Continuous Distribution Of Dislocations Research Articles

Modelling cracks in arbitrarily shaped finite bodies by distribution of dislocation

This paper presents an analytical method based on the principle of continuous distribution of dislocation to model curved cracks in solids of arbitrarily shaped finite geometries. Both the boundary of the finite body and the curved crack are modelled by distributed dislocation. In this method the influence function of the dislocation along the finite body boundary is reduced to a product of the Hilbert kernel with a normal function. Similarly the influence function for the curved cracks is reduced to the product of Cauchy kernel and a normal function. This approach results in a system of singular integral equations. Using the order decreasing method, the system is reduced to normal integral equations, which are solved numerically. Stress intensity factors are evaluated for a well-known crack problem and two railhead crack problems with a view to assessing the capability of the developed method to solve complex engineering problems.

Mechanics of compressive stress evolution during thin film growth

Based on recent in situ measurements, Chason et al. (2002) proposed that the evolution of compressive stress during thin film growth by vapor deposition is due to an increase in surface chemical potential in the presence of growth flux and the consequent exchange of adatoms between the free surface and the grain boundaries. Based on this hypothesis, we present a model for grain boundary stress evolution during thin film growth. To illustrate the mechanics of the problem, first it is assumed that the local normal stress on the grain boundary is proportional to the local grain boundary opening only. The resulting “linear spring” model captures all essential features of the experimental observations. A more accurate description of the grain boundary stress evolution is presented by modeling the stress field due to the material inserted into the grain boundary as that resulting from a continuous distribution of dislocations along the grain boundary. The adatom flux between the grain boundary and the free surface is assumed to be proportional to the difference in chemical potential between the two. This model successfully explains a wide range of experimental observations, including the development of compressive stress during room temperature growth, effect of growth rate on the kinetics of compressive stress evolution and the continued tensile stress generation during low-temperature growth.

Significance of the deviations of the crack front into the plane perpendicular to the crack propagation direction – I. Crack-front dislocation generation

A static crack front deviates more and more from straight line in a solid as I. the number of dislocations generated from the crack front increases and/or II. the temperature increases. The significance of these deviations into the plane perpendicular to the crack propagation direction is the subject of the present study, which we divide into two parts. In this paper (Part I), the influence of dislocation generation on the shape of a static crack front and on the conditions for crack motion are investigated. We have considered an elastic-plastic crack model in which, due to dislocation generation during mode I loading, the initially straight crack front deviates in the sinusoidal form in a plane perpendicular to both the average crack plane and the direction of fracture propagation. No crack opening displacement is allowed. The dislocations generated form a plastic zone separated from the crack by a dislocation free zone. Both the crack and the plastic zone are described in terms of continuous distributions of dislocations that are sinusoidal and straight edges, respectively. Expressions for the dislocation distributions, the relative displacement of the faces of the crack, the number of dislocations in the plastic region, and the crack opening force G per unit length of the crack front are evaluated. The similarities with isolated cracks (planar and wavy) are emphasized. It is shown that the stress at the front of the sinusoidal crack is unbounded in the mean fracture plane but bounded outside. Consequently, only the crack front sites located on the average crack plane are possible sites for the initiation of crack motion. G differs from that of the planar crack by a geometrical factor that depends on a new parameter, the crack front inclination angle θ. This is an acute angle, measured in the plane perpendicular to the crack propagation direction, between the crack front and the average fracture plane. As θ increases with the number of dislocations generated, G decreases and is ultimately zero for a critical value θc=tan −1(1/sqrtν) where ν is the Poisson ratio. This is a new condition for crack arrest in solids. Applying the theory to a steel, it is found that this condition could be achieved under localized plastic yielding at crack tips.

직교이방성 판 내의 다중 곡선균열 해석

The interactions between curved cracks are examined in an orthotropic plate and the effects of rectilinear anisotropy on the stress intensity factors are analyzed. The finite element alternating method (FEAM) is used in this study to get the stress intensity factors for the multiple curved cracks. To obtain analytical solutions, which is necessary in FEAM, the curved cracks are modeled as continuous distributions of dislocations, and integral equations are formulated for unknown dislocation density functions to satisfy the given resultant forces on the crack surfaces. Several basic problems are solved to verify the accuracy and efficiency of the proposed method and it can be found that present results show good agreements with the previously published results.

Continuous Distributions of Dislocations in Bodies with Microstructure

Continuous Distributions of Dislocations in Bodies with Microstructure

Modelling cracks in finite bodies by distributed dislocation dipoles

The method of continuous distribution of dislocations is extended here to model cracks in finite geometries. The cracks themselves are still modelled by distributed dislocations, whereas the finite boundaries are represented by a continuous distribution of dislocation dipoles. The use of dislocation dipoles, instead of dislocations, provides a unified formulation to treat both simple and arbitrary boundaries in a numerical solution. The method gives a set of singular integral equations with Cauchy kernels, which can be readily solved using Gauss–Chebyshev quadratures for finite bodies of simple shapes. When applied to arbitrary geometries, the continuous distribution of infinitesimal dislocation dipoles is approximated by a discrete distribution of finite dislocation dipoles. Both the stress intensity factor and the T‐stress are evaluated for some well‐known crack problems, in an attempt to assess the performance of the methods and to provide some new engineering data.

