Discrete Dislocation Model Research Articles

An FFT based adaptive grid framework to represent non-singular dislocations

Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics requires very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated in the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed into a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

Construction of a Two-Dimensional Discrete Dislocation Model to Describe the Plastic Deformation Process of a Single Crystal

Construction of a Two-Dimensional Discrete Dislocation Model to Describe the Plastic Deformation Process of a Single Crystal

Dynamic length scale and weakest link behavior in crystal plasticity

Irreversible deformation of crystals is often characterized by stochastic scale-free distributed intermittent local plastic bursts. Quenched obstacles with short-range interaction were found to limit the size of these events, which was termed as a transition from wild to mild fluctuations. Here, we show by analyzing the local yield thresholds in a discrete dislocation model that a dynamic length scale can be introduced based on weakest link principles, and this scale characterizes the extension of plastic events. The interplay between long-range dislocation interactions and short-range quenched disorder is found to destroy scale-free dynamical correlations, thus leading to event localization (that is, shortening of the length scale) which explains the crossover between the wild and mild regimes. Several methods are presented to determine the dynamic length scale which can be generalized to other types of heterogeneous materials.

Discrete dislocation modelling of δ hydrides in Zr: towards an understanding of the importance of interfacial stresses for crack initiation

We carried out discrete dislocation plasticity (DDP) simulations in the infinite-matrix approximation of δ Zr micro-hydrides – including their elastic field - first precipitating in Zr and then subjected to an external stress. We calculate and present the dislocation density, maximum principal shear and stresses, configurational energy density and plastic displacement. It was found that even hydrides as thin as 20 nm have a large impact on dislocation density, plastic deformation and stress build-up, and that this effect increases for increasing hydride thickness. It also appears that the critical location for crack nucleation is at the Zr/hydride interface. In fact, slip bands that develop on one side of the interface, and are stopped by it, are accompanied by the formation of high stresses. The observation that the interfacial stress is the only way in which hydrides are represented in the simulations, with no point obstacles or change in the slip systems at the interface, combined with the observed strong influence of the hydrides on the results, suggests a possible mechanistic explanation for the embrittlement caused by the precipitation of intragranular hydrides. The mechanism is rooted in the volumetric expansion upon precipitation and naturally includes a thickness dependence that is also experimentally observed.

Exploration of the dislocation-electrochemistry relation in LiFePO4 cathode materials

Defects, such as dislocations, in electrode materials play a significant role in the performance of lithium-ion batteries. The dislocation-electrochemistry relation has only been observed experimentally and not been fully clarified. Computational studies on this mechanism were also very limited, especially the altered cyclic voltammetry behaviors and associated effective diffusivity. This work focuses on the influences of few characteristics of dislocations on the electrochemical performance of an anisotropic cathode material, lithium iron phosphate (LiFePO4). Utilizing linear elastic mechanics and the superposition principle, we study stress and displacement fields of a LiFePO4 particle containing different densities and orientations of dislocations. With the mechanical-electrochemical coupling effects expressed by the modified Butler-Volmer equation and using the finite different method, the cyclic voltammetry curves for different dislocation configurations in the particle are investigated. Our results show that introducing dislocations can shift and distort the cyclic voltammetry curves, especially at one specific dislocation orientation. It is also found that the Li-ion molar fraction-dependent partial molar volume is an important prerequisite of the distortion in cyclic voltammetry curves. Moreover, the altered cyclic voltammetry curves at different scanning rates indicate the improvements of electrical power, stored electrical energy, and the effective diffusivity of lithium. Our discrete dislocation model indicates that the capacity loss of LiFePO4 nanoparticles can be alleviated by introducing tailored dislocations. This study assists the understanding of electrode materials with pre-existing dislocations and provides strategies of using defect engineering to improve the kinetic performance in lithium-ion batteries.

Modeling of precipitate strengthening with near-chemical accuracy: case study of Al-6xxx alloys

Many metal alloys are strengthened by controlling precipitation to achieve an optimal peak-aged condition where the strength-limiting processes of precipitate shearing and Orowan looping are thought to be comparable. Qualitative models have long captured the basic mechanisms but realistic predictions have been challenging due to both the lack of accurate material parameters and an inability to quantitatively validate the models. Here, dislocation/precipitate interaction mechanisms are studied in Al-6xxx Al–Mg–Si alloys using atomistic simulations in tandem with a near-chemically-accurate Al–Mg–Si neural network interatomic potentials. Results show that a given precipitate can exhibit shearing or looping depending on the relative orientation of the precipitate and dislocation, as influenced by the matrix and precipitate coherency stresses, direction-dependence of precipitate shearing energies, and dislocation line tension. Analytic models for shearing and calibrated discrete dislocation models of looping accurately capture the trends and magnitudes of strengthening in most cases. Good agreement with experiments is then approached by using the theories together with the more-accurate first-principles material properties. The combination of theories and simulations demonstrated here constitutes a path for understanding and predicting the role of chemistry and microstructure on alloy strength that can be applied in many different alloys.

