Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in Al

The interaction of dislocations with grain boundaries (GBs) determines a number of important aspects of the mechanical performance of materials, including strengthening and fatigue resistance. Here, the coupled atomistic/discrete-dislocation (CADD) multiscale method, which couples a discrete dislocation continuum region to a fully atomistic region, is used to study screw-dislocations interacting with Σ3, Σ11, and Σ9 symmetric tilt boundaries in Al. The low-energy Σ3 and Σ11 boundaries absorb lattice dislocations and generate extrinsic grain boundary dislocations (GBDs). As multiple screw dislocations impinge on the GB, the GBDs form a pile-up along the GB and provide a back stress that requires increasing applied load to push the lattice dislocations into the GB. Dislocation transmission is never observed, even with large GBD pile-ups near the dislocation/GB intersection. Results are compared with experiments and previous, related simulations. The Σ9 grain boundary, composed from a more complex set of structural units, absorbs screw dislocations that remain localized, with no GBD formation. With increasing applied stress, new screw dislocations are then nucleated into the opposite grain from structural units in the GB that are nearby but not at the location where the original dislocation intersected the boundary. The detailed behaviour depends on the precise location of the incident dislocations and the extent of the pile-up. Transmission can occur on both Schmid and non-Schmid planes and can depend on the shear stresses on the GB plane. A continuum yield locus for transmission is formulated. In general, the overall dissociation and/or transmission behaviour is also determined by the Burgers vectors and associated steps of the primitive vectors of the grain boundary, and the criteria for dislocation transmission formulated by Lee et al . [Scripta Metall. 23 799 (1989); Phil. Mag. A 62 131 (1990); Metall. Trans. A 21 2437 (1990)] are extended to account for these factors.     

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.     

Nanoscale continuum calculation of basal dislocation core structures in graphite

In this work, we calculate the core structures of basal dislocations in graphite in a nanoscale continuum framework. The model consists of a stack of buffered Kirchhoff plates where the plates represent the covalent interactions within individual graphene sheets and the buffer layers represent the secondary interactions between them. In the mid-plane of the buffer layers, cohesive surfaces are introduced to account for the nonlinear deformations due to basal dislocations. The cohesive surface separation is governed by using an empirical 4-8 Lennard–Jones potential. Meanwhile, their relative shear sliding is governed by using a newly proposed empirical periodic stacking-fault potential. With these potentials, the core structures of full dislocations and partials are calculated and examined. It is shown that the full dislocations automatically split into partials that repel each other. The core sizes of individual partials, measured between peak stresses, are about 5?nm wide for the edge component and slightly narrower for the screw component. Since these sizes are about 10 times the lattice constant, they lend credence to our continuum model of basal dislocation cores in graphite. It is also shown that when the dislocations are densely packed on the same glide plane, i.e. in a pile-up, with spacing one to two times the core size, the split partials retain their individual identity with well-defined and well-separated stress peaks. Meanwhile, the membrane normal stresses in the graphene sheets rise considerably at the pile-up tips which, in turn, may provoke further deformation and damage modes such as kinking and delamination.     

Multiscale dislocation dynamics simulations of shock-induced plasticity in small volumes

Multiscale dislocation dynamics plasticity (MDDP) was used to investigate shock-induced deformation in monocrystalline copper. In order to enhance the numerical simulations, a periodic boundary condition was implemented in the continuum finite element (FE) scale so that the uniaxial compression of shocks could be attained. Additionally, lattice rotation was accounted for by modifying the dislocation dynamics (DD) code to update the dislocations’ slip systems. The dislocation microstructures were examined in detail and a mechanism of microband formation is proposed for single- and multiple-slip deformation. The simulation results show that lattice rotation enhances microband formation in single slip by locally reorienting the slip plane. It is also illustrated that both confined and periodic boundary conditions can be used to achieve uniaxial compression; however, a periodic boundary condition yields a disturbed wave profile due to edge effects. Moreover, the boundary conditions and the loading rise time show no significant effects on shock–dislocations interaction and the resulting microstructures. MDDP results of high strain rate calculations are also compared with the predictions of the Armstrong–Zerilli model of dislocation generation and movement. This work confirms that the effect of resident dislocations on the strain rate can be neglected when a homogeneous nucleation mechanism is included.     

Strain induced grain boundary premelting in bulk copper bicrystals

In bulk bicrystals strain induced grain boundary premelting (SIGBPM) occurs when heavy screw dislocation pileup can be held up to a certain high temperature, approximately 0.6T _M, where T _M is the melting point of bulk material in Kelvin. SIGBPM occurs at grain boundaries to which new twist component is added due to the rotation of both component crystals toward opposite direction about the axis perpendicular to the grain boundary plane. At the original grain boundary, grain boundary sliding takes place due to this relative rotation. In f.c.c. metals with relatively low stacking fault energies such as copper, nickel, brass(30Zn) and silver, dislocations dissociate into partials. Therefore high density tangled dislocations introduced during plastic deformation hardly loose. If these dislocations can be held to high temperatures, SIGBPM is promoted. Formation of static or dynamic recrystallized grains suppresses SIGBPM itself and the propagation of grain boundary cracks formed by SIGBPM.     

