Special rotational deformation effect on the emission of dislocations from a crack tip in deformed nanocrystalline solids

The interaction between the special rotational deformation and the crack in deformed nanocrystalline ceramics and metals is investigated by the complex variable method. We have theoretically described the effects of the disclination quadrupole produced by the special rotational deformation on the emission of lattice dislocations from the crack tip. The complex form expressions of the stress field and the force field are divided. The critical stress intensity factors for the first dislocation emission are calculated. The influence of the disclination strength, the grain size (the quadrupole arms), the location of the disclinations and the angle between the crack plane and one of the quadrupole arms, as well as the crack length on the critical stress intensity factors is discussed in detail. And the results show that, the special rotational deformation has great influence on the dislocation emission from the crack tip.     

An elasto-plastic theory of dislocation and disclination fields

A linear theory of the elasto-plasticity of crystalline solids based on a continuous representation of crystal defects – dislocations and disclinations – is presented. The model accounts for the translational and rotational aspects of lattice incompatibility, respectively associated with the presence of dislocations and disclinations. The defects content relates to the incompatible plastic strain and curvature tensors. The stress state is described by using the conjugate variables to strain and curvature, i.e., the stress and couple-stress tensors. Defect motion is described by two transport equations. A dynamic interplay between dislocations and disclinations results from a disclination-induced source term in the transport of dislocations. Thermodynamic guidance provides the driving forces conjugate to dislocation and disclination velocity in a continuous context, as well as admissible constitutive relations for the latter. When dislocation and disclination velocity vanish, the model reduces to deWit’s elasto-static theory of crystal defects. It also reduces to Acharya’s linear elasto-plastic theory for dislocation fields when the disclination density is ignored. The theory is intended for use in instances where rotational defects matter, such as grain boundaries. To illustrate its applicability, a finite high-angle tilt boundary is modeled using a disclination dipole and its behavior under tensile loading normal to the boundary is shown.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

Deformation of silicon – Insights from microcompression testing at 25–500 °C

The plastic deformation of silicon and other brittle materials near room temperature has conventionally been studied under high confining pressures, although it has been suggested that these may modify the dislocation core structure. Here, the possibility of using microcompression has been studied. Using this method the yield stress of silicon micropillars was measured for different pillar diameters and between 25 and 500 °C for a constant diameter of 2 μm. No pronounced effect of size on the yield stress was found, but the transition from failure by cracking to predominately plastic deformation was shown to be consistent with a previously proposed simple model for axial splitting. Deformed specimens were analysed by transmission electron microscopy to elucidate the operative dislocation mechanisms. This showed that at 500 °C deformation occurs by twinning and formation of partial dislocations, whereas at 100 °C it is associated with micro-cracking and only weakly dissociated dislocations.     

Continuum theory of dislocations and disclinations in nonlinearly elastic micropolar media

A nonlinear theory of continuously distributed dislocation and disclination type defects in elastic media with intrinsic rotational degrees of freedom and couple stresses is proposed. The mediumstrains are assumed to be finite. The solving equations of the continuum theory of defects are obtained by passing to the limit from a discrete set of isolated dislocations and disclinations to their continuous distribution. The notions of dislocation and disclination densities in a micropolar body under large deformations are introduced. Incompatibility equations are obtained and a boundaryvalue problem of equilibriumis posed for an elastic micropolar body with a given density of distributed defects. A nonlinear problem of determining the intrinsic stresses in a hollow circular cylinder due to a given distribution of disclinations is solved.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Computational modeling of domain wall interactions with dislocations in ferroelectric crystals

This paper provides a theoretical and numerical framework to investigate the interactions between domain walls and arrays of dislocations in ferroelectric single crystals. A phase-field approach is implemented in a non-linear finite element method to determine equilibrium solutions for the coupled electromechanical interactions between a domain wall and a dislocation array. The numerical simulations demonstrate the effect of the relative size and orientation of dislocations on 180° and 90° domain wall configurations. In addition, results for the pinning strength of dislocations in the case that domain walls move due the application of external electric field and shear stress are computed. The presented numerical results are compared with the findings reported for charged defects and it is shown that non-charged defects, such as dislocations, can also interact strongly with domain walls, and therefore affect the ferroelectric material behavior.     

