Length-scale-dependent deformation mechanisms in incoherent metallic multilayered composites

Nano-indentation hardness as a function of bilayer period has been measured for sputter-deposited Cu–Nb multilayers. For this face-centered cubic/body-centered cubic system with incoherent interfaces, we develop dislocation models for the multilayer flow strength as a function of length scale from greater than a micrometer to less than a nanometer. A dislocation pile-up-based Hall–Petch model is found applicable at the sub-micrometer length scales and the Hall–Petch slope is used to estimate the peak strength of the multilayers. At the few to a few tens of nanometers length scales, confined layer slip of single dislocations is treated as the operative mechanism. The effects of dislocation core spreading along the interface, interface stress and interface dislocation arrays on the confined layer slip stress are incorporated in the model to correctly predict the strength increase with decreasing layer thickness. At layer thicknesses of a few nanometers or less, the strength reaches a peak. We postulate that this peak strength is set by the interface resistance to single dislocation transmission, and calculate the transition from confined layer slip to an interface cutting mechanism.     

A comparison of the strength of multilayers,thin films and nanocrystalline compacts

A survey of the yield strength and hardness of copper-based materials shows a progressive lowering of the strength with respect to the extrapolation of the Hall–Petch relation as the length scale of the microstructure decreases. The overall trend can be modeled by scaling the elastic screening length for the dislocation line tension with the microstructural length scale, as proposed by Scattergood and Koch.     

Towards a full analytic treatment of the Hall–Petch behavior in multilayers: putting the pieces together

A full analytic treatment of the Hall–Petch-like size-effect in multilayers must account for anomalous stress-concentration, dislocation source characteristics, and a variable dislocation pinning stress. From scaling arguments, the criteria for such an equation are described, and an approximate analytic formula is suggested that meets these criteria.     

Electron theory based explanation and experimental study on deformation behavior of Cu/Ni multilayers

Cu/Ni multilayers with various defined thickness of Cu and Ni layers were electrodeposited on low carbon steel substrates. Hardness measurements indicated that the increase in yield strength (one-third of hardness) with a decrease of layer thickness for Cu/Ni multilayers with single layer thickness at sub-micron length scale could be described by the Hall-Petch formula of the dislocation pile-up model. In the regime of few tens to a hundred nanometers of single layer thickness, the dislocation pileup-based Hall-Petch model broke down. This could be explained quantitatively according to the criterion condition on the limit size of dislocation derived from a modified Thomas-Fermi-Dirac electron theory.     

纯铁的细晶强化和加工硬化

提出一个考虑晶粒尺寸影响和形变中位错密度演化行为的理论模型，用于模拟纯铁的变形行为，特别是描述扩展的Ｈａｌ－Ｐｅｔｃｈ关系，与文献中的试验结果有较好的一致性。这表明晶粒尺寸对流动应力的影响是间接的，是通过其对变形后位错密度的影响来实现对流动应力的影响；而位错密度在形变中的演化有自己的内在规律，与晶粒尺寸无多大关系。     

调制波长对Cu／Ni金属多层膜力学性能的影响

用电沉积法在低碳钢基体上制备了具有不同调制波长(一个调制波长等于单层Cu膜与单层Ni膜厚度之和)的Cu/Ni金属多层膜，研究了多层膜硬度与其中单层膜厚度之间的关系。结果表明，当膜厚在亚微米范围内时，Cu/Ni多层膜的屈服强度(为硬度值的1/3)与单层膜厚之间符合基于位错塞积模型的Hall-Pctch(H-P)关系式；而当单层膜厚小于100nm时，屈服强度与膜厚的关系偏离了H-P线性关系。基于程开甲等人位错稳定性理论首次对金属多层膜变形行为偏离Hall-Petch关系的现象作了定量解释。     

The effect of alpha platelet thickness on plastic flow during hot working of TI–6Al–4V with a transformed microstructure

