materials.     

A constitutive model of nanocrystalline metals based on competing grain boundary and grain interior deformation mechanisms

In this work, a viscoplastic constitutive model for nanocrystalline metals is presented. The model is based on competing grain boundary and grain interior deformation mechanisms. In particular, inelastic deformations caused by grain boundary diffusion, grain boundary sliding and dislocation activities are considered. Effects of pressure on the grain boundary diffusion and sliding mechanisms are taken into account. Furthermore, the influence of grain size distribution on macroscopic response is studied. The model is shown to capture the fundamental mechanical characteristics of nanocrystalline metals. These include grain size dependence of the strength, i.e., both the traditional and the inverse Hall–Petch effects, the tension–compression asymmetry and the enhanced rate sensitivity.     

Information on deformation mechanisms in nanocrystalline Pd–10% Au inferred from texture analysis

There is still a lack of understanding of deformation mechanisms in nanocrystalline (nc) materials. Studies on microstructures formed in nc Pd–10% Au (grain size about 15 nm) after high pressure torsion revealed signatures of various deformation processes as cooperative grain boundary sliding (GBS), shear banding, dislocation slip and twinning. In order to estimate contributions of these processes to total strain, a comparison was made between torsion textures formed in nc and coarse grained (cg) samples after identical shear strain. The textures were measured with synchrotron radiation. Intensities of characteristic components of the shear texture are two times stronger in the cg sample than in the nc one, indicating that dislocation slip is less significant in the nc sample. It is proposed that numerous planes of cooperative GBS revealed by TEM contribute to plasticity of nc alloy.     

Deformation twinning in nanocrystalline materials

Nanocrystalline (nc) materials can be defined as solids with grain sizes in the range of 1-100 nm. Contrary to coarse-grained metals, which become more difficult to twin with decreasing grain size, nanocrystalline face-centered-cubic (fcc) metals become easier to twin with decreasing grain size, reaching a maximum twinning probability, and then become more difficult to twin when the grain size decreases further, i.e. exhibiting an inverse grain-size effect on twinning. Molecular dynamics simulations and experimental observations have revealed that the mechanisms of deformation twinning in nanocrystalline metals are different from those in their coarse-grained counterparts. Consequently, there are several types of deformation twins that are observed in nanocrystalline materials, but not in coarse-grained metals. It has also been reported that deformation twinning can be utilized to enhance the strength and ductility of nanocrystalline materials. This paper reviews all aspects of deformation twinning in nanocrystalline metals, including deformation twins observed by molecular dynamics simulations and experiments, twinning mechanisms, factors affecting the twinning, analytical models on the nucleation and growth of deformation twins, interactions between twins and dislocations, and the effects of twins on mechanical and other properties. It is the authors’ intention for this review paper to serve not only as a valuable reference for researchers in the field of nanocrystalline metals and alloys, but also as a textbook for the education of graduate students.     

Grain growth and stabilisation of nanostructured aluminium at high temperatures: review

Nanostructured (NS) materials have a large stored energy due to their large grain boundary area and thus tend to be unstable with respect to grain growth during high temperature annealing or deformation. This problem can limit the application of NS materials at high temperatures (>0·5T_m, absolute melting temperature), especially Al alloys owing to their low melting points. Restoration processes and grain growth in NS Al based materials are critically reviewed, with emphasis on nanostructure grain stabilisation at high temperatures. The mechanisms of normal and abnormal grain growth during isothermal annealing are presented, followed by consideration of thermal stabilisation by the addition of solute atoms/impurities and/or dispersion of second phase particles. Grain growth is significantly facilitated by applying deformation at elevated temperatures during preparation or further processing of semifinished NS materials. The dynamic restoration processes, dynamic grain growth and dynamic particle coarsening are addressed in NS Al. Finally, grain growth during consolidation of nanocrystalline powders (one of the principal methods to fabricate bulk NS Al) is presented, and the effects of processing parameters on grain size stabilisation are discussed.     

Size dependent grain-boundary conductivity in doped zirconia

The oxygen vacancy distributions at a grain boundary and resulting grain-boundary conductivities were calculated as a function of grain size, with an emphasis on nanocrystalline materials. The cause for the “intrinsic” grain-boundary blocking effect is analyzed, and the variation of the grain-boundary conductivity with grain size is discussed. Compared with literature experimental results, the calculation results are reasonable. Finally, the feasibility of Bauerle's equivalent electrical circuit for nanocrystalline materials is evaluated.     

