Continuous Distribution Of Dislocations Research Articles

On the mechanics of crystalline solids with a continuous distribution of dislocations

We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We then consider the static case, which describes crystalline bodies in equilibrium in free space. The mathematical problem in this case is the free minimization of an energy integral, and the associated Euler-Lagrange equations constitute a nonlinear elliptic system of partial differential equations. We solve the problem in the simplest cases of interest.

Continuum model for dislocation dynamics in a slip plane

In this paper, we present a continuum model for dislocation dynamics in a slip plane, which accurately incorporates both the long-range interaction and the local line tension effect of dislocations. Unlike the continuum models in the literature using dislocation densities, we use the disregistry across the slip plane to represent the continuous distribution of dislocations in the slip plane, which has the advantage of including the orientation dependence of dislocations in a very simple way. The continuum dislocation dynamics model is validated by linear instability analysis of a uniform dislocation array to small perturbations and comparisons of the results with those of the discrete dislocation dynamics model. Numerical examples for the evolution of distributions of dislocations and plastic slips in a slip plane are presented.

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.

The influence of dislocation distribution density on curvature and interface stress in epitaxial thin films on a flexible substrate

We consider the problem of relaxation of coherency stresses by lattice misfit dislocations, which can be represented as edge dislocations uniformly distributed along the interface, and can be associated with the concept of dislocation density according to the mathematical theory of continuous distributions of dislocations. The orientation of dislocation line density is constrained within the plane of epitaxial layer and the Burgers vector density within the same plane is considered as a local variable of the problem. Under the external stress-free conditions the system is then enforced to satisfy the boundary conditions according to the geometry of the epitaxial film and the substrate together with the corresponding compatibility conditions. The dislocation density tensor is then connected with the deformation and rotation tensors and subsequently with the curvature of the system. The closed form solution of the problem is given for a generic one-dimensional case.

Calculation of the stress fields in a polysynthetic twin located near the crystal surface

Without using the thin twin approximation, assuming a continuous distribution of dislocations on the boundaries, a procedure to calculate the stress fields in a polysynthetic twin located near the crystal surface has been developed. It is shown that the stresses in the polysynthetic twin are localized at the boundaries and apices of the twins that enter into the composition of the former twin. Examples of calculations of the cleavage stresses of the polysynthetic twin having rectilinear and curvilinear boundaries with uniform and nonuniform distribution of dislocations at them are given.

The Analysis of the Stress and Displacement Field near the Surface Crack Tip Terminating Perpendicular to the Interface between Two Orthotropic Materials

The evaluation of the generalized stress intensity factor (GSIF) and T-stress for the case of the surface crack terminating perpendicular to the interface between two orthotropic materials is considered. The combination of the discretization, numerical and analytic methods is used. The discretization method, such as common finite element method (FEM), is served to include the boundary condition to the GSIF solution and to describe the remote stress and displacement field region with the low influence of the singularity of the crack tip. The Lekhnickii-Eshelby-Stroh (LES) formalism is used to derive the approach solution for the near stress field of the crack tip and the singularity problem in an orthotropic 'trimaterial' using the Schwartz-Neumann's alternating technique. The problem of the stress singularity is treated as a non-linear eigenvalue problem, which leads to the characteristic equation for the stress singularities of the form rδ −1 , 0 <δ <1. Two ways of the evaluation of the GSIF are presented, using the reciprocal theorem ψ -integral) and the crack model by means of continuous distribution of dislocations. Both results are compared for a specific material. The continuous distribution of dislocations technique is also used for determination of the T-stress.

