Non-singular dislocation continuum theories: strain gradient elasticity vs. Peierls–Nabarro model

Abstract

Non-singular dislocation continuum theories are studied. A comparison between Peierls–Nabarro dislocations and straight dislocations in strain gradient elasticity is given. The non-singular displacement fields, non-singular stresses, plastic distortions and dislocation core shapes are analysed and compared for the two models. The main conclusion of this study is that due to their characteristic properties, the non-singular displacement fields, non-singular stresses and dislocation core shape of screw and edge dislocations obtained in the framework of strain gradient elasticity are more realistic and physical than the corresponding fields of the Peierls–Nabarro model. Strain gradient elasticity of dislocations is a continuum dislocation theory including a weak non-locality within the dislocation core and predicting the size and shape of the dislocation core. The dislocation core is narrower in the strain gradient elasticity dislocation model than in the Peierls–Nabarro model and more evenly distributed in two dimensions. The present analysis shows that for the modelling of the dislocation core structure the non-singular dislocation fields of strain gradient elasticity are the suitable ones.     

On gradient field theories: gradient magnetostatics and gradient elasticity

In this work, the fundamentals of gradient field theories are presented and reviewed. In particular, the theories of gradient magnetostatics and gradient elasticity are investigated and compared. For gradient magnetostatics, non-singular expressions for the magnetic vector gauge potential, the Biot–Savart law, the Lorentz force and the mutual interaction energy of two electric current loops are derived and discussed. For gradient elasticity, non-singular forms of all dislocation key formulas (Burgers equation, Mura equation, Peach–Koehler stress equation, Peach–Koehler force equation, and mutual interaction energy of two dislocation loops) are presented. In addition, similarities between an electric current loop and a dislocation loop are pointed out. The obtained fields for both gradient theories are non-singular due to a straightforward and self-consistent regularization.     

Elastic displacement field of point defects in anisotropic cubic crystals

The elastic displacement field of point defects in cubic crystals is calculated for weak anisotropy by second order perturbation theory and by a variational procedure. The results are compared with numerical calculations for Cu. Further analytical approximations are given for the volume change in an infinite crystal and for the interaction energy of two point defects.     

The reciprocal theorem and rigid spherical inclusion vis-à-vis certain point singularities

The elastic interaction of certain point singularities with a rigid spherical inclusion embedded into an otherwise infinite elastic medium is investigated. The particular singularities considered are point force, force-dipole (with and without moment), centre of dilatation and concentrated moment. In each case, simple, closed form expressions are deduced by application of Betti's reciprocal theorem for the net force and net torque acting on the inclusion.     

The interaction of a screw dislocation with point defects in bcc iron

In this study, we calculate the interaction energy of intrinsic point defects vacancies and interstitials) with screw dislocations in body-centered cubic iron. First (we calculate the dipole tensor of a defect in the bulk crystal using molecular statics. Using a formulation based on linear elasticity theory, we calculate the interaction energy of the defect and the dislocation using both isotropic and anisotropic strain fields. Second, we perform atomistic calculations using molecular statics methods to directly calculate the interaction energy. Results from these two methods are compared. We verify that continuum methods alone are unable to correctly predict the interactions of defects and dislocations near the core. Although anisotropic theory agrees qualitatively with atomistics far from the core, it cannot predict which dumbbell orientations are stable and any continuum calculations must be used with caution. Spontaneous absorption by the core of both vacancies and dumbbells is seen. This paper demonstrates and discusses the differences between continuum and atomistic calculations of interaction energy between a dislocation core and a point defect.     

Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

Behavior of screw dislocations near phase boundaries in the gradient theory of elasticity

The solution of the boundary-value problem on a rectilinear screw dislocation parallel to the interface between phases with different elastic moduli and gradient coefficients is obtained in one of the versions of the gradient theory of elasticity. The stress field of the dislocation and the force of its interaction with the interface (image force) are presented in integral form. Peculiarities of the short-range interaction between the dislocation and the interface are described, which is impossible in the classical linear theory of elasticity. It is shown that neither component of the stress field has singularities on the dislocation line and remains continuous at the interface in contrast to the classical solution, which has a singularity on the dislocation line and permits a discontinuity of one of the stress components at the interface. This results in the removal of the classical singularity of the image force for the dislocation at the interface. An additional elastic image force associated with the difference in the gradient coefficients of contacting phases is also determined. It is found that this force, which has a short range and a maximum value at the interface, expels a screw dislocation into the material with a larger gradient coefficient. At the same time, new gradient solutions for the stress field and the image force coincide with the classical solutions at distances from the dislocation line and the interface, which exceed several atomic spacings.     

On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions

We investigate linear theories of incompatible micromorphic elasticity, incompatible microstretch elasticity, incompatible micropolar elasticity and the incompatible dilatation theory of elasticity (elasticity with voids). The incompatibility conditions and Bianchi identities are derived and discussed. The Eshelby stress tensor (static energy momentum) is calculated for such inhomogeneous media with microstructure. Its divergence gives the driving forces for dislocations, disclinations, point defects and inhomogeneities which are called configurational forces.     

