Coarse-grained atomistic simulation of dislocations

This paper presents a new methodology for coarse-grained atomistic simulation of dislocation dynamics. The methodology combines an atomistic formulation of balance equations and a modified finite element method employing rhombohedral-shaped 3D solid elements suitable for fcc crystals. With significantly less degrees of freedom than that of a fully atomistic model and without additional constitutive rules to govern dislocation activities, this new coarse-graining (CG) method is shown to be able to reproduce key phenomena of dislocation dynamics for fcc crystals, including dislocation nucleation and migration, formation of stacking faults and Lomer-Cottrell locks, and splitting of stacking faults, all comparable with fully resolved molecular dynamics simulations. Using a uniform coarse mesh, the CG method is then applied to simulate an initially dislocation-free submicron-sized thin Cu sheet. The results show that the CG simulation has captured the nucleation and migration of large number of dislocations, formation of multiple stacking fault ribbons, and the occurrence of complex dislocation phenomena such as dislocation annihilation, cutting, and passing through the stacking faults. The distinctions of this method from existing coarse-graining or multiscale methods and its potential applications and limitations are also discussed.     

A constitutive model for creep-plasticity interaction based on dislocation dynamics

A glide-plus-climb micromechanism of dislocation evolution with the formation of subgrains is proposed for modelling of the creep-plasticity interaction (CPI). The long-range internal stress can be divided into the resistance for dislocation climb in subgrain boundaries and that for dislocation glide within grains or subgrains. Their evolution equations are then derived based on dislocation dynamics. Furthermore, a unified constitutuve model for CPI is developed from Orowan's formula. Theoretical calculations on the basis of this model show a very good agreement between the model prediction and experimental results of benchmark tests for 2 1/4 Cr-1 Mo steel at 600°C.     

A multi-scale computational model of crystal plasticity at submicron-to-nanometer scales

Plastic flow in crystal at submicron-to-nanometer scales involves many new interesting problems. In this paper, a unified computational model which directly combines 3D discrete dislocation dynamics (DDD) and continuum mechanics is developed to investigate the plastic behaviors at these scales. In this model, the discrete dislocation plasticity in a finite crystal is solved under a completed continuum mechanics framework: (1) an initial internal stress field is introduced to represent the preexisting stationary dislocations in the crystal; (2) the external boundary condition is handled by finite element method spontaneously; and (3) the constitutive relationship is based on the finite deformation theory of crystal plasticity, but the discrete plastic strains induced by the slip of the newly nucleated or propagating dislocations are calculated by dislocation dynamics methodology instead of phenomenological evolution equations used in conventional crystal plasticity. These discrete plastic strains are then localized to the continuum material points by a Burgers vector density function proposed by us. Various processes, such as loop dislocation evolution, dislocation junction formation etc., are simulated to verify the reliability of this computational model. Specifically, a uniaxial compression test for micro-pillars of Cu is simulated by this model to investigate the ‘dislocation starvation hardening’ observed in the recent experiment.     

A computational method for dislocation-precipitate interaction

A new computational method for the elastic interaction between dislocations and precipitates is developed and applied to the solution of problems involving dislocation cutting and looping around precipitates. Based on the superposition principle, the solution to the dislocation-precipitate interaction problem is obtained as the sum of two solutions: (1) a dislocation problem with image stresses from interfaces between the dislocation and the precipitate, and (2) a correction solution for the elastic problem of a precipitate with an initial strain distribution. The current development is based on a combination of the parametric dislocation dynamics (PDD) and the boundary element method (BEM) with volume integrals.The method allows us to calculate the stress field both inside and outside precipitates of elastic moduli different from the matrix, and that may have initial coherency strain fields. The numerical results of the present method show good convergence and high accuracy when compared to a known analytical solution, and they are also in good agreement with molecular dynamics (MD) simulations. Sheared copper precipitates (2.5 nm in diameter) are shown to lose some of their resistance to dislocation motion after they are cut by leading dislocations in a pileup. Successive cutting of precipitates by the passage of a dislocation pileup reduces the resistance to about half its original value, when the number of dislocations in the pileup exceeds about 10. The transition from the shearable precipitate regime to the Orowan looping regime occurs for precipitate-to-matrix elastic modulus ratios above approximately 3-4, with some dependence on the precipitate size. The effects of precipitate size, spacing, and elastic modulus mismatch with the host matrix on the critical shear stress (CSS) to dislocation motion are presented.     

A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes

We present a self-consistent formulation of 3-D parametric dislocation dynamics (PDD) with the boundary element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. The error in the image force on a screw dislocation approaching a free surface is shown to increase as the dislocation approaches the surface, but is nevertheless controllable. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000×2000 (in units of lattice parameter), and 64 quadrature points. The Eshelby twist angle in a finite-length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Finally, large scale 3-D simulation results of single slip behavior in cylindrical microcrystals are presented. Plastic flow characteristics and the stress-strain behavior of cylindrical microcrystals under compression are shown to be in agreement with experimental observations. It is shown that the mean length of dislocations trapped at the surface is the dominant factor in determining the size effects on hardening of single crystals. The influence of surface image fields on the flow stress is finally explored. It is shown that the flow stress is reduced by as much as 20% for small single crystals of size less than .     

