Continuum Theory Of Dislocations Research Articles

Slip-discreteness-corrected strain gradient crystal plasticity (SDC-SGCP) theory

Strain gradient plasticity theory addresses the plastic strain gradient induced hardening by considering the internal stress and Taylor hardening associated with the geometrically necessary dislocations (GNDs). However, the continuum description of internal stress associated with GNDs is inaccurate due to the coarsening of discrete dislocations. Corrections are thus derived as the difference between the stresses produced by the continuous configuration and the discrete configuration. We further demonstrate the capability of this correction in effectively capturing the internal stress induced strengthening effect associated with GNDs, and elucidate that its role in strengthening is to homogenize the deformation and extend the influence of grain boundaries into the interior of grains within polycrystals. This capability to capture intragranular slip distribution is validated through the simulation of a polycrystalline tensile experiment. This work explains the limitations of classical crystal plasticity theory under high strain gradients and offers a straightforward yet robust slip discreteness correction to crystal plasticity with transparent input from dislocation theory, opening a new perspective for the connections between continuum crystal plasticity theory and dislocation theory.

Symmetry breaking induced asymmetric dislocation-planar fault interactions in ordered intermetallic alloys

In this study, we present the first comprehensive examination of symmetry breaking in the interactions between dislocation and superlattice planar faults, including anti-phase boundary (APB), complex stacking fault (CSF), superlattice intrinsic stacking fault (SISF), to reveal the underlying asymmetric dislocation reaction mechanisms depending on the sense of applied stress, employing both large-scale atomistic simulations and continuum dislocation theory. Four ordered intermetallic alloy systems including γ−Ni/γ′−Ni3Al and γ−Ni/γ′−Ni3Fe, γ−Al/γ−TiAl, and α−Ti/α2−Ti3Al were selected as the representative model systems, with two primary symmetry breaking effects, i.e., translational and three-fold rotational symmetry breaking considered. Detailed atomic steps of asymmetrical dislocation reactions and the corresponding asymmetrical dislocation bypassing mechanisms of precipitation have been elucidated, shown to be highly dependent on the geometrical configuration of the precipitate and the relative magnitudes of APB, CSF and SISF fault energies. A continuum model framework was then developed, which, for the first time, provides accurate and quantitative predictions of the threshold conditions triggering critical asymmetrical dislocation slips, verified to be in good agreement with the simulation results. Our study also successfully reproduced the experimentally observed dislocation-induced APB-SISF transformation, with a new dislocation reaction mechanism proposed to explain the transformation process. The findings are expected to be a key enabling stepstone for future innovation in intermetallic alloys strengthened through ordered phases for advanced applications in aeronautic and automotive industries.

Phase-field simulation of misfit dislocations in two-phase electrode particles: Driving force calculation and stability analysis

Dislocations in lithium-ion battery materials significantly influence the performance of the battery. A chemo-mechanical phase-field model combined with a non-singular continuum theory of dislocations is developed to simulate misfit dislocations at the phase boundary of two-phase LiFePO4 particles. The configurational mechanics of dislocations in the mechanically coupled phase transformation problem is proposed and then used to formulate a dislocation formation criterion. The model is numerically implemented using the finite element method to compute driving forces on misfit dislocations. The driving force is further used to study the stability of the misfit dislocation to predict the critical size of dislocation-free two-phase electrode particles. The orthotropic and non-homogeneous elastic constants in two phases, anisotropic misfit strain, different aspect ratios of the particle, and different positions of the phase boundary in equiaxed particles are considered. The large misfit strain at the phase boundary is found to cause a redistribution of the concentration and thus a relaxation of the stress field at the interface. Stress relaxation is crucial to the driving force on the dislocation and ignoring the stress relaxation leads to an underestimation of the critical particle size. The critical particle size decreases with increasing aspect ratios and shows a saturation after aspect ratios above 2. An optimum aspect ratio of the dislocation-free electrode particle is around 1.5 which features the highest specific surface area.

