On the traction boundary value problem in the linearized gauge theory of dislocations

The static traction boundary value problem for finite material bodies is shown to be well posed in the linearized gauge theory of dislocations. The dislocation field variables assume the roles of generalized stress potentials that satisfy a system of fourth order linear partial differential equations. Accordingly, the stress distributions may be calculated directly from the traction boundary data without solving for the elastic displacement fields. Satisfaction of appropriate gauge conditions are shown to lead to significant simplifications and certain systems of first integrals of the governing equations are exhibited. The important thing here is that the gauge theory of dislocations provides direct means of calculating the distributions of dislocations that arise from given systems of boundary tractions. This is in sharp contrast with previous theories in which the distributions of dislocations are calculated from given distributions of dislocation densities.     

Effective stress in the gauge theory of dislocations

Standard practices of the calculus of variations are used to modify the Lagrangian function of elasticity so that boundary tractions and initial linear momentum densities can be specified. Taken in conjunction with the Yang-Mills type minimal coupling theory of dislocations, these practices lead directly to a demonstration that the effective stress and linear momentum are what drive the dislocation fields in finite material bodies without disclinations.     

The Filon construct for moving dislocations

Filon’s construct, originally developed for plane isotropic linear elastostatics, examines the difference between solutions to the same boundary value problem but for two different values of the elastic moduli, and relates the difference solution to one occurring in a corresponding stationary dislocation problem. This paper generalises the result to three-dimensional linear elastodynamics with particular reference to moving dislocations. Also studied is the inverse procedure which derives the pair of elastodynamical problems from a given distribution of dislocations. Body-forces and an auxiliary plastic distortion tensor are novel and essential features of the argument. Specialisation to the static and quasi-static theories is straightforward. The inverse Filon construct is illustrated by the stationary edge dislocation and uniformly moving screw dislocation in a homogeneous isotropic linear elastic whole space.     

A yang-mills type minimal coupling theory for materials with dislocations and disclinations

Noting that the group $\text{S0(3) ? T(3)}$ may be viewed as a 6-parameter gauge group that leaves the Lagrangian of elasticity theory invariant, the Yang-Mills universal gauge theory construction is used to erect a complete continuum theory of material bodies with dislocation and disclination fields. Breaking of the homogeneity of the action of $\text{S0(3)}$ is shown to give rise to disclinations and rotational dislocations while homogeneity breaking of $\text{T(3)}$ gives rise to translational dislocations. A rigorous justification for replacing displacement gradients by the components of the distortion tensor and Newtonian kinematic velocity by distortional velocity is obtained. Exact determinations are made of the elastic excess forces, the forces on dislocations and the forces on disclinations, and these forces are shown to be totally equilibrating in all instances. Implications of the theory are given and an analysis is made of the field equations and associated dispersion relations that obtain in a disclination free material in the linear elasticity approximation.     

Dislocation dipoles in nanocrystalline films

A theoretical model is suggested that describes the behavioral features and energetic characteristics of dipoles of grain boundary dislocations in nanocrystalline films. Such dislocation dipoles in nanocrystalline films are shown to play the role of misfit defect configurations that compensate, in part, for misfit stresses that occur due to a mismatch between crystal lattice parameters of films and substrates. Ranges of parameters (misfit parameter, grain size, etc.) are revealed at which the formation of dislocation dipoles is energetically favorable in nanocrystalline films. It is demonstrated that dislocation dipoles are typical structural elements of nanocrystalline films fabricated at highly nonequilibrium conditions.     

加层压电半空间界面裂纹力电联合冲击动态响应

基于线性压电弹性理论,研究电绝缘和电接触两种电边界条件下,任意多个任意排列的压电加层半空间界面裂纹在力电联合冲击作用下的动态响问题。通过引进Laplace变换和 Fourier 变换及位错密度函数,将问题首先转化为Cauchy 奇异积分方程,进而转化为代数方程进行数值求解。数值算例表明,电载荷、加层厚度及材料参数对动能量释放率具有重要的但不同的影响。研究结果对结构设计及结构失效的预防具有理论和应用价值。     

The study of grain boundary density effect on multi-grain thin film under tension

The grain-size effect on the yield strength and strain hardening of thin film at sub-micron and nanometer scale closely relates to the interactions between grain boundary and dislocation. Based on higher-order gradient plasticity theory, we have systematically investigated the size effect of multi-grain thin film arising from the grain boundary density under tensile stress. The developed formulations employing dislocation density and slip resistance have been implemented into the finite element program, in which grain boundary is treated as impenetrable interface for dislocations. The numerical simulation results reasonably show that plastic hardening rate and yield strength are linear to the grain boundary density of multi-grain thin film. The aspect ratio of grain size and orientation of slip system have distinct influence on the grain plastic properties. The research of slip system including homogeneous and nonhomogeneous distribution patterns reveals that the hardening effect of low-angle slip system is greater than that of high-angle slip system. The results agree well with the experimentally measured data and the solutions by discrete dislocation dynamics simulation.     