On the nucleation of a Zener crack from a wedge disclination dipole in the presence of a circular inhomogeneity

The nucleation of a Mode-I Zener crack from a wedge disclination dipole in the presence of a circular inhomogeneity is investigated. It is assumed that the disclination dipole and the nucleated Zener crack are along the radial direction of the inhomogeneity. Two cases are studied herein, i.e., the positive or negative wedge disclination of the dipole locates nearer to the inhomogeneity respectively. In order to investigate how various factors such as the elastic mismatch between the inhomogeneity and the matrix influence the nucleation of the Zener crack, the Stress Intensity Factor (SIF) at the sharp tip of the Zener crack is determined for different sets of geometric and material parameters with the distributed dislocation technique. The formulated singular integral equations are then solved numerically. Our results indicate that a nearby ‘hard’ inhomogeneity (having a higher shear modulus than the matrix) is beneficial to the crack nucleation for the first case (the positive disclination locates nearer to the inhomogeneity) while it retards the crack nucleation for the second case (the negative disclination locates nearer to the inhomogeneity). A nearby ‘soft’ inhomogeneity is helpful to the crack nucleation for the second case while it has inverse effects on the crack nucleation for the first case. This phenomenon can be explained with the concept of material force. The characteristics of the crack nucleation and the effects of the disclination strength, the distance between the inhomogeneity and the dipole, the disclination dipole arm length and inhomogeneity size on the crack nucleation are also systematically studied. The obtained results are helpful to characterize and enhance the strength of precipitate alloys and particle reinforced composites.     

Dislocation shielding of a cohesive crack

Dislocation interaction with a cohesive crack is of increasing importance to computational modelling of crack nucleation/growth and related toughening mechanisms in confined structures and under cyclic fatigue conditions. Here, dislocation shielding of a Dugdale cohesive crack described by a rectangular traction-separation law is studied. The shielding is completely characterized by three non-dimensional parameters representing the effective fracture toughness, the cohesive strength, and the distance between the dislocations and the crack tip. A closed form analytical solution shows that, while the classical singular crack model predicts that a dislocation can shield or anti-shield a crack depending on the sign of its Burgers vector, at low cohesive strengths a dislocation always shields the cohesive crack irrespective of the Burgers vector. A numerical study shows the transition in shielding from the classical solution of Lin and Thomson (1986) in the high strength limit to the solution in the low strength limit. An asymptotic analysis yields an approximate analytical model for the shielding over the full range of cohesive strengths. A discrete dislocation (DD) simulation of a large (>10³) number of edge dislocations interacting with a cohesive crack described by a trapezoidal traction-separation law confirms the transition in shielding, showing that the cohesive crack does behave like a singular crack at very high cohesive strengths (∼7 GPa), but that significant deviations in shielding between singular and cohesive crack predictions arise at cohesive strengths around 1GPa, consistent with the analytic models. Both analytical and numerical studies indicate that an appropriate crack tip model is essential for accurately quantifying dislocation shielding for cohesive strengths in the GPa range.     

Piezoelectric screw dislocations interacting with a circular inclusion with imperfect interface

The electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated. The appointed screw dislocation may be located either outside or inside the inclusion and is subjected to a line charge and a line force at the core. The analytic solutions of electroelastic fields are obtained by means of the complex-variable method. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces exerted on the piezoelectric screw dislocations are derived. The motion and the equilibrium position of the appointed screw dislocation near the circular interface are discussed for variable parameters (interface imperfection, material electroelastic mismatch, and dislocation position), and the influence of the nearby parallel screw dislocations is also considered. It is found that the piezoelectric screw dislocation is always attracted by the electromechanical imperfect interface. When the interface imperfection is strong, the impact of material electroelastic mismatch on the image force and the equilibrium position of the dislocation becomes weak. Additionally, the effect of the nearby dislocations on the mobility of the appointed dislocation is very important.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Size and strain rate effects in tensile strength of penta-twinned Ag nanowires

Penta-twinned Ag nanowires(pt-AgNWs) have recently attracted much attention due to their interesting mechanical and physical properties. Here we perform largescale atomistic simulations to investigate the influence of sample size and strain rate on the tensile strength of pt-AgNWs. The simulation results show an apparent size effect in that the nanowire strength(defined as the critical stress for dislocation nucleation) increases with decreasing wire diameter. To account for such size effect, a theoretical model involving the interaction between an emerging dislocation and the twin boundary has been developed for the surface nucleation of dislocations. It is shown that the model predictions are in quantitative agreement with the results from atomistic simulations and previous experimental studies in the literatures. The simulations also reveal that nanowire strength is strain-rate dependent, which predicts an activation volume for dislocation nucleation in the range of 1–10b~3,where b is the magnitude of the Burgers vector for a full dislocation.     

The effect of path cut on Somigliana ring dislocation elastic fields

In this paper we look at ring dislocations (circular loops) in an infinite isotropic full-space. The dislocation direction is either axial or radial. Unlike dislocations in plane analysis the path cut has a significant effect on the elastic fields. Solutions for the dislocations are given for a variety of path cuts with closed form expressions for the displacement and stress fields. When considered alone these dislocations do not obey Frank’s rule; these anomalies and other fundamental properties are discussed.     

