Discrete Dislocation Simulations Research Articles

Interaction between 1/2[formula omitted]110[formula omitted]{001} dislocations and {110} prismatic loops in uranium dioxide: Implications for strain-hardening under irradiation

Plasticity of irradiated UO2 is of major interest to improve the risk assessment of the nuclear fuel cladding failure in the case of design basis accidents. In this study, we investigate the main irradiation-hardening processes induced by {110} irradiation loops interacting with glissile dislocations of the primary slip system, 1/2<110>{001}, of UO2. The interactions are simulated at two scales using molecular dynamics and discrete dislocation dynamics, to characterise local interactions and identify strengthening configurations as a function of the dislocation-irradiation loop geometry. In particular, we show that 1/2<110>{001} screw dislocations can be strongly pinned by helical turn configurations. Statistical large-scale discrete dislocation simulations are performed to investigate the collective behaviour of a large density of irradiation defects and quantify irradiation hardening. Several microstructural processes including loop drag and shovelling are observed and their involvement in clear band formation and hardening of UO2 fuel at high temperature is discussed.

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.

Dislocation structure analysis in the strain gradient of torsion loading: a comparison between modelling and experiment

Complex stress states due to torsion lead to dislocation structures characteristic for the chosen torsion axis. The formation mechanism of these structures and the link to the overall plastic deformation are unclear. Experiments allow the analysis of cross sections only ex situ or are limited in spacial resolution which prohibits the identification of the substructures which form within the volume. Discrete dislocation dynamics simulations give full access to the dislocation structure and their evolution in time. By combining both approaches and comparing similar measures the dislocation structure formation in torsion loading of micro wires is explained. For the ⟨100⟩ torsion axis, slip traces spanning the entire sample in both simulation and experiment are observed. They are caused by collective motion of dislocations on adjacent slip planes. Thus these slip traces are not atomically sharp. Torsion loading around a ⟨111⟩ axis favors plasticity on the primary slip planes perpendicular to the torsion axis and dislocation storage through cross-slip and subsequent collinear junction formation. Resulting hexagonal dislocation networks patches are small angle grain boundaries. Both, experiments and discrete dislocation simulations show that dislocations cross the neutral fiber. This feature is discussed in light of the limits of continuum descriptions of plasticity.

Analysing discrete dislocation data using alignment and curvature tensors

Analysis of large scale discrete dislocation data requires the characterisation of complex dislocation networks by suitable average quantities. In the current work, we suggest dislocation alignment tensors and closely related curvature tensors as easily extractable and intelligible measures of geometrical and topological characteristics of dislocation distributions. We provide formulae for extracting these measures from discrete dislocation data based on straight segments. Examples for interpreting and visualising these measures are provided for a simple configuration and two more involved results from discrete dislocation simulations. We suggest the alignment and curvature tensors for wider use in plasticity research.

Shear stress relaxation through the motion of edge dislocations in Cu and Cu–Ni solid solution: A molecular dynamics and discrete dislocation study

Shear stress relaxation through the motion of multiple edge dislocations in a periodic cell is studied by molecular dynamics and discrete dislocation methods. In both methods, we consider systems with an initial edge dislocation on the parallel slip planes. The initial shear strain is applied to the system along the Burgers vector of the dislocations. The velocity profile for a single edge dislocation fitted from a molecular dynamics (Bryukhanov, 2020) is used as the input to discrete dislocation dynamics. The dependence of shear stress, plastic strain, and plastic strain rate on time are analyzed. We show that when dislocations are sufficiently far from each other, relaxation occurs due to dislocation motion, and dislocation velocity can exceed the anisotropic speed of sound. If the dislocations are closer together, relaxation is due to the growth of stacking faults. We find that the stress at which the stress relaxation mechanism changes decreases with increasing dislocation density. Stress relaxation in a Cu–Ni solid solution occurs faster than in pure Cu due to a higher dislocation velocity in the high-velocity regime. However, the shear stress at which nickel atoms increase the plastic strain rate increases with an increase in dislocation density. The dependence of the relaxation time on the dislocation density obtained in discrete dislocation simulations is approximated using the power law. The elastic interaction between dislocations reduces the exponent of the power law from 1/ρ to approximately 1/ρ0.8 for both pure Cu and Cu–Ni solid solutions.