The effect of alpha platelet thickness on the plastic flow of Ti–6Al–4V with a transformed microstructure was established by conducting isothermal, hot compression tests at hot working temperatures on samples with identical crystallographic texture and beta grain size. Microstructures containing alpha laths/platelets ranging in thickness from approximately 0.4 to 10 μm were produced by various sequences consisting of hot rolling and heat treatment. Constant-strain-rate and strain-rate-jump compression tests were conducted at subtransus temperatures of 815, 900, and 955°C in the strain rate regime between 10⁻³ and 10 s⁻¹. The rate-jump tests suggested that plastic flow is controlled by a power-law creep (dislocation glide/climb) mechanism in all cases except the low-strain rate deformation of material with the thinnest alpha laths. All of the constant-strain-rate compression tests yielded flow curves consisting of a peak stress at low strains (≤0.03), extensive flow softening, and a steady-state flow stress at large strains. The peak stress results indicated a significant Hall–Petch dependence on alpha lath/platelet thickness at the two lower test temperatures. The magnitude of this dependence was predicted by the classical Eshelby expression for grain-size strengthening. In addition, a first-order analysis demonstrated that the observed flow softening is of the same magnitude as that which would be associated with the loss of Hall–Petch strengthening (due to alpha–beta interfaces) during hot working.     

What is behind the inverse Hall–Petch effect in nanocrystalline materials?

An inverse Hall–Petch effect has been observed for nanocrystalline materials by a large number of researchers. This effect implies that nanocrystalline materials get softer as grain size is reduced below a critical value. Postulated explanations for this behavior include dislocation-based models, diffusion-based models, grain-boundary-shearing models and two-phase-based models. In this paper, we report an explanation for the inverse Hall–Petch effect based on the statistical absorption of dislocations by grain boundaries, showing that the yield strength is dependent on strain rate and temperature and deviates from the Hall–Petch relationship below a critical grain size.     

The Hall–Petch breakdown in nanocrystalline metals: A crossover to glass-like deformation

The deformation behavior of nanocrystalline Ni–W alloys is evaluated by nanoindentation techniques for grain sizes of 3–150 nm, spanning both the range of classical Hall–Petch behavior as well as the regime where deviations from the Hall–Petch trend are observed. The breakdown in strength scaling, observed at a grain size of 10–20 nm, is accompanied by a marked transition to inhomogeneous, glass-like flow (i.e. shear banding) at the finest grain sizes approaching the amorphous limit. As a consequence of this mechanistic crossover, additional inflections arise in the mechanical properties; maxima are observed in both the rate and pressure dependence of deformation at approximately the same grain size as the onset of the Hall–Petch breakdown. These data experimentally connect the mechanical properties of nanocrystalline alloys to the well-known behavior of amorphous metals.     

Effects of Nb and Al on the microstructures and mechanical properties of high Nb containing TiAl base alloys

The influence of Nb and Al contents on the microstructure and yield strength of high Nb containing TiAl base alloys was investigated. The experimental results show that the yield strength at 900 °C of the alloys with the same type of microstructure, such as fully lamellar (FL), nearly lamellar (NL) and degraded fully lamellar (DFL), increases with increasing Nb content and decreasing Al content in the composition range of 0–10 at.% Nb and 44–49 at.% Al. DFL is the degraded form of FL microstructure after exposure at 1050 °C for 30 h. It is shown that the Nb addition in the alloys increases the value of the σ₀ term in the Hall–Petch relation of yield stress vs. lamellar spacing. This result has been related to TEM observations of dislocation structure in deformed specimens. The observations indicated that high level of Nb solute in the γ-TiAl matrix leads to a high critical resolved shear stress (CRSS) of dislocation loops. High Nb addition also reduces the degradation rate of FL microstructure after exposure at 1050 °C for 30 h. Both effects of high Nb addition are related to the change of the directionality of Ti–Ti (Nb) and Nb–Al bonds in the lattice. The decrease in Al content results in an increase in the volume fraction of α₂ phase, which leads to a decrease in the lamellar spacing of the lamellar structure. The high temperature strength of the alloys is determined by the lamellar spacing λ through the Hall–Petch equation k_λλ^−1/2.     

Sputter-deposited Cu/Cu(O) multilayers exhibiting enhanced strength and tunable modulus

By means of brief pauses in radiofrequency (RF) sputter deposition between individual layers, ultrathin copper oxide layers were formed through adsorption in the Cu/Cu multilayers. Their mechanical properties were compared with the Cu/Cu(O) multilayers whose oxide layers were deliberately deposited between copper layers. The mechanical hardness value of the Cu/Cu(O) multilayers approached that of nanostructured copper thin films. The Young’s modulus of the multilayers was tunable, in accordance with the elasticity theories of composites. In addition, the Hall–Petch slope of the RF sputter-deposited Cu monolayers indicated that their theoretical strength approached the shear modulus of copper.     