Hybrid Green's functions for multiple cracks: Numerical extension of single crack analytical Green's functions to multiple cracks

The Green's functions for the line force and dislocation that satisfy the traction free boundary condition on the surfaces of arbitrary multiple straight cracks in the isotropic solids are obtained. We develop the hybrid Green's functions combining the analytical and numerical Green's function methods. The Green's function is split into the singular and the image terms. The crack opening displacement, represented as the continuous distribution of dislocation dipoles over each crack, serves as the source of the image term. It is proposed that the image term from each crack is further split into two parts: analytical image term for the individual single crack and the additional numerical image term caused by the presence of other cracks. Although, for the single crack problem, the analytical image term is the only image term needed to satisfy the traction free boundary condition, additional image term is needed to satisfy the boundary condition for multiple cracks. The advantage of the analytical image term is its ability to absorb high stress gradient when the singularity is located near one of the cracks. The additional image term, given in terms of the additional dislocation dipole distribution, needs only to provide small and smooth perturbation, due to the presence of other cracks, to satisfy the traction free boundary condition.

Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part II. Application to In xGa 1− xAs/InP system

Cracking phenomena in tensile-strained In x Ga 1− x As epitaxial film on an InP substrate are analyzed via the formulation given in Part I [Lee, S., Choi, S.T., Earmme, Y.Y., 2006. Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part I. Mathematical formulation. International Journal of Solids and Structures 43, 3401–3413], where the solution for a dislocation in an anisotropic trimaterial is used as a fundamental solution and the crack is modeled by the continuous distribution of dislocations. Misfit strains and stresses are evaluated as a function of indium content x in an In x Ga 1− x As/InP system. A single crack and periodic cracks, respectively, induced by the misfit stresses are considered. The crack opening profile, the crack mouth displacement, and the energy release rate as a function of the crack length are obtained. The critical conditions for a single crack and periodic cracks, respectively, are thus obtained, and are found to depend on the film thickness, the crack length, and the period of the cracks. The results of these analyses are also compared with published data obtained from experiments.

Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial half-space

The problem of an edge-bridged crack terminating perpendicular to a bimaterial interface in a half-space is analyzed for a general case of elastic anisotropic bimaterials and specialized for the case of orthotropic bimaterials. The edge crack lies in the surface layer of thickness h bonded to semi-infinite substrate. It is assumed that long fibres bridge the crack. Bridging model follows from the assumption of “large” slip lengths adjacent to the crack faces and neglect of initial stresses. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. With respect to the bridged crack problems in finite dissimilar bodies, the reciprocal theorem (Ψ-integral) is demonstrated as to compute, in the present context, the generalized stress intensity factor through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. Also the application of the configurational force mechanics is discussed within the context of the investigated problem.

Calculation of K-factor and T-stress for cracks in anisotropic bimaterials

The problem of a crack in a thin layer terminating perpendicular to a layer/substrate interface is analyzed for a general case of elastic anisotropy. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. A system of simultaneous functional equations is obtained that enables to find the singularity exponent λ. The reciprocal theorem ( ψ-integral) is used to compute the generalized stress intensity factor (GSIF) through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. The results obtained are compared with the evaluation of GSIF based upon the dislocation arrays technique. Existing semi-analytical solution for singularities in anisotropic trimaterials is applied and its validity for the specimen investigated is checked by FEM. The evaluation of T-stress using the dislocation arrays technique is performed.

Incompatible deformation in rock mechanics

Compatibility, i.e., continuous displacement field, is often taken for granted but experience teaches that rock deformation is in most cases discontinuous. The relation between incompatible deformation and continuous distribution of dislocations is established since long and is recalled in this paper. Contrary to several applications of Cosserat theory to already laminated (jointed) rock, the main goal of this paper is to give an explanation for the formation of discontinuities (joints) in rock as a result of a particular type of buckling which leads the body outside of the Euclidean space of its original configuration.

Analysis of Edge Bridged Crack in Bimaterial Anisotropic Half-Space

The problem of an edge-bridged crack terminating perpendicular to a bimaterial interface in a half- space is analyzed for a general case of elastic anisotropy bimaterials and specialized for the case of orthotropic bimaterials. The edge crack lies in the surface layer of thickness h bonded to semi-infinite substrate. It is assumed that long fibres bridge the crack. Bridging model follows from the assumption of “large” slip lengths adjacent to the crack faces and neglect of initial stresses. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. With respect to the bridged crack problems in finite dissimilar bodies, the reciprocal theorem (ψ - integral) is discussed as to compute, in the present context, the generalized stress intensity factor through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM.

