Discrete Dislocation Dynamics Model Research Articles

Defect dynamics modeling of mesoscale plasticity

The collective motion of defects and their interaction are the basic building blocks for plastic deformation and corresponding mechanical behaviors of crystalline metals. Especially, dislocations among various defects are the “carrier” of plastic deformation in many crystalline materials, particularly ductile materials. To get a fundamental understanding of plastic deformation mechanisms, it calls for an integrated computational platform to simultaneously capture detailed defects characteristics across several length scales together with corresponding macroscopic mechanical response. In this paper, we present a three-dimensional mesoscale defect dynamics model to directly couple the three dimensional discrete dislocation dynamics model with continuum finite element method, aiming at capturing both size dependent plasticity at micron-, and submicron scale and constitutive behaviors at larger scales where such size-dependence disappear. Using non-singular dislocation theories, our model could accurately consider both short- and long-range elastic interactions between multiple dislocation segments with even higher computational efficiency than traditional dislocation dynamics simulations, together with the careful consideration of crystal/material rotation in the coupled framework. In addition, our model could directly model dislocation nucleation from stress concentrators such as a void, crack and indentor tip, which could allow us to investigate various defects’ motion and their mutual interactions, predicting macroscopic mechanical response of complex structures. The developed concurrently coupled model could also consider multiphysical phenomena by solving coupled governing equations in finite element framework, which could shed light on complex defect behaviors under various physical environments.

Advanced modeling of higher-order kinematic hardening in strain gradient crystal plasticity based on discrete dislocation dynamics

An extensive study of size effects on the small-scale behavior of crystalline materials is carried out through discrete dislocation dynamics (DDD) simulations, intended to enrich strain gradient crystal plasticity (SGCP) theories. These simulations include cyclic shearing and tension-compression tests on two-dimensional (2D) constrained crystalline plates, with single- and double-slip systems. The results show significant material strengthening and pronounced kinematic hardening effects. DDD modeling allows for a detailed examination of the physical origin of the strengthening. The stress–strain responses show a two-stage behavior, starting with a micro-plasticity regime with a steep hardening slope leading to strengthening, and followed by a well-established hardening stage. The scaling exponent between the apparent (higher-order) yield stress and the geometrical size h varies depending on the test type. Scaling relationships of h−0.2 and h−0.3 are obtained for respectively constrained shearing and constrained tension-compression, aligning with some experimental observations. Notably, the DDD simulations reveal the occurrence of the uncommon type III (KIII) kinematic hardening of Asaro in both single- and double-slip cases, emphasizing the relevance of this hardening type in the realm of small-scale plasticity. Inspired by insights from DDD, two advanced SGCP models incorporating alternative descriptions of higher-order kinematic hardening mechanisms are proposed. The first model uses a Prager-type higher-order kinematic hardening formulation, and the second employs a Chaboche-type (multi-kinematic) formulation. Comparison of these models with DDD simulation results underscores their ability to effectively capture the observed strengthening and hardening effects. The multi-kinematic model, through the use of quadratic and non-quadratic higher-order potentials, shows a notably better qualitative congruence with DDD findings. This represents a significant step towards accurate modeling of small-scale material behaviors. However, it is noted that the proposed models still have limitations, especially in matching the DDD scaling exponents, with both models producing h−1 scaling relationships (i.e., Orowan relationship for precipitate size effects). This indicates the need for further improvements in gradient-enhanced theories in order to guarantee their suitability for practical engineering applications.