Grain boundary sliding during ambient-temperature creep in hexagonal close-packed metals

Even at ambient temperature or less, below their 0.2% proof stresses all hexagonal close-packed metals and alloys show creep behaviour because they have dislocation arrays lying on a single slip system with no tangled dislocation inside each grain. In this case, lattice dislocations move without obstacles and pile-up in front of a grain boundary. Then these dislocations must be accommodated at the grain boundary to continue creep deformation. Atomic force microscopy revealed the occurrence of grain boundary sliding (GBS) in the ambient-temperature creep region. Lattice rotation of 5° was observed near grain boundaries by electron backscatter diffraction pattern analyses. Because of an extra low apparent activation energy of 20 kJ/mol, conventional diffusion processes are not activated. To accommodate these piled-up dislocations without diffusion processes, lattice dislocations must be absorbed by grain boundaries through a slip-induced GBS mechanism.     

Kinetics of Grain Boundary Recovery in Deformed Polycrystals

Annealing kinetics are studied for nonequilibrium ensembles of dislocations occurring in grain boundaries during plastic deformation. Two types of dislocation ensembles are considered: 1) walls of sessile extrinsic grain boundary dislocations (EGBDs), which cause a change of the GB misorientation angle, and 2) arrays of glissile EGBDs having a Burgers vector tangential to the grain boundary plane. For both types similar exponential relationships are obtained for the relaxation of the average EGBD density, with approximately the same characteristic time proportional to the cube of grain size.     

温度和应变速率耦合作用下纳米晶Ni压缩行为研究

本文利用低温力学测试系统研究了电化学沉积纳米晶Ni在不同温度和宽应变速率条件下的压缩行为. 借助应变速率敏感指数、激活体积、扫描电子显微镜及高分辨透射电子显微镜方法, 对纳米晶Ni的压缩塑性变形机理进行了表征. 研究表明, 在较低温度条件下, 纳米晶Ni的塑性变形主要是由晶界位错协调变形主导, 晶界本征位错引出后无阻碍的在晶粒内无位错区运动, 直至在相对晶界发生类似切割林位错行为. 并且, 在协调塑性变形时引出位错的残留位错能够增加应变相容性和减小应力集中; 在室温条件下, 纳米晶Ni的塑性变形机理主要是晶界-位错协调变形与晶粒滑移/旋转共同主导. 利用晶界位错协调变形机理和残留位错运动与温度及缺陷的相关性揭示了纳米晶Ni在不同温度、不同应变速率条件下力学压缩性能差异的内在原因.     

Strongly non-local modelling of dislocation transport and pile-up

The purpose of this work is the continuum modelling of transport and pile-up of infinite discrete dislocation walls driven by non-local interaction and external loading. To this end, the underlying model for dislocation wall interaction is based on the non-singular Peierls–Nabarro (PN) model for the dislocation stress field. For simplicity, attention is restricted to walls consisting of single-sign dislocations and to continuous wall distributions on a single glide plane. In this context, the influence of strongly non-local (SNL; long-range) interaction, and its approximation as weakly non-local (WNL; short-range) are studied in the context of interaction- and external-load-driven wall pile-up at a boundary. The pile-up boundary is modelled via a spatially dependent dislocation mobility which decreases to zero at the boundary. The pile-up behaviour predicted by the current SNL-based continuous wall distribution modelling is consistent with that predicted by discrete wall distribution modelling. Both deviate substantially from the pile-up behaviour predicted by WNL-based continuous wall distribution modelling. As such, it is clearly essential to account in continuum models for the intrinsic SNL character of the interaction between same-sign dislocations ‘close’ to the boundary. Gradient-based WNL ‘approximation’ of this interaction is not justified.     

A Revision of the Models for the Accommodation of Extrinsic Grain Boundary Dislocations

The kinetics of the spreading of electron diffraction contrast from extrinsic grain boundary dislocations (EGBDs) is studied in terms of the dissociation model. For the spreading time an equation similar to those given by Lojkowski-Grabski (W. Lojkowski and M.W. Grabski, in Deformation of Polycrystals: Mechanisms and Microstructures, edited by N. Hansen, A. Horswell, T. Leffers, and H. Lilholt (RisØ Nat. Lab., Roskilde, Denmark, 1981), p. 329.) and Johannesson-Thölen (T. Johannesson and A. Thölen, Metal Sci. J. 6, 189 (1972).) formulae is obtained, but with a much less numeric coefficient which is approximately equal to the coefficient calculated in the incorporation model (A.A. Nazarov, A.E. Romanov, and R.Z. Valiev, Scripta Metall. Mater. 24, 1929 (1990).). A comparison with experimental data shows that the new equation describes the spreading kinetics better than the common formulae.     