Variational arguments and Noether’s theorem on the nonlinear continuum theory of dislocations

A formulation concerning the hyperelastic simple bodies with a continuous distribution of dislocations at finite strain is proposed. The presence of dislocations within the body renders the stored energy function non-homogeneous. Based on the notion of uniform reference, as developed by Noll, the stored energy density per unit uniform reference configuration is introduced. The main property of the latter is that it receives all the explicit dependence of the standard stored energy function on the material variables. Variational formulations for the direct and inverse deformation descriptions are established and it is shown that the Euler–Lagrange equation for the direct deformation description provides the standard balance of physical forces as in the classical case. Moreover, the lack of invariance under translations in material space results in a non-conservation law describing the balance of material forces. The additional source term appearing in it, i.e., an additional material force, is due to the presence of dislocations. Thus, one may conclude that the force acting on dislocations is a material one. Repeating the aforementioned procedure for the inverse deformation description, one obtains again the same two equations in a different order, that is, the balance of material forces is derived from the Euler–Lagrange equation, whereas the balance of physical forces is obtained from the invariance of the energy functional under translations in the physical space.

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional, anisotropic, elastic, and piezoelectric media. II. Infinite straight dislocation and Frank disclination

The fields of stress and electric displacement caused by infinitely extended straight dislocations and Frank disclinations are deduced from the author's statements for the fields caused by a continuous distribution of dislocations and disclinations (S. Minagawa, Phil. Mag. 84 2229 (2004)). The multiple integrals in the original statements are converted into functions of space coordinates. Cauchy's theorem plays an important part. The improper integral that appears in computations of the fields around a Frank disclination is interpreted as its finite part by Hadamard. Examples are the fields around an infinite straight defect in caesium copper chloride, as well as those in gallium arsenide. The contours and zero lines are plotted to illustrate the fields caused by a dislocation and a disclination dipole.

Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part I. Mathematical formulation

This paper presents an analysis of a single vertical crack and periodically distributed vertical cracks in an epitaxial film on a semi-infinite substrate where the cracks penetrate into the substrate. The film and substrate materials have different anisotropic elastic constants, necessitating Stroh formalism in the analysis. The misfit strain due to the lattice mismatch between the film and the substrate serves as the driving force for crack formation. The solution for a dislocation in an anisotropic trimaterial is used as a Green function, so that the cracks are modeled as the continuous distributions of dislocations to yield the singular integral equations of Cauchy-type. The Gauss–Chebyshev quadrature formula is adopted to solve the singular integral equations numerically. Energy arguments provide the critical condition for crack formation, at which the cracks are energetically favorable configurations, in terms of the ratio of the penetration depth into the substrate to the film thickness, the ratio of the spacing of the periodic cracks to the film thickness, and the generalized Dundurs parameters between the film and substrate materials.

A solution to the thermo-elastic interface crack branching in dissimilar anisotropic bi-material media

An interface crack or delamination may often branch out of the interface in a laminated composite due to thermal stresses developing around the delamination/crack tip when the media is exposed to heat flow induced by environmental events such as a sudden short-duration fire. In this paper, the thermo-elastic problem of interface crack branching in dissimilar anisotropic bi-media is studied by using the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. Based on the complex variable method and the analytical continuation principle, the thermo-elastic interface crack/delamination problem is examined and a general solution in compact form is derived for dissimilar anisotropic bi-media. A set of Green’s functions is proposed for the dislocations (conventional dislocation and thermal dislocation/heat vortex) in anisotropic bi-media. These functions may be more suitable than those which have appeared in the literature on addressing thermo-elastic interface crack branching in dissimilar anisotropic bi-materials. Using the contour integral method, a closed form solution to the interaction between the dislocations and the interface crack is obtained. Within the scope of linear fracture mechanics, the thermo-elastic problem of interface crack branching is then solved by modelling the branched portion as a continuous distribution of dislocations. The influence of thermal loading and thermal properties on the branching behavior is examined, and criteria for predicting interface crack branching are suggested, based on the extensive numerical results from the study of various cases.