Learning dislocation dynamics mobility laws from large-scale MD simulations

By dispensing with all the atoms and only focusing on dislocation lines, the computational method of Discrete Dislocation Dynamics (DDD) gains greatly over Molecular Dynamics (MD) in simulation efficiency of metal plasticity. But whereas in MD dislocations follow natural dynamics of atomic motion, DDD must rely on a dislocation mobility function to prescribe how a dislocation line should respond to the driving force exerted on it. However, reflecting our still incomplete understanding of ways in which dislocations move, mobility functions presently employed in DDD simulations entail simplifications and approximations of limited or, worse still, unknown accuracy and applicability. Here we introduce a data-driven approach in which the dislocation mobility function is modeled as a graph neural network (GNN) trained on large-scale MD simulations of crystal plasticity. We apply our proposed approach to predicting plastic strength of body-centered-cubic (BCC) metal tungsten and show that, once implemented in a DDD model, our GNN dislocation mobility function accurately reproduces the challenging tension/compression asymmetry of plastic flow observed both in ground-truth MD simulations and in experiment. Furthermore, subsequently validated by MD simulations, the same function accurately predicts plastic response of tungsten under conditions not previously seen in training. By demonstrating its ability to learn relevant physics of dislocation motion, our DDD+ML approach opens a promising avenue to bringing fidelity of DDD models closer in line with direct MD simulations at a much reduced computational cost.

Research on microplastic deformation of Inconel718 under the conditions of high temperature and high strain rate

The compression experiments of the nickel-based superalloy Inconel718 were carried out with the help of split Hopkinson pressure bar device, and the microstructure evolution law in the compression deformation of the alloy was studied by means of optical microscopy, electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM). Based on the dislocation theory, a three-dimensional discrete dislocation dynamics model is established. By simulating the dynamic evolution of the discrete dislocation, the interaction of the dislocation itself and the formation of the dislocation microstructure were captured, and further explore the microplastic deformation process of Inconel718. The results show that deformation conditions significantly affect the rheological behavior and dynamic recrystallization behaviour. The cross slip of dislocations is closely related to temperature and strain rate, which in turn affect the dislocation configuration. The deformation mechanism of Inconel718 under the conditions of high temperature and high strain rate is jointly dominated by dislocation slip and twin deformation. As the temperature increases, the average Schmidt factor increases, and more dislocation slip systems are activated. As the strain rate increases, the number of deformation twins increases, while the thickness of deformation twins decreases.

Dislocation descriptors of low and high angle grain boundaries with convolutional neural networks

Multiscale modeling of plasticity in polycrystalline metals is a long-standing challenge in part because of the lack of an accepted grain boundary descriptor, which has hindered the bridging of scales between atomistic simulations and meso-scale discrete dislocation dynamics (DDD) models. While grain boundary dislocations (GBDs) for low angle grain boundaries can be ascertained by Burgers circuit analyses, the dislocation structures of high angle grain boundaries have remained elusive because of overlapping dislocation core fields. Here, we use convolutional neural networks (CNNs) to establish the locations of GBDs responsible for the misorientations of <001> symmetrical-tilt Cu grain boundaries, from the local atomistic stress fields modeled with molecular dynamics (MD) simulations. We achieve accurate CNN predictions of GBDs with sub-angstrom resolution across an extensive set of low- to high-tilt-angle grain boundaries through a training dataset, generated from superposing continuum-representations of the MD stress fields of single dislocations that consider the contributions of both the Volterra and dislocation core fields. The approach paves the way for dislocation representations of the atomistic grain boundary structures modeled by MD simulations or density functional theory (DFT) calculations in mesoscale DDD models to elucidate multiscale plasticity effects.

An Investigation of Spiral Dislocation Sources Using Discrete Dislocation Dynamics (DDD) Simulations

Discrete Dislocation Dynamics (DDD) simulations are a powerful simulation methodology that can predict a crystalline material’s constitutive behavior based on its loading conditions and micro-constituent population/distribution. In this paper, a 3D DDD model with spiral dislocation sources is developed to study size-dependent plasticity in a pure metal material (taken here as Aluminum). It also shows, for the first time, multipole simulations of spirals and how they interact with one another. In addition, this paper also discusses how the free surface of a crystalline material affects the plasticity generation of the spiral dislocation. The surface effect is implemented using the Distributed Dislocation Method. One of the main results from this work, shown here for the first time, is that spiral dislocations can result in traditional Frank–Read sources (edge or screw character) in a crystal. Another important result from this paper is that with more dislocation sources, the plastic flow inside the material is more continuous, which results in a lowering of the flow stress. Lastly, the multipole interaction of the spiral dislocations resulted in a steady-state fan-shaped action for these dislocation sources.