Transformation of Slip Dislocations in Σ3 Grain Boundary

Grain boundary processes during plastic deformation of bicrystals were studied by TEM. Two methods were used. In situ straining in the electron microscope followed by post mortem examination and post mortem observation of specimens previously deformed by in situ synchrotron radiation X-ray topography. Two mechanisms governing slip propagation across a coherent twin boundary in a Fe-Si alloy bicrystal were identified. The first mechanism is a dissociation of a slip dislocation with the Burgers vector lying parallel to the boundary into three equal grain boundary dislocations. The second mechanism is a decomposition of a slip dislocation with Burgers vector inclined to the boundary into a dislocation mobile in the other grain and two screw grain boundary dislocations.     

单向拉伸作用下Cu(100)扭转晶界塑性行为研究

应用分子动力学方法研究了在不同扭转角度下的Cu（100）失配晶界位错结构,以及不同位错结构对晶界强度的影响.模拟结果表明：小角度扭转晶界上将形成失配位错网,失配位错密度随着晶粒之间的失配扭转角度的增加而增加.变形过程中,位错网每个单元中均产生位错形核扩展.位错之间的塞积作用影响晶界的屈服强度：随着位错网格密度的增加,位错之间的塞积作用增强,界面的屈服强度得到提高.大角度扭转晶界将形成面缺陷,在变形中位错由晶界角点处形核扩展,此时由于面缺陷位错开动应力趋于一致,因此晶界的临界屈服强度趋于定值. 关键词： 扭转晶界 失配位错网 强化机理 分子动力学     

Interplay between elastic fields due to gravity and a partial dislocation for a hard-sphere crystal coherently grown under gravity: driving force for defect disappearance

We previously observed that an intrinsic staking fault shrunk through a glide of a Shockley partial dislocation terminating its lower end in a hard-sphere crystal under gravity coherently grown in ?001? by Monte Carlo simulations [Mori et al., Molec. Phys. 105, 1377 (2007)]; it was an answer to a one-decade long standing question why the stacking disorder in colloidal crystals reduced under gravity [Zhu et al., Nature 387, 883 (1997)]. Here, we present an elastic energy calculation; in addition to the self-energy of the partial dislocation [Mori et al., Prog. Theor. Phys. Suppl. 178, 33 (2009)] we calculate the cross-coupling term between elastic field due to gravity and that due to a Shockley partial dislocation. The cross-term is an increasing function of the linear dimension R over which the elastic field expands, showing that a driving force arises for the partial dislocation moving toward the upper boundary of a grain.     

On the non-uniform motion of dislocations: the retarded elastic fields,the retarded dislocation tensor potentials and the Liénard–Wiechert tensor potentials

The topic of this paper is the fundamental theory of the non-uniform motion of dislocations in two and three space dimensions. We investigate the non-uniform motion of an arbitrary distribution of dislocations, a dislocation loop and straight dislocations in infinite media using the theory of incompatible elastodynamics. The equations of motion are derived for non-uniformly moving dislocations. The retarded elastic fields produced by a distribution of dislocations and the retarded dislocation tensor potentials are determined. New fundamental key formulae for the dynamics of dislocations are derived (Jefimenko type and Heaviside–Feynman type equations of dislocations). In addition, exact closed-form solutions of the elastic fields produced by a dislocation loop are calculated as retarded line integral expressions for subsonic motion. The fields of the elastic velocity and elastic distortion surrounding the arbitrarily moving dislocation loop are given explicitly in terms of the so-called three-dimensional elastodynamic Liénard–Wiechert tensor potentials. The two-dimensional elastodynamic Liénard–Wiechert tensor potentials and the near-field approximation of the elastic fields for straight dislocations are calculated. The singularities of the near-fields of accelerating screw and edge dislocations are determined.     

Two-Point Resistance of a Cobweb Network with a 2r Boundary

We consider the problem of the two-point resistance on an m × n cobweb network with a 2r boundary,which has never been solved before. Up to now researchers just only solved the cases with free boundary or null resistor boundary. This paper gives the general formulae of the resistance between any two nodes in both finite and infinite cases using a method of direct summation pioneered by Tan [Z. Z. Tan, et al., J. Phys. A 46(2013) 195202], which is simpler and can be easier to use in practice. This method contrasts the Green's function technique and the Laplacian matrix approach, which is difficult to apply to the geometry of a cobweb with a 2r boundary. We deduce several interesting results according to our general formula. In the end we compare and illuminate our formulae with two examples. Our analysis gives the result directly as a single summation, and the result is mainly composed of the characteristic roots.     

