Effect of source strength on dislocation pileups in the presence of stress gradients

The behaviour of a dislocation pileup with a finite-strength source is investigated in the presence of various stress gradients within a continuum model where a free-dislocation region exists around the source. Expressions for dislocation density and stress field within the pileup are derived for the situation where there are first and second spatial gradients in applied stress. For a pileup configuration under an applied stress, yielding occurs when the force acting on the leading dislocations at the pileup tips reaches the obstacle strength, and at the same time, it is required that the source be at the threshold stress for dislocation production. A numerical methodology is presented to solve the underlying equations that represent the yielding conditions. The yield stress calculated for a pileup configuration is found to depend on stress gradients, obstacle spacing and source/obstacle strengths. It increases with increasing the first stress gradient, yet dependent on the second stress gradient. Furthermore, while the dependency of yield stress on the obstacle spacing intensifies with increasing the first stress gradient, it diminishes with an increase of second stress gradient. Therefore, the second stress gradient, as a newly introduced parameter, can provide a new physical insight into the size-dependent plasticity phenomena at small length scales.     

Towards a further understanding of dislocation pileups in the presence of stress gradients

This study is an essential complement and extension to the stress-gradient concept recently proposed by Hirth. An analytic method is presented for studying the behaviour of double-ended dislocation pileup in the presence of various stress gradients by solving a singular integral equation based on the continuous approximation of dislocations. Four special cases of double-ended pileup in the presence of stress gradients are discussed in detail. The corresponding dislocation distribution, the length of pileup, the total number of dislocations within the pileup and the force on the leading dislocations at the pileup ends are derived, respectively. It is shown that both the number of dislocations and the force on the leading dislocation in a pileup are sensitive to the relative magnitude of stress near the dislocation source and both are less than that in constant stress case. Of particular importance, it is indicated that the small-scaled materials subjected to a stress involving a gradient would be stronger than that under a constant stress. Applied to wire torsion and foil bending, the stress gradient model predicts an increase in the initial yielding, which is in reasonable agreement with the recent experimental data. The proposed stress gradient concept may provide a new physical insight into the size-dependent plasticity phenomena at small length scale.     

颗粒增强金属基复合材料变形强化中的应变梯度效应

 针对单轴压缩实验,根据颗粒增强金属基复合材料中颗粒和基体两相的局部变形协调条件,并通过简单的位错模型,确定出与变形协调相应的几何必需位错密度,进而导出一种颗粒强化-应变梯度律。从中可以清楚地看出,颗粒增强金属基复合材料的强化由材料的微结构特征几何参数l和基体应变梯度联合控制。对于颗粒含量一定的复合材料,颗粒越小,应变梯度越高,强化效果越好。这一结果揭示了,颗粒强化及尺寸效应主要是通过应变梯度效应来表现的。这也同时说明,应变梯度可能是控制材料变形与断裂的重要因素之一。     

Non-singular dislocation continuum theories: strain gradient elasticity vs. Peierls–Nabarro model

Abstract

Non-singular dislocation continuum theories are studied. A comparison between Peierls–Nabarro dislocations and straight dislocations in strain gradient elasticity is given. The non-singular displacement fields, non-singular stresses, plastic distortions and dislocation core shapes are analysed and compared for the two models. The main conclusion of this study is that due to their characteristic properties, the non-singular displacement fields, non-singular stresses and dislocation core shape of screw and edge dislocations obtained in the framework of strain gradient elasticity are more realistic and physical than the corresponding fields of the Peierls–Nabarro model. Strain gradient elasticity of dislocations is a continuum dislocation theory including a weak non-locality within the dislocation core and predicting the size and shape of the dislocation core. The dislocation core is narrower in the strain gradient elasticity dislocation model than in the Peierls–Nabarro model and more evenly distributed in two dimensions. The present analysis shows that for the modelling of the dislocation core structure the non-singular dislocation fields of strain gradient elasticity are the suitable ones.     

