Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in Al

The interaction of dislocations with grain boundaries (GBs) determines a number of important aspects of the mechanical performance of materials, including strengthening and fatigue resistance. Here, the coupled atomistic/discrete-dislocation (CADD) multiscale method, which couples a discrete dislocation continuum region to a fully atomistic region, is used to study screw-dislocations interacting with Σ3, Σ11, and Σ9 symmetric tilt boundaries in Al. The low-energy Σ3 and Σ11 boundaries absorb lattice dislocations and generate extrinsic grain boundary dislocations (GBDs). As multiple screw dislocations impinge on the GB, the GBDs form a pile-up along the GB and provide a back stress that requires increasing applied load to push the lattice dislocations into the GB. Dislocation transmission is never observed, even with large GBD pile-ups near the dislocation/GB intersection. Results are compared with experiments and previous, related simulations. The Σ9 grain boundary, composed from a more complex set of structural units, absorbs screw dislocations that remain localized, with no GBD formation. With increasing applied stress, new screw dislocations are then nucleated into the opposite grain from structural units in the GB that are nearby but not at the location where the original dislocation intersected the boundary. The detailed behaviour depends on the precise location of the incident dislocations and the extent of the pile-up. Transmission can occur on both Schmid and non-Schmid planes and can depend on the shear stresses on the GB plane. A continuum yield locus for transmission is formulated. In general, the overall dissociation and/or transmission behaviour is also determined by the Burgers vectors and associated steps of the primitive vectors of the grain boundary, and the criteria for dislocation transmission formulated by Lee et al . [Scripta Metall. 23 799 (1989); Phil. Mag. A 62 131 (1990); Metall. Trans. A 21 2437 (1990)] are extended to account for these factors.     

Interaction of dislocation pile-up with a low-angle tilt boundary: a discrete dislocation dynamics study

Abstract

The pile-up of dislocations between two low-angle tilt boundaries (LATB) in an fcc crystal was simulated using three-dimensional discrete dislocation dynamics. The LATB was constructed using glissile edge dislocations stacked on each other. The dislocations in the pile-up were chosen such that their reactions with the dislocations in the LATB resulted in glissile junctions. Parallel pairs of dislocations were inserted to a maximum allowable value estimated from theoretical expressions. A resolved shear stress was applied and increased in steps so as to move the dislocations in the pile-up towards the boundaries. The shear stress required to break the lead dislocation from the wall was determined for varying spacings between the two boundaries. The shear stress and boundary spacing followed the Hall–Petch type relation. Dislocation pile-ups without a LATB were also simulated. The spacing of the dislocations in the pile-up with LATB was found to be closer (ie higher dislocation density) than that without LATB. It was shown through analytical expressions that LATB exerts an attractive force on the dislocations in the pile-up thereby creating a denser pile-up.     

Continuum model for dislocation dynamics in a slip plane

In this paper, we present a continuum model for dislocation dynamics in a slip plane, which accurately incorporates both the long-range interaction and the local line tension effect of dislocations. Unlike the continuum models in the literature using dislocation densities, we use the disregistry across the slip plane to represent the continuous distribution of dislocations in the slip plane, which has the advantage of including the orientation dependence of dislocations in a very simple way. The continuum dislocation dynamics model is validated by linear instability analysis of a uniform dislocation array to small perturbations and comparisons of the results with those of the discrete dislocation dynamics model. Numerical examples for the evolution of distributions of dislocations and plastic slips in a slip plane are presented.     

A computational approach towards modelling dislocation transmission across phase boundaries

To study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls–Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation–phase boundary interaction in two-phase microstructures.     

Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary

In 1965, Armstrong and Head explored the problem of a pile-up of screw dislocations against a grain boundary. They used numerical methods to determine the positions of the dislocations in the pile-up and they were able to fit approximate formulae for the locations of the first and last dislocations. These formulae were used to gain insights into the Hall–Petch relationship. More recently, Voskoboinikov et al. used asymptotic techniques to study the equivalent problem of a pile-up of a large number of screw dislocations against a bimetallic interface. In this paper, we extend the work of Voskoboinikov et al. to construct systematic asymptotic expressions for the formulae proposed by Armstrong and Head. The further extension of these techniques to more general pile-ups is also outlined. As a result of this work, we show that a pile-up against a grain boundary can become equivalent to a pile-up against a locked dislocation in the case where the mismatch across the boundary is small.     

Grain boundary sliding during ambient-temperature creep in hexagonal close-packed metals

Even at ambient temperature or less, below their 0.2% proof stresses all hexagonal close-packed metals and alloys show creep behaviour because they have dislocation arrays lying on a single slip system with no tangled dislocation inside each grain. In this case, lattice dislocations move without obstacles and pile-up in front of a grain boundary. Then these dislocations must be accommodated at the grain boundary to continue creep deformation. Atomic force microscopy revealed the occurrence of grain boundary sliding (GBS) in the ambient-temperature creep region. Lattice rotation of 5° was observed near grain boundaries by electron backscatter diffraction pattern analyses. Because of an extra low apparent activation energy of 20 kJ/mol, conventional diffusion processes are not activated. To accommodate these piled-up dislocations without diffusion processes, lattice dislocations must be absorbed by grain boundaries through a slip-induced GBS mechanism.     