A discrete dislocation dynamics model of creep in polycrystals

The dislocation creep represents high temperature plastic deformation of metallic materials. However, the subject is usually approached with constitutive and phenomenological formulations. The notable exceptions are the recent use of numerical modeling. For example, single-crystal discrete dislocation dynamics (DDD) has been adopted to involve dislocation glide and climb together with diffusion of vacancies. This naturally provides a physics-based framework for dislocation creep. However, the existing model(s) are limited to single crystals and hence are limited in scope. This study extended a single crystal dislocation creep model into polycrystalline ensemble. This was then used to simulate steady state creep rate (ε˙) with respect to creep stress (σ), temperature (T) and grain size (d) in polycrystalline aluminum. Our virtual experiments showed an Arrhenius relationship between ε˙ and T, and a power law scaling with σ. Further, our simulations also revealed a power law increase in ε˙ with d. This is striking, as available experimental data indicate both grain size dependence as well as independence. The answer to this apparent contradiction also emerged from our numerical simulations. It was shown that ε˙ is controlled by interactions of dislocations with static obstacles and grain boundaries. Dominance of static obstacles, in particular, significantly diminished the grain size effect. The role of dislocation mean free path on the pinning and dislocation creep behavior of polycrystalline metallic material was thus mechanistically established.

Bayesian optimization of discrete dislocation plasticity of two-dimensional precipitation-hardened crystals

Discovering relationships between materials' microstructures and mechanical properties is a key goal of materials science. Here, we outline a strategy exploiting Bayesian optimization to efficiently search the multidimensional space of microstructures, defined here by the size distribution of precipitates (fixed impurities or inclusions acting as obstacles for dislocation motion) within a simple two-dimensional discrete dislocation dynamics model. The aim is to design a microstructure optimizing a given mechanical property, e.g., maximizing the expected value of shear stress for a given strain. The problem of finding the optimal discretized shape for a distribution involves a norm constraint, and we find that sampling the space of possible solutions should be done in a specific way in order to avoid convergence problems. To this end, we propose a general mathematical approach that can be used to generate trial solutions uniformly at random while enforcing an Euclidean norm constraint. Both equality and inequality constraints are considered. A simple technique can then be used to convert between Euclidean and other Lebesgue $p$-norm (the 1-norm in particular) constrained representations. Considering different dislocation-precipitate interaction potentials, we demonstrate the convergence of the algorithm to the optimal solution and discuss its possible extensions to the more complex and realistic case of three-dimensional dislocation systems where also the optimization of precipitate shapes could be considered.

Modeling of microscale internal stresses in additively manufactured stainless steel

Additively manufactured (AM) metallic materials often comprise as-printed dislocation cells inside grains. These dislocation cells can give rise to substantial microscale internal stresses in both initial undeformed and plastically deformed samples, thereby affecting the mechanical properties of AM metallic materials. Here we develop models of microscale internal stresses in AM stainless steel by focusing on their back stress components. Three sources of microscale back stresses are considered, including the printing and deformation-induced back stresses associated with as-printed dislocation cells as well as the deformation-induced back stresses associated with grain boundaries. We use a three-dimensional discrete dislocation dynamics model to demonstrate the manifestation of printing-induced back stresses. We adopt a dislocation pile-up model to evaluate the deformation-induced back stresses associated with as-printed dislocation cells. The extracted back stress relation from the pile-up model is incorporated into a crystal plasticity (CP) model that accounts for the other two sources of back stresses as well. The CP finite element simulation results agree with the experimentally measured tension–compression asymmetry and macroscopic back stress, the latter of which represents the effective resultant of microscale back stresses of different origins. Our results provide an in-depth understanding of the origins and evolution of microscale internal stresses in AM metallic materials.