Discrete dislocation simulation of the ultrasonic relaxation of non-equilibrium grain boundaries in a deformed polycrystal

For the first time, the relaxation of disordered dislocation arrays in a model 3 × 3 columnar polycrystal under ultrasonic action is studied using the discrete dislocation approach. All grains contain three non-parallel slip systems located at an angle of 60° to each other. The non-equilibrium state of the grain boundaries is modeled using two finite edge dislocation walls with Burgers vector of opposite signs, which are equivalent to a wedge junction disclination quadrupole. It is shown that ultrasonic treatment causes a significant rearrangement of the lattice dislocations and their gliding towards the grain boundaries. It results in a decrease in the internal stress fields associated with the presence of non-equilibrium grain boundaries and relaxation of dislocation structure. The model predicts an existence of optimal amplitude, at which the maximum relaxing effect can be achieved. Dependence of the relaxation of dislocation structure on the grain size is also investigated.

A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty

Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distribution of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). The outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error.

Improved Peierls-Nabarro model for asymmetric nonplanar 1/2

The 1/2 < 11_0]{111} ordinary dislocations play an important role in the mechanical behaviors of L10 alloys. The core of this kind of dislocations has asymmetric four-fold nonplanar structure spreading onto two equivalent {111} planes, which makes the traditional planar core structure based Peierls-Nabarro (P–N) model inappropriate to describe the dislocation properties. Under this context, an improved Peierls-Nabarro (P–N) model is developed in the present work to investigate the dislocation width, stress field and Peierls stress of the ordinary 1/2 < 11_0]{111} screw dislocation in L10 alloys (TiAl, TiGa and CuAu). This extended P–N model considers the nonplanar core structure with physical information obtained by the density functional theory (DFT) simulations. By the variational method, the corresponding dislocation equation is solved. It is found that the dislocation width of the ordinary 1/2 < 11_0]{111} screw dislocation is as narrow as 0.5b. Compared to the results predicted for the planar core structure or the assumed Volterra dislocation, the long-range stress field induced by the real asymmetric four-fold core presents a clear rotation. When increasing the applied stress, the distribution of the Burgers vector along the four different core folds differs more and more from each other. When the applied stress attains a critical value (i.e., the Peierls stress), the dislocation core turns to be a planar structure and a fully discrete method is used to calculate the Peierls stress. Different from the original P–N model that underestimates the Peierls stress by several orders of magnitude, the Peierls stress predicted by the improved P–N model is well consistent with the previous DFT simulations. Apparently, when the long-range interaction between dislocations is taken into account for discrete dislocation simulations or theoretical analysis and when the Peierls stress is calculated for L10 alloys, the consideration of the asymmetric nonplanar core structure by the improved P–N model is necessary to have more accurate results.

Learning to Predict Crystal Plasticity at the Nanoscale: Deep Residual Networks and Size Effects in Uniaxial Compression Discrete Dislocation Simulations

The density and configurational changes of crystal dislocations during plastic deformation influence the mechanical properties of materials. These influences have become clearest in nanoscale experiments, in terms of strength, hardness and work hardening size effects in small volumes. The mechanical characterization of a model crystal may be cast as an inverse problem of deducing the defect population characteristics (density, correlations) in small volumes from the mechanical behavior. In this work, we demonstrate how a deep residual network can be used to deduce the dislocation characteristics of a sample of interest using only its surface strain profiles at small deformations, and then statistically predict the mechanical response of size-affected samples at larger deformations. As a testbed of our approach, we utilize high-throughput discrete dislocation simulations for systems of widths that range from nano- to micro- meters. We show that the proposed deep learning model significantly outperforms a traditional machine learning model, as well as accurately produces statistical predictions of the size effects in samples of various widths. By visualizing the filters in convolutional layers and saliency maps, we find that the proposed model is able to learn the significant features of sample strain profiles.

On the effects of hardening models and lattice rotations in strain gradient crystal plasticity simulations

This work presents numerical simulations considering two hardening models for higher order strain gradient crystal plasticity: energetic and dissipative. These models are defined as a function of plastic slip gradients. Energetic hardening, in particular, is defined based on a quadratic uncoupled defect energy. Considering wedge indentation simulations, it is observed that the effect of the hardening models depends on the indenter angle, with the energetic hardening effect being less relevant for flatter indenters. Comparisons with discrete dislocation simulations suggest also a link between the presence of dislocation obstacles and energetic hardening. Microstructures obtained with each hardening model are different, with the dissipative case presenting a geometrically necessary dislocation distribution that exhibits some of the characteristics seen in “cell-wall” structures. Changes in the geometrically necessary dislocations with lattice rotations are also influenced by hardening definition. This work shows that these changes are important to determine the evolution of plastic slip distributions.

