Low-strain fatigue in AISI 316L steel surface grains: a three-dimensional discrete dislocation dynamics modelling of the early cycles I. Dislocation microstructures and mechanical behaviour

The early stages of the formation of dislocation microstructures in low-strain fatigue are analysed, using three-dimensional discrete dislocation dynamics modelling. Simulations under various conditions of loading amplitude and grain size have been performed. Both the dislocation microstructures and the associated mechanical behaviour are accurately reproduced in single-slip as well as in double-slip loading conditions. The microstructures thus obtained are analysed quantitatively, in terms of number of slip bands per grain, band thickness and band spacing. The simulations show the crucial role of cross-slip both for the initial spreading of strain inside the grain and for the subsequent strain localization in the form of slip bands. A complete and detailed scheme for the persistent slip band formation is proposed, from the observation of the numerical dislocation arrangements.     

Interaction of dislocation pile-up with a low-angle tilt boundary: a discrete dislocation dynamics study

Abstract

The pile-up of dislocations between two low-angle tilt boundaries (LATB) in an fcc crystal was simulated using three-dimensional discrete dislocation dynamics. The LATB was constructed using glissile edge dislocations stacked on each other. The dislocations in the pile-up were chosen such that their reactions with the dislocations in the LATB resulted in glissile junctions. Parallel pairs of dislocations were inserted to a maximum allowable value estimated from theoretical expressions. A resolved shear stress was applied and increased in steps so as to move the dislocations in the pile-up towards the boundaries. The shear stress required to break the lead dislocation from the wall was determined for varying spacings between the two boundaries. The shear stress and boundary spacing followed the Hall–Petch type relation. Dislocation pile-ups without a LATB were also simulated. The spacing of the dislocations in the pile-up with LATB was found to be closer (ie higher dislocation density) than that without LATB. It was shown through analytical expressions that LATB exerts an attractive force on the dislocations in the pile-up thereby creating a denser pile-up.     

Introducing dislocation climb by bulk diffusion in discrete dislocation dynamics

We report a method to incorporate dislocation climb controlled by bulk diffusion in a three-dimensional discrete dislocation dynamics (DDD) simulation for fcc metals. In this model we couple the vacancy diffusion theory to the DDD in order to obtain the climb rate of the dislocation segments. The capability of the model to reproduce the motion of climbing dislocations is examined by calculating several test-cases of pure climb-related phenomena and comparing the results with existing analytical predictions and experimental observations. As test-cases, the DDD is used to study the activation of Bardeen–Herring sources upon the application of an external stress or under vacancy supersaturation. Loop shrinkage and expansion due to vacancy emission or absorption is shown to be well described by our model. In particular, the model naturally describes the coarsening of a population of loops having different sizes.     

Discrete dislocation dynamics: an important recent break-through in the modelling of dislocation collective behaviour

Recent results obtained by 3D discrete Dislocation Dynamics (DD) simulations are reviewed. Firstly, in the case of fatigued AISI 316L stainless steel, it is shown how DD simulations can both explain the formation of persistent slip bands and give a criterion for crack initiation. The same study is performed in the case of precipitate hardened metals where the precipitate size plays a crucial role. Secondly, we show how molecular dynamics (MD) simulations can feed the DD simulations for two applications. The first concerns the modelling of BCC Fe for which the dislocation mobility is derived from MD simulations. The second considers the modelling of irradiated stainless steels (FCC), where MD is used to define the local rules of interactions between dislocations and Frank loops. To cite this article: M.C. Fivel, C. R. Physique 9 (2008).     

