Straight Edge Dislocations Research Articles

A Modern Approach to the Complicated Line Integrals in Linear Elastic Dislocation Theory

The standard textbook analysis of dislocations is generally limited to the case of infinitely straight screw or edge dislocations, which do not exist. This is due to the complexity of the formulas for arbitrary dislocation loops, i.e., Burger’s equation for the displacement field, the Peach-Köhler equation for the stress field and Blin’s equation for the interaction energy, which involve line integrals along the dislocation loop. The integrands are complex, and integration often involves non-elementary functions. Elaboration of the integrands with symbolic mathematical software produces tensor formulas which can be reused at will. By formulating convenient parametric expressions for the configuration studied and using superposition, mathematical software can be used to perform the integrations for arbitrary Burgers vectors. Often, the resulting expressions for the tensorial fields are very long, but they can be easily incorporated as user-defined formulas for plotting, parametric analysis, and incorporation into routines for energy minimisation or the non-linear equations for force equilibrium. The effectiveness of this approach will be illustrated by the example of short straight dislocations, circular dislocations, the interaction between a pileup and dissociated dislocations in the grain boundary, and the nucleation of dislocations at grain boundaries.

Statistical properties of fractal type dislocation cell structures

The dislocation microstructure developing during plastic deformation strongly influences the stress-strain properties of crystalline materials. Resent theoretical investigations based on the 2D continuum theory of straight parallel edge dislocations were able to predict a periodic dislocation microstructure. The results obtained, however, can only be considered as a very first step toward the understanding of the origin of dislocation patterning. One of the most challenging problems is the modeling of the formation of the fractal like dislocation microstructure. So, it is crucial to determine the statistical properties of such a structure developing at ideal multiple slip orientation. In the paper, by x-ray line profile analysis and the method of high resolution electron backscatter diffraction (HR-EBSD) a complex experimental characterization of dislocation microstructure developing in uniaxially compressed Cu single crystals is presented. With these methods, the maps of the internal stress, the Nye tensor, and the geometrically necessary dislocation (GND) density were determined at different load levels. It is found from the fractal analysis of the GND maps that the fractal dimension of the cell structure is decreasing with increasing average spatial dislocation density fluctuation. Moreover, it is shown that the evolution of different types of dislocations can be successfully monitored with the HR-EBSD-based technique.

Combined modeling and experimental characterization of Mn segregation and spinodal decomposition along dislocation lines in Fe–Mn alloys

In the current work, Mn enrichment at dislocations in Fe–Mn alloys due to segregation and spinodal decomposition along the dislocation line is studied via modeling and experimental characterization. To model these phenomena, both finite-deformation microscopic phase-field chemomechanics (MPFCM) and Monte Carlo molecular dynamics (MCMD) are employed. MPFCM calibration is carried out with the same Fe–Mn MEAM-based potential used in MCMD, as well as CALPHAD data. Simulation results for Mn segregation to, and spinodal decomposition along, straight screw and edge dislocations as well as dislocation loops, are compared with characterization results from atom probe tomography (APT) for two Fe–Mn alloy compositions. In contrast to classical Volterra (linear elastic) dislocation theory, both MPFCM and MCMD predict a non-zero hydrostatic stress field in screw cores. Being of much smaller magnitude than the hydrostatic stress in straight edge cores, much less solute segregates to screw than to edge cores. In addition, the segregated amount in screw cores is below the critical concentration of 0.157 for the onset of spinodal decomposition along the line. On the other hand, results from MPFCM-based modeling imply that the concentration dependence of the solute misfit distortion and resulting dependence of the elastic energy density on concentration have the strongest effect. The maximum amount of Mn segregating to straight edge dislocations predicted by MPFCM agrees well with APT results. On the other hand, the current MPFCM model for Fe–Mn predicts little or no variation in Mn concentration along a straight dislocation line, in contrast to the APT results. As shown by the example of a dislocation loop in the current work, a change in the hydrostatic stress along the line due to changing character of dislocation does lead to a corresponding variation in Mn concentration. Such a variation in Mn concentration can also then be expected along a dislocation line with kinks or jogs.