A microstructure-sensitive location-specific design tool for predicting the yield and creep behavior of LSHR Ni-base superalloy

A microstructure-sensitive model that predicts the strain-rate sensitive flow behavior as well as creep-strain life of a refractory Ni-base superalloy, Low Solvus High Refractory (LSHR), is presented. The model is based on discrete dislocation simulations that are computationally expensive, but the design tool derived from the simulations is fast acting. The model employs experimental data on microstructure and mechanical behavior, as well as the thermodynamic model PANDAT™, to calibrate, validate and verify the use of the model as a design tool. The mechanical properties predicted include flow stress as a function of temperature and strain-rate, as well as time for 0.1-0.2% creep strain as a function of stress and temperature. The model extends prior work of a strain rate sensitive flow stress model developed for IN100 alloy, by calibrating the model to data on LSHR, and by including dislocation creep as well as grain size dependent diffusional creep behavior and predicting time to reach a design creep life in terms of creep strain. The resultant model was found to capture reported data on LSHR, in both subsolvus and supersolvus heat treated conditions. The calibrated model was validated using additional data on yield and creep of LSHR with two other microstructures. The validation makes the model a promising design tool in the engineering of complex heat treat disks, where location-specific properties are desired.

A crystal plasticity model for slip in hexagonal close packed metals based on discrete dislocation simulations

This work develops a method for calibrating a crystal plasticity model to the results of discrete dislocation (DD) simulations. The crystal model explicitly represents junction formation and annihilation mechanisms and applies these mechanisms to describe hardening in hexagonal close packed metals. The model treats these dislocation mechanisms separately from elastic interactions among populations of dislocations, which the model represents through a conventional strength-interaction matrix. This split between elastic interactions and junction formation mechanisms more accurately reproduces the DD data and results in a multi-scale model that better represents the lower scale physics. The fitting procedure employs concepts of machine learning—feature selection by regularized regression and cross-validation—to develop a robust, physically accurate crystal model. The work also presents a method for ensuring the final, calibrated crystal model respects the physical symmetries of the crystal system. Calibrating the crystal model requires fitting two linear operators: one describing elastic dislocation interactions and another describing junction formation and annihilation dislocation reactions. The structure of these operators in the final, calibrated model reflect the crystal symmetry and slip system geometry of the DD simulations.

A generalized cohesive zone model and a grain boundary yield criterion for gradient plasticity derived from surface- and interface-related arguments

In this work, new generalized interface models for gradient plasticity based on the dislocation density tensor are presented. The first one is a new generalized cohesive zone model which can be used for interface damage and delamination in gradient plasticity applications. The second one is a new grain boundary yield criterion with isotropic and kinematic hardening and a related flow rule, being formulated based on results from discrete dislocation simulations. The derivation starts from surface-related considerations, like in the work of Del Piero (2009), who starts with the virtual power of the external forces and then derives the principle of virtual power (Povp) rather than postulating it. Here, however, the formulation is carried out without using the notion of ‘virtual power’, although the derivation may as well be based on the Povp. Moreover, the derivation does not involve new micro force balance equations. In addition, the dislocation density tensor occurs rather as an outcome of the approach than as an ingredient. The discussion is closed by an example.

Discrete dislocation simulations of compression of tapered micropillars

The effect of taper on the plastic response of micropillars with a relatively high density of dislocation sources (1.5×1014m−2) is analyzed. The large number of dislocation sources and dislocations in the simulations rule out many of the mechanisms that govern size effects in pillars with a low dislocation source density. The mechanical response of compressed pillars with mean widths of W=0.4, 0.8, 1.6, 3.2 µm and taper angles of 0°, 2° and 5° is analyzed using 2.5D discrete dislocation plasticity. For all taper angles, large scatter is found in the stress strain response for the submicron, W=0.4 and 0.8 µm pillars, and relatively little scatter for the larger pillars. Taper leads to an increased average hardening rate for the submicron pillars, although this increase is within the scatter band of the stress strain response. Little sensitivity of the plastic response to taper is found for the larger pillars. The effect of size and taper on the stress strain response stems from the build up of geometrically necessary dislocations (GNDs). The reduced number of dislocation sources in the submicron pillars is identified as the origin of the large scatter in the predicted mechanical response.