Contact between two plastically deformable crystals: a discrete dislocation dynamics study

It is customary to simplify the analysis of contact between two elastically deformable bodies by treating an equivalent problem where only one body is deformable and the other is rigid. This is possible provided that the gap geometry and the effective elastic modulus of the bodies in the simplified problem are the same as in the original problem. However, the question arises on whether – and to which extent – the simplification is still valid even when (size-dependent) plasticity occurs. Studies using discrete dislocation plasticity have also, so far, addressed simple contact problems where only one body can deform plastically. Here, we extend the analysis to two bodies in contact that can both deform by dislocation plasticity and investigate under which conditions the response agrees with that of an equivalent simplified problem. The bodies in contact are metal single crystals with sinusoidal and flat surface. It is found that the response of two plastically deformable bodies in contact can be simplified to an equivalent problem where one body is rigid and the other can deform plastically. Also, a plasticity size effect is observed, but the effect fades when the platen becomes more plastically deformable.     

Investigation of precipitation hardening by dislocation dynamics simulations

Dislocation dynamics (DD) simulations are used to investigate precipitation-induced strengthening in a Zr–1% Nb alloy. A method is proposed to carry out simulations under dynamical conditions in connection with the microstructure of the investigated alloy. First, a sensitivity study of simulation parameters, suspected of altering simulation results, is presented. It allows setting up simulation conditions ensuring statistical representativeness. The effect of the strain rate is then investigated and analyzed in connection with the random distribution adopted to describe precipitation distribution. The strengthening induced by two different families of Nb precipitates is estimated from simulations whose simulation box is reduced to the volume of a single grain. It is shown that the obtained strengthening values are smaller than those predicted by most published models. Determination of total strengthening shows that usual superposition rules do not apply. A mixture law, fitting DD results well, is proposed in this paper.     

Characterization of precipitation hardening mechanisms by investigating dislocation dynamics in the HVEM

MgO single crystals and an Al-Zn-Mg alloy were deformed inside the HVEM. The static dislocation configurations and the dynamical character of the dislocation motion yield information on the applied effective stress and on the stress to surmount the array of dispersed obstacles in the athermal mode.     

Quasi-coarse-grained dynamics: modelling of metallic materials at mesoscales

A computationally efficient modelling method called quasi-coarse-grained dynamics (QCGD) is developed to expand the capabilities of molecular dynamics (MD) simulations to model behaviour of metallic materials at the mesoscales. This mesoscale method is based on solving the equations of motion for a chosen set of representative atoms from an atomistic microstructure and using scaling relationships for the atomic-scale interatomic potentials in MD simulations to define the interactions between representative atoms. The scaling relationships retain the atomic-scale degrees of freedom and therefore energetics of the representative atoms as would be predicted in MD simulations. The total energetics of the system is retained by scaling the energetics and the atomic-scale degrees of freedom of these representative atoms to account for the missing atoms in the microstructure. This scaling of the energetics renders improved time steps for the QCGD simulations. The success of the QCGD method is demonstrated by the prediction of the structural energetics, high-temperature thermodynamics, deformation behaviour of interfaces, phase transformation behaviour, plastic deformation behaviour, heat generation during plastic deformation, as well as the wave propagation behaviour, as would be predicted using MD simulations for a reduced number of representative atoms. The reduced number of atoms and the improved time steps enables the modelling of metallic materials at the mesoscale in extreme environments.     

Plasticity localization ahead of a crack tip modeled by 2D discrete dislocation dynamics

The formation of a plastic zone in the presence of a crack, in a bicrystal, is modeled by two-dimensional discrete dislocation dynamics. The influence of a large number of parameters is studied: the misorientation, the emission from the crack tip and the obstacle density which modifies the internal stress (hardening) and degree of localization of mobile dislocations (slip bands). Emphasis is put on obtaining the best ductility, since the focus is on face centered cubic materials. Finally, a strong effect of the plastic model on the dynamics and on the secondary slip (intensity and localization), beyond the influence of the initial distribution of sources and obstacles, is reported.     

Sound dispersion in a discrete dislocation system

The Granato-Lucke theory of dislocation internal friction is modified for the case in which the discreteness of the dislocation positions is taken into account. In this case a dislocation moves in an effective stress field differing from the excitation field, and the Granato-Lucke dispersion relations are altered accordingly.Translated from Ivestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 56–58, March, 1984.     