Relationship between surface tension and energy,interfacial energy and lattice friction

Any surface, in order to decrease its surface energy, contracts. It is shown for the first time that this contraction is formally equivalent to the introduction of a continuous distribution of surface dislocations. Equilibrium is attained when the increase in strain energy associated with these surface dislocations just balances the corresponding decrease due to the reduction in free surface area. Numerical calculations have been carried out for finite solid and liquid bodies as well as for liquid droplets in contact with solids. These findings suggest that it is possible to reformulate the behaviour of liquids in terms of dislocation theory in a much more general way than has hitherto been done.     

A mesh-independent method for planar three-dimensional crack growth in finite domains

The discrete crack mechanics (DCM) method is a dislocation-based crack modeling technique where cracks are constructed using Volterra dislocation loops. The method allows for the natural introduction of displacement discontinuities, avoiding numerically expensive techniques. Mesh dependence in existing computational modeling of crack growth is eliminated by utilizing a superposition procedure. The elastic field of cracks in finite bodies is separated into two parts: the infinite-medium solution of discrete dislocations and an finite element method solution of a correction problem that satisfies external boundary conditions. In the DCM, a crack is represented by a dislocation array with a fixed outer loop determining the crack tip position encompassing additional concentric loops free to expand or contract. Solving for the equilibrium positions of the inner loops gives the crack shape and stress field. The equation of motion governing the crack tip is developed for quasi-static growth problems. Convergence and accuracy of the DCM method are verified with two- and three-dimensional problems with well-known solutions. Crack growth is simulated under load and displacement (rotation) control. In the latter case, a semicircular surface crack in a bent prismatic beam is shown to change shape as it propagates inward, stopping as the imposed rotation is accommodated.     

Continuous plastic cracks in ordered alloys

A discrete dislocation analysis of the continuous plastic crack is carried out for ordered alloys. The crack is assumed to nucleate and reach a size where it will emit a set of lattice dislocations in order to decrease its energy. Further growth of the crack takes place elastically until it can emit the next set of lattice dislocations. Repeated emission of lattice dislocations, with elastic crack growth in between, leads to the Griffith configuration where the energy variation with size of the crack is zero. It is shown that a crack, either tensile or shear, can be stabilized by the presence of antiphase boundary energy alone. In the absence of frictional stress or with the very low frictional stresses encountered in real materials, the lattice dislocations are generated in pairs on each slip plane. However, when the frictional stress is high, the lattice dislocations are generated as single ones, giving rise to an antiphase boundary between the crack and the lattice dislocation.     

Strengthening via deformation twinning in a nickel alloy

Nanograins and nanotwins are produced in specimens using one processing technique to allow direct comparison in their nanohardnesses. It is shown that the hardness of nanotwins can be close to the lower end of the hardness of nanograins. The resistance of nanotwins to dislocation movement is explained based on elastic interactions between the incident 60° dislocation and the product dislocations. The latter includes one Shockley partial at the twin boundary and one 60° dislocation in the twinned region. The analysis indicates that a resolved shear stress of at least 1.24 GPa is required for a 60° dislocation to pass across a twin boundary in the nickel alloy investigated. It is this high level of the required shear stress coupled with a limited number of dislocations that can be present between two adjacent twin boundaries that provides nanotwins with high resistance to dislocation movement. The model proposed is corroborated by the detailed analysis of high-resolution transmission electron microscopy.     

The brittle-ductile transition in silicon: The influence of pre-existing dislocation arrangements

It has been shown that the value of the brittle-ductile transition temperature in specimens of silicon containing pre-cracks at the surface is strongly dependent on the pre-existing dislocation arrangement close to the crack tip. In particular, removing dislocations from the vicinity of the crack tip has the effect of raising the transition temperature, while introducing more surface dislocations by grinding reduces the transition temperature. In both cases the transition temperature remains sharp, implying that the crack tip sources have to be nucleated in the test before effective shielding occurs. The differences in the transition temperatures reflect the differences in the distances of the dislocations from the crack tips, and in the source lengths of the pre-existing sources which send dislocations to the crack tip.Specimens which are pre-stressed at the brittle-ductile transition to a K value at which crack tip sources are expected to be nucleated exhibit a gradual “soft” transition over a wide range of temperature. This result is in agreement with the predictions of the computer modelling of dislocation emission from crack tips, developed by Hirsch et.al [3,7], and together with the results on the dependence of T_c on dislocation arrangements, provides strong support for their theory of the brittle-ductile transition.     