Renormalized Energy and Forces on Dislocations

In this work we discuss, from a variational viewpoint, the equilibrium problem for a finite number of Volterra dislocations in a plane domain. For a given set of singularities at fixed locations, we characterize elastic equilibrium as the limit of the minimizers of a family of energy functionals, obtained by a finite-core regularization of the elastic-energy functional. We give a sharpasymptotic estimate of the minimum energy as the core radius tends to zero, which allows one to eliminate this internal length scale from the problem. The energy content of a set of dislocations is fully characterized by the regular part of the asymptotic expansion, the so-called renormalized energy, which contains all information regarding self- and mutual interactions between the defects. Thus our result may be considered as the analogue for dislocations of the classical result of Bethuel, Brezis and Hélein for Ginzburg--Landau vortices. We view the renormalized energy as the basic tool for the study of the discrete-to-continuum limit in plasticity of crystals, i.e., the passage from models of isolated defects to theories of continuous distributions of dislocations. The renormalized energy is a function of the defect positions only: we prove that its derivative with respect to the position of a given dislocation is the resultant of the Eshelby stress on that dislocation, which can be identified in turn with the classical Peach--Köhler force.

A method for tracking internal crack propagation in railhead

This paper presents a method for tracking two-dimensional propagation of internal cracks, in particular vertical split head (VSH) defects, due to contact loading in railhead. Generalised curved crack is assumed to have present in the railhead and its propagation is simulated by solving interaction of arbitrary shaped cracks successively. Furthermore, the finite shape of the rail section is also modelled as the continuous distribution of dislocation within an infinite plane. Crack propagation is simulated using the criterion of vanishing of Mode II stress intensity factors (SIF) at crack tips. Examples are provided for tracking the propagation of pre-existing internal cracks in railheads turning into VSH defects under centric contact loading.

Thermo-elastic crack branching in general anisotropic media

Thermal fields may exist in addition to mechanical loading, for example, due to short term exposure to fire. In this paper, the branching of cracks in the presence of combined thermal and mechanical loads is investigated for general anisotropic media by employing the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. A general solution to the thermo-elastic crack problem for an anisotropic material under arbitrary loading is obtained in a compact form. Green’s functions are also presented for a thermal dislocation (heat vortex) and a conventional dislocation (or, referred as mechanical dislocation), which are formulated considering the cuts located at an arbitrary angle with respect to the x 1 axis of the coordinate system ( x 1, x 2, x 3). Using the derived compact expressions, the interaction between the crack and the dislocation is studied and a closed form solution for this interaction is obtained. The branching portion of the thermo-elastic crack is modelled as a continuous distribution of dislocations. This problem is then converted into a set of singular integral equations. Numerical results are presented to illustrate the possible effects of thermal loading on the propagation of the branched crack.

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional, anisotropic, elastic and piezoelectric media: elliptical Frank disclination

A statement is made on the theory of continuous distributions of dislocations and disclinations in anisotropic elastopiezoelectric media. The basic field equations governing the fields of stress functions, electric vector potential and incompatibility are presented and solved to give the fields of stress and electric displacement caused by a distribution of dislocations and disclinations. They are expressed in terms of the dislocation- and disclination-density tensors by means of the convolution integrals, extended throughout the medium, and the Fourier integrals. To treat the fields around discrete defects, that is dislocation and/or Frank disclination, the convolution integrals are replaced by the line integrals belonging to the loop of the defect. The fields of stress and electric displacement are given in terms of three quadruple integrals, which are converted into single integrals of explicitly given functions, in the case where the loop of the defect is elliptical. Numerical computations are carried out to estimate the fields in gallium arsenide. The values of those fields at a certain point of the body are presented. The contours and zero lines of the fields of dilatational stress and electric displacement in the plane placed parallel to and at a certain distance from the loop are illustrated.