Elastodiffusion calculation of asymptotic absorption efficiencies and bias factors of dislocations in Al, Ni and Fe

The elastic interactions between dislocations and lattice point defects (PDs) control the rate of PD elimination at dislocations, and thereby the PD spatial distribution within the microstructure. By relying on the Onsager formalism, we introduce a new partition of the PD diffusion driving force in a strain field. We present DFT-based calculations of straight and loop-form edge dislocation absorption efficiencies and the resulting biases of vacancy and self-interstitial elimination in pure Fe, Ni, and Al metals. We highlight the effects of the dislocation density, the temperature and the dislocation orientation on the elastodiffusion behavior and the resulting PD redistribution near dislocations. Based on a detailed study of the sensitivity of absorption efficiencies to the boundary conditions, i.e., the capture radius and the associated PD concentrations, we introduce the notion of asymptotic absorption efficiencies and bias.

Using lifetime of point defects for dislocation bias in bcc Fe

The interaction between dislocations and point defects is key to deformation processes and microstructural evolution of structural materials. In this work, we compute the lifetime of point defects to describe their interaction with dislocations. This approach can accurately account for the effects of the dislocation core and anisotropic defect dynamics to accumulatively determine the capture efficiency, sink strength, and dislocation bias at different temperatures and dislocation densities. Particularly, the absorption of point defects by straight screw and edge dislocations in a model bcc iron system is studied. The maximum swelling rates based on the obtained bias factors are in close agreement with a variety of experimental measurements, including both neutron and ion-irradiation data, especially when considering the survival fraction for point defects from displacement cascades. This approach applies to many other processes and sinks, such as dislocation loops and interfaces, providing a powerful means to develop fundamental insights critical for improving radiation resistance and mechanical properties of structural materials through controlling defect interaction and evolution.

A Combined Atomistic-Continuum Study on the Unfaulting of Single and Multi-layer Interstitial Dislocation Loops in Irradiated FCC and HCP Metals

Unfaulting of single and multi-layer interstitial loops in face centered cubic (FCC) and hexagonal close-packed (HCP) metals, with Ni, Al, Mg and Zr as model systems, have been comprehensively studied employing molecular dynamics simulations and continuum modeling. A versatile approach properly accounting for atom insertion/removal and displacement field was developed, able to correctly construct single and multi-layer interstitial loop structures of arbitrary geometries. The unfaulting mechanisms and corresponding dislocation reaction intermediates in different unfaulting scenarios were clarified. Critical conditions necessary for unfaulting have been identified, and an analytical model has been developed to accurately predict these conditions. Particularly for unfaulting in FCC materials, the role of Double Shockley (D-Shockley) partial in unfaulting has been explicitly elucidated, and the mechanisms of unfaulting initiating from periphery of the Frank loop and induced by a moving straight edge dislocation have been clarified. For unfaulting in HCP, despite certain similarity to its FCC counterpart, clear difference in unfaulting behaviors due to the HCP stacking sequence has been demonstrated. A comprehensive mapping of unfaulting routes for double-layer interstitial loops in HCP has been established to reveal the relationships between the various loop morphologies observed in experiments. The present work provides important mechanistic insights to address key knowledge deficits regarding interstitial Frank loop unfaulting, essential for better understanding of deformation and irradiation-induced failure in structural metals. The new modeling techniques and predictive tools offered are expected to be of great merits to not only the study of unfaulting of dislocation loops, but also future investigation of other complex dislocation or defect ensembles.

Elastic interaction-induced anisotropic growth of dislocation loop arrays

The elastic interactions and reactions of dislocations lead to the formation of complex dislocation substructures, which is critical to the strain hardening and fatigue failure. Phase field dislocation dynamics simulations are conducted as a first step to understand the elastic interactions between dislocation loops. When the interloop spacing is small, the elastic interactions with neighboring loops become strong, rendering the edge segments strongly pinned, while allowing for the screw segments to propagate more easily. The interactions are found to result in an anisotropic stress distribution around the dislocation loops, leading to the formation of arrays of long, straight edge dislocations that could act as barriers to subsequent slip. Furthermore, the effect of initial loop size and applied strain rate on the elastic interaction-induced anisotropic pinning effect is investigated and discussed. The results are important for coarse-graining dislocation substructures formation into continuum level models of deformation in crystalline solids.