Deformation-mechanism map for nanocrystalline metals by molecular-dynamics simulation

Molecular-dynamics simulations have recently been used to elucidate the transition with decreasing grain size from a dislocation-based to a grain-boundary-based deformation mechanism in nanocrystalline f.c.c. metals. This transition in the deformation mechanism results in a maximum yield strength at a grain size (the 'strongest size') that depends strongly on the stacking-fault energy, the elastic properties of the metal, and the magnitude of the applied stress. Here, by exploring the role of the stacking-fault energy in this crossover, we elucidate how the size of the extended dislocations nucleated from the grain boundaries affects the mechanical behaviour. Building on the fundamental physics of deformation as exposed by these simulations, we propose a two-dimensional stress-grain size deformation-mechanism map for the mechanical behaviour of nanocrystalline f.c.c. metals at low temperature. The map captures this transition in both the deformation mechanism and the related mechanical behaviour with decreasing grain size, as well as its dependence on the stacking-fault energy, the elastic properties of the material, and the applied stress level.     

A micro-continuum model for the creep behavior of complex nanocrystalline materials

A nanocrystalline material which has an average grain size of less than 100 nm is characterized with a significant portion of atoms residing in the grain boundaries or in the grain-boundary affected zone (GBAZ), while nanocrystalline materials with a more complex structure may contain additional strengthening nanoparticles or nano pores. In this article we develop a micro-continuum model to capture the creep response of such a complex nanocrystalline system. We make use of the concept of a three-phase composite with the GBAZ serving as the matrix, and grain interiors and dispresed particles (or voids) as two distinct types of inclusions. Both the grain interior and the GB zone are capable of undergoing the rate-dependent plastic deformation, but the strengthening nanoparticles or pores are taken to deform only elastically. During deformation the porosity will continue to evolve; its evolution is also addressed. In addition, the effect of temperature on the overall creep response is also accounted for. Several important features of creep characteristics in light of grain size, and nanoparticle and nanopore concentrations, are illustrated, and it is also demonstrated that the calculated results are in reasonable agreement with available experimental data.     

Effect of microstructure on thermal expansion behaviour of nanocrystalline metallic materials

Materials properties, among which thermodynamic ones, are influenced by microstructural features. This is so also in the case of nanocrystalline materials, featuring average grain size below 100 nm. A reduced grain size involves that significant fractions of atoms are localised in grain boundary regions and this has remarkable effects on the resulting thermodynamic properties, like heat capacity, transition temperatures, coefficient of thermal expansion, etc. In the present work we consider the thermal expansion behaviour of ball-milled nanocrystalline metallic powders using dilatometric measurements. High-energy ball-milling, that is capable to achieve extremely high deformation degrees, induces in the milled powders microstructural defects, like vacancies, antisites, dislocations and planar faults. Another effect of milling is the reduction of the crystallite size, that, in the long run, may reach the nanometric range. In view of the microstructural changes that can be brought about by milling and of the numerous transformations occurring during the dilatometric runs, a comparative study has been conducted on intermetallic, NiAl and Ni₃Al, and on a pure metal, nickel, powders. The results emerging from the experimental investigation are quite complex, owing to the complex defect structures that are present in the ball-milled powders. It turns out that the thermal expansion coefficient of the nanocrystalline powders increases as the average grain size is reduced. However, when the average grain size achieves very low values, the strain relaxation of the crystalline lattice and the rearrangement of grain boundary regions result in a reduction of the thermal expansion coefficient. Another aspect that emerges from the dilatometric curves is the interplay between recrystallization and reordering, i.e. the re-establishment of the long-range order in the intermetallic powders, that had been partially or fully eliminated by ball-milling.     

形变诱发纳米晶局域固态非晶化的研究进展

形变诱发纳米晶金属材料局域固态非晶化转变是近年来提出的获得局域固态非晶化组织的一种新途径,这种转变机制使得以位错、变形孪晶、晶界滑动和晶粒转动为主要变形机制的纳米晶材料中可能存在一种全新的塑性变形机制,并且,局域固态非晶化的临界转变条件和转变机制可为材料的结构优化设计提供依据。概括了国内外实验及数值模拟手段关于形变诱发局域固态非晶化转变的研究,例如采用机械球磨、高压扭转变形、经典力场和分子动力学等方法,证明了形变诱发局域固态非晶化转变的存在。此外,还分析了发生局域固态非晶化转变的内在机制。基于晶体相场模型的优势,提出用该方法模拟局域固态非晶化转变的突出之处,表明了晶体相场法能够有效研究局域固态非晶化转变过程。     

Nonlinear microstructural constitutive equation of nanocrystalline metals

Summary. Based on experimental observations, nanocrystalline materials are modeled as composite systems in which the amorphous interfacial phase is treated as the matrix, whereas the nano-scale single crystals are modeled as inclusions. Generally speaking, the elastic moduli of nanoscale crystals are higher than those of the amorphous matrix phase, and the deformation mechanism of nanocrystalline materials depends heavily on the size of the crystals. For conventional macro size crystal materials, such as coarse-grained polycrystalline materials, the deformation mechanism due to dislocation movement is dominant. When the crystal size is reduced to a certain critical value, plastic deformation is caused by shear banding in the amorphous matrix. In order to model such a deformation mechanism in nanocrystalline materials, constitutive equations are established based on internal variable theory. The proposed model reveals the relation between the yield strength and the grain size of the material.     

Grain boundary engineering and superstrength of nanocrystals

A new paradigm of hardening of nanocrystals is proposed based on the competing influence of various mechanisms of plastic deformation, i.e., dislocation sliding and grain-boundary slip. It has been confirmed using the results of computer modeling and the experimental data that the use of grain boundary engineering on the basis of the proposed ideas makes it possible to enhance substantially the strength of titaniumbased materials up to ultimate (theoretical) values.     

Effect of Mo on microstructure and mechanical properties of bulk nanocrystalline Fe3Al materials prepared by aluminothermic reaction

Abstract

Bulk nanocrystalline Fe₃Al based materials with 5, 10 and 15 wt-%Mo were prepared by aluminothermic reaction. The microstructure and mechanical properties of the materials were investigated. It was shown that the materials consisted of a nanocrystalline matrix phase that was composed of Fe, Al and Mo and a little Al₂O₃ contamination phase. The nanocrystalline phase had a disordered bcc crystal structure. Average grain sizes of the nanocrystalline phase of the materials with 5, 10 and 15 wt-%Mo were 19, 31 and 24 nm respectively and that of the material with 5 wt-%Mo was the smallest. The materials with 10 and 15 wt-%Mo exhibited brittle behaviour in compression, whereas the material with 5 wt-%Mo had a large plastic deformation. The material with 5 wt-%Mo had the highest bending strength and the lowest compressive yield strength.     

In-situ TEM straining experiments of Al films on polyimide using a novel FIB design for specimen preparation

In-situ transmission electron microscopy (TEM) straining experiments are tedious to perform but give invaluable insight into the deformation processes of materials. With the current interest in mechanical size-effects of nanocrystalline materials and thin metallic films, in-situ tensile testing in the TEM is the key method for identifying underlying deformation mechanisms. In-situ TEM experiments can be significantly simplified using well-designed specimens. The advantages of a novel focussed ion beam design and first in-situ straining results of 500-nm thick single-crystalline Al films on polyimide are reported and compared to conventionally prepared Al films on polyimide.     

A micromechanical approach to the stress–strain relations, strain-rate sensitivity and activation volume of nanocrystalline materials

This review highlights a secant viscosity approach that has wide applicability for the determination of mechanical properties of nanocrystalline materials. Along the way we also add some new elements and provide fresh perspectives. This approach was originally proposed for the nonlinear, time-dependent, work-hardening creep of dual-phase composites (Li and Weng, J Mech Phys Solids 45:1069–1083, 1997a), but by conceiving a nanocrystalline material as a composite of the stronger grain interior and the softer grain-boundary (GB, or grain-boundary affected zone GBAZ), it becomes possible to extend it to calculate the grain-size dependence of their flow stress, strain-rate sensitivity, and activation volume. We also use it to explain how the flow stress first increases and then decreases as the grain size decreases from the coarse grain to the nanometer range, leading to the Hall–Petch and the inverse Hall–Petch relations. The critical state at which the slope of the strength variation with respect to the grain size becomes zero also yields the strongest material state. In this way the two most important parameters for material design—the maximum strength and the critical grain at which it occurs—can be obtained. The strain-rate sensitivity parameters are also shown to follow a similar pattern as the flow stress, but the activation volume varies in an exactly reverse way.