Effects of size on the dynamics of dislocations in ice single crystals

Single crystals of ice subjected to primary creep in torsion exhibit a softening behavior: the plastic strain rate increases with time. In a cylindrical sample, the size of the radius affects this response. The smaller the radius of the sample becomes while keeping constant the average shear stress across a section, the softer the response. The size-dependent behavior is interpreted by using a field dislocation theory, in terms of the coupled dynamics of excess screw dislocations gliding in basal planes and statistical dislocations developed through cross slip occurring in prismatic planes. The differences in the results caused by sample height effects and variations in the initial dislocation microstructure are discussed.     

Two hardening mechanisms in single crystal thin films studied by discrete dislocation plasticity

Two-dimensional discrete dislocation plasticity simulations of the evolution of thermal stress in single crystal thin films on a rigid substrate are used to study size effects. The relation between the residual stress and the dislocation structure in the films after cooling is analyzed using dislocation dynamics. A boundary layer characterized by a high stress gradient and a high dislocation density is found close to the impenetrable film-substrate interface. There is a material-dependent threshold film thickness above which the dislocation density together with the boundary layer thickness and stress state are independent of film thickness. In such films the stress outside the boundary layer is on average very low, so that the film-thickness-independent boundary layer is responsible for the size effect. A larger size effect is found for films thinner than the threshold thickness. The origin of this size effect stems from nucleation activity being hindered by the geometrical constraint of the small film thickness, so that by decreasing film thickness, the dislocation density decreases while the stress in the film increases. The size dependence is only described by a Hall–Petch type relation for films thicker than the threshold value.     

Evaluating the effects of loading parameters on single-crystal slip in tantalum using molecular mechanics

This study is aimed at developing a physics-based crystal plasticity finite element model for body-centred cubic (BCC) metals, through the introduction of atomic-level deformation information from molecular dynamics (MD) investigations of dislocation motion at the onset of plastic flow. In this study, three critical variables governing crystal plasticity mediated by dislocation motion are considered. MD simulations are first performed across a range of finite temperatures up to 600K to quantify the temperature dependence of critical stress required for slip initiation. An important feature of slip in BCC metals is that it is not solely dependent on the Schmid law measure of resolved shear stress, commonly employed in crystal plasticity models. The configuration of a screw dislocation and its subsequent motion is studied under different load orientations to quantify these non-Schmid effects. Finally, the influence of strain rates on thermal activation is studied by inducing higher stresses during activation at higher applied strain rates. Functional dependence of the critical resolved shear stress on temperature, loading orientation and strain rate is determined from the MD simulation results. The functional forms are derived from the thermal activation mechanisms that govern the plastic behaviour and quantification of relevant deformation variables. The resulting physics-based rate-dependent crystal plasticity model is implemented in a crystal plasticity finite element code. Uniaxial simulations reveal orientation-dependent tension–compression asymmetry of yield that more accurately represents single-crystal experimental results than standard models.     

Dislocation density-based constitutive model for the mechanical behaviour of irradiated Cu

Performance degradation of structural steels in nuclear environments results from the formation of a high number density of nanometre-scale defects. The defects observed in copper-based alloys are composed of vacancy clusters in the form of stacking fault tetrahedra and/or prismatic dislocation loops that impede the motion of dislocations. The mechanical behaviour of irradiated copper alloys exhibits increased yield strength, decreased total strain to failure and decreased work hardening as compared to their unirradiated behaviour. Above certain critical defect concentrations (neutron doses), the mechanical behaviour exhibits distinct upper yield points. In this paper, we describe the formulation of an internal state variable model for the mechanical behaviour of such materials subject to these (irradiation) environments. This model has been developed within a multiscale materials-modelling framework, in which molecular dynamics simulations of dislocation–radiation defect interactions inform the final coarse-grained continuum model. The plasticity model includes mechanisms for dislocation density growth and multiplication and for irradiation defect density evolution with dislocation interaction. The general behaviour of the constitutive (homogeneous material point) model shows that as the defect density increases, the initial yield point increases and the initial strain hardening decreases. The final coarse-grained model is implemented into a finite element framework and used to simulate the behaviour of tensile specimens with varying levels of irradiation-induced material damage. The simulation results compare favourably with the experimentally observed mechanical behaviour of irradiated materials.     