Nonlocal hcp kernel functions based on ab initio calculations: Pertinent dislocation problems revisited

Eringen's nonlocal theory and an accurate determination of the nonlocal kernel functions for hexagonal close-packed (hcp) crystals are of interest. The kernel functions are closely related to the anisotropy as well as any crystalline symmetries. To this end, five new distinct nonlocal kernel functions which have the characteristics of discrete atomistic Green's functions in the stress space are obtained through consideration of the nonlocal dispersion relations associated with certain directions combined with ab initio Density Functional Perturbation Theory (DFPT) calculations of the pertinent phonon frequencies. This is the first work which provides the nonlocal hcp kernel functions pertinent to each component of the elastic moduli tensor of Eringen's nonlocal continuum theory based on quantum mechanical considerations. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameters associated with the hcp crystals of Mg, Cd, Ti, and Zn have been computed using ab initio Density Functional Theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a straight screw dislocation and a straight edge dislocation inside an infinitely extended hcp medium have been examined for the case where the dislocation lies in the basal plane. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. In contrast to the classical anisotropy theory that predicts singular shear stress on the slip plane at the core of the dislocation, the corresponding nonlocal shear stress component predicted by the present theory is zero at the dislocation core. It is noteworthy to mention that the normalized nonlocal shear stress distribution and its maximum value are greatly influenced by the anisotropy of the crystalline solids.

Geometrical modeling and numerical analysis of edge dislocation

In this study, we conduct modeling and numerical analysis of a straight edge dislocation in a three-dimensional geometrically nonlinear elastic medium. Based on the fundamental framework of geometrical elasto-plasticity, kinematics of the continuum is represented by the reference R, intermediate B and current S states and local deformations are described by the multiplicative decomposition of the deformation gradient; F = Fe·Fp. Here, the reference and current states are Euclidean submanifolds while the intermediate state B is represented by Weitzenbock manifold with non-zero torsion in the affine connection. Following to the equivalence of torsion and dislocation density tensor through Hodge star operation, a plastic deformation gradient Fp is obtained from the Cartan first structure equation using the homotopy operator. The current state S is determined so that it minimizes the strain energy functional. We solve the variational problem by using isogeometric analysis; Galerkin method with non-uniform B-spline basis function. The present analysis shows that all stress components agree quantitatively well with those of the Volterra dislocation model. Around the dislocation core, stress fields are non-singular on the contrary to the Volterra dislocation. In addition, we found that all stress components distribute asymmetrically due to the geometrical nonlinearity of the model.

Uniform motion of an edge dislocation within Mindlin’s first strain gradient elasticity

Motion of dislocations plays the fundamental role in plastic deformations of crystalline solids. According to the prediction of classical elasticity, the strain and stress fields due to a moving dislocation suffer from singularities. In the present paper, it is shown that the employment of Mindlin’s first strain gradient theory of elasticity leads to the removal of the singularities from all the elastic fields induced by a straight edge dislocation that moves uniformly in an infinitely extended isotropic body. However, at distances far enough away from the dislocation core, the predictions of such theory and classical elasticity coincide with each other. The present analysis also shows that the corresponding plastic strain field of the moving dislocation is, likewise, free from any singularity. Moreover, explanations are provided in the current paper for the roles of the micro-inertia and the characteristic lengths of the constituent material of the body as well as the effect of the velocity of the dislocation on its induced field quantities.

Mobility of dislocations in aluminum: The role of non-Schmid stress state

The extent that non-Schmid stresses influence the mobility of dislocations in Al is studied using molecular dynamics (MD) simulations. Specifically, mobility laws for straight screw, 30°, 60° and edge dislocations are atomistically derived for different combinations of Escaig stress, τe, and stress normal to the (111) slip plane, σn. MD simulations show that the mobility of each dislocation is distinctly influenced by non-Schmid stresses. Furthermore, MD simulations of dislocation shear loop expansion show that mobility laws derived from straight dislocations are applicable to describe the more complex behavior of stress-state dependent expansion of a dislocation loop. Data describing the dislocation mobility laws are incorporated into discrete dislocation dynamics (DDD) simulations using a hierarchical multiscale approach. DDD simulations of isolated dislocation loop expansion show strong agreement with MD simulation results, validating the nonlinear multiscale implementation. DDD simulations of dislocation network evolution show that the use of stress state dependent dislocation mobility laws provides quantifiable changes to the plastic deformation path, which leads to different final microstructures. The results presented here demonstrate the key role that local stresses, experienced by the dislocation core, play on the evolution of a dislocation network during plastic deformation.