Making sense of dislocation correlations

Since crystal plasticity is the result of moving and interacting dislocations, it seems self-evident that continuum plasticity should in principle be derivable as a statistical continuum theory of dislocations, though in practice we are still far from doing so. One key to any statistical continuum theory of interacting particles is the consideration of spatial correlations. However, because dislocations are extended one-dimensional defects, the classical definition of correlations for point particles is not readily applicable to dislocation systems: the line-like nature of dislocations entails that a scalar pair correlation function does not suffice for characterizing spatial correlations and a hierarchy of two-point tensors is required in general. The extended nature of dislocations as closed curves leads to strong self-correlations along the dislocation line. In the current contribution, we thoroughly introduce the concept of pair correlations for general averaged dislocation systems and illustrate self-correlations as well as the content of low order correlation tensors using a simple model system. We furthermore detail how pair correlation information may be obtained from three-dimensional discrete dislocation simulations and provide a first analysis of correlations from such simulations. We briefly discuss how the pair correlation information may be employed to improve existing continuum dislocation theories and why we think it is important for analyzing discrete dislocation data.

Influence of dislocations on domain walls in perovskite ferroelectrics: Phase-field simulation and driving force calculation

Dislocations in ferroelectrics cause domain wall pinning and nucleation of ferroelectric domains, which significantly influence the electromechanical properties of ferroelectrics. To have an in-depth understanding of the role of dislocations in ferroelectrics, this work presents a finite element analysis of the driving force on domain walls and their interaction with dislocations. The phase-field model coupled with the non-singular continuum dislocation theory and the generalized configurational force theory are derived for dislocated ferroelectrics. In particular, by making use of the driving force on domain walls and dislocations, we propose a new approach to evaluate the critical electric field for domain walls to break through dislocations. It is shown to be computationally much more efficient than the brute force approach of scanning all electric fields. The influences of dislocations on the 180° and 90° domain walls are analyzed numerically. The dislocation-induced stress field causes fluctuation of the polarization, which results in multiple equilibrium positions of the domain wall. Increased polarization was observed on the tensile side of the dislocation, which is responsible for the strong pinning strength of the dislocation. The needle domain nucleated from the dislocation core also causes the pinning of domain walls, and the interaction between the domain wall and the needle domain results in nonlinear dependency of the critical electric field on the Burgers vector.

Chemo-mechanical study of dislocation mediated ion diffusion in lithium-ion battery materials

A mechanically coupled diffusion model combined with finite element formulation is developed to study the influence of dislocations on ion diffusion in lithium-ion batteries. The dislocation is modeled by the regularized eigenstrain based on a non-singular continuum dislocation theory. The model is validated with the analytical solution of the stress field of edge dislocations and the solution for the stress-dependent equilibrium concentration around the dislocation. Simulation results on LiMn2O4 demonstrate strong ion enrichment and depletion on the tensile and compressive sides of an edge dislocation, respectively. A stronger influence of the edge dislocation on diffusion is found at a lower state-of-charge, which verifies the experimental observation reported in the literature. The diffusion-induced stress compensates partially the stress field of the edge dislocation and is ascertained to have a state-of-charge dependency. The existence of dislocation does not introduce obvious mobility anisotropy in the bulk material but it results in local mobility heterogeneity around the dislocation. A three-dimensional simulation of the diffusion along the edge dislocation line reveals that the pipe diffusion can be initiated or accelerated on the tensile side of the edge dislocation.

Is crystal plasticity non-conservative? Lessons from large deformation continuum dislocation theory

Crystal plasticity for large plastic deformations is usually modeled as volume conserving. However, there is experimental evidence that huge amounts of vacancies are produced during plastic deformation. The only regularly considered mechanism of non-conservative dislocation motion is climb of edge dislocations, but this is supposed to be enabled by high vacancy concentrations rather than producing them. In the current contribution we show that large deformation kinematics applied to evolving fields of geometrically necessary dislocations accounts for kinking and jogging of dislocations when cutting through one another. Therefore, dislocation fields in multiple slip situations will not be confined to slip planes and their motion will necessarily involve non-conservative motion of jogs. Because inelastic volume changes in crystals are only possible in the presence of point defects, we find that large deformation continuum dislocation theory must be coupled to an evolving vacancy concentration. We suggest to treat vacancy diffusion as an independent inelastic deformation mechanism yielding a three-way multiplicative decomposition of the deformation gradient. Small numerical examples are used to illustrate the kinematics of emerging and moving kinks and jogs, and the ensuing production of vacancy concentration. Continuum thermodynamics is used to derive coupled evolution equations for geometrically necessary dislocation density vectors and vacancy concentration. We conclude that large deformation crystal plasticity as an averaged theory of dislocations may not be conservative and that the usual kinematic assumption Lp=∑sγ˙sms⊗ns does not hold in multiple slip deformation.

