Conservation Of Burgers Vector Research Articles

Application of rigorous interface boundary conditions in mesoscale plasticity simulations

The interactions between dislocations and interface/grain boundaries, including dislocation absorption, transmission, and reflection, have garnered significant attention from the research community for their impact on the mechanical properties of materials. However, the traditional approaches used to simulate grain boundaries lack physical fidelity and are often incompatible across different simulation methods. We review a new mesoscale interface boundary condition based on Burgers vector conservation and kinetic dislocation reaction processes. The main focus of the paper is to demonstrate how to unify this boundary condition with different plasticity simulation approaches such as the crystal plasticity finite element (CPFEM), continuum dislocation dynamics (CDD), and discrete dislocation dynamics (DDD) methods. In DDD and CDD, plasticity is simulated based on dislocation activity; in the former, dislocations are described as discrete lines while in the latter in terms of dislocation density. CPFEM simulates plasticity in terms of slip on each slip system, without explicit treatment of dislocations; it is suitable for larger scale simulations. To validate our interface boundary condition, we implemented simulations using both the CPFEM method and a two-dimensional CDD model. Our results show that our compact and physically realistic interface boundary condition can be easily integrated into multiscale simulation methods and yield novel results consistent with experimental observations.

The structure of deformation twins in BCC transition metals during nucleation and growth

Nucleation and growth of deformation twins in BCC transition metals have been investigated in many computational and experimental studies without conclusively describing the critical twin nuclei thickness and its subsequent growth. While most studies conclude that the thickness is three layers by assuming conservation of Burgers vector and nucleation from a screw dislocation, recent experimental studies have contradicted this, suggesting a minimum of six layers. Here we demonstrate, under nonzero finite pure shear stresses, that the minimum thickness during twin nucleation is two layers in Group VB and VIB metals whereas it is three layers in iron. The twinning dislocations that constitute the nuclei have been determined which are dependent on metal's electronic structure i.e, position in periodic table. Growth of twins occurs in a layer-by-fashion maintaining isosceles twin boundary structures in Group VB metals and Fe whereas the more commonly acknowledged reflection twin boundaries are favored in Group VIB.

Dislocation nucleation in the phase-field crystal model

We use the phase field crystal model to study nucleation of edge dislocations in two dimensions under an applied stress field. A dislocation dipole nucleates under the applied stress, consistent with Burgers vector conservation. The phase field correctly accounts for elastic energy storage prior to nucleation, and for dissipative relaxation during the nucleation event. We show that a lattice incompatibility field is a sensitive diagnostic of the location of the nucleation event, and of the Burgers vector and slip direction of the dislocations that will be nucleated above threshold. A direct calculation of the phase field energy accurately correlates with the nucleation event, as signaled by the lattice incompatibility field. We show that a Schmid-like criterion concerning the resolved stress at the nucleation site correctly predicts the critical nucleation stress. Finally, we present preliminary results for a three-dimensional, body-centered cubic lattice. The phase field allows a direct computation of the lattice incompatibility tensor for both dislocation lines and loops.

Curvature effects on boundary migration

We investigate continuity constraints on the plastic distortion and distortion rate tensors across surfaces of discontinuity moving through elastoplastic bodies. Tangential continuity of these tensors across interfaces moving normally to themselves derive from the conservation of the Burgers vector over patches bridging the interface, in the limit where such patches shrink onto the interface. Along interfaces whose motion involves a rotation, Burgers vector conservation follows from a trade-off between any tangential discontinuity of the dislocation-mediated plastic distortion rate relative to the interface and a plastic distortion rate arising from the rotation of the interface. The interface rotation dynamics follows from this balance. Additionally using the thermodynamic requirement of positive interface dissipation allows formulating mobility laws for the motion of the interface. The simplest admissible constitutive relationship relates the interface velocity to the traction vector. This mobility relationship recovers and develops in a tensorial context the conventional relations for grain boundary migration and grain growth in recovery/dynamic recrystallization processes, but challenges the constitutive character of the mobility parameter.