Continuum dynamics of the formation,migration and dissociation of self-locked dislocation structures on parallel slip planes

In continuum models of dislocations, proper formulations of short-range elastic interactions of dislocations are crucial for capturing various types of dislocation patterns formed in crystalline materials. In this article, the continuum dynamics of straight dislocations distributed on two parallel slip planes is modelled through upscaling the underlying discrete dislocation dynamics. Two continuum velocity field quantities are introduced to facilitate the discrete-to-continuum transition. The first one is the local migration velocity of dislocation ensembles which is found fully independent of the short-range dislocation correlations. The second one is the decoupling velocity of dislocation pairs controlled by a threshold stress value, which is proposed to be the effective flow stress for single slip systems. Compared to the almost ubiquitously adopted Taylor relationship, the derived flow stress formula exhibits two features that are more consistent with the underlying discrete dislocation dynamics: (i) the flow stress increases with the in-plane component of the dislocation density only up to a certain value, hence the derived formula admits a minimum inter-dislocation distance within slip planes; (ii) the flow stress smoothly transits to zero when all dislocations become geometrically necessary dislocations. A regime under which inhomogeneities in dislocation density grow is identified, and is further validated through comparison with discrete dislocation dynamical simulation results. Based on the findings in this article and in our previous works, a general strategy for incorporating short-range dislocation correlations into continuum models of dislocations is proposed.     

Modeling of thermally activated dislocation glide and plastic flow through local obstacles

A unified phenomenological model is developed to study the dislocation glide through weak obstacles during the first stage of plastic deformation in metals. This model takes into account both the dynamical responses of dislocations during the flight process and thermal activations while dislocations are bound by obstacle arrays. The average thermal activation rate is estimated using an analytical model based on the generalized Friedel relations. Then, the average flight velocity after an activation event is obtained numerically by discrete dislocation dynamics (DD). To simulate the dynamical dislocation behavior, the inertia term is implemented into the equation of dislocation motion within the DD code. The results from the DD simulations, coupled with the analytical model, determine the total dislocation velocity as a function of the stress and temperatures. By choosing parameters typical of the face centered cubic metals, the model reproduces both obstacle control and drag control motion in low and high velocity regimes, respectively. As expected by other string models, dislocation overshoots of obstacles caused by the dislocation inertia at the collisions are enhanced as temperature goes down.     

Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics

A new numerical dislocation climb model based on incorporating the pipe diffusion theory (PDT) of vacancies with 3D discrete dislocation dynamics (DDD) is developed. In this model we hold that the climb rate of dislocations is determined by the gradient of the vacancy concentration on the segment, but not by the mechanical climb force as traditionally believed. The nodal forces on discrete dislocation segments in DDD simulation are transferred to PDT to calculate the vacancy concentration gradient. This transfer establishes a bridge connecting the DDD and PDT. The model is highly efficient and accurate. As verifications, two typical climb-involved examples are predicted, e.g. the activation of a Bardeen-Herring source as well as the shrinkage and annihilation of prismatic loops. Finally, the model is applied to study the breakup process of an infinite edge dislocation dipole into prismatic loops. This coupling methodology provides us a useful tool to intensively study the evolution of dislocation microstructures at high temperatures.     

FeCrNi合金静动态物理本构模型研究

以金属材料塑性变形的位错动力学为基础,将FeCrNi合金的流动应力分解为非热应力和热激活应力两部分.通过对该合金屈服应力随温度变化特性、屈服应力的应变速率特性、孪晶组织的温度特性及位错组态的应变速率特性进行分析,认为非热应力不只是应变的函数,还与温度和应变速率相关,因此对Johnson-Cook模型方程形式进行修正以描述非热应力. 同时认为影响热激活应力的微结构参数主要为位错阻碍间距\Deltal, 定义并推导出表征\Delta l演化的g函数的表达式,将其引入Kocks的热激活方程,从而建立FeCrNi合金的物理型本构模型.该模型初步实现了对FeCrNi合金从室温到高温、从准静态到动态塑性变形行为的描述.      

关联参照模型和位错发射过程的分子动力学模拟

提出关联参照模型和随位错位置变化的柔性位移边界条件．提供了一个在固定位移边界条件下位错穿越边界的方法．应用三维分子动力学方法研究了体心立方（ＢＣＣ）金属晶体钼裂尖发射位错的力学行为．     

A single theory for some quasi-static,supersonic, atomic,and tectonic scale applications of dislocations

We describe a model based on continuum mechanics that reduces the study of a significant class of problems of discrete dislocation dynamics to questions of the modern theory of continuum plasticity. As applications, we explore the questions of the existence of a Peierls stress in a continuum theory, dislocation annihilation, dislocation dissociation, finite-speed-of-propagation effects of elastic waves vis-a-vis dynamic dislocation fields, supersonic dislocation motion, and short-slip duration in rupture dynamics.     