Plasticity of zirconium hydrides: a coupled edge and screw discrete dislocation model

Understanding the plastic behaviour of thin zirconium hydrides is important for its implications on crack nucleation in the Zirconium alloy cladding used in fission reactors. Microvoids originate at fractured hydrides, and their coalescence may lead to the failure of the component. In this work, an innovative discrete dislocation framework is presented together with the preliminary results. The aim is to develop a model that is significantly faster than existing 3D formulations, to make it possible to run a statistical analysis on a simulated microstructure. This comes with limitations, which are discussed together with the planned developments. The model combines two planar and orthogonal simulations. In one only edge, in the other only screw dislocations are allowed, thereby describing all sides of a dislocation loop approximated as a rectangle. The two families of dislocations interact via their elastic stress, and this coupling proved to be important and significantly enhanced the dislocation density. The proposed model enables us to implement a 3D stress analysis of the hydrides. The simulations show that the most critical scenario is when neighbouring slip planes become populated with opposite-signed dislocations. This was observed both in the edge and in the screw case, and was reflected in the principal stress calculated by combining the two. It was also observed that the degree of permeability of the interface to dislocation crossing is inversely correlated to the stress inside the hydride and to the dislocation source activation.

Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram

Crack growth kinetics in crystalline materials is examined both from the point of continuum mechanics and discrete dislocation dynamics. Kinetics ranging from the Griffith crack to continuous elastic-plastic cracks are analyzed. Initiation and propagation of incipient cracks require very high stresses and appropriate stress gradients. These can be obtained either by pre-existing notches, as is done in a typical American Society of Testing and Materials (ASTM) fatigue and fracture tests, or by in situ generated stress concentrations via dislocation pile-ups. Crack growth kinetics are also examined using the modified Kitagawa–Takahashi diagram to show the role of internal stresses and their gradients needed to sustain continuous crack growth. Incipient crack initiation and growth are also examined using discrete dislocation modeling. The analysis is supported by the experimental data available in the literature.

Discrete dislocation, crystal plasticity and experimental studies of fatigue crack nucleation in single-crystal nickel

Dislocation configurational energy and stored energy densities are determined in discrete dislocation and crystal plasticity modelling respectively and assessed with respect to experiments on single crystal nickel fatigue crack nucleation. Direct comparisons between the three techniques are provided for two crystal orientation fatigue tests. These provide confirmation that both quantities correctly identify the sites of fatigue crack nucleation and that stored energy density is a reasonable approximation to the more rigorous dislocation configurational energy. GND density is shown to be important in locating crack nucleation sites because of its role in the local configurational energy density.

Demand‐based coupling of the scales in the finite temperature CADD method

AbstractConcurrent finite temperature atomistic‐to‐continuum models commonly use a fixed time interval coupling of the atomistic and continuum scale. With special focus on the continuum, this means, that its solution fields are updated in fixed intervals of time. However, in times, when the change in the continuum boundary conditions is negligible, these computations are unnecessary. We present the novel idea of a demand‐based coupling in this work, in which the continuum solution fields are only updated if needed, to save computational effort. To demonstrate the approach, we apply the coupling to a prototypical representative of the mentioned models, namely the coupled atomistic and discrete dislocation (CADD) model. We test the basic functionality of the approach in a 1‐dimensional pulse propagation problem.

Mimicking complex dislocation dynamics by interaction networks

Two-dimensional discrete dislocation models exhibit complex dynamics in relaxation and under external loading. This is manifested both in the time-dependent velocities of individual dislocations and in the ensemble response, the strain rate. Here we study how well this complexity may be reproduced using so-called Interaction Networks, an Artificial Intelligence method for learning the dynamics of complex interacting systems. We test how to learn such networks using creep data, and show results on reproducing individual and collective dislocation velocities. The quality of reproducing the interaction kernel is discussed.