Modelling the effects of texture on the statistics of stage I fatigue crack growth

Abstract A physically based model for the growth of short fatigue cracks through a local grain structure is presented. The growth rate is taken to be proportional to the cyclic plastic displacement at the crack tip, which is evaluated using an existing solution for the continuous distribution of dislocations ahead of the crack in a slip band that is blocked by a grain boundary. The effect of the local microstructure and texture are incorporated using orientation factors relating the applied far field stress to the shear stress which causes a grain to grain variation in the mode II loading of the crack tip. The orientation factor for each grain is established by assigning the crystallographic orientation of the grain using a probabilistic description of the overall texture of the polycrystal. The effects of idealized fibre (〈111〉 and 〈100〉), double fibre and random textures are examined. The growth rates of many individual cracks through different local microtextures were calculated within a Monte Carlo fr...

Modelling the effects of texture on the statistics of stage I fatigue crack growth

Abstract A physically based model for the growth of short fatigue cracks through a local grain structure is presented. The growth rate is taken to be proportional to the cyclic plastic displacement at the crack tip, which is evaluated using an existing solution for the continuous distribution of dislocations ahead of the crack in a slip band that is blocked by a grain boundary. The effect of the local microstructure and texture are incorporated using orientation factors relating the applied far field stress to the shear stress which causes a grain to grain variation in the mode II loading of the crack tip. The orientation factor for each grain is established by assigning the crystallographic orientation of the grain using a probabilistic description of the overall texture of the polycrystal. The effects of idealized fibre (〈111〉 and 〈100〉), double fibre and random textures are examined. The growth rates of many individual cracks through different local microtextures were calculated within a Monte Carlo framework. This allows the influence of texture on not only the average behaviour of the short cracks but also the statistical variation in the behaviour to be examined. Curves showing the probability of propagating cracks reaching a given crack length in a given number of cycles are shown.

On the dynamics of continuous distributions of dislocations

For materials with continuous distributions of dislocations a configuration space which unifies the continuum theory of defects with classical elasticity is given. Weak equations of motion are derived from the principle of virtual work. Using the Helmholtz decomposition theorem, this yields a coupled system of equations for the dynamics of dislocations and classical elasticity.

Congruences of Dislocations in Continuously Dislocated Crystals

The time-dependent congruences of Volterra-type dislocations are investigated based on the generalized formulas of Frenet in a Riemannian space. The analysis is applied to the description of congruences of edge and mixed dislocations consistent with a continuous distribution of dislocations for which its material space is an equidistant Riemannian space. In particular, the principal congruences of dislocations are considered. The kinematics of congruences of mixed dislocations endowed with univocally defined local slip planes is discussed. It is shown that the geometry of such congruences of dislocations admits a class of nonlinear evolution equations describing the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions imposed on the derived system of equations in order to describe the evolution of curvature and torsion of congruences of edge dislocations are proposed. In the static case, an expression is given for shear stresses required to bend prismatic edge dislocations of torsion zero located on the totally geodesic crystal surfaces. It follows that the congruence of these dislocations is endowed with a finite self-energy function.

A Zener–Stroh crack in a fiber-reinforced composite material

A Zener–Stroh crack is formed by coalescing a dislocation pileup and therefore is a net dislocation loaded crack. In this paper, the interaction between a Zener–Stroch crack and surrounding cylindrical fibers in a fiber-reinforced composite material has been investigated. The crack is located in the matrix and near a fiber, while the influence of other fibers on the crack is considered through the three-phase composite cylinder model. Using the solution of stress field for a single dislocation in a three-phase cylinder as the Green’s function, the crack is represented by a continuous distribution of dislocations. A set of singular integral equations are thus formulated and solved numerically. The stress intensity factor (SIF) on the crack tip due to crack–inclusions interaction is calculated. The results show that the volume concentration of fibers has non-negligible influence on the SIF of the crack.

The distributed dislocation method applied to the analysis of elastoplastic strain concentrations

Conventionally, the use of continuous distributions of dislocations to model plasticity has been confined to the analysis of crack tip plasticity using linear arrays of dislocations, within the framework of plane analysis. By expanding this technique into a distribution of dislocation over an area, a method is developed to model the plasticity at stress raising features such as notches or holes under plane strain conditions. The method explicitly takes account of the boundary conditions by using a dislocation solution which accounts for the presence of the stress-raise itself. Other free boundaries may be modelled more approximately using boundary elements which also correctly include the presence of the stress raiser. The dislocations are distributed over finite sized cells, and the solutions found for the strain fields compare favourably with both finite element and bounding Neuber and Glinka results.