Microstructural instabilities: dislocation substructures with pronounced mechanical effects

Microstructure evolution is largely dominated by the internal stress fields that appear upon the appearance of inhomogeneous structures in a material. The hardening behaviour of metals physically originates from such a complex microstructure evolution. As deformation proceeds, statistically homogeneous distributions of dislocations in grains become unstable, which constitutes the driving force for the development of a pronounced dislocation substructure. The dislocation structure already appears at early stages of deformation due to the statistical trapping of dislocations. Cell walls contain dislocation dipoles and multipoles with high dislocation densities and enclose cell-interior regions with a considerably smaller dislocation density. The presence and evolution of such a dislocation arrangement in the material influence the mechanical response of the material and is commonly associated with the transient hardening after strain path changes. This contribution introduces a micromechanical continuum model of the dislocation cell structure based on the physics of the dislocation interactions. The approximation of the internal stress field in such a microstructure and the impact on the macroscopic mechanical response are the main items investigated here.     

Plastic slip distribution in two-phase laminate microstructures: Dislocation-based versus generalized-continuum approaches

The size-dependent mechanical response of a simple model microstructure is investigated using continuum dislocation-based, Cosserat and strain-gradient models of crystal plasticity. The governing equations and closed-form analytical solutions for plastic slip and lattice rotation are directly compared. The microstructure consists of a periodic succession of hard (elastic) and soft (elastoplastic single-crystal) layers, subjected to single glide perpendicular to the layers. In the dislocation-based approach, inhomogeneous plastic deformation and lattice rotation are shown to develop in the soft channels, either because of bowing of dislocations or owing to pile-up formation. The generalized continuum non-local models are found to be able to reproduce the plastic slip and lattice rotation distribution. In particular, a correspondence was found between the generalized-continuum results and line tension effects; the additional or higher- order balance equations introduced in the non-local models turn out to be the counterparts of the equilibrium equation for bowed dislocations. The relevance and possible physical interpretation of additional or higher-order interface conditions responsible for the inhomogeneous distribution of plastic slip and lattice rotations are discussed.     

Properties of solitons in the Frenkel-Kontorova model

An exact numerical calculation is presented for the properties of solitons in the discrete model of Frenkel and Kontorova. We have calculated the single soliton energy, the Peierls pinning energy and the soliton-soliton interaction energy. The exponential behaviour predicted for the soliton-soliton interaction by the continuum approximation is found to hold even for large soliton densities and very strong substrate potentials though in the latter case, the coefficients differ substantially from those of the continuum approximation.     

Climb via vacancy diffusion of edge dislocations in 2D dislocation microstructures

The prevention of strength degradation of components is one of the great challenges in solid mechanics. In particular, at high temperatures material may deform even at low stresses, a deformation mode known as deformation creep. One of the microstructural mechanisms that governs deformation creep is dislocation motion due to the absorption or emission of vacancies, which results in motion perpendicular to the glide plane, called dislocation climb. However, the importance of the dislocation network for the deformation creep remains far from being understood. In this study, a climb model that accounts for the dislocation network is developed, by solving the diffusion equation for vacancies in a region with a general dislocation distribution. The definition of the sink strength is extended, to account for the contributions of neighbouring dislocations to the climb rate. The model is then applied to dislocation dipoles and dislocation pile-ups, which are dense dislocation structures and it is found that the sink strength of dislocations in a pile-up is reduced since the vacancy field is distributed between the dislocations. Finally, the importance of the results for modelling deformation creep is discussed.     

Periodic image effects in dislocation modelling

The use of periodic boundary conditions for modelling crystal dislocations is predicated on one's ability to handle the inevitable image effects. This communication deals with an often overlooked mathematical subtlety involved in dealing with the periodic dislocation arrays, that is conditional convergence of the lattice sums of image fields. By analysing the origin of conditional convergence and the numerical artefacts associated with it, we establish a mathematically consistent and numerically efficient procedure for regularization of the lattice sums and the corresponding image fields. The regularized solutions are free from the artefacts caused by conditional convergence and regain periodicity and translational invariance of the periodic supercells. Unlike the other existing methods, our approach is applicable to general anisotropic elasticity and arbitrary dislocation arrangements. The capabilities of this general methodology are demonstrated by application to a variety of situations encountered in atomistic and continuum modelling of crystal dislocations. The applications include introduction of dislocations in the periodic supercell for subsequent atomistic simulations, atomistic calculations of the core energies and the Peierls stress and continuum dislocation dynamics simulations in three dimensions.     