Plastic shear response of a single asperity: a discrete dislocation plasticity analysis

We investigate the plastic shear response during static friction of an asperity protruding from a large FCC single crystal. The asperity is in perfectly adhesive contact with a rigid platen and is sheared by tangentially moving the platen. Using discrete dislocation plasticity simulations, we elucidate the plastic shear behaviour of single asperities of various size and shape, in search for the length scale that controls the plastic behaviour. Since plasticity can occur also in the crystal, identification of the length scale that controls a possible size-dependent plastic behaviour is far from being trivial. It is found that scaling down the dimensions of an asperity results in a higher contact shear strength. The contact area is dominant in controlling the plastic shear response, because it determines the size of the zone, in and below the asperity, where dislocation nucleation can occur. For a specific contact area, there is still a dependence on asperity volume and shape, but this is weaker than the dependence on contact area alone.     

Modeling the deformation behavior of nanocrystalline alloy with hierarchical microstructures

A mechanism-based plasticity model based on dislocation theory is developed to describe the mechanical behavior of the hierarchical nanocrystalline alloys. The stress–strain relationship is derived by invoking the impeding effect of the intra-granular solute clusters and the inter-granular nanostructures on the dislocation movements along the sliding path. We found that the interaction between dislocations and the hierarchical microstructures contributes to the strain hardening property and greatly influence the ductility of nanocrystalline metals. The analysis indicates that the proposed model can successfully describe the enhanced strength of the nanocrystalline hierarchical alloy. Moreover, the strain hardening rate is sensitive to the volume fraction of the hierarchical microstructures. The present model provides a new perspective to design the microstructures for optimizing the mechanical properties in nanostructural metals.     

Deformation of a nanocube with a single incoherent precipitate: role of precipitate size and dislocation looping

ABSTRACT

Precipitate hardening is a key strengthening mechanism in metallic alloys. Classical models for precipitate hardening are based on the average behaviour of an ensemble of precipitates, and fail to capture the complexity of dislocation-precipitate interactions that have recently been observed at individual precipitates in simulations and in-situ electron microscopy. In order to achieve tailored mechanical properties, detailed deformation mechanisms at specific precipitates that account for precipitate size, crystallography, and defect structure must be understood, but has been challenging to achieve experimentally. Here, in-situ scanning electron microscope mechanical testing is used to obtain the compressive stress–strain behaviour at an individual, incoherent Au precipitate within a Cu nanocube, and determine the influence of precipitate and cube size on yield strength and strain hardening. TEM imaging and strain mapping of the initial structure shows misfit dislocations at the Au precipitate, threading dislocations that traverse the Cu shell, and localised and anisotropic strain near the precipitate and threading dislocation. These nanocubes have yield strengths of 800–1000?MPa and strain hardening rate of 1–4?GPa. Yield strength is found to depend on the distance from the precipitate interface to the cube edge, while strain hardening depends on both cube size and precipitate size. An analytical model is developed to quantify the contribution of Orowan looping, Orowan stress, back stress and image stress to plasticity at the Au precipitate. Orowan stress is found to be the largest contributor, followed by back stress and image stress.     

Contact between two plastically deformable crystals: a discrete dislocation dynamics study

It is customary to simplify the analysis of contact between two elastically deformable bodies by treating an equivalent problem where only one body is deformable and the other is rigid. This is possible provided that the gap geometry and the effective elastic modulus of the bodies in the simplified problem are the same as in the original problem. However, the question arises on whether – and to which extent – the simplification is still valid even when (size-dependent) plasticity occurs. Studies using discrete dislocation plasticity have also, so far, addressed simple contact problems where only one body can deform plastically. Here, we extend the analysis to two bodies in contact that can both deform by dislocation plasticity and investigate under which conditions the response agrees with that of an equivalent simplified problem. The bodies in contact are metal single crystals with sinusoidal and flat surface. It is found that the response of two plastically deformable bodies in contact can be simplified to an equivalent problem where one body is rigid and the other can deform plastically. Also, a plasticity size effect is observed, but the effect fades when the platen becomes more plastically deformable.     