Scaling of discrete dislocation predictions for near-threshold fatigue crack growth

Analyses of the growth of a plane strain crack subject to remote mode I cyclic loading under small scale yielding are carried out using discrete dislocation dynamics. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with the lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation being incorporated through a set of constitutive rules. An irreversible relation is specified between the opening traction and the displacement jump across a cohesive surface ahead of the initial crack tip in order to simulate cyclic loading in an oxidizing environment. Calculations are carried out with different material parameters so that values of yield strength, cohesive strength and elastic moduli varying by factors of three to four are considered. The fatigue crack growth predictions are found to be insensitive to the yield strength of the material despite the number of dislocations and the plastic zone size varying by approximately an order of magnitude. The fatigue threshold scales with the fracture toughness of the purely elastic solid, with the experimentally observed linear scaling with Young’s modulus an outcome when the cohesive strength scales with Young’s modulus.     

On plasticity and fracture of nanostructured Cu/X (X = Au,Cr) multilayers: The effects of length scale and interface/boundary

Plastic deformation and fracture behavior of two different types of Cu/X (X = Au, Cr) multilayers subjected to tensile stress were investigated via three-point bending experiments. It was found that the plastic deformation ability and fracture mode depended on layer thickness and interface/boundary. The Cu/Au multilayer showed significant features of plastic flow before fracture, and such plasticity was gradually suppressed by premature unstable shearing across the layer interface with decreasing layer thickness. In comparison, Cu/Cr multilayers were prone to a quasi-brittle normal fracture with decreasing layer thickness. Both experimental observations and theoretical analyses revealed differences in plasticity and fracture mode between the two types of metallic multilayers and the relevant physical mechanism transition due to length scale constraint and interface/boundary blocking of dislocation motion.     

The grain size dependence of flow stress in a Cu–26Ni–17Zn alloy

The effect of grain size in the range 15–120 μm on flow stress was studied at room temperature to investigate the Hall–Petch relationship in a Cu–26Ni–17Zn alloy. It was found that the Hall–Petch relation is valid for the alloy. The Hall–Petch constants, σ_oε and k_ε are related to true strain (ε) in such a way that σ_oε is proportional to ε and k_ε to ε^1/2. An equation for flow stress as a function of true strain and grain size has been derived from these results. Dislocation density model for grain size strengthening is found valid for this alloy. Solid solution strengthening in the Cu–26Ni–17Zn alloy is attributed to the interaction of nickel and zinc atoms with screw dislocations and the effective interaction is more due to modulus mismatch than size misfit.     

The stress field of an array of parallel dislocation pile-ups: Implications for grain boundary hardening and excess dislocation distributions

Dislocation pile-ups at grain boundaries determine the back-stress opposing plastic deformation while promoting yielding in neighbouring grains. A regular array of parallel pile-ups of edge dislocations was analysed numerically to determine the equilibrium positions of all dislocations and compared with the result of various simplifications. A model based on infinite low angle boundaries could not reproduce the long-range stress field of the numerical calculations; an analytical correction for this simplification is presented. Infinite parallel dislocations overestimated the long-range stresses compared with finite segments. The effect of randomness in dislocation distributions was studied and average stress fields calculated, which were used to estimate the stress fields of more complex pile-up arrangements. Results for multiple pile-ups do not support classical arguments for the Hall–Petch relationship. Distributions of excess dislocations produce considerable long-range stresses which are not effectively screened by pile-ups of the opposite sign.     

A closer look at the local responses of twin and grain boundaries in Cu to stress at the nanoscale with possible transition from the P–H to the inverse P–H relation

Nanocrystalline copper is considered to be a candidate for electrical contacts for dynamic systems because of its intrinsic conductivity and enhanced fretting resistance. However, the enhanced electron scattering at high-density grain boundaries significantly deteriorates the overall conductivity of nanocrystalline copper. Recent studies suggest that nanosized twin boundaries in copper might be a solution to such a dilemma. To better understand the general mechanical behavior of nanotwin boundaries, we conducted molecular dynamics simulation studies to investigate responses of both nanotwin and nanograin boundaries in copper to stress at the nanoscale, particularly in the critical range of 5–25 nm where the inverse Petch–Hall relation (P–H) may occur in nanocrystalline copper. The obtained results suggest that the twin boundary blocks dislocation movement more effectively and the degree of emitting dislocations under stress is considerably lower than that of grain boundary, leading to superior mechanical behavior. The inverse P–H relation is not applicable to the nanotwinned system. It is also demonstrated that the inverse P–H relation occurring in nanograined materials does not necessarily result from grain boundary sliding.     