An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes

Barnett’s coordinate-free expression of the displacement field of a triangular loop in an isotropic media is revisited in a view to be implemented in 3D discrete dislocation dynamics codes. A general meshing procedure solving the problems of non-planar loops is presented. The method is user-friendly and can be used in numerical simulations since it gives the contribution of each dislocation segment to the global displacement field without defining the connectivity of closed loops. Easy to implement in parallel calculations, this method is successfully applied to large-scale simulations.     

Static friction of sinusoidal surfaces: a discrete dislocation plasticity analysis

Discrete dislocation plasticity simulations are carried out to investigate the static frictional response of sinusoidal asperities with (sub)-microscale wavelength. The surfaces are first flattened and then sheared by a perfectly adhesive platen. Both bodies are explicitly modelled, and the external loading is applied on the top surface of the platen. Plastic deformation by dislocation glide is the only dissipation mechanism active. The tangential force obtained at the contact when displacing the platen horizontally first increases with applied displacement, then reaches a constant value. This constant is here taken to be the friction force. In agreement with several experiments and continuum simulation studies, the friction coefficient is found to decrease with the applied normal load. However, at odds with continuum simulations, the friction force is also found to decrease with the normal load. The decrease is caused by an increased availability of dislocations to initiate and sustain plastic flow during shearing. Again in contrast to continuum studies, the friction coefficient is found to vary stochastically across the contact surface, and to reach locally values up to several times the average friction coefficient. Moreover, the friction force and the friction coefficient are found to be size-dependent.     

基于点缺陷扩散理论与离散位错动力学耦合的位错攀移模型研究

位错的攀移运动对高温下晶体材料的塑性行为有重要影响,为了能够有效揭示攀移的物理本质及其对塑性行为的作用,本文基于点缺陷扩散理论,通过将体扩散和管扩散机理的共同作用与三维离散位错动力学耦合,建立了适用条件更广的位错攀移模型. 利用此模型我们模拟了单个及多个棱柱型位错环的收缩变形过程,发现影响位错攀移速率的决定因素不是传统理论认为的机械攀移力,而是位错周围(体扩散)及位错段上(管扩散)的空位浓度梯度. 该模型也能够完全重现棱柱型位错环群的粗化过程中不同位错环半径及晶体内平均空位浓度随时间变化的三个阶段. 关键词： 位错攀移 点缺陷扩散理论 位错动力学 棱柱位错环     

Plastic shear response of a single asperity: a discrete dislocation plasticity analysis

We investigate the plastic shear response during static friction of an asperity protruding from a large FCC single crystal. The asperity is in perfectly adhesive contact with a rigid platen and is sheared by tangentially moving the platen. Using discrete dislocation plasticity simulations, we elucidate the plastic shear behaviour of single asperities of various size and shape, in search for the length scale that controls the plastic behaviour. Since plasticity can occur also in the crystal, identification of the length scale that controls a possible size-dependent plastic behaviour is far from being trivial. It is found that scaling down the dimensions of an asperity results in a higher contact shear strength. The contact area is dominant in controlling the plastic shear response, because it determines the size of the zone, in and below the asperity, where dislocation nucleation can occur. For a specific contact area, there is still a dependence on asperity volume and shape, but this is weaker than the dependence on contact area alone.     

Coercivity of precipitation hardened cobalt rare earth 17:2 permanent magnets

The influence of the microstructure on the coercivity has been investigated by means of transmission electron microscopy. It is shown that a thin coherent (CoCu)₅Sm-cell boumdary phase, separating cells of 17:2-crystal structure, acts as a pinning centre for magnetic domain walls. The attractive interaction force is interpreted in terms of a micromagnetic theory for domain wall pinning. The coercive force is determined by the domain wall energy gradient and by the magnetoelastic coupling energy between domain wall stresses and lattice deformation strains. The calculated coercive force due to the lattice mismatch, originated by the cellular coherent precipitation structure, is comparable to the experimentally obtained values.     