Crysal Plasticity Finite Element Simulations based on Continuum Dislocation Dynamics

AbstractPlastic deformation of crystalline materials is the result of the motion and interaction of dislocations. Continuum dislocation dynamics (CDD) defines flux‐type evolution equations of dislocation variables which can capture the kinematics of moving curved dislocations. Coupled with Orowan's law, which connects the plastic shear rate to the dislocation flux, CDD defines a dislocation density based material law for crystal plasticity.In the current work we provide simulations of a micro‐bending experiment of a single crystal and compare the results qualitatively to those from discrete dislocation simulations from the literature. We show that CDD reproduces salient features from discrete dislocation simulations regarding the stress distribution, the dislocation density and the accumulated plastic shear, which would be hard to obtain from more traditional crystal plasticity constitutive laws. © 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Three-Dimensional Continuum Dislocation Dynamics Simulations of Dislocation Structure Evolution in Bending of a Micro-Beam

ABSTRACTA persistent challenge in multi-scale modeling of materials is the prediction of plastic materials behavior based on the evolution of the dislocation state. An important step towards a dislocation based continuum description was recently achieved with the so called continuum dislocation dynamics (CDD). CDD captures the kinematics of moving curved dislocations in flux-type evolution equations for dislocation density variables, coupled to the stress field via average dislocation velocity-laws based on the Peach-Koehler force. The lowest order closure of CDD employs three internal variables per slip system, namely the total dislocation density, the classical dislocation density tensor and a so called curvature density.In the current work we present a three-dimensional implementation of the lowest order CDD theory as a materials sub-routine for Abaqus®in conjunction with the crystal plasticity framework DAMASK. We simulate bending of a micro-beam and qualitatively compare the plastic shear and the dislocation distribution on a given slip system to results from the literature. The CDD simulations reproduce a zone of reduced plastic shear close to the surfaces and dislocation pile-ups towards the center of the beam, which have been similarly observed in discrete dislocation simulations.

Extended modelling of dislocation transport-formulation and finite element implementation

In the past decade, several higher-order crystal plasticity models have been developed to properly capture size effects, dislocation pile-up and patterning. Here we consider a formulation which accounts for the presence and behavior of both positive and negative dislocations in terms of densities. We derive an implicit finite element implementation for the continuum crystal plasticity model including dislocation transport, using a generalised continuum expression for the short-range dislocation interactions, by discretizing the two governing non-linear transport equations in time and space. The resulting non-linear algebraic equations are solved by an incremental-iterative solution scheme. We compare the resulting numerical solutions with discrete dislocation simulations. This analysis shows the capabilities of the implicit FEM framework to solve continuum dislocation transport in crystal plasticity with the added energetic dislocation interactions.

On the basic relation between mean free slip length and work hardening of metals

The basic theory for athermal dislocation storage and work hardening is revisited. The mean free slip length is redefined enabling a straightforward geometrical interpretation that can be directly compared to the grain size. It is realized that this geometrical mean free slip length is larger than the typical grain size of commercial alloys, i.e. linear stage II work hardening is predicted only for single crystals or very coarse-grained pure metals, in the athermal limit at sufficiently low temperatures. The new theory suggests that the distribution of obstacles in the slip plane can be represented by two parameters, a fractal dimension D and the density of obstacles n per area. For a given geometrical mean free slip length it is derived that the athermal storage rate of dislocations is proportional to and to . A new natural definition of a fractal dimension is proposed, enabling its calculation directly from discrete dislocation simulations. An estimate of D is given, based on line-tension simulations of a dislocation loop expanding through an array of obstacles of random cutting strengths.

The influence of thermal residual stresses and thermal generated dislocation on the mechanical response of particulate-reinforced metal matrix nanocomposites

Due to thermal expansion mismatch between reinforcing particles and matrix, thermal induced dislocations are generated in metal matrix nanocomposites (MMNCs) during cooling down from the processing temperature. These dislocations have been identified as an important strengthening mechanism in particulate-reinforced MMNCs. In this study, the development of thermal residual stresses and thermal induced dislocations in MMNCs are predicted using discrete dislocation simulation, assuming the whole material is under uniform temperature change. Shear deformation is applied after the composites are cooled down to room temperature and the influence of thermal residual stresses and thermal generated dislocation on the overall response of particulate-reinforced MMNCs are investigated. The results show that the thermal residual stresses are high enough to generate dislocations and the dislocation density is higher in the interfacial region than the rest of the matrix. The predicted mechanical behavior of the MMNCs matches the experimental results better when thermal residual stresses are included in the simulations.

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.