Solving the Puzzle of⟨100⟩Interstitial Loop Formation in bcc Iron

The interstitial loop is a unique signature of radiation damage in structural materials for nuclear and other advanced energy systems. Unlike other bcc metals, two types of interstitial loops, 1/2<111> and <100>, are formed in bcc iron and its alloys. However, the mechanism by which <100> interstitial dislocation loops are formed has remained undetermined since they were first observed more than fifty years ago. We describe our atomistic simulations that have provided the first direct observation of <100> loop formation. The process was initially observed using our self-evolving atomistic kinetic Monte Carlo method, and subsequently confirmed using molecular dynamics simulations. Formation of <100> loops involves a distinctly atomistic interaction between two 1/2<111> loops, and does not follow the conventional assumption of dislocation theory, which is Burgers vector conservation between the reactants and the product. The process observed is different from all previously proposed mechanisms. Thus, our observations might provide a direct link between experiments and simulations and new insights into defect formation that may provide a basis to increase the radiation resistance of these strategic materials.

Interactions between edge lattice dislocations and Σ11 symmetrical tilt grain boundaries in copper: A quasi-continuum method study

“Positive” and “negative” interactions between specified numbers of piled-up lattice dislocations and +Σ11(3¯11¯) and -Σ11(1¯31) symmetrical tilt grain boundaries in two bi-grain models of copper have been studied using the quasi-continuum method. The mechanisms of both interactions in representative loading stages are analyzed in terms of the evolution of grain boundary (GB) configurations and Burgers vector conservation. A unified geometrical criterion is proposed to predict “hard” and “easy” dislocation transmission without explicitly defining the “positive” and “negative” dislocation/GB interactions. The pile-up of incoming dislocations has a remarkable influence on both “positive” and “negative” interactions, and can activate dislocation transmission at smaller external loadings. Critical stress analysis for relatively easier dislocation transmission in “negative” interactions reveals that they depend on the ratio of total shear stresses on the incoming and outgoing slip planes and the ratio of total normal and shear stresses on the outgoing slip plane. An analysis of the Burgers vectors of residuals produced from dislocation/GB interactions shows that GBs impeding dislocation motion have larger residual values.

Extracting dislocations and non-dislocation crystal defects from atomistic simulation data

We describe a novel method for extracting dislocation lines from atomistic simulation data in a fully automated way. The dislocation extraction algorithm (DXA) generates a geometric description of dislocation lines contained in an arbitrary crystalline model structure. Burgers vectors are determined reliably, and the extracted dislocation network fulfills the Burgers vector conservation rule at each node. All remaining crystal defects (grain boundaries, surfaces, etc), which cannot be represented by one-dimensional dislocation lines, are output as triangulated surfaces. This geometric representation is ideal for visualization of complex defect structures, even if they are not related to dislocation activity. In contrast to the recently proposed on-the-fly dislocation detection algorithm (ODDA) Stukowski (2010 Modelling Simul. Mater. Sci. Eng. 18 015012) the new method is extremely robust. While the ODDA was designed for a computationally efficient on-the-fly analysis, the DXA method enables a detailed analysis of dislocation lines even in highly distorted crystal regions, as they occur, for instance, close to grain boundaries or in dense dislocation networks.

Statistical foundation of continuum dislocation plasticity

A continuum model of mesoscale plasticity intended to cover dislocation pattering and size effects is proposed. The statistical approach to dislocation dynamics put forward by Groma et al. [Acta Mater. 51, 1271 (2003)] is applied to curved glide dislocations and dipolar dislocation loops appearing in a crystal deformed by single slip. Due to the very different size and mobile properties of the glide dislocations and the loops, the two entities are differently treated. The statistics of Groma et al. is directly applicable to the loops, which behave as rigid objects of restricted mobility. The glide dislocations are divided into groups, which in the averaging process lead to so-called single-valued dislocation fields. The averaged continuous distribution of the glide dislocations is treated as a superposition of these fields. The system of equations of the continuum model consists of (i) equations for the flow of glide dislocations of each single-valued field and the balance equation for the density of dipolar loops (these equations are derived from the discrete equations of motion and contain constitutive functionals representing short range interactions), (ii) the conservation of Burgers vector in the single-valued fields, (iii) the equations of continuum crystal plasticity, and (iv) the Orowan law for the shear strain rate caused by the glide dislocations. The possibilities for specification and evaluation of the constitutive functionals, namely, the short range correlations among dislocations (both the glide dislocations and the loops), are indicated.

