On the stress field and crack nucleation behavior of a disclinated nanowire with surface stress effects

This paper studies the stress field and crack nucleation behavior in a disclinated nanowire with a continuum model. The surface stress effects of the nanowire is accounted for with the Gurtin-Murdoch model. The Green’s functions for the stress fields of a single wedge disclination and a single edge dislocation in a cylindrical nanowire are solved respectively with the complex variable method. To make the superposition principle valid, the stress field induced by the residual surface tension is properly handled in the Green’s functions. After that, the distributed dislocation method is applied to simulate the crack nucleation behavior. The influences of the surface stress effects on the stress fields of the wedge disclination and edge dislocation as well as on the Griffith crack nucleation behavior are systematically discussed.     

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.     

Dislocation dynamics in an isotropic,paramagnetic and electrically conducting solid in a large magnetic field

The authors are concerned with the mechanical and electrical fields produced by moving dislocations in a paramagnetic solid with electric conduction in the presence of a large magnetic field. It is assumed that the magnetic field is constant throughout the body, and the materials are isotropic in both the mechanical and electromagnetic features. The fields of displacement and electric scalar potential are given in terms of Green's functions and the plastic distortion tensor. Finally, the expressions for the fields produced by an infinite straight dislocation, which is moved by a uniform subsonic velocity, are given. Numerical computations are made to estimate the electrical fields.     

Damage Identification in Plate-type Structures Using 2-D Spatial Wavelet Transform and Flexibility-Based Methods

As dislocation emission is regarded as the critical point of brittle-to-ductile transition of fracture mechanism and the defects interact with domain walls motions due to their self-stress fields, the investigations on dislocation forces, especially the interaction force and the reaction between domain switching and dislocations, are the most important fundamentals of dislocation slip in the brittle-to-ductile transition and the pinning effect on domain wall motion mechanisms. In this paper, an innovative method base on the strain nucleus simulated by an assembly of four dislocations and the Green’s function integration is used to solve generalized stress field arising from domain switching. Then, the accurate expressions of the interaction forces, pinning forces and the image forces on dislocation are obtained and analyzed. At last, the lattice resistance is also discussed. The numerical results show that the dislocation emission is possible when temperature and load are proper, and the domain switching interaction force is the main driving force of dislocation slip in ferroelectric material under negative electric field load. The curve and equation of the lattice resistance correlated with temperature can also be fitted by the results of the proposed analytical method and corresponding experiments.     

Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow

In the framework of plane thermoelastic problems is discussed the thermal stress field near the tips of an arbitrarily inclined crack in an isotropic semi-infinite medium with the thermally insulated edge surface under uniform heat flow. The crack is replaced by continuous distributions of quasi-Volterra dislocations corresponding to line heat sources and edge dislocations, and we obtain a set of simultaneous singular integral equations for dislocation density functions, whose solution is given in the forms of series in terms of Tchebycheff polynomials of the first kind. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. Numerical results are given for the particular case where the surface of the inclined crack is maintained at constant temperature and the heat supplied across the surface of the crack vanishes as a whole. The effects of the distance from the crack tip to the edge surface of the semi-infinite medium and the angle of inclination of the crack on the stress intensity factors and the initial direction of crack extension are shown graphically.     

Elastic fields due to an edge dislocation in an isotropic film-substrate by the image method

The stress and displacement fields due to an edge dislocation in a linearly elastic, isotropic film-substrate are derived using the method of image dislocations. The key features of the developed method are (1) a decomposition scheme in which the interfacial conditions are satisfied a priori, (2) all singular integrals are eliminated from the governing equations, (3) the elastic fields are obtained in the form of series, the terms of which can be exactly evaluated via integral transforms, and (4) the solutions converge satisfactorily with only three terms of the series. It is shown that the film thickness and dislocation position have a significant influence on the image force acting on the dislocation, and that the film thickness variation due to an accumulation of dislocations may degrade the performance of optical films.     