On the Nonlinear Mechanics of Bravais Crystals with Continuous Distributions of Defects

Basic ideas in the mechanics of Bravais crystals with continuous distributions of point and line defects are revisited from a perspective that emphasizes the role of configurational forces. In addition, the creation, destruction, and transport of defects are presented in detail. The formulation of the balance laws is given over time-dependent spatial control volumes; furthermore, independent configurational and deformational force balances are postulated. Such a formulation is more versatile than one with standard fixed control volumes and provides the correct Eshelby relation from simple invariance arguments independent of constitutive relations. Two alternative but equivalent constitutive models are proposed, and the corresponding restrictions following from a dissipation inequality are determined. Relations to some previous formulations of continuous distributions of dislocations and vacancies are discussed.

A kinematic model for continuous distributions of dislocations

In continuum theory of defects the notion of a flat connection is employed. This paper gives a characterisation of these connections via injective R 3-valued differential forms. For material structures with continuous distributions of dislocations, a configuration space in the sense of global analysis is introduced and analysed. A kinematics for these dislocations is formulated which generalises from elasticity.

A Mesoscale Study of the Interaction between Grain Boundaries and the Deformation Field

The interaction between grain boundaries and the deformation field was investigated for high-purity aluminum bicrystals in terms of continuous distributions of dislocations observed at the mesoscale. At differing strain levels the influence of the boundary is observed to vary from that of a dislocation trap, with relatively high levels of geometrically necessary dislocations (GNDs) accumulating near the boundary, to that of a sink, where a reduction of GNDs was observed relative to the levels observed in the adjacent grains. Comparisons of these observations with results obtained by finite element simulation show that the sink-like behavior is poorly understood within the current framework of continuum crystal plasticity.

Dynamics of Martensitic Interfaces

The basic dynamic behavior of martensitic interfaces has been analyzed within the framework of lattice dislocation dynamics. Two limiting cases of the martensitic interface structure have been considered: (a) the case when the interface can be appropriately described in terms of an array of non-interacting (well-spaced) interfacial dislocations and; (b) the case when the interfacial dislocations are so closely spaced that the interface can be approximated by a continuous distribution of dislocations. In the first case, it was demonstrated that, after the inclusion of a "chemical" driving force in the equation of motion, the dynamics of lattice dislocations can be directly applied to analyze the interfacial dynamics. In the second case, on the other hand, while the lattice dislocation dynamics is still quite relevant, several parameters in the equation of motion have to be redefined to reflect the fact that the interface now acts as a planar defect. For both of the cases of interfacial dislocation structure, we have analyzed the two basic modes of interfacial motion: (a) the continuous mode in which the motion is controlled by various energy-dissipative processes (e.g., phonon and electron drag) and; (b) the discontinuous or jerky mode in which the motion is controlled by the thermal activation of the interface/obstacle interactions.

Stress Concentrations in an Intermingled Hybrid Composite

This paper studies the stress redistribution in a tensile hybrid composite sheet due to the breakage of a high modulus fiber. Employing a continuous distribution of dislocations, a set of singular integral equations is established to analyze the fiber crack impinging upon weakly bonded fiber-matrix interfaces. After solving the integral equations numerically, the stress concentration factors of both high modulus and low modulus fibers are evaluated as a function of loading stress and interfacial parameters. The results are compared with those obtained from shear-lag model solution.

Kinematics of edge dislocations. I. Involutive distributions of local slip planes

The geometry of continuous distributions of dislocations and secondary point defects created by these distributions is considered. Particularly, the dependence of a distribution of dislocations on the existence of secondary point defects is modeled by treating dislocations as those located in a time-dependent Riemannian material space describing, in a continuous limit, the influence of these point defects on metric properties of a crystal structure. The notions of local glide systems and involutive distributions of local slip planes are introduced in order to describe, in terms of differential geometry, some aspects of the kinematics of the motion of edge dislocations. The analysis leads, among others, to the definition of a class of distributions of dislocations with a distinguished involutive distribution of local slip planes and such that a formula of mesoscale character describing the influence of edge dislocations on the mean curvature of glide surfaces is valid.

Force between two parallel screw dislocations and application to linear screw dislocation pileups—Gauge theory results

An analytic expression for the force between two parallel screw dislocations, derived earlier on the basis of the gauge theory of dislocations, has been used to investigate the static distribution of a given numberN of parallel screw dislocations confined between two immobile dislocation obstacles. It is shown that in the limit of a continuous distribution of dislocations the equilibrium condition leads to a Fredholm integral equation of first type which does not admit any nontrivial solution. Implication of this result is discussed. For a finite number of dislocations, the ratio (η) of the obstacle separation to the core radius is an important parameter governing the nature of solution of the discrete equation. It is found that for a givenN, there is a critical valueη c below which there does not exist any solution.

On the isotropy of continuized dislocated crystals. I. The isotropic lattice distortion

The geometrical theory of continuous distributions of dislocations traditionally neglects the dependence of a distribution of dislocations on the existence of point defects created by this distribution (e.