An analysis of dislocation nucleation near a free surface

Molecular dynamics analyses of defect-free aluminum single crystals subject to bending are carried out to investigate dislocation nucleation from free surfaces. A principal aim of the analyses is to provide background for the development of dislocation nucleation criteria for use in discrete dislocation plasticity calculations. The molecular dynamics simulations use an embedded atom potential for aluminum. Bending is imposed on a strip by specifying a linear variation of displacement rate on opposite edges. The overall bending response is determined and the character of the dislocations nucleated is identified. It is found that the stress magnitudes at the instant of dislocation nucleation are nearly an order of magnitude smaller than for homogeneous bulk dislocation nucleation. The characterization of dislocation nucleation in terms of various phenomenological nucleation criteria is explored, in particular: (i) a critical resolved shear stress; (ii) the onset of an elastic instability; and (iii) a critical stress-gradient criterion. It is found that dislocation nucleation is not well-represented by a critical value of the resolved shear stress but is reasonably well-represented by the critical stress-gradient criterion.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Determination of interaction forces between parallel dislocations by the evaluation of J integrals of plane elasticity

The Peach–Koehler expressions for the glide and climb components of the force exerted on a straight dislocation in an infinite isotropic medium by another straight dislocation are derived by evaluating the plane and antiplane strain versions of J integrals around the center of the dislocation. After expressing the elastic fields as the sums of elastic fields of each dislocation, the energy momentum tensor is decomposed into three parts. It is shown that only one part, involving mixed products from the two dislocation fields, makes a nonvanishing contribution to J integrals and the corresponding dislocation forces. Three examples are considered, with dislocations on parallel or intersecting slip planes. For two edge dislocations on orthogonal slip planes, there are two equilibrium configurations in which the glide and climb components of the dislocation force simultaneously vanish. The interactions between two different types of screw dislocations and a nearby circular void, as well as between parallel line forces in an infinite or semi-infinite medium, are then evaluated.     

Transient dislocation emission from a crack tip under dynamic mode III loading

Summary  Transient dislocation emission from a crack tip under dynamic mode III loading is analyzed. By taking into account the dynamic interaction between the crack and dislocation, the governing equation for the dislocation motion is derived under the quasi-steady assumption. The behavior of dislocation emission is explored in detail by solving this equation numerically. A critical initial speed can be determined, which must be exceeded by dislocations to escape from the crack tip. The dislocation emission process is found to be completed in such a short time period that the applied load may be approximately treated as constant during dislocation emission. Based on this fact, an asymptotic criterion for transient dislocation emission is developed, from which the critical initial speed can be evaluated. In the case that the dislocation is emitted from rest, we recover the quasi-static criterion of dislocation emission. Received 22 November 2000; accepted for publication 20 March 2001     

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.     

Transient studies of screw and edge dislocation generation at crack edges

Exact transient analyses of the generation of screw and edge dislocations at the edges of stationary cracks subjected to the diffraction of, respectively, plane SH- and SV-waves, and their subsequent arrest are performed. The solutions are examined in light of a dislocation emission criterion which is based, simultaneously, on standard dislocation force concepts and quasi-static emission studies.This examination allows expressions for the times of emission and arrest, the distances traveled by the dislocations, and the dynamic stress intensity factors to be derived in terms of parameters such as dislocation speed and yield stress. These expressions exhibit distinctive dynamic effects and reveal several features of the generation process:In particular, the times and distances are, while on a micromechanical scale, not necessarily insignificant, and imply that purely brittle fracture may not easily occur. Then, edge dislocation emission would appear to occur at a preferred speed, while screw dislocation emission apparently prefers to take place quasi-statically.Examination of a general incident waveform class shows that a continuous wave could cause dislocation generation to occur before a step-stress wave can. Moreover, the emission process depends upon a weighted time history of the incident wave stress, not its instantaneous value. This, in turn, implies that dislocation emission does not necessarily shield the crack edge by lowering the dynamic stress intensity factor. Finally, unless the dislocations are allowed to decelerate to zero speed upon arrest, a repetitious process of start-stop motion can in principle take place.     

颗粒增强复合材料压缩行为的位错动力学模拟

颗粒增强铜基复合材料因具有极高的强度和弹性模量, 优异的导电、导热性能和抗磨损能力, 被广泛应用于航天航空、汽车、电子工业等领域. 第二相强化是其主要的强化方式, 其通过合金中弥散的微粒阻碍位错运动, 可有效提高金属材料的力学性能, 提高其服役安全. 针对该问题本文采用三维离散位错动力学(three-dimensional discrete dislocation dynamics, 3D-DDD)方法, 对微尺度颗粒增强铜基复合材料进行了微柱压缩模拟, 分析了位错与第二相颗粒交互作用对材料力学响应的影响, 揭示第二相颗粒强化的微观机理. 本研究将第二相颗粒视为位错不可穿透的球形微粒, 采用位错绕过机制模拟颗粒与位错的交互作用过程. 通过调控滑移面相对于第二相颗粒中心的距离发现: 屈服应力和应变硬化率均随距离的增大而减小. 研究也发现Schmid因子越高的滑移系, 屈服应力越低, 后续应变硬化率越低. 多位错与颗粒交互作用的模拟发现, 同一滑移面中位错间的反应和不同滑移系中位错的交互作用可能是导致屈服应力和应变硬化率降低的关键.