Discrete dislocation dynamics simulations of nanoindentation with pre-stress: Hardness and statistics of abrupt plastic events

Nanoindentation of crystalline materials has been thought as a primarily surface-driven technique that is not able to probe bulk mechanical properties directly, such as material yield strength or bulk plastic flow rates. We elucidate this question through extensive discrete dislocation plasticity simulations of nanoindentation on a single crystal. We consider the competition between nanoindentation and tensile loading (pre-stress) towards crystal plasticity. For this purpose, we study a two-dimensional discrete dislocation model where indentation is performed by using cylindrical indentation with varying radius under both displacement and load control. We focus on the behavior of the hardness and pop-in event statistics during nanoindentation under various pre-stress levels and we correlate them to the spatially correlated dislocation microstructure behavior. At small indentation depths (relative to other microstructural or tip length scales), we find that the hardness is inversely dependent on the plastic strain/dislocation density induced by the applied tensile pre-stress; consequently, we argue that small-depth indentation may be useful for identifying bulk plastic yielding behavior prior to indentation. In contrast, for larger indentation depths, pre-stress has no effect on hardness. However, effect of pre-stress can be revealed through plastic events statistics, both in load and displacement controlled protocols. Moreover, post-indentation surface morphology clearly shows the effect of the pre-stress.

On the nucleation of planar faults during low temperature and high stress creep of single crystal Ni-base superalloys

The present work studies the nucleation of planar faults in the early stages of low temperature (750 °C) and high stress (800 MPa) creep of a Ni-base single crystal superalloy (SX). Two families of 60° dislocations with different Burgers vectors were detected in the transmission electron microscope (TEM). These can react and form a planar fault in the γ′ phase. A 2D discrete dislocation model helps to rationalize a sequence of events which lead to the nucleation of a planar fault. First, one 60° channel dislocation approaches another 60° interface dislocation with a different Burgers vector. At a distance of 5 nm, it splits up into two Shockley partials. The interface dislocation is pushed into the γ′-phase where it creates a small antiphase boundary. It can only move on when the leading Shockley partial joins it and creates an overall 1/3<112> superdislocation. This process is fast and therefore is difficult to observe. The results obtained in the present work contribute to a better understanding of the processes which govern the early stages of low temperature and high stress primary creep of SX.

Subcritical crack growth and crack tip driving forces in relation to material resistance

AbstractBasic concepts, related to the crack tip driving forces in relation to the material resistance, are analyzed for the elastic and elastic-plastic crack growth condition. This defines the crack initiation and growth conditions, as well as for crack arrest. Environment provides an additional driving force, thereby reducing the mechanical driving force required for the crack to grow. It is shown that (a) crack initiation and its growth are inseparable and (b) the magnitude of the applied and/or internal stresses; their gradients are also important for initiation and continuous growth of a crack. Elastic-plastic crack growth is also analyzed using the discrete dislocation models. The results show that its behavior is similar to that of an elastic crack. These concepts are valid for all subcritical crack growth. Mechanical and mechanical equivalent of chemical forces are defined for estimating the life prediction of a component in service. Failure diagrams are defined based on the extension of classical Kitagawa-Takahashi diagram that bridges the behavior of smooth and fracture mechanics specimens. Connections between crack initiation, growth, arrest, and overload fracture are established via these failure diagrams. Application of these diagrams for engineering components in service is outlined for diagnostic and prognostic purposes.

Role of weakest links and system-size scaling in multiscale modeling of stochastic plasticity

Plastic deformation of crystalline and amorphous matter often involves intermittent local strain burst events. To understand the physical background of the phenomenon a minimal stochastic mesoscopic model was introduced, where microstructural details are represented by a fluctuating local yielding threshold. In the present paper, we propose a method for determining this yield stress distribution by lower scale discrete dislocation dynamics simulations and using a weakest link argument. The success of scale-linking is demonstrated on the stress-strain curves obtained by the resulting mesoscopic and the discrete dislocation models. As shown by various scaling relations they are statistically equivalent and behave identically in the thermodynamic limit. The proposed technique is expected to be applicable for different microstructures and amorphous materials, too.

Coarse-grained elastodynamics of fast moving dislocations

The fundamental mechanism of dynamic plasticity in metallic materials subjected to shock loading remains unclear because it is difficult to obtain the precise information of individual fast moving dislocations in metals from the state-of-the-art experiments. In this work, the dynamics of sonic dislocations in anisotropic crystalline materials is explored through a concurrent atomistic-continuum modeling method. We make a first attempt to characterize the complexity of nonuniformly moving dislocations in anisotropic crystals from atomistic to microscale, including the energy intensities as well as the wavelengths of acoustic phonons emitted from sonic dislocations, and the velocity-dependent stress fluctuations around the core of nonuniformly moving dislocations. Instantaneous dislocation velocities and phonon drag effects on the dislocation motions are quantified and analyzed. Mach cones in a V-shaped pattern of the phonon wave-fronts are observed in the wake of the sonic dislocations. Analysis of simulation results based on a wavelet transform show that the faster a dislocation is moving, the longer the emitted phonon wavelength. The dislocation velocity drops dramatically with the occurrence of the interactions between dislocations and phonon waves reflected from the boundaries of specimens. The concurrent atomistic-continuum modeling framework is demonstrated to be the first multiscale method that explicitly treats the strong coupling between the long-range elastic fields away from the dislocation core, the highly nonlinear time-dependent stress field within the core, and the evolutions of the atomic-scale dislocation core structures. As such, it is shown that this method is capable in predicting elastodynamics of dislocations in the presence of inertia effects associated with sonic dislocations in micron-sized anisotropic crystalline materials from the atomic level, which is not directly accessible to the recent elastodynamic discrete dislocation model.