Superscrew dislocations in silicon carbide: Dissociation, aggregation, and formation

An approach to precisely determine the magnitude and the sign of Burgers vectors of superscrew dislocations (including elementary screw dislocations and micropipes) in silicon carbide (SiC) single crystals has been developed. The histogram of Burgers vector values associated with micropipes follows Gaussian distribution with most possible absolute Burgers vector value of about 100Å for both 4H and 6H SiC. Superscrew dislocations are quasistable. Their transformations, such as dissociation, coalescence, and aggregation, follow the principle of Burgers vector conservation and energetically favored behavior. The coalescence of dislocations with same-sign Burgers vectors is less likely to occur during the growth. Finally, a possible mechanism of mismatched coalescence between multiple nucleation sites is proposed to understand the formation of micropipes with large Burgers vectors at the early stage of SiC bulk growth.

Discrete Crystal Elasticity and Discrete Dislocations in Crystals

This article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals. The theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups, differential forms and operators, and a theory of integration of forms. In particular, we define the lattice complex of a number of commonly encountered lattices, including body-centered cubic and face-centered cubic lattices. We show that material frame indifference naturally leads to discrete notions of stress and strain in lattices. Lattice defects such as dislocations are introduced by means of locally lattice-invariant (but globally incompatible) eigendeformations. The geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations, such as the dislocation density tensor, the equation of conservation of Burgers vector, Kroner's relation and Mura's formula for the stored energy. We additionally supply conditions for the existence of equilibrium displacement fields; we show that linear elasticity is recovered as the Γ-limit of harmonic lattice statics as the lattice parameter becomes vanishingly small; we compute the Γ-limit of dilute dislocation distributions of dislocations; and we show that the theory of continuously distributed linear elastic dislocations is recovered as the Γ-limit of the stored energy as the lattice parameter and Burgers vectors become vanishingly small.

Modeling the evolution of crystallographic dislocation density in crystal plasticity

Dislocations are the most important material defects in crystal plasticity, and although dislocation mechanics has long been understood as the underlying physical basis for continuum crystal plasticity formulations, explicit consideration of crystallographic dislocation mechanics has been largely absent in working constitutive models. Here, dislocation density state variables evolve from initial conditions according to equations based on fundamental concepts in dislocation mechanics such as the conservation of Burgers vector in multiplication and annihilation processes. The model is implemented to investigate the polyslip behavior of single-crystal aluminum. The results not only capture the mechanical stress/strain response, but also detail the development of underlying dislocation structure responsible for the plastic behavior.

Modelling of dislocation structures in materials

A phenomenological model of the evolution of an ensemble of interacting dislocations in an isotropic elastic medium is formulated. The line-defect microstructure is described in terms of a spatially coarse-grained order parameter, the dislocation density tensor. The tensor field satisfies a conservation law that derives from the conservation of Burgers vector. Dislocation motion is entirely dissipative and is assumed to be locally driven by the minimization of plastic free energy. We first outline the method and resulting equations of motion to linear order in the dislocation density tensor, obtain various stationary solutions, and give their geometric interpretation. The coupling of the dislocation density to an externally imposed stress field is also addressed, as well as the impact of the field on the stationary solutions.

Cooperative Particle Motions and Dynamical Behaviors of Free Dislocations in Strongly Coupled Quasi-2D Dusty Plasmas.

We report the optical study of microscopic particle motions and the dynamical behavior of free dislocations (FDs) in the strongly coupled quasi-two-dimensional dusty plasmas with SiO{sub 2} fine particle suspensions. The continuous excitation and relaxation of rotational cooperative particle motions with varying scales and lifetimes are observed, which distort the lattice and induce the FD processes such as generation, dissociation, propagation, recombination, and annihilation under the conservation of Burgers vector. {copyright} {ital 1996 The American Physical Society.}

Burgers vector conservation laws

A set of rules has been developed that allows for an exact description of various types of dislocation configurations in solids. These rules are in the form of conservation laws that exist between six uniquely defined Burgers vectors. Whereas four such laws are required for a lattice dislocation, the number may increase to as many as seven for a grain boundary dislocation. These findings are shown to be directly applicable to recent high-resolution electron microscopy observations of grain boundaries.

The continuity of dislocations

Dislocation continuity is derived from the Bilby–Kondo theory of dislocations using exterior calculus. Dislocation density is represented by the torsion vector-valued two-form. Burgers vectors are associated with the vector part of the torsion while dislocation lines are associated with the two-form part. The exterior derivative of the torsion is shown to vanish when the crystal curvature vanishes. This implies two simultaneous continuity conditions: Burgers vector conservation and continuity of dislocation lines. On the other hand, dislocation continuity is violated when the curvature does not vanish. Since this can occur on grain boundaries it is inferred that grain boundaries are regions where crystal curvature is concentrated.