Quantifying the grain boundary resistance against slip transfer by experimental combination of geometric and stress approach using stage-I-fatigue cracks

In recent studies, many groups have investigated the interaction of dislocations and grain boundaries by bi-crystals and micro-specimen experiments. Partially, these experiments were combined with supplementary simulations by discrete dislocation dynamics, but quantitative data for the grain boundary resistance against slip transfer is still missing. In this feasibility study with first results, we use stage-I-fatigue cracks as highly localised sources for dislocations with well-known Burgers vectors to study the interaction between dislocations in the plastic zone in front of the crack tip and selected grain boundaries. The stress concentration at the grain boundary is calculated with the dislocation-free zone model of fracture using the dislocation density distribution in the plastic zone from slip trace height profile measurements by atomic force microscopy. The grain boundary resistance values calculated from common geometric models are compared to the local stress distribution at the grain boundaries. Hence, it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer to a self-contained concept. As a result, the prediction of the grain boundary resistance effect based on a critical stress concept is possible with knowledge of the geometric parameters of the grain boundary only, namely the orientations of both participating grains and the orientation of the grain boundary plane.     

剪切作用下Cu(100)扭转晶界塑性行为研究

本文采用分子动力学方法研究了在剪切载荷作用下,Cu(100)扭转晶界对Cu柱屈服强度的影响.模拟结果发现,在加载过程中,低角度扭转晶界形成的位错网发生位错形核与扩展,位错之间的塞积作用提高了Cu柱的屈服强度;对于高角度扭转晶界,晶界发生滑动降低了Cu柱的屈服强度.同时发现,随着扭转角度的增加,Cu柱的屈服强度先增大,当扭转角度大于临界角度时,Cu柱的屈服应力逐渐减小.这表明剪切载荷作用下,两种不同的机理主导Cu柱的屈服,对于小于临界角度的扭转晶界,Cu柱的屈服由晶界位错形核和扩展机理主导,对于大于临界角度 关键词： 扭转晶界 分子动力学 位错形核 晶界滑移     

Interaction of Lomer–Cottrell locks with screw dislocations

In fcc crystals, dislocations are dissociated into partial dislocations and, therefore, restricted to move on {111} glide planes. By junction reactions with dislocations on two intersecting {111} planes, Lomer–Cottrell dislocations along ?110? directions can be formed which are barriers for approaching screw dislocations. Treating the interaction between a dissociated screw dislocation and a LC lock conventionally, using classical continuum theory and assuming the partials to be Volterra dislocations, leads to erroneous conclusions. A realistic result can only be obtained in the framework of the Peierls model, treating the partials as Peierls dislocations and explicitly taking account of the change in atomic misfit energy in the glide plane. At even moderate stresses (at less than 3 × 10^?3 µ in Cu), the screw will combine with the LC lock to form a Hirth lock. As a result, the nature of the repulsive force will change drastically.     

硅中位错增殖的实验观察

用化学侵蚀法研究了在机械应力和热应力作用下硅中位错的增殖和非均匀成核。结果表明,在使位错增殖和成核作用上,热应力同机械应力是等效的。硅中小角晶界中的位错,原生孤立位错都能成为位错源;晶体内部的缺陷及表面蚀斑处的应力集中能够引起位错成核;硅中螺型位错能够通过交叉滑移机制发生增殖。对新生位错环空间分布的研究表明,Frank-Read机制可能是位错增殖的主要形式。位错能否发生增殖,主要决定于位错源所受分切应力的数值、晶体温度、位错本身的结构特点以及钉扎情况等。     

Dislocation content of geometrically necessary boundaries aligned with slip planes in rolled aluminium

Previous studies have revealed that dislocation structures in metals with medium-to-high stacking fault energy, depend on the grain orientation and therefore on the slip systems. In the present work, the dislocations in eight slip-plane-aligned geometrically necessary boundaries (GNBs) in three grains of near 45° ND rotated cube orientation in lightly rolled pure aluminium are characterized in great detail using transmission electron microscopy. Dislocations with all six Burgers vectors of the ½?1?1?0? type expected for fcc crystals were observed but dislocations from the four slip systems expected active dominate. The dislocations predicted inactive are primarily attributed to dislocation reactions in the boundary. Two main types of dislocation networks in the boundaries were identified: (1) a hexagonal network of the three dislocations in the slip plane with which the boundary was aligned; two of these come from the active slip systems, the third is attributed to dislocation reactions (2) a network of three dislocations from both of the active slip planes; two of these react to form Lomer locks. The results indicate a systematic boundary formation process for the GNBs. Redundant dislocations are not observed in significant densities.