Quantifying the grain boundary resistance against slip transfer by experimental combination of geometric and stress approach using stage-I-fatigue cracks

In recent studies, many groups have investigated the interaction of dislocations and grain boundaries by bi-crystals and micro-specimen experiments. Partially, these experiments were combined with supplementary simulations by discrete dislocation dynamics, but quantitative data for the grain boundary resistance against slip transfer is still missing. In this feasibility study with first results, we use stage-I-fatigue cracks as highly localised sources for dislocations with well-known Burgers vectors to study the interaction between dislocations in the plastic zone in front of the crack tip and selected grain boundaries. The stress concentration at the grain boundary is calculated with the dislocation-free zone model of fracture using the dislocation density distribution in the plastic zone from slip trace height profile measurements by atomic force microscopy. The grain boundary resistance values calculated from common geometric models are compared to the local stress distribution at the grain boundaries. Hence, it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer to a self-contained concept. As a result, the prediction of the grain boundary resistance effect based on a critical stress concept is possible with knowledge of the geometric parameters of the grain boundary only, namely the orientations of both participating grains and the orientation of the grain boundary plane.     

Dislocation Plasticity Effects on Interfacial Fracture

Analyses are reviewed where plastic flow in the vicinity of an interfacial crack is represented in terms of the nucleation and glide of discrete dislocations. Attention is confined to cracks along a metal-ceramic interface, with the ceramic idealized as being rigid. Both monotonic and fatigue loading are considered. The main focus is on the stress and deformation fields near the crack tip predicted by discrete dislocation plasticity, in comparison with those obtained from conventional continuum plasticity theory. The role that discrete dislocation plasticity can play in interpreting interface fracture properties in the presence of plastic flow is discussed.     

Grain-boundary dislocations and enhanced diffusion in nanocrystalline bulk materials and films

A theoretical model is suggested which describes the transformations of grain-boundary dislocation walls and their influence on diffusion processes in nanocrystalline materials fabricated under highly non-equilibrium conditions. It is shown that the decay of boundary dislocation walls of finite extent, occurring via the climb of boundary dislocations and the corresponding emission of vacancies, is capable of highly enhancing the grain-boundary diffusion in nanocrystalline materials. The enhanced diffusion, in turn, strongly affects the deformation behaviour of nanocrystalline materials. In the case of nanocrystalline films deposited on to substrates, the effects of misfit stresses on the transformations of boundary dislocation walls and the diffusion are analysed. It is demonstrated that the mean diffusion coefficient in a nanocrystalline film may increase by approximately several orders of magnitude owing to misfit stresses.     

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.     

Continuous description of a grain boundary in forsterite from atomic scale simulations: the role of disclinations

We present continuous modelling at inter-atomic scale of a high-angle symmetric tilt boundary in forsterite. The model is grounded in periodic arrays of dislocation and disclination dipoles built on information gathered from discrete atomistic configurations generated by molecular dynamics simulations. The displacement, distortion (strain and rotation), curvature, dislocation and disclination density fields are determined in the boundary area using finite difference and interpolation techniques between atomic sites. The distortion fields of the O, Si and Mg sub-lattices are detailed to compare their roles in the accommodation of lattice incompatibility along the boundary. It is shown that the strain and curvature fields associated with the dislocation and disclination fields in the ‘skeleton’ O and Si sub-lattices accommodate the tilt incompatibility, whereas the elastic strain and rotation fields of the Mg sub-lattice are essentially compatible and induce stresses balancing the incompatibility stresses in the overall equilibrium.     

Absorption and generation of electromagnetic waves by structural inhomogeneities of solid

The integrability conditions of the electromagnetic field equations in a continuum with defects and their wave solutions are found. The following dislocation effects on the electromagnetic wave propagation in a continuous medium are investigated: the change in the direction of the electromagnetic wave propagation in a continuous medium; the rotation of the polarization plane of electromagnetic field wave in a continuous medium; the excitation of longitudinal components of the electromagnetic wave in a continuous medium and the change in the electromagnetic wave intensity related to this phenomenon. The energy balance equation for the electromagnetic field in a continuum with a stationary distribution of dislocation is found and it is shown that an electromagnetic wave excites exciton modes localized at dislocations in the solid.     

Dislocation emission at the onset of plasticity during nanoindentation in gold

Nanoindentations and the subsequent plastic damage in the form of dislocation configurations have been both generated and imaged with scanning tunnelling microscopy on a reconstructed Au(001) surface, the resulting observations being interpreted in terms of the elastic theory of dislocations in a continuum. The rearranged pileup material around the nanoindentation is described in terms of dislocation emission and glide involving, in particular, multiple cross-slip. ‘Mesas’, shallow protusions stemming from a special dislocation configuration consisting of Schockley partial dislocations encompassing two stacking faults, are shown to glide parallel to the surface under the stress generated by further nanoindentations. The spatial distribution of ‘mesas’ around the nanoindentation traces is shown to be controlled by a balance between the interactions between the different ‘mesas’ and the stresses arising from the nanoindentation itself.     

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.