Multiscale modelling of precipitation in concentrated alloys: from atomistic Monte Carlo simulations to cluster dynamics I thermodynamics

In a recent paper, the authors showed how the clusters free energies are constrained by the coagulation probability, and explained various anomalies observed during the precipitation kinetics in concentrated alloys. This coagulation probability appeared to be a too complex function to be accurately predicted knowing only the cluster distribution in Cluster Dynamics (CD). Using atomistic Monte Carlo (MC) simulations, it is shown that during a transformation at constant temperature, after a short transient regime, the transformation occurs at quasi-equilibrium. It is proposed to use MC simulations until the system quasi-equilibrates then to switch to CD which is mean field but not limited by a box size like MC. In this paper, we explain how to take into account the information available before the quasi-equilibrium state to establish guidelines to safely predict the cluster free energies.     

Modeling of a dislocation source in the field of activated and unactivated discrete barriers

In this work, computer-modeling methods have been used to consider the formation of a dislocation loop. The barrier strengths correspond to the reacting (unactivated) and unreacting (activated) forest dislocations. It is determined that the minimum operating stress of the source coincides with the classical, critical Frank-Read stress only for a rather narrow range of length of the sources. In a majority of the cases, it is comparable to the critical stress for flow past a network of barriers. Beyond the front of the dislocations propagating from the source, dislocation loops formed by the Orowan mechanism remain, the total length of which is equal to 12–14% of the length of the loop formed.Tomsk Structural Engineering Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 20–24, October, 1992.     

Discrete dislocation simulations of precipitation hardening in superalloys

The low-temperature yield stress of a nickel-based superalloy, containing up to 40% Ni₃A1 precipitates (γ′), is calculated by discrete dislocation simulations. A pair of screw or 60°(a/2) ?110? dislocation glides under external stress across a {111} plane of γ phase, intersected by a random distribution of either spherical or cubic γ′ precipitates. The stress is raised until the dislocations can cut or bow round all the obstacles. In this paper the emphasis is on the cutting regime which is prevalent when the precipitates are small and/or have low antiphase-boundary (APB) energies. From a large number of simulations in the cutting regime, the effects of size, shape, volume fraction and APB energy are found to be as follows: The yield stress is proportional to the square root of the volume fraction of γ′. The yield stress depends weakly on the precipitate size in the size range 20–400?nm, for APB energies of 150, 250 and 320?mJ?m^?2. The yield stress depends linearly on the APB energy for APB energies up to 320?mJ?m^?2 in the size range 50–200?nm. At a precipitate size of 100?nm, cubes are weaker obstacles than equivalent spheres by about 25% for an APB energy of 320?mJ?m^?2; however, the shape effect on strengthening decreases with decreasing APB energy and decreasing precipitate size. When a coherency stress (from a lattice parameter mismatch of 0.3%) is added, the yield stress increases by about 10%. When solid-solution strenthening is added, it is potent when the solute is in the γ matrix, but much less potent when the solute is in γ′. When the γ′ precipitates are larger than 400?nm across and the APB energy greater than 250?mJ?m^?2, significant Orowan looping occurs. The yield stress drops inversely as the precipitate size and becomes insensitive to the APB energy but sensitive to the shear modulus. Many of these results from the full simulations differ from the analytical models of strengthening in superalloys but they can be rationalized from the results of simulations on simple homogenized precipitate structures.     

Continuum model for dislocation dynamics in a slip plane

In this paper, we present a continuum model for dislocation dynamics in a slip plane, which accurately incorporates both the long-range interaction and the local line tension effect of dislocations. Unlike the continuum models in the literature using dislocation densities, we use the disregistry across the slip plane to represent the continuous distribution of dislocations in the slip plane, which has the advantage of including the orientation dependence of dislocations in a very simple way. The continuum dislocation dynamics model is validated by linear instability analysis of a uniform dislocation array to small perturbations and comparisons of the results with those of the discrete dislocation dynamics model. Numerical examples for the evolution of distributions of dislocations and plastic slips in a slip plane are presented.