Study of Geometrically Necessary Dislocations of a Partially Recrystallized Aluminum Alloy Using 2D EBSD.

During recrystallization, the growth of fresh grains initiated within a deformed microstructure causes dramatic changes in the dislocation structure and density of a heavily deformed matrix. In this paper, the microstructure of a cold rolled and partially recrystallized Al-Mg alloy (AA5052) was studied via electron backscattered diffraction (EBSD) analysis. The structure and density of the geometrically necessary dislocations (GNDs) were predicted using a combination of continuum mechanics and dislocation theory. Accordingly, the Nye dislocation tensor, which determines the GND structure, was estimated by calculation of the lattice curvature. To do so, five components of the Nye dislocation tensor were directly calculated from the local orientation of surface points of the specimen, which was determined by two-dimensional EBSD. The remaining components of GNDs were determined by minimizing a normalized Hamiltonian equation based on dislocation energy. The results show the elimination of low angle boundaries, lattice curvature, and GNDs in recrystallized regions and the formation of low angle boundaries with orientation discontinuities in deformed grains, which may be due to static recovery.

A continuum dislocation-based model of wedge microindentation of single crystals

Recent Electron Backscatter Diffraction (EBSD) experiments have revealed the emergence of heterogeneous dislocation microstructures forming under a wedge indenter in fcc crystals, where micro-meter dislocation patterns challenge the predictions of traditional models of plasticity. In order to explain the formation of these features and develop a relationship between the force-displacement curve and the dislocation substructure, we present here a model of wedge indentation based on the continuum theory of dislocations. The model accounts for large deformation kinematics through the multiplicative split of the deformation gradient tensor, where the incompatible plastic component of deformation results from the flux of dislocations on different and interacting slips systems. Constitutive equations for dislocation fluxes are determined from a dissipative variational principle. As a result, each dislocation density satisfies an initial-boundary value problem with convective-diffusive character, which is coupled to the macroscopic stress and displacement fields governing the deformation process. Solution to the self-consistent continuum formulation is found using the finite element method. Computer simulations mimic the experimental conditions of wedge micro-indentation experiments into Ni single-crystals used by Kysar et al. (2010a). A comparison of overall dislocation density distribution and macroscopic mechanical response shows good overall agreement with the experimental results in terms of the detailed features of dislocation patterns and lattice rotations as well as the macroscopic force-displacement response.

Critical thickness phenomenon in single-crystalline wires under torsion

Analysis of idealised thin single-crystal wires under torsion based on the continuum theory of dislocations gives results in accordance with the critical thickness theory. The dislocation-free zone near the wire surface and the nearly-zero stress around the wire axis are predicted by both the continuum dislocation theory and critical thickness theory. It is demonstrated that the size effect at the onset of yielding, the distributions of stress and geometrically necessary dislocations in the thin wires in torsion, simply result from the critical thickness effect. A continuous increase of plastic strain from the neutral axis toward the wire surface is indicated. The plastic strain becomes (nearly) flat around the wire surface. Such a phenomenon is attributed to the fact that this is the region in which dislocations sources can operate, to provide the geometrically necessary dislocations required by the plastic strain gradient beneath. The results of continuum dislocation theory quantitatively elucidate the critical thickness phenomenon occurred in single-crystal wires under torsion. This links the continuum dislocation theory to the underlying physical picture of Matthews' critical thickness theory.

Numerical solution of plane constrained shear problem for single crystals within continuum dislocation theory

AbstractWithin the recently proposed Continuum Dislocation Theory (CDT), numerical solutions of simple shear test for single crystals strip using finite elements are performed. Under the assumption of plane constrained deformation of crystal having only one active slip system, the plastic slip, the dislocation density as well as the stress‐strain curve are computed. The results of numerical simulations are compared with those obtained from analytical solutions which show good agreement. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

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.