A New dynamic model for study of dislocation pattern formation

Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expessed in terms of not only the dislocation density itself but also their spatial gradients. The linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns. The project supported by the National Natural Science Foundation of China, Grant No.19392300     

On dislocation patterning: Multiple slip effects in the rate equation approach

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development of various types of dislocation microstructures (dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc.), depending on the externally applied loading and the intrinsic lattice constraints. The term “dislocation patterning” was introduced over 20 years ago by the third author and a corresponding “gradient dislocation dynamics” framework was suggested to describe such phenomena. In the W–A model proposed at that time by the last two authors, it was shown how coupled nonlinear evolution equations of the reaction-diffusion type for the forest (immobile) and gliding (mobile) dislocation densities can generate dislocation microstructures which correspond to walls perpendicular to the slip direction for Cu-crystals oriented for single slip under cyclic loading conditions. This model is adapted to the multiple slip case here. Weakly nonlinear analysis predicts that dislocation patterns should correspond to domains of walls perpendicular to each slip direction and separated by domain walls in the same orientations. This result is confirmed by numerical analysis and experimental observations. The present model generalizes the original W–A model to the case of multiple slip and considers also explicitly gradient effects by allowing for non-uniform dislocation velocities and internal stress effects.     

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation simulation in finite body: 2. Numerical implementation

In the previous paper by Yu and Diab (2013), several sets of boundary integral equations are derived for general anisotropic materials and corresponding equations for materials with different classes of symmetry are deduced. The work presented herein implements two sets of boundary element schemes to numerically solve the stress field. The integration on the element that has the singular point of the kernel is bounded and can be evaluated analytically. Four benchmark elastic problems are solved numerically to show the advantage of the two schemes over the conventional boundary element formulation in eliminating the boundary layer effect. The one with the weaker singularity has better convergence and gives more accurate results. The presented formulation also provides a direct approach to solve for stress field in a finite solid body in the presence of dislocations. Combined with discrete dislocations dynamics, boundary value problems with dislocations in finite bodies can be solved. Two examples, bending of a single crystal beam and pure shearing of a polycrystalline solid, are simulated by discrete dislocation dynamics using the scheme that has the weaker singularity. The comparisons with the published results using the well-established superposition technique validate the proposed formulation and show its quick convergence.     

Atomistically informed dislocation dynamics in fcc crystals

We develop a nodal dislocation dynamics (DD) model to simulate plastic processes in fcc crystals. The model explicitly accounts for all slip systems and Burgers vectors observed in fcc systems, including stacking faults and partial dislocations. We derive simple conservation rules that describe all partial dislocation interactions rigorously and allow us to model and quantify cross-slip processes, the structure and strength of dislocation junctions, and the formation of fcc-specific structures such as stacking fault tetrahedra. The DD framework is built upon isotropic non-singular linear elasticity and supports itself on information transmitted from the atomistic scale. In this fashion, connection between the meso and micro scales is attained self-consistently, with all material parameters fitted to atomistic data. We perform a series of targeted simulations to demonstrate the capabilities of the model, including dislocation reactions and dissociations and dislocation junction strength. Additionally we map the four-dimensional stress space relevant for cross-slip and relate our findings to the plastic behavior of monocrystalline fcc metals.     

Discrete dislocation dynamics simulation of cutting of γ′ precipitate and interfacial dislocation network in Ni-based superalloys

We proposed a back force model for simulating dislocations cutting into a γ′ precipitate, from the physical viewpoint of work for making or recovering an antiphase boundary (APB). The first dislocation, or a leading partial of a superdislocation, is acted upon by a back force whose magnitude is equal to the APB energy. The second dislocation, or a trailing partial of a superdislocation, is attracted by the APB with a force of the same magnitude. The model is encoded in a 3D discrete dislocation dynamics (DDD) code and demonstrates that a superdislocation nucleates after two dislocations pile up at the interface and that the width of dislocations is naturally balanced by the APB energy and repulsion of dislocations. The APB energy adopted here is calculated by ab initio analysis on the basis of the density functional theory (DFT). Then we applied our DDD simulations to more complicated cases, namely, dislocations near the edges of a cuboidal precipitate and those at the γ/γ′ interface covered by an interfacial dislocation network. The former simulation shows that dislocations penetrate into a γ′ precipitate as a superdislocation from the edge of the cube, when running around the cube to form Orowan loops. The latter reveals that dislocations become wavy at the interface due to the stress field of the dislocation network, then cut into the γ′ precipitate through the interspace of the network. Our model proposed here can be applied to study the dependence of the cutting resistance on the spacing of dislocations in the interfacial dislocation network.     

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system.