Elastic continuum theories of lattice defects: a review

The presently available elastic continuum theories of lattice defects are reviewed. After introducing a few elementary concepts and the basic equations of elasticity the Eshelby’s theory of misfitting inclusions and inhomogeneities is outlined. Kovács’ result that any lattice defect can be described by a surface distribution of elastic dipoles is described. The generalization of the isotropic continuum approach to anisotropic models and to Eringen’s isotropic but non-local model is discussed. Kröner’s theroy (where a defect is viewed as a lack of strain compatibility in the medium) and the elastic field equations (formulated in a way analogous to Maxwell’s field equations of magnetostatics) are described. The concept of the dislocation density tensor is introduced and the utility of higher-order dislocation density correlation tensors is discussed. The beautiful theory of the affine differential geometry of stationary lattice defects developed by Kondo and Kröner is outlined. Kosevich’s attempt to include dynamics in the elastic field equations is described. Wadati’s quantum field theory of extended objects is mentioned qualitatively. Some potential areas of research are identified.     

A new integral equation formulation for the analysis of crack-inclusion interactions

A new integral equation method for the analysis of the interactions between cracks and elastic inclusions embedded in a two-dimensional, linearly elastic, isotropic infinite medium subjected to in-plane force is presented. By distributing dislocations along the crack lines and forces along the matrix-inclusion interfaces, a set of coupled integral equations is obtained. The discretization procedure of the integrals involved is discussed and the relations between the stress intensity factors and the values of the dislocation functions at the respective crack tips are derived. Several sample problems are presented in order to determine the versatility and the accuracy of this approach.     

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.     

A nonlinear complementarity approach for elastoplastic problems by BEM without internal cells

A nonlinear complementarity approach is presented to solve elastoplastic problems by the boundary element method, in which the equations are formulated by stress equations and complementarity function obtained from the plasticity constitutive law. The domain integrals involved are transformed into boundary integrals by radial integration method, using compactly supported radial basis functions. Two numerical examples demonstrate the algorithm’s applicability and effectiveness.     

Electroelastic analysis for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.     

Interaction between crack and arbitrarily shaped hole with stress and displacement boundaries

The interaction between a crack and an arbitrarily shaped hole under stress and displacement boundaries in an infinite plane subjected to a remote uniform load is studied. The Green's functions of a point dislocation for the problems are derived and are then used to analyze the interaction problem. The superposition principle is employed to reduce the original problem to two subsidiary problems. The complex stress functions of each problem are composed of two parts, in which the second parts are always holomorphic. Using analytical continuation in conjunction with rational mapping function, the stress functions are obtained in closed form. The interaction of a hole or an inclusion with a crack is solved using dislocations to model the crack and solving a system of singular integral equations. Stress intensity factors for crack tips and stress concentration factors for inclusion corner are determined and plotted for various cases. The affecting ranges of hole and inclusion are investigated.     

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.     

A distributed dislocation stress analysis for crazes and plastic zones at crack tips

A distributed dislocation method is developed to obtain analytically the applied stress as well as the surface stress profile along narrow plastic zones at the tip of a crack in a homogeneous tensile stress field. Replacing the plastic zone by a continuous array of mathematical dislocations, the stress field solution of this mixed boundary value problem (the displacement profile of the plastic zone is fixed while the tensile stresses are zero across the crack) can be solved. A computer program based on this stress field solution has been constructed and tested using the analytical results of the Dugdale model. The method is then applied to determining the surface stress profiles of crazes and plane-stress plastic deformation zones grown from electron microprobe cracks in polystyrene and polycarbonate respectively. The necessary craze and zone surface displacement profiles are determined by quantitative analysis of transmission electron micrographs. The surface stress profiles, which show small stress concentrations at the craze or zone tip falling to an approximately constant value which is maintained to the crack tip, are compared with those previously computed using an approximate Fourier transform method involving estimation of the displacement profile in the crack. The agreement between the approximate method and the exact distributed dislocation method is satisfactory.     

Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity

The aim of this paper is to study the elastic stress and strain fields of dislocations and disclinations in the framework of Mindlin’s gradient elasticity. We consider simple but rigorous versions of Mindlin’s first gradient elasticity with one material length (gradient coefficient). Using the stress function method, we find modified stress functions for all six types of Volterra defects (dislocations and disclinations) situated in an isotropic and infinitely extended medium. By means of these stress functions, we obtain exact analytical solutions for the stress and strain fields of dislocations and disclinations. An advantage of these solutions for the elastic strain and stress is that they have no singularities at the defect line. They are finite and have maxima or minima in the defect core region. The stresses and strains are either zero or have a finite maximum value at the defect line. The maximum value of stresses may serve as a measure of the critical stress level when fracture and failure may occur. Thus, both the stress and elastic strain singularities are removed in such a simple gradient theory. In addition, we give the relation to the nonlocal stresses in Eringen’s nonlocal elasticity for the nonsingular stresses.     

电磁材料中广义螺型位错与圆形界面刚性线夹杂的干涉效应

研究了广义螺型位错和圆形界面刚性导体线夹杂的磁电弹耦合干涉效应.采用Riemann-Schwarz对称原理并结合复势函数奇性主部分析,得到该问题的一般解答.当界面只含一条刚性线时,获得了封闭形式解.运用扰动技术,求解了位错点的扰动应力、电位移和磁感应强度场.由推广的Peach-Koehler公式求出了作用在位错上的位错力,讨论了圆弧形刚性线几何条件和材料失配对位错力的影响规律.解答不但可作为格林函数获得任意分布位错的相应解答,而且可以用于研究无穷远纵向剪切和面内电磁场作用下界面刚性线夹杂和任意形状裂纹的磁电弹耦合干涉效应问题.     

Computation of green’s tensor and wave propagation in the random granular elastic medium

Abstract

A linear differential operator equation involving randomly variable field parameters, characterising the heterogeneous granular elastic medium is considered. The appropriate Green’s tensor is evaluated for the non-deterministic operator equations in the form of Fourier integrals in the frequency space; the exact evaluation is carried out to obtain the 36 components of Green’s tensor. The problem of wave propagation in the random granular elastic medium is then carried out with the help of the associated Green’s tensor. The effect of random variation of parameters on wave propagation in the granular elastic medium is examined. Dispersion equations have been analysed in details.     

A view of the relation between the continuum theory of lattice defects and non-euclidean geometry in the linear approximation

A view is presented of the relation between the continuum theory of defects in crystals and the mathematical theory of non-metric, non-Riemannian geometry. Both theories are treated in the linear approximation. The lattice defects consist of disclinations, dislocations, and extra-matter, which are identified with the following three important tensors from non-Euclidean geometry: the Riemann-Christoffel curvature tensor, the Cartan torsion tensor and the non-metric Q-tensor. The correspondence between the two theories is established by finding a relation between the coefficients of linear connection of non-Euclidean geometry and the elastic strain, bend-twist, and quasi-plastic strain of defect theory. The definitions of the important tensors from non-Euclidean geometry then generally correspond to the field equations of defect theory. The identities for the curvature tensor generally correspond to the continuity equations of defect theory. The relation to the conventional formulation of defect theory is pointed out. Two examples are given to illustrate the concepts of the paper. One example is related to the deformations associated with constant dislocation distribution and the other to the deformations of a constant disclination distribution.     

Integral Equation Method for Evaluation of Eddy-Current Impedance of a Rectangular,Near Surface Crack Inside a Cylindrical Hole

An integral equation method for solving the eddy-current nondestructive evaluation problem of a flat, rectangular, near surface crack inside of a cylindrical hole in a conducting material is presented. The method involves expanding the Green’s tensor, the incoming field, and the jump in electric potential over the crack in suitable basis functions. Here, plane waves, cylindrical waves, and basis functions related to the Chebyshev polynomials, are used. The way of discretization in this method leads to a formulation where the scattering is defined by a scattering matrix, independent of the incoming field. This presents an advantage, when conducting numerical simulations, since the scattering matrix does not have to be recalculated for every probe position. The numerical calculations are straightforward to perform and model predictions are compared with finite element results.