Generalized Kanzaki force field of extended defects in crystals with applications to the modeling of edge dislocations

The Kanzaki forces and their associated multipolar moments are standard ways of representing point defects in an atomistically informed way in the continuum. In this article, the Kanzaki force approach is extended to other crystalline defects. The article shows how the resulting Kanzaki force fields are to be computed for any general extended defect by first computing the relaxed defect's structure and then defining an affine mapping between the said defect structure and the original perfect lattice. This methodology can be employed to compute the Kanzaki force field of any mass-conserving defect, including dislocations, grain and twin boundaries, or cracks. Particular focus is then placed on straight edge dislocation in face-centered cubic (fcc) and body-centered cubic (bcc) pure metals, which are studied along different crystallographic directions. The particular characteristics of these force fields are discussed, drawing a distinction between the slip Kanzaki force field associated with the Volterra disregistry that characterizes the dislocation, and the core Kanzaki force field associated with the specific topology of the dislocation's core. The resulting force fields can be employed to create elastic models of the dislocation that, unlike other regularization procedures, offer a geometrically true representation of the core and the elastic fields in its environs, capturing all three-dimensional effects associated with the core.

Dislocation dynamics formulation for self-climb of dislocation loops by vacancy pipe diffusion

It has been shown in experiments that self-climb of prismatic dislocation loops by pipe diffusion plays important roles in their dynamical behaviors, e.g., coarsening of prismatic loops upon annealing, as well as the physical and mechanical properties of materials with irradiation. In this paper, we show that the dislocation dynamics self-climb formulation that we derived in (Niu et al., 2017) is able to quantitatively describe the properties of self-climb of prismatic loops that were observed in experiments and atomistic simulations. This dislocation dynamics formulation applies to self-climb by pipe diffusion for any configurations of dislocations, and is able to recover the available models in the literature for rigid self-climb motion of small prismatic loops. We also present implementation method of this self-climb formulation. Simulations performed show evolution, translation and coalescence of prismatic loops as well as prismatic loops driven by an edge dislocation by self-climb motion and the elastic interaction between them. These results are in excellent agreement with available experimental and atomistic results. We have also performed systematic analyses of the behaviors of a prismatic loop under the elastic interaction with an infinite, straight edge dislocation by motions of self-climb and glide.

Plasticity as the $${\Gamma}$$ Γ -Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of disloactions is not assumed. In order to prove meaningful lower bounds the ball construction technique, which was developed in the context of Ginzburg-Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of a loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.

Mobility of dislocations in Aluminum: Faceting and asymmetry during nanoscale dislocation shear loop expansion

Dislocation loop expansion in Al is studied with a hierarchical multiscale approach via atomistic and discrete dislocation dynamics (DDD) simulations. First, mobility laws for straight screw, 30°, 60°, and edge dislocations are calculated using molecular dynamics (MD) simulations. Each atomistic mobility law is described by an empirical piecewise function with a character angle dependent power law exponent to capture the nonlinear damping regime. For the dislocation velocity range considered in this work, the mobilities of screw and 60° dislocations are lower than edge and 30° dislocations. The mobilities of mixed dislocations, such as 30° and 60°, are found to depend on the character of the leading Shockley partial. Second, MD simulations of dislocation loop expansion under a constant external Schmid stress are performed. The expanded dislocation loop is found to facet in the 60° and screw segments due to low mobilities. The dislocation loop is also found to have reflection asymmetry about the edge axis due to mixed dislocation mobility asymmetry. Finally, the nonlinear atomistic mobility laws are implemented into DDD simulations to illustrate that atomistic mobility laws for straight dislocations can describe faceting and asymmetric behaviors observed during dislocation loop expansion in higher length scale simulations.

A new insight into the J‐, M‐, and L‐integrals of dislocations

AbstractThe explicit formulas of the J‐, M‐, and L‐integrals of straight (screw and edge) dislocations in isotropic materials are presented. The obtained results reveal the physical interpretation and significance of the M‐, and L‐integrals for straight dislocations. The M‐integral between two straight dislocations (per unit length) is equal to half the interaction energy of the two dislocations (per unit length) depending on the distance and on the angle, plus twice the corresponding pre‐logarithmic energy factor. The L3‐integral between two straight dislocations is the z‐component of the configurational vector moment or the rotational moment (torque) about the z‐axis caused by the interaction between the two dislocations.

Formation of Cottrell atmosphere studied by Langevin equation coupled with the first strain gradient elasticity

This study investigate the formation of Cottrell atmosphere around a dislocation on the basis of Langevin equation coupled with the first strain gradient elasticity (SGE) of Cauchy type. Following to the eigen strain theory of Eshelby, we introduced a straight edge dislocation at the center of a two-dimensional rectangular SGE medium. Non-singular stress field around the dislocation is obtained using the isogeometric analysis (IGA) solved with the 3rd-order NURBS basis functions. Long-time diffusion of solute atoms is governed by the overdamped Langevin equation which equips the Peach-Koehler potential, obtained from stress field around the dislocation, in addition to the conventional thermal noise. Present numerical analysis revealed the formation process of Cottrell atmosphere in a realistic time scale.