Discrete dislocation modeling of stress corrosion cracking in an iron

Abstract Material strengthening and embrittlement are controlled by interactions between dislocations and hydrogen that alter the observed deformation mechanisms. In this work, we used an energetics approach to differentiate two fundamental stress corrosion mechanisms in iron, namely, hydrogen-enhanced localized plasticity and hydrogen-enhanced decohesion. Considering the small-scale yielding condition, we use a discrete dislocation framework with line dislocations to simulate the crack-tip plastic behavior. The crack growth was modeled using the change in surface energies (cohesive zone laws) due to hydrogen segregation. The changes in the surface energies as a function of hydrogen concentration are computed using atomistic simulations. Results indicate that, when hydrogen concentrations are low, crack growth occurs by alternating mechanisms of cleavage and slip. However, as the hydrogen concentrations increased above some critical value, the crack grows predominately by the cleavage-based decohesion process.

Stress and strain fluctuations in plastic deformation of crystals with disordered microstructure

.We investigate the spatial structure of stress and strain patterns in crystal plasticity. To this end, we combine theoretical arguments with plasticity simulations using three different models: (i) a generic model of bulk crystal plasticity with stochastic evolution of the local microstructure, (ii) a 2D discrete dislocation simulation assuming single-slip deformation in a bulk crystal, and (iii) a 3D discrete dislocation model for deformation of micropillars in multiple slip. For all three models we investigate the scale-dependent magnitude of local fluctuations of internal stress and plastic strain, and we determine the spatial structure of the respective auto- and cross-correlation functions. The investigations show that, in the course of deformation, nontrivial long range correlations emerge in the stress and strain patterns. We investigate the influence of boundary conditions on the observed spatial patterns of stress and strain, and discuss implications of our findings for larger-scale plasticity models.

A link between microstructure evolution and macroscopic response in elasto-plasticity: Formulation and numerical approximation of the higher-dimensional continuum dislocation dynamics theory

Micro-plasticity theories and models are suitable to explain and predict mechanical response of devices on length scales where the influence of the carrier of plastic deformation – the dislocations – cannot be neglected or completely averaged out. To consider these effects without resolving each single dislocation a large variety of continuum descriptions has been developed, amongst which the higher-dimensional continuum dislocation dynamics (hdCDD) theory by Hochrainer et al. (Phil. Mag. 87, pp. 1261–1282) takes a different, statistical approach and contains information that are usually only contained in discrete dislocation models. We present a concise formulation of hdCDD in a general single-crystal plasticity context together with a discontinuous Galerkin scheme for the numerical implementation which we evaluate by numerical examples: a thin film under tensile and shear loads. We study the influence of different realistic boundary conditions and demonstrate that dislocation fluxes and their lines' curvature are key features in small-scale plasticity.

Discrete dislocation modeling on interaction between type-I blunt crack and cylindrical void in single crystals

The interaction between a type-I blunt crack and a near-tip void were modeled by the two-dimensional discrete dislocation dynamics (2-D DDD), with special emphasis on the void growth, crack-tip deformation and their size effect. Different from the classical crystal plasticity-based FE modeling, the present 2-D DDD modeling clearly indicates that, even for the type-I opening crack, the horizontal slip bands connecting the crack tip and the void surface are easily formed, especially when the ligament between the crack-tip and void surface is not too long. Besides the horizontal slip bands, some asymmetric slip bands which are at an oblique angle to the horizontal x-axis can also be formed and intersect with the micro-void. On two coupled parallel slip bands, some oppositely signed dislocations, which can be regarded as equivalent prismatic dislocation loops, glide toward the void/crack-tip surface and thus drive the void growth and the crack tip further blunting. In addition, the size effect by ligament length and void radius on void growth and crack-void interaction has also been investigated carefully. It was found that the size effect is closely related to the formation of horizontal shear slip bands. There exists a critical void radius, under which the near-tip void tends to change into a flat shape and its area decreases gradually with increasing the applied stress intensity factor (SIF) K. These results might be helpful for a better understanding of the fracture and damage mechanisms in the crack-tip process zone.