Non-singular dislocation loops in gradient elasticity

Using gradient elasticity, we give in this Letter the non-singular fields produced by arbitrary dislocation loops in isotropic media. We present the ‘modified’ Mura, Peach–Koehler and Burgers formulae in the framework of gradient elasticity theory.     

Origin of Classical Singularities

We briefly review some results concerning theproblem of classical singularities in generalrelativity, obtained with the help of the theory ofdifferential spaces. In this theory one studies a givenspace in terms of functional algebras defined on it.Then we present a generalization of this methodconsisting in changing from functional (commutative)algebras to noncommutative algebras. By representingsuch an algebra as a space of operators on a Hilbertspace we study the existence and properties of variouskinds of singular space-times. The results obtainedsuggest that in the noncommutative regime, supposedly reigning in the Planck era, there is nodistinction between singular and non-singular states ofthe universe, and that classical singularities areproduced in the transition process from thenoncommutative geometry to the standard space-timephysics.     

Punktdefekte im kubischen elastischen Kontinuum

The determination of the displacement and strain fields of a point defect in a cubic crystal requires even in the framework of continuum elasticity theory numerical calculation. These fields of elastic dipoles are expanded in suitable vector and tensor fields. The coefficients of this expansion are calculated up to polynomials of 5th and 4th order in the direction cosines using the ratios of elastic constants as parameters. With this expansion the interaction of elastic dipoles in a cubic medium can be calculated. The results have been applied to the interaction of F-centres and of O₂ ^?-centres in alkali halides.     

Edge dislocations near phase boundaries in the gradient theory of elasticity

A solution of the boundary-value problem in the gradient theory of elasticity concerning a rectilinear edge dislocation parallel to the interface between phases with different elastic moduli and gradient coefficients is obtained. The interaction between the dislocation and the interface is considered on a nanoscopic level. It is shown that the stress field has no singularities on the dislocation line and remains continuous at the interface, unlike the classical solution, which is singular at the dislocation line and allows a discontinuity of two stress components at the interface. The gradient solution also removes the classical singularity of the image force for the dislocation on the interface. An additional elastic image force associated with the difference in the gradient coefficients of contacting phases is also determined. It is found that this force, which has a short range and a maximum at the interface, expels the edge dislocation into the material with a smaller gradient coefficient.     

The elastic interaction of a dislocation dipole with dilatation centres

The elastic interaction of a dilatation centre with edge dislocation dipole is investigated and the share of this elastic interaction in the time dependence of the diffusion of point defects to dislocations in plastically deformed crystalline materials is discussed.Reported on in brief at the International Conference on Electron Diffraction and Crysta Defects, Melbourne, August 1965.     

Interaction of point defects with an edge dislocation in the gradient theory of elasticity

Interaction between a point defect and an edge dislocation is studied in the framework of the gradient theory of elasticity. The change in the energy of the system caused by a displacement of the point defect relative to the dislocation line is calculated. The results of the theoretical analysis are used to describe edge dislocation pinning by impurity atoms.     

Lattice model with power-law spatial dispersion for fractional elasticity

A lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.     

On correct nonlocal generalized theories of elasticity

The paper discusses nonlocal elasticity theories among which are models of media with defect fields, gradient elasticity theories, and hybrid nonlocal elasticity theories. Gradient theories are analyzed, and their correctness properties are examined. Applied theories that satisfy the correctness conditions are developed, and known applied gradient theories are verified for the correctness properties. A new nonlocal generalized theory has been developed for which the operator of balance equations is represented as the product of the equilibrium operator of classical elasticity theory and the Helmholtz operator. It is shown that this theory is one-parameter and is the only representative of hybrid models constructed by a complete system of equations for forces and moments. Unlike classical elasticity that is free from scale parameters characterizing the internal material structure, nonlocal elasticity theories naturally incorporate these parameters. That is why they are suitable for the modeling of scale effects and find application in the solution of numerous applied problems for heterogeneous structures with developed phase interfaces where the degree of influence of scale effects depends on the density of phase boundaries. Nonlocal continuum models are especially attractive for modeling the properties of various micro/nanostructures, elastic properties of composites and structured materials with submicron- and nanosized internal structures in which effective properties are to a great extent defined by the scale effects (short-range interaction effects of cohesion and adhesion). Generalized elasticity theories even for isotropic materials contain many additional physical constants that are difficult or impossible to determine experimentally. Applied models with a small number of additional physical parameters are therefore of great interest. However, the reduction of nonlocal theories aimed at reducing the number of additional parameters is a nontrivial task and may lead to incorrect theories. The goal of this paper is to study the symmetry properties in gradient theories, to analyze the correctness of gradient theories, and to develop applied one-parameter elasticity theories.     

Diffuse x-ray scattering near Bragg peaks from point defects in a general lattice

Using the continuum theory of linear elasticity, diffuse x-ray scattering has been calculated in the immediate neighbourhood of Bragg peaks from point defects in a lattice containing more than one atom in the unit cell. General expressions are obtained for the Debye-Waller factor, Huang diffuse scattering and the asymmetric scattering due to the defect. For lattices with one atom per unit cell, these expressions reduce to the well-known formulae of diffuse scattering.