Micromechanics of dislocations in solids: J-, M-, and L-integrals and their fundamental relations

The aim of the present work is the unification of incompatible elasticity theory of dislocations and Eshelbian mechanics leading naturally to Eshelbian dislocation mechanics. In such a unified framework, we explore the utility of the J-, M-, and L-integrals. We give the physical interpretation of the M-, and L-integrals for dislocations, connecting them with established quantities in dislocation theory such as the interaction energy and the J-integral of dislocations, which is equivalent to the well-known Peach–Koehler force. The J-, M-, and L-integrals for dislocations have been studied in the framework of three-dimensional, incompatible, linear elasticity. First of all, the general formulas of the J-, M-, and L-integrals for dislocations are given. Next, the examined integrals are specified for straight dislocations. Finally, the explicit formulas of the J-, M-, and L-integrals are calculated for straight (screw and edge) dislocations in isotropic materials. The obtained results reveal the physical interpretation and significance of the M-, and L-integrals for straight dislocations. The M-integral between two straight dislocations (per unit dislocation length) is equal to the half of the interaction energy of the two dislocations (per unit dislocation length) depending on the distance and on the angle, plus twice the corresponding pre-logarithmic energy factor. The L3-integral between two straight dislocations is the z-component of the configurational vector moment or the rotational moment about the z-axis caused by the interaction between the two dislocations. Fundamental relations between the J-, M-, and L3-integrals are derived showing the inherent connection between them. The relations connecting directly the J-, M-, and L3-integrals with the interaction energy are obtained. These relations have been proven to be of great significance. Since based on them; the interpretation of the J-, and L3-integrals as translational and rotational energy-release, respectively, is achieved, and secondly a stability criterion for straight edge dislocations is formulated in terms of the Jk-integral, revealing the physical importance of the considered integrals.

Dislocation climb models from atomistic scheme to dislocation dynamics

We develop a mesoscopic dislocation dynamics model for vacancy-assisted dislocation climb by upscalings from a stochastic model on the atomistic scale. Our models incorporate microscopic mechanisms of (i) bulk diffusion of vacancies, (ii) vacancy exchange dynamics between bulk and dislocation core, (iii) vacancy pipe diffusion along the dislocation core, and (iv) vacancy attachment-detachment kinetics at jogs leading to the motion of jogs. Our mesoscopic model consists of the vacancy bulk diffusion equation and a dislocation climb velocity formula. The effects of these microscopic mechanisms are incorporated by a Robin boundary condition near the dislocations for the bulk diffusion equation and a new contribution in the dislocation climb velocity due to vacancy pipe diffusion driven by the stress variation along the dislocation. Our climb formulation is able to quantitatively describe the translation of prismatic loops at low temperatures when the bulk diffusion is negligible. Using this new formulation, we derive analytical formulas for the climb velocity of a straight edge dislocation and a prismatic circular loop. Our dislocation climb formulation can be implemented in dislocation dynamics simulations to incorporate all the above four microscopic mechanisms of dislocation climb.

Effect of collision cascades on dislocations in tungsten: A molecular dynamics study

Tungsten (W) is the prime candidate material for the divertor and other plasma-facing components in DEMO. The point defects (i.e. vacancies and self-interstitials) produced in collision cascades caused by incident neutrons aggregate into dislocation loops (and voids), which strongly affect the mechanical properties. The point defects also interact with existing microstructural features, and understanding these processes is crucial for modelling the long term microstructural evolution of the material under fusion conditions. In this work, we performed molecular dynamics simulations of cascades interacting with initially straight edge dislocation dipoles. It was found that the residual vacancy number usually exceeds the residual interstitial number for cascades interacting with vacancy type dipoles, but for interstitial type dipoles these are close. We observed that a cascade near a dislocation promotes climb, i.e. it facilitates the movement of point defects along the climb direction. We also observed that the dislocations move easily along the glide direction, and that kinks are formed near the centre of the cascade, which then facilitate the movement of the dipoles. Some dipoles are sheared off by the cascade, and this is dependent on PKA energy, position, direction, and the width of dipole.