Continuum Model and Numerical Method for Dislocation Structure and Energy of Grain Boundaries

We present a continuum model to determine the dislocation structure and energy of low angle grain boundaries in three dimensions. The equilibrium dislocation structure is obtained by minimizing the grain boundary energy that is associated with the constituent dislocations subject to the constraint of Frank's formula. The orientation-dependent continuous distributions of dislocation lines on grain boundaries are described conveniently using the dislocation density potential functions, whose contour lines on the grain boundaries represent the dislocations. The energy of a grain boundary is the total energy of the constituent dislocations derived from discrete dislocation dynamics model, incorporating both the dislocation line energy and reactions of dislocations. The constrained energy minimization problem is solved by the augmented Lagrangian method and projection method. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of the energy and dislocation densities of both planar and curved low angle grain boundaries.

Development of mean-field continuum dislocation kinematics with junction reactions using de Rham currents and graph theory

An accurate description of the evolution of dislocation networks is an essential part of discrete and continuum dislocation dynamics models. These networks evolve by motion of the dislocation lines and by forming junctions between these lines via cross slip, annihilation and junction reactions. In this work, we introduce these dislocation reactions into continuum dislocation models using the theory of de Rham currents. We introduce dislocations on each slip system as potentially open lines whose boundaries are associated with junction points and, therefore, still create a network of collectively closed lines that satisfy the classical relations α=curlβp and divα=0 for the dislocation density tensor α and the plastic distortion βp. To ensure this, we leverage Frank’s second rule at the junction nodes and the concept of virtual dislocation segments. We introduce the junction point density as a new state variable that represents the distribution of junction points within the crystal containing the dislocation network. Adding this information requires knowledge of the global structure of the dislocation network, which we obtain from its representation as a graph. We derive transport relations for the dislocation line density on each slip system in the crystal, which now includes a term that corresponds to the motion of junction points. We also derive the transport relations for junction points, which include source terms that reflect the topology changes of the dislocation network due to junction formation.

Avalanche correlations and stress-strain curves in discrete dislocation plasticity

The sequence of deformation bursts during plastic deformation exhibits scale-free features. In addition to the burst or avalanche sizes and the rate of avalanches the process is characterized by correlations in the series which become manifest in the resulting shape of the stress-strain curve. We analyze such features of plastic deformation with two-dimensional and three-dimensional simulations of discrete dislocation dynamics models and we show that only with severe plastic deformation the ensuing memory effects become negligible. The role of past deformation history and dislocation pinning by disorder are studied. In general, the correlations have the effect of reducing the scatter of the individual stress-strain curves around the mean one.

In-Situ SEM High Strain Rate Testing of Large Diameter Micropillars Followed by TEM and EBSD Postmortem Analysis

BackgroundDislocation dynamic simulations are intended as a tool to understand and predict the mechanical behavior of metallic materials, but its prediction has never been directly verified by experiments due to differences in specimen strain rate and size.ObjectiveIn this work, a comprehensive experimental framework is proposed to attempt direct comparison between experiments and discrete dislocation dynamics (DDD) modelling. MethodsBy integrating high-throughput sample fabrication and a customized testing apparatus, the sample size and strain rate typically employed in DDD simulations are explored experimentally. Constitutive properties such as stress-strain response are measured, and microstructural information is obtained from transmission electron microscopy (TEM) imaging, electron backscatter diffraction (EBSD), and TEM-based orientation mapping. ResultsMagnesium and copper were selected, as case studies, to demonstrate the newly developed experimental procedure. Measured stress-strain responses for Mg are consistent with those obtained with a miniaturized Hopkison bar experiments. By exploiting the validated workflow, the effect of strain rate on micropillar heterogeneous deformation and associated dislocation plasticity were revealed.ConclusionThe work establishes a methodology for the systematic study of not only metals but also other materials and structures at the microscale and high strain rates.

