Dislocation Dynamics Model Research Articles

Statistics of internal stress fluctuations in dislocated crystals and relevance to density-based dislocation dynamics models

A statistical analysis of internal stress fluctuations, defined as the difference between the local mean stress and stress on dislocations, is presented for deforming crystals with 3D discrete dislocation systems. Dislocation realizations are generated using dislocation dynamics simulations and the associated stress field is computed as a superposition of a regularized stress field of dislocation lines within the domain of the solution and a complementary stress field computed via a finite-element boundary value problem. The internal stress fluctuations of interest are defined by an ensemble of the difference between the stress on dislocation lines and the local mean field stress in the crystal. The latter is established in a piecewise fashion over small voxels in the crystal thus allowing the difference between the local average stress and stress on segments to be easily estimated. The results show that the Schmid stress (resolved shear stress) and Escaig stress fluctuations on various slip systems sampled over a random set of points follow a Cauchy (Lorentz) distribution at all strain levels, with the amplitude and width of the distribution being dependent on the strain. The implications of the Schmid and Escaig internal stress fluctuations are discussed from the points of view of dislocation cross-slip and the dislocation motion in continuum dislocation dynamics.

On the similitude relation for dislocation wall thickness under cyclic deformation

Dislocation substructures have been extensively characterized to explain the origin of strain hardening. Regions of high dislocation densities (walls) constrain the glide of mobile dislocations in regions of lower dislocation density (channels). Transmission electron microscopy (TEM) has shown that the distance in between walls is inversely proportional to flow stress, which is often referred as to the similitude principle. However, it still remains unclear whether a similitude scaling law exists for the dislocation wall thickness or wall fraction. The understanding of such a scaling law is instrumental to support substructure-based crystal plasticity, validate dislocation dynamics models, and explain substructure formation processes. Hence, this work surveys TEM images from various FCC metallic materials under cyclic deformation at various temperatures to assess the existence of the similitude principle for dislocation walls. The results demonstrate that the wall thickness does follow similitude, but the proportionality constant depends on the temperature for some structures.

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.

Transitions in the morphology and critical stresses of gliding dislocations in multiprincipal element alloys

Refractory multiprincipal element alloys (MPEAs) are promising material candidates for high-temperature, high-strength applications. However, their deformation mechanisms, particularly at the dislocation scale, are unusual compared to those of conventional alloys, making it challenging to understand the origin of their high strength. Here, using an atomistically informed phase-field dislocation dynamics model, we study transitions in the morphology and critical stresses of long gliding screw dislocations over extended distances in a refractory MPEA. The model MPEA crystal accounts for the atomic-scale fluctuations in chemical composition across the glide planes via spatially correlated lattice energies and for the differences in glide resistances between screw and edge dislocations of unit length. We show that the dislocation moves in a stop-start motion, alternating between wavy morphology in free flight and nearly recovered straight screw orientation in full arrest. The periods of wavy glide are due to variable kink-pair formation and migration rates along the length, where portions with higher rates glide more quickly. The critical stress to initiate motion corresponds to the stress required to form and migrate a kink pair at the weakest region along the length of the dislocation. Heterogeneity in lattice energy leads to variability in the local stress-strain response and to a strain hardening-like response, in which the critical stress to reactivate glide increases with glide distance. Statistical assessment of hundreds of realizations of dislocations indicates that the amount of hardening directly scales with the dispersion in underlying lattice energy.

Dislocation transmission across Σ3{112} incoherent twin boundary: a combined atomistic and phase-field study

Grain boundaries (GBs) in polycrystalline materials act as impediments to dislocation motion and result in strengthening. Understanding slip transmission through GBs, specifically twin boundaries, is essential to understand the plastic deformation behavior of polycrystalline fcc materials. In this study the interaction between a glide dislocation and Σ3{112} incoherent twin boundary (ITB) in copper is investigated using a combined atomistic and mesoscale approach. The material parameters and structure of the GB in the mesoscale phase field dislocation dynamics (PFDD) model are informed from Molecular Statics (MS) simulations. The structural unit of the ITB consists of an array of three partial dislocations. The interaction between a glide dislocation impinging on each of the GB partial dislocations is investigated using both PFDD and Molecular Dynamics (MD) with two boundary conditions. Transmission planes predicted by both PFDD and MD (NVT) are in agreement, and show that not all transmission events are direct. Critical transmission stresses predicted by PFDD are in the range of 276 MPa to 1380 MPa, while MD predictions are in the range from 100 MPa to 700 MPa. The PFDD and MD predictions of slip transmission are explained using dislocation theory based on isotropic linear elasticity.

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.

The Effect of Multipoles on the Elasto-Plastic Properties of a Crystal: Theory and Three-Dimensional Dislocation Dynamics Modeling

Abstract Plastic deformation in metals is dominated by the interactions among dislocations and other defects inside the crystal. A large number of dislocation multipoles (dipoles, tripoles, quadrupoles, etc.) can form during plastic deformation. Depending on the relative position and the orientation of the dislocations, interactions in and between multipoles can change the elastoplastic properties of a material. The authors of this article investigate the effect of dislocation multipoles on the elastoplastic properties of a material. This is performed analytically under different multipole configurations (i.e., the distance between active glide planes and the signs of the dislocations) as well as using a three-dimensional discrete dislocation dynamics (DDD) code. The simulations show that multipoles exhibit a hardening/softening effect when the sign of the dislocations involved is the same, and a hardening effect only when the dislocations are of opposite sign to nearby ones. The distance between the two neighboring dislocations was also affecting the proportional limit (PL) for the material. Such hardening or flow stress (FS) results, as in this study, can be incorporated into larger-scale modeling work.

Temperature dependent plasticity in BCC micropillars

Recent experiments in micron- or sub-micron metallic pillars have displayed size dependent plasticity. Dislocation dynamics (DD) simulations can be a useful tool in studying dislocation-mediated plasticity, which is widely believed to contribute to observed size effects at small length scale. In particular, the plasticity in body-centered cubic (BCC) metals is known to have a strong temperature dependence, which stems from the thermally activated motion of screw dislocations. In this study, we develop a DD model based on the atomistic characterization of dislocation mobility and potential source mechanisms to investigate the temperature dependent plasticity in BCC micropillars. Our models show the dislocation source mechanism changes with respect to temperature due to the change in mobility of screw dislocations, and these results are compared with experimental results and theoretical models. In addition, the size dependence increases with temperature, which agrees with recent experimental observations.

A novel multiscale material plasticity simulation model for high-performance cutting AISI 4140 steel

The achievable machined surface quality relies significantly on the material behavior during the high-performance cutting process. In this paper, a multiscale material plasticity simulation framework is developed to predict the deformation behaviors of AISI 4140 steel under various high-performance cutting conditions. The framework was built by coupling a three-dimensional discrete dislocation dynamic (3D-DDD) model with a finite element method (FEM) through the optimization of a dislocation density-based (DDB) constitutive equation (compiled as a user-defined subroutine in ABAQUS). The movement of edge and screw dislocations such as generation, propagation, siding, and their interactions, was performed by 3D-DDD, and the statistical features of dislocations were used to optimize the critical constants of the DDB constitutive equation. For validation, a classic FEM cutting model (Johnson-Cook constitutive equation) was employed as a reference. The simulation results indicated that the proposed multiscale model not only can precisely predict the stress, strain, cutting force, and temperature as those predicted by the classic FEM simulations, but also capture the microstructure characteristics such as grain size and dislocation density distributions under the tested cutting conditions. Severe dynamic recrystallization phenomena were found at the core shear zones. The recrystallization process reached a dynamic equilibrium at the machined surfaces when the cutting speed is larger than 280 m/min or the external-assisted temperature is between 200 and 350°, indicating an optimal range of machining parameters for improved surface integrity.

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.

Dislocation dynamics modelling of the creep behaviour of particle-strengthened materials.

Plastic deformation in crystalline materials occurs through dislocation slip and strengthening is achieved with obstacles that hinder the motion of dislocations. At relatively low temperatures, dislocations bypass the particles by Orowan looping, particle shearing, cross-slip or a combination of these mechanisms. At elevated temperatures, atomic diffusivity becomes appreciable, so that dislocations can bypass the particles by climb processes. Climb plays a crucial role in the long-term durability or creep resistance of many structural materials, particularly under extreme conditions of load, temperature and radiation. Here we systematically examine dislocation-particle interaction mechanisms. The analysis is based on three-dimensional discrete dislocation dynamics simulations incorporating impenetrable particles, elastic interactions, dislocation self-climb, cross-slip and glide. The core diffusion dominated dislocation self-climb process is modelled based on a variational principle for the evolution of microstructures, and is coupled with dislocation glide and cross-slip by an adaptive time-stepping scheme to bridge the time scale separation. The stress field caused by particles is implemented based on the particle–matrix mismatch. This model is helpful for understanding the fundamental particle bypass mechanisms and clarifying the effects of dislocation glide, climb and cross-slip on creep deformation.

Effect of size and orientation on stability of dislocation networks upon torsion loading and unloading in FCC metallic micropillars

At the continuum length scale, mechanical properties of metals show relatively weak orientation dependence; however, they exhibit strong anisotropic behaviors as the size of sample decreases to micron and nanometer length scales. In this study, three-dimensional dislocation dynamics (DD) simulations are performed to investigate the orientation-dependent plasticity in submicron face-centered cubic (FCC) micropillars subjected to torsion. Accommodating results from atomistic modeling, updated surface nucleation schemes in DD models have been developed for three orientations ([001], [101], and [111]), allowing investigation of the dislocation microstructure evolution and the corresponding anisotropic mechanical response upon torsional loading and unloading. The DD simulation results show that the coaxial and hexagonal networks formed in [101] and [111] oriented nanopillars, respectively, exhibited excellent plastic recovery, while the rectangular network formed in the [001] crystal orientation was more stable and did not experience as much plastic recovery.

Cell structure formation in a two-dimensional density-based dislocation dynamics model

Cellular patterns formed by self-organization of dislocations are a most conspicuous feature of dislocation microstructure evolution during plastic deformation. To elucidate the physical mechanisms underlying dislocation cell structure formation, we use a minimal model for the evolution of dislocation densities under load. By considering only two slip systems in a plane strain setting, we arrive at a model which is amenable to analytical stability analysis and numerical simulation. We use this model to establish analytical stability criteria for cell structures to emerge, to investigate the dynamics of the patterning process and establish the mechanism of pattern wavelength selection. This analysis demonstrates an intimate relationship between hardening and cell structure formation, which appears as an almost inevitable corollary to dislocation dominated strain hardening. Specific mechanisms such as cross slip, by contrast, turn out to be incidental to the formation of cellular patterns.

Phase field dislocation dynamics (PFDD) modeling of non-Schmid behavior in BCC metals informed by atomistic simulations

Body-Centered Cubic (BCC) metals exhibit anomalous mechanical properties such as twinning/anti-twinning and tension/compression asymmetry, which are attributed to asymmetric behavior of screw dislocations. Unlike Face-Centered Cubic (FCC) metals, the Critical Resolved Shear Stress (CRSS) of BCC metals deviates from the Schmid law. We use a mesoscale modeling approach called Phase Field Dislocation Dynamics (PFDD) to understand the critical factors that control this non-Schmid behavior and reproduce atomistic predictions of the CRSS (Peierls stress). All inputs to the PFDD model are obtained from Molecular Statics (MS) simulations. Multiple pathways for modeling the non-Schmid behavior are investigated by incorporating the representative dislocation properties into different energy terms in the PFDD model. One way to understand non-Schmid behavior is to incorporate stress components projected on inclined planes into the external energy term within PFDD. Alternatively, we propose that non-Schmid behavior can also be accounted for by considering the variation of the {110} unstable stacking fault energy and the dislocation core width as a function of the applied tensorial stress field. The CRSS predicted using PFDD modeling is in excellent agreement with MS predictions.

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.

On the computational solution of vector-density based continuum dislocation dynamics models: A comparison of two plastic distortion and stress update algorithms

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test was first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. The simulations show that dislocations exhibit some self-organized structures as the strain is increased.

On the implementation of dislocation reactions in continuum dislocation dynamics modeling of mesoscale plasticity

The continuum dislocation dynamics framework for mesoscale plasticity is intended to capture the dislocation density evolution and the deformation of crystals when subjected to mechanical loading. It does so by solving a set of transport equations for dislocations concurrently with crystal mechanics equations, with the latter being cast in the form of an eigenstrain problem. Incorporating dislocation reactions in the dislocation transport equations is essential for making such continuum dislocation dynamics predictive. A formulation is proposed to incorporate dislocation reactions in the transport equations of the vector density-based continuum dislocation dynamics. This formulation aims to rigorously enforce dislocation line continuity using the concept of virtual dislocations that close all dislocation loops involved in cross slip, annihilation, and glissile and sessile junction reactions. The addition of virtual dislocations enables us to accurately enforce the divergence free condition upon the numerical solution of the dislocation transport equations for all slip systems individually. A set of tests were performed to illustrate the accuracy of the formulation and the solution of the transport equations within the vector density-based continuum dislocation dynamics. Comparing the results from these tests with an earlier approach in which the divergence free constraint was enforced on the total dislocation density tensor or the sum of two densities when only cross slip is considered shows that the new approach yields highly accurate results. Bulk simulations were performed for a face centered cubic crystal based on the new formulation and the results were compared with discrete dislocation dynamics predictions of the same. The microstructural features obtained from continuum dislocation dynamics were also analyzed with reference to relevant experimental observations.

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).

Modeling of the ECAP Induced Strain Hardening Behavior in FCC Metals

In this work, a multi-scale framework that couples a crystal plasticity (CP) scheme with a continuum dislocation dynamics (CDD) model is proposed to predict the material behavior, microstructure and texture during equal channel angular pressing (ECAP) processes. The strain hardening in the model is considered to result from both the increase in the dislocation density and the grain fragmentation. The grain fragmentation process is modeled by accounting for the grain-grain interaction and incorporating the concept of the geometrically necessary dislocations (GNDs) into the mean free path of the dislocations. GNDs result from grain boundaries restricting the free deformation of a grain, causing an internal plastic deformation gradient that subsequently leads to grain fragmentation. A commercial Al-1100 billet, with rolling texture, is ECAP processed under Route C for different numbers of passes. Mechanical, microstructure, and texture characterization is achieved for the received and ECAPed materials. The proposed model parameters are calibrated using the tensile true-stress true-strain curves of the unprocessed material at two strain rates. The ECAP-processed aluminum microstructure, texture, dislocation densities and the mechanical properties are predicted.