Projecting low and extensive dimensional chaos from spatio-temporal dynamics

We review the spatio-temporal dynamical features of the Ananthakrishna model for the Portevin-Le Chatelier effect, a kind of plastic instability observed under constant strain rate deformation conditions. We then establish a qualitative correspondence between the spatio-temporal structures that evolve continuously in the instability domain and the nature of the irregularity of the scalar stress signal. Rest of the study is on quantifying the dynamical information contained in the stress signals about the spatio-temporal dynamics of the model. We show that at low applied strain rates, there is a one-to-one correspondence with the randomly nucleated isolated bursts of mobile dislocation density and the stress drops. We then show that the model equations are spatio-temporally chaotic by demonstrating the number of positive Lyapunov exponents and Lyapunov dimension scale with the system size at low and high strain rates. Using a modified algorithm for calculating correlation dimension density, we show that the stress-strain signals at low applied strain rates corresponding to spatially uncorrelated dislocation bands exhibit features of low dimensional chaos. This is made quantitative by demonstrating that the model equations can be approximately reduced to space independent model equations for the average dislocation densities, which is known to be low-dimensionally chaotic. However, the scaling regime for the correlation dimension shrinks with increasing applied strain rate due to increasing propensity for propagation of the dislocation bands. The stress signals in the partially propagating to fully propagating bands turn to have features of extensive chaos.     

A dislocation-based stress-strain gradient plasticity model for strength and ductility in materials with gradient microstructures

Although metallic materials with gradient microstructure exhibit notable performance in harsh environmental conditions, they can also exhibit unusual mechanical behaviour. This is attributed to both grain size and the gradient of grain size distribution in the structure. Metallic materials with a homogenous distribution of grain size follow the traditional Hall-Petch relationship, in which strength increases with decreasing grain size at the expense of ductility. However, studies show that materials with a gradient of grain size microstructure do not follow the Hall-Petch relationship, and thus have improved strength and ductility. This suggests that with creative design and engineering of microstructure, the strength-ductility trade-off can be reduced or prevented.

In this study, we developed and implemented a dislocation density based model to investigate the mechanical behaviour of nano-microstructure. We designed a multi-scale modelling framework, coupling VPSC (Viscoplastic Self Consistent model) with CDD (Continuum Dislocation Dynamics), applying crystal plasticity equations to simulate dislocation interaction in polycrystalline metallic materials. We also developed design parameters and a model to predict the strength and ductility of materials with gradient microstructure.     

Dislocation and defect density-based micromechanical modeling of the mechanical behavior of fcc metals under neutron irradiation

A rate-independent dislocation and defect density-based evolution model is presented that captures the pre- and post-yield material behavior of fcc metals subjected to different doses of neutron radiation. Unlike previously developed phenomenological models, this model is capable of capturing the salient features of irradiation-induced hardening, including increase in yield stress followed by yield drop and non-zero stress offset from the unirradiated stress–strain curve. The key contribution is a model for the critical resolved slip resistance that depends on both dislocation and defect densities, which are governed by evolution equations based on physical observations. The result is an orientation-dependent non-homogeneous deformation model, which accounts for defect annihilation on active slip planes. Results for both single and polycrystalline simulations of OFHC copper are presented and are observed to be in reasonably good agreement with experimental data. Extension of the model to other fcc metals is straightforward and is currently being developed for bcc metals.     