Spectral discrete dislocation dynamics with anisotropic short range interactions

Discrete dislocation dynamics models dislocation-mediated plastic deformation and microstructural evolution in solids whilst discretely representing and tracking all line defects present in the medium. Hybrid methodologies for evaluating mechanical fields using continuum approaches while retaining the discrete formalism for evaluating dislocation kinematics and interactions have greatly improved the numerical efficiency of the method. In the present work, we propose a spectrally-based continuum solution of the mechanical fields based on the field dislocation mechanics framework which produces highly accurate elastic fields near dislocation cores in both isotropic and anisotropic media. Application of the model to a few elementary cases (i.e. dislocation junction formation, dislocation migration) show that the model proposed predicts accurate elastic fields both on the short and long range.

Dislocation–grain boundary interaction-based discrete dislocation dynamics modeling and its application to bicrystals with different misorientations

Grain boundaries (GBs) have a significant influence on the mechanical properties of metallic materials. It has been a great challenge to describe dislocation interactions with various GBs. In the present article, a generalized dislocation–GB interaction model was constructed and then implemented in the three-dimensional multiscale discrete dislocation dynamics (DDD) framework. In the model, two dislocation–GB interaction mechanisms, i.e., dislocation absorption at GBs and dislocation emission from GBs, were considered. In order to make the dislocation–GB interaction model suitable for various GB types, a ‘coarse-graining’ approach was applied to deal with the process of dislocation absorption and emission. As the validations and applications of the proposed dislocation–GB interaction model, nanopillars containing a non-sigma large-angle GB and subjected to uniaxial compression were studied. The simulated results show that the bi-crystalline nanopillars possess a higher yield strength and flow stress, smaller stress-drop size than single-crystalline counterparts, which is consistent with earlier experimental observations in the literature. Afterward, the DDD simulation was employed to reveal the effect of GB misorientation on the mechanical responses of bicrystals with a large-angle-symmetric-tilt GB. Simulations indicate that the mechanical responses of bicrystals are affected by the GB structures and complex dislocation–dislocation and dislocation–GB interactions. In contrast, the dislocation absorption and emission events, as well as the evolution of resolved shear stress and dislocation density, do not depend on the GB misorientation angles or the GB strength (or the GB energy).

Analyses of Dislocation Effects on Plastic Deformation

This paper reviews the dislocation based models that have been used until now to consider dislocation effects on the plastic deformation in the crystalline material. We separate two general strategies: discrete dislocation dynamics models that consider individual dislocations and continuum dislocation dynamics models that consider dislocation density as a state variable to analyze the plastic deformation. Since these methods are very widespread due to suggested new approaches and computational advances they need to be reviewed. The objective of this review is to introduce these methods and comparing existing results in two categories to know which method can be used to reach more accurate results for considering dislocation in crystalline metals. Seventeen papers and thesis were chosen that predicted the stress–strain curve of different crystalline material by emphasizing dislocation effect on their results. Advantages and lacks of both methods are mentioned and it is discussed which method is suitable for stress–strain prediction. It is observed that continuum dislocation dynamics methods can predict the stress–strain curves more efficiently than discreet dislocation dynamics.

Elucidating the dual role of grain boundaries as dislocation sources and obstacles and its impact on toughness and brittle-to-ductile transition

In this paper, we resolve the role of grain boundaries on toughness and the brittle-to-ductile transition. On the one hand, grain boundaries are obstacles for dislocation glide. On the other hand, the intersection points of grain boundaries with the crack front are assumed to be preferred dislocation nucleation sites. Here, we will show that the single contributions of grain boundaries (obstacles vs. source) on toughness and the brittle-to-ductile transition are contradicting, and we will weight the single contributions by performing carefully designed numerical experiments by means of two-dimensional discrete dislocation dynamics modelling. In our parameter studies, we vary the following parameters: (i) the mean free path for dislocation glide, δ, combined with (ii) the (obstacle) force of the grain boundary, ϕ, and (iii) the dislocation source spacing along the crack front, λ. Our results show that for materials or microstructures for which the mean distance of the intersection points of grain boundaries with the crack front is the relevant measure for λ, a decrease of grain size results in an increase of toughness. The positive impact of grain boundaries outweighs the negative consequences of dislocation blocking. Furthermore, our results explain the evolving anisotropy of toughness in cold-worked metals and give further insight into the question of why the grain-size-dependent fracture toughness passes through a minimum (and the brittle-to-ductile transition temperature passes through a maximum) at an intermediate grain size. Finally, a relation of the grain-size-dependence of fracture toughness in the form of K(dδ, dλ) = KIC + kdδ0.5/dλ is deduced.