Shielding effects and residual stresses at cleavage due to pre-existing dislocations

The motion of pre-existing edge dislocations in an infinite linear elastic body is studied at initiation of crack growth and at quasi-static steady-state crack growth. Dislocation nucleation is assumed not to occur. Thus, the study concerns only dislocations that are present in the virgin material. A dislocation is assumed to glide if its driving force exceeds a critical value. Changes in dislocation density, crack tip shielding and residual stresses are obtained. The shielding of a stationary crack tip is found to be small compared with the shielding of a growing crack tip. At steady-state the residual stresses far behind the crack tip are tensile near the crack, decreasing to zero at a certain distance from the crack plane. It is shown that the shielding due to pre-existing dislocations, e.g., for cleavage in α-iron crystals may be considerable. This revised version was published online in July 2006 with corrections to the Cover Date.     

The role of partial grain boundary dislocations in grain boundary sliding and coupled grain boundary motion

We study the process of grain boundary sliding through the motion of grain boundary dislocations, utilizing molecular dynamics and embedded atom method (EAM) interatomic potentials. For a Σ = 5 [001]{310} symmetrical tilt boundary in bcc Fe, the sliding process was found to occur through the nucleation and glide of partial grain boundary dislocations, with a secondary grain boundary structure playing an important role in the sliding process. While the homogeneous nucleation of these grain boundary dislocations requires shear strain levels higher than 7%, preexisting grain boundary dislocations are shown to glide at applied shear levels of 1.5%. The glide of the dislocations results in coupled motion of the boundary in the directions parallel and perpendicular to itself. Finally, interstitial impurities and vacancies were introduced in the grain boundary to study the effects on the sliding resistance of the boundary. While vacancies and H interstitials act as preferred nucleation sites, C interstitials do not. Both hydrogen and C interstitials stop dislocation glide whereas vacancies do not. A detailed study of the dynamic properties of these dislocations is also presented.     

Influence from geometrical features on the growth rate of a micro-structurally short fatigue crack

The influence on the crack growth rate on a micro-structurally short edge crack subjected to fatigue loading from changes in crack length, distance to grain boundaries and applied load has been investigated. The crack is assumed to grow in a single shear mechanism due to nucleation, glide and annihilation of dislocations along preferred slip planes in the material. The external geometry is modelled by distributed dislocation dipole elements in a boundary element approach under quasi-static and plane strain conditions. The evolving plasticity is described by individual discrete dislocations along a slip plane emanating from the crack in the crack direction. The crack growth rate is shown to be controlled by the plasticity, which in turn is controlled by geometrical parameters in combination with the external load.     

Surface dislocation model of a dislocation in a two-phase medium

The surface dislocation method developed earlier for solving the free surface boundary problem is now extended to the two-phase interface boundary problem wherein a lattice dislocation is situated in one of the phases. The interface is planar where two semi-infinite half spaces of different elastic properties are joined. The interface consists of four surface arrays of dislocations, two in each phase, so that the continuity of two stress components and two displacement components is maintained. The continuous distribution of dislocations is employed to arrive at the distribution function representing the surface arrays. The Airy stress functions for the two phases are derived and shown to give the same result as that obtained earlier by other methods. The distortions involved across the interface are represented in terms of simple surface arrays to show the advantage of the surface dislocation model. The stress field around the dislocation in the two-phase medium is plotted and the effect of the shear modulus of the second phase and of Poisson's ratio discussed. The advantages of applying the surface dislocation model either by the continuous distribution method or the discrete dislocation method are indicated.     

Dislocational pile-up—grain boundary interaction at elevated temperature

The nucleation and growth of grain boundary voids on the grain boundary facets, transverse to the direction of applied load, is one of the widely accepted mechanisms of damage development and material failure at high temperature.The aim of this work is the analysis of certain sources of stress concentration on the grain boundary which lead to time dependent high stress gradients.Specifically, the interaction of a dislocational pile-up with the grain boundary at high temperature is studied in order to model a variety of practical possible grain defects which can be approached on the basis of the theory of continuously distributed dislocations. Assuming the existence of the pile-up before the temperature rises, one investigates the time dependent stress field along the grain boundary, stress relaxation parameters and critical time. The diffusive absorption of the dislocations from the pile-up as well as the dislocation redistribution in the pile-up versus time are represented.This data leads to the estimation of possible grain boundary void nucleation, material damping due to diffusive matter redistribution and evaluation of plastic energy dissipation.     

A new approach to the theory of grain boundaries

It is shown that the classical picture of a grain boundary in terms of a single array of lattice dislocations is incomplete. In addition, it is also necessary to incorporate into the boundary a second array of continuously distributed surface dislocations of infinitesimal strength and of opposite sign to those of the lattice dislocations, but with the same total magnitude. Furthermore, the lattice dislocations can also dissociate into a continuous distribution by the formation of cores. As the angular misorientation of the grain boundary increases, more and more of the surface dislocations combine with the lattice dislocations, in turn resulting in the formation of a larger and larger stress-free ledge, with a consequent overall reduction in the strength of the remaining lattice dislocation.