Structural states and specific features of the deformation behavior of metals and alloys with mixed nano-and micrograin structure

A classification of the structural states of materials with a mixed nano-and microcrystalline structure is proposed. Theoretical analysis of the structural mechanisms and peculiarities of plastic flow of singlephase and two-phase nanostructured metals and alloys with a bimodal size distribution of grains and phases is performed. The effect of grain-boundary and dislocation mechanisms of plastic flow on the specific features of the deformation behavior and plasticity of nanocrystalline materials is analyzed. A microstructural model of strain hardening of a material with two-scale nano-and micrograin structure is proposed and the condition for the loss of plastic flow stability of such a material is investigated. The dependence of the yield strength and uniform strain of nanocrystalline materials with a two-scale structure on the grain size and the ratio of the volume fractions of the nano-and microstructural components is calculated.     

Indenter size effect on the reversible incipient plasticity of Al(001) surface:Quasicontinuum study

Indenter size effect on the reversible incipient plasticity of Al(001) surface is studied by quasicontinuum simulations.Results show that the incipient plasticity under small indenter, the radius of which is less than ten nanometers, is dominated by a simple planar fault defect that can be fully removed after withdrawal of the indenter; otherwise, irreversible incipient plastic deformation driven by a complex dislocation activity is preferred, and the debris of deformation twins, dislocations,and stacking fault ribbons still remain beneath the surface when the indenter has been completely retracted. Based on stress distributions calculated at an atomic level, the reason why the dislocation burst instead of a simple fault ribbon is observed under a large indenter is the release of the intensely accumulated shear stress. Finally, the critical load analysis implies that there exists a reversible-irreversible transition of incipient plasticity induced by indenter size. Our findings provide a further insight into the incipient surface plasticity of face-centered-cubic metals in nano-sized contact issues.     

Size dependence of the yield strength of fcc and bcc metallic micropillars with diameters of a few micrometers

Recent micropillar compression tests of fcc and bcc single crystals have shown that ‘Smaller is Stronger’ even in the absence of significant strain gradients, an effect that is empirically characterised by a power-law relation. When a micropillar contains a dislocation network, this power-law relation has been explained in terms of the size-dependent operation stress of the weakest single arm dislocation sources. This single arm dislocation source model has successfully captured the power-law relation for the strength of a few fcc micropillars, but a physical interpretation has not been made by comparing different materials. We applied the model, not only to fcc but also to bcc micropillars, to understand quantitatively why different materials have different power-law exponents. Here, the different power-law exponents are interpreted by comparing material parameters that are size-independent properties. Also, by rearranging these parameters such that the formulation becomes independent of material parameters, we found an alternate form of the scaling law that is a unique function of micropillar diameter. Furthermore, recent experimental studies of the effects of increasing the dislocation density, which show hardening for large micropillars and softening for small micropillars, are interpreted in terms of the statistics of dislocation source distribution. The strengths and limitations of this statistical approach are discussed. The effects of temperature on the power-law exponents are also studied.     

Effect of passivation on higher order gradient plasticity models for non-proportional loading: energetic and dissipative gradient components

Abstract

In this work, a new class of thermodynamic-based higher order gradient plasticity theory is proposed and applied to the stretch-surface passivation problem for investigating the material behaviour under the non-proportional loading condition. This paper incorporates the thermal and mechanical responses of microsystems. It addresses phenomena such as size and boundary effects and in particular microscale heat transfer in fast-transient processes. The stored energy of cold work is considered in the development of the recoverable counterpart of the free energy. The main distinction in this formulation is the presence of the dissipative higher order microstress quantity that is known to give rise to the stress jump phenomenon, which causes a controversial dispute in the field of strain gradient plasticity theory with respect to whether it is physically acceptable or not. The finite element solution for the stretch-surface passivation problem is developed and validated by comparing with three sets of small-scale experiments. Based on the validated finite element solution, the stress jump phenomenon under the stretch-surface passivation condition is investigated with the effects of the various material parameters. The evolution of the free energy and dissipation potentials is investigated at elevated temperatures. The two-dimensional simulation is also given to examine the gradient and grain boundary effect, the mesh sensitivity of the two-dimensional model and the stress jump phenomenon. Finally, some significant conclusions are presented.