A New Formulation of Coupling and Sliding Motions of Grain Boundaries Based on Dislocation Structure

A continuum model of the two dimensional low angle grain boundary motion and the dislocation structure evolution on the grain boundaries has been developed in [L. Zhang and Y. Xiang, J. Mech. Phys. Solids, 117 (2018), pp. 157--178]. The model is based on the motion and reaction of the constituent dislocations of the grain boundaries. The long-range elastic interaction between dislocations is included in the continuum model, and it maintains a stable dislocation structure described by Frank's formula for grain boundaries. In this paper, we develop a new continuum model for the coupling and sliding motions of grain boundaries that avoids the time-consuming calculation of the long-range elastic interaction. In this model, the long-range elastic interaction is replaced by a constraint of Frank's formula. The constrained evolution problem in our new continuum model is further solved by using the projection method. Effects of the coupling and sliding motions in our new continuum model and relationship with the classical motion by curvature model are discussed. The continuum model is validated by comparisons with discrete dislocation dynamics model and the early continuum model [L. Zhang and Y. Xiang, J. Mech. Phys. Solids, 117 (2018), pp. 157--178] in which the long-range dislocation interaction is explicitly calculated.

Influence of rafted microstructures on creep in Ni-base single crystal superalloys: a 3D discrete dislocation dynamics study

Ni-base single-crystal superalloys exhibit a dynamic evolution of their microstructure during operation at elevated temperatures. The rafting of γ′ precipitates changes the mechanical behavior in a way that was understood insufficiently. In this work, we combine a phase-field method with a discrete dislocation dynamics model to clarify the influence of different rafted microstructures with the same initial dislocation density and configuration on creep behavior. The unrafted and rafted microstructures of Ni-base single crystal superalloys are simulated by a phase-field crystal plasticity method. By introducing these microstructures into a 3D discrete dislocation dynamics (DDD) model, the creep behavior under uniaxial loads of 350 and 250 MPa along [100] direction at 950 °C is studied. Due to the negative lattice mismatch of Ni-base superalloys, the N-type rafting with the formation of plate-like γ′ precipitates occurs under uniaxial tensile loads along {100} direction at high temperatures, while the P-type rafting with the formation of rod-like γ′ precipitates occurs under compressive loads. Taking the cuboidal, N-type rafted and P-type rafted microstructures as the initial and fixed microstructures for the same loading conditions, it is found from DDD simulations that the rafted microstructures result in smaller creep deformation than the cuboidal microstructure. The reason for this is that the coalescence of γ′ precipitates during the rafting diminishes the width of some γ channels, so as to increase the local Orowan stresses which retard the dislocation glide. For tensile loads, the N-type rafted microstructure has the best creep resistance. For a low compressive load, the P-type rafting shows a better creep resistance than N-type rafting.

On the assessment of creep damage evolution in nickel-based superalloys through correlative HR-EBSD and cECCI studies

The evolution of dislocation density with creep strain in single-crystal superalloys is studied quantitatively using high-resolution electron backscatter diffraction (HR-EBSD) and electron channelling contrast imaging under controlled diffraction conditions (cECCI). Data regarding dislocation density/structure is measured for deformation at 900 °C and 450 MPa up to ≈ 1% plastic strain. Effects of chemical composition are elucidated via three purpose-designed superalloys of differing rhenium and ruthenium contents. The evidence indicates that dislocation avalanching is already prevalent at plastic strains of ≈ 0.1%; thereafter, an exponential decay in the dislocation multiplication rate is indicative of self-hardening due to dislocation constriction within the matrix channels, as confirmed by the imaging. The results are rationalised using discrete dislocation dynamics modelling: a universal dislocation evolution law emerges, which will be useful for alloy design efforts.