Temperature dependence of the flow stress of Fe–18Mn–0.6C–xAl twinning-induced plasticity steel

The tensile properties of Fe–18Mn–0.6C with Al-alloying additions of 0, 1.5 and 2.5 wt.% were investigated in the temperature range from 213 K (?60 °C) to 413 K (140 °C). The addition of Al resulted in an increase of the yield strength due to solid solution hardening and a decrease of the work hardening due to the suppression of deformation twinning. Both the decrease of the deformation temperature and the addition of Al suppressed the dynamic strain aging, clearly indicating an interaction between the stacking fault region of the mobile dislocation and Mn–C point defect complexes. A constitutive model for the temperature dependence of the flow stress, taking into account the thermally activated dislocation glide, Al solid solution hardening and the dynamic Hall–Petch effect caused by deformation twinning, was developed.     

The relationships of microstructure and properties of a fully lamellar TiAl alloy

The relationship between the yield strength and microstructure parameters of a fully lamellar TiAl alloy has been studied systematically. The grain size and the lamellar spacing were chosen as microstructure parameters. The experimental results showed that the yield strength increases with the decrease of grain size and more obviously with the decrease of the lamellar spacing. The relationship between yield strength and grain size and lamellar spacing can be approximately described by Hall–Petch relation.     

Analytical and computational description of effect of grain size on yield stress of metals

Four principal factors contribute to grain-boundary strengthening: (a) the grain boundaries act as barriers to plastic flow; (b) the grain boundaries act as dislocation sources; (c) elastic anisotropy causes additional stresses in grain-boundary surroundings; (d) multislip is activated in the grain-boundary regions, whereas grain interiors are initially dominated by single slip, if properly oriented. As a result, the regions adjoining grain boundaries harden at a rate much higher than grain interiors. A phenomenological constitutive equation predicting the effect of grain size on the yield stress of metals is discussed and extended to the nanocrystalline regime. At large grain sizes, it has the Hall–Petch form, and in the nanocrystalline domain the slope gradually decreases until it asymptotically approaches the flow stress of the grain boundaries. The material is envisaged as a composite, comprised of the grain interior, with flow stress σ_fG, and grain boundary work-hardened layer, with flow stress σ_fGB. The predictions of this model are compared with experimental measurements over the mono, micro, and nanocrystalline domains. Computational predictions are made of plastic flow as a function of grain size incorporating differences of dislocation accumulation rate in grain-boundary regions and grain interiors. The material is modeled as a monocrystalline core surrounded by a mantle (grain-boundary region) with a high work hardening rate response. This is the first computational plasticity calculation that accounts for grain size effects in a physically-based manner. A discussion of statistically stored and geometrically necessary dislocations in the framework of strain-gradient plasticity is introduced to describe these effects. Grain-boundary sliding in the nanocrystalline regime is predicted from calculations using the Raj–Ashby model and incorporated into the computations; it is shown to predispose the material to shear localization.     

Effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces – Part I: Theory

A phase field model corresponding to vacancy-mediated interdiffusion in coherent multilayers with completely miscible constituents is developed to explore the effects of several factors on interdiffusion across coherent multilayer interfaces, such as: (1) the dependence of diffusion potentials and mobilities on coherency stress; (2) the dependence of diffusion potentials and mobilities on composition; (3) the elastic constant inhomogeneity resulting from a inhomogeneous composition distribution; and (4) the properties of vacancy sources/sinks. The Gibbs free energy of the system consists of chemical and elastic energies. The gradient energy is neglected as the multilayers under consideration can be chemically well approximated by an ideal substitutional solution model. Elastic energy is a function of the stress-free strain and inhomogeneous elastic moduli distributions, while the stress is solved by anisotropic phase field microelasticity theory. The diffusion potentials are obtained straightforwardly as functional derivatives of the free energy with respect to composition and are in keeping with previous derivations that involved many mathematical manipulations or quite advanced theories. The diffusion mobilities are affected by the stress through modification of the vacancy formation and migration energies. Two limiting cases of vacancy sources/sinks are taken into account: ideal vacancy sources/sinks are uniformly and densely distributed, or not present at all, so the vacancy concentration is in equilibrium all the times, as determined by the local stress and composition in the former case, but deviates from the equilibrium concentration in the latter. The model can be conveniently extended to consider the non-ideal activity of vacancy sources/sinks by introducing a general kinetic relation for the vacancy creation rate.