Geometrically linear continuum theory of dislocations revisited from a thermodynamical perspective

Continuum dislocation theory (CDT) allows the consideration of dislocation ensembles by introducing the dislocation density tensor. Though the kinematics of geometrically linear CDT are well established, the closure of governing field equations is not finished yet. The present study now brings together different principles for such a closure: It is shown how the field equations for the CDT can be obtained from potential energy minimization and from the phase field approach. These two energetic methods are integrated into a generic thermodynamic framework with twofold benefit: First, the rigorous thermodynamic treatment allows clarifying physical consequences of the energetic methods, among them the proof of thermodynamic consistency. Second, the framework provides a basis for consistent extensions of CDT. In this way, a new dynamic formulation of CDT is presented, which enables the analysis of the evolution of dislocation structures during plastic deformation. Moreover, a variety of possible dissipative phenomena is considered and the mechanical balance laws are deduced. For two special cases, the field equations are derived in the strong form and the stability of the solution is analyzed. Next, a flexible numerical solution algorithm is presented using the finite difference method. Solutions of various initial boundary value problems are presented for the case of plane deformations. Therefore, some of the dissipative phenomena are further investigated and two distinct sources of the Bauschinger effect are identified. Special attention is also given to different boundary conditions and their effect on the solution. For the case of uniaxial compression, the numerical results are confronted with experimental data. Thus, the simulations are validated and a new consistent interpretation of the experimental results is achieved.

Analogies between continuum dislocation theory, continuum mechanics and fluid mechanics

Continuum Dislocation Theory (CDT) relates gradients of plastic deformation in crystals with the presence of geometrically necessary dislocations. Interestingly, CDT shows striking analogies to other branches of continuum mechanics. The present contribution demonstrates this on two essential kinematical quantities which reflect tensorial dislocation properties: the (resultant) Burgers vector and the dislocation density tensor. First, the limiting process for the (resultant) Burgers vector from an integral to a local quantity is performed analogously to the limiting process from the force vector to the traction vector. By evaluating the balance of forces on a tetrahedral volume element, Cauchy found his famous formula relating traction vector and stress tensor. It is shown how this procedure may be adopted to a continuously dislocated tetrahedron. Here, the conservation of Burger’s vector implicates the introduction of the dislocation density tensor. Second, analogies between the plastic flow of a continuously dislocated solid and the liquid flow of a fluid are highlighted: the resultant Burgers vector of a dislocation ensemble plays a similar role as the (resultant) circulation of a vortex tube. Moreover, both vortices within flowing fluids and dislocations within deforming solids induce discontinuities in the velocity field and the plastic distortion field, respectively. Beyond the analogies, some peculiar properties of the dislocation density tensor are presented as well.

The impact of dislocations on coercivity in L10-MnAl

Abstract Two different transformation routes, quenching followed by heat treatment (route 1), and cooling at intermediate rate (route 2), were used to obtain the L1 0 phase in Mn-Al alloys. The resulting materials showed remarkable differences in microstructure and magnetic properties. In addition, fully transformed material was cold-worked and then recovered by heat treatment. The microstructure of both cold-worked and recovered samples appeared similar in backscattered electron images, however, the magnetic properties differed greatly. Large numbers of electron backscatter diffraction (EBSD) patterns were recorded from all the materials and the pattern quality was quantified, yielding a measure of the dislocation density. The results indicated that both the dislocation density and the coercivity of the cold-worked sample were much higher than in the recovered sample. In addition to other microstructural changes, the route 1 sample had higher dislocation density and coercivity compared to the route 2 sample. For the as-transformed materials, the results were supported by EBSD misorientation mapping and strain analysis of x-ray diffraction data. The continuum theory of dislocations was used to show that the local stress fields of dislocations will cause distortions in the crystal structure leading to local changes in the intrinsic magnetic properties. Such features are likely to act as pinning centres for magnetic domain walls and therefore a higher dislocation density implies a higher coercivity. This conclusion was additionally supported by initial magnetisation curves, which showed evidence for an active pinning mechanism in the cold-worked sample but not for the recovered sample.

The Impact of the Dislocation Density, Lattice and Impurity Friction on the Dynamics of Expansion of a Dislocation Loop in FCC Metals

The study of the role of various factors in plastic behavior of materials is carried out using a mathematical model that takes into account fundamental properties of deformation defects in a crystal lattice based on the continuum theory of dislocations. Calculations were performed for copper, nickel, aluminum, and lead using a specialized software system Dislocation Dynamics of Crystallographic Slip. It has been shown that a decrease in the density of dislocations from 1012 m-2 to 1011 m-2 leads to an increase in the dislocation path in 10−16 times, and the maximum velocity in 1.5−2 times in copper and nickel, by nearly 20% in aluminum, and practically remains unchanged in lead. A decrease in the lattice and impurity friction from 2 MPa to 0.1 MPa leads to a linear increase in the path and the maximum velocity of the dislocation by 10−25%.

Continuum dislocation theory accounting for redundant dislocations and Taylor hardening

This paper develops the phenomenological continuum dislocation theory accounting for the density of redundant dislocations and Taylor hardening for single crystals. As illustration, the problem of anti-plane constrained shear of single crystal deforming in single slip is solved within the proposed theory. The distribution of excess dislocations in the final state of equilibrium as well as the stress-strain curve exhibiting the Bauschinger translational work hardening and the size effect are found. Comparison with the stress-strain curve obtained from the continuum dislocation theory without the density of redundant dislocations and Taylor hardening is provided.

Dislocation patterning in a two-dimensional continuum theory of dislocations

Understanding the spontaneous emergence of dislocation patterns during plastic deformation is a long standing challenge in dislocation theory. During the past decades several phenomenological continuum models of dislocation patterning were proposed, but few of them (if any) are derived from microscopic considerations through systematic and controlled averaging procedures. In this paper we present a 2D continuum theory that is obtained by systematic averaging of the equations of motion of discrete dislocations. It is shown that in the evolution equations of the dislocation densities diffusion like terms neglected in earlier considerations play a crucial role in the length scale selection of the dislocation density fluctuations. It is also shown that the formulated continuum theory can be derived from an averaged energy functional using the framework of phase field theories. However, in order to account for the flow stress one has in that case to introduce a nontrivial dislocation mobility function, which proves to be crucial for the instability leading to patterning.

On a continuum theory of dislocation equilibrium

A continuum theory of dislocations is suggested which is capable of predicting the equilibrium distributions of a large number of screw dislocations in anisotropic beams with arbitrary cross-section. The theory leads to a boundary value problem with unknown boundary. The problem is solved for isotropic beams with circular cross-sections and thin rectangular cross-sections. The solution for circular cross-sections is compared with the results of numerical simulations by [Weinberger, C. R., 2011. The structure and energetics of, and the plasticity caused by, Eshelby dislocations, International Journal of Plasticity, 27, 1391–1408]. An extension of the theory to equilibrium of edge dislocations is discussed.

Continuum representation of systems of dislocation lines: A general method for deriving closed-form evolution equations

Plasticity is governed by the evolution of, in general anisotropic, systems of dislocations. We seek to faithfully represent this evolution in terms of density-like variables which average over the discrete dislocation microstructure. Starting from T. Hochrainer's continuum theory of dislocations (CDD) (Hochrainer, 2015), we introduce a methodology based on the ‘Maximum Information Entropy Principle’ (MIEP) for deriving closed-form evolution equations for dislocation density measures of different order. These equations provide an optimum representation of the kinematic properties of systems of curved and connected dislocation lines with the information contained in a given set of density measures. The performance of the derived equations is benchmarked against other models proposed in the literature, using discrete dislocation dynamics simulations as a reference. As a benchmark problem we study dislocations moving in a highly heterogeneous, persistent-slip-band like geometry. We demonstrate that excellent agreement with discrete simulations can be obtained in terms of a very small number of averaged dislocation fields containing information about the edge and screw components of the total and excess (geometrically necessary) dislocation densities. From these the full dislocation orientation distribution which emerges as dislocations move through a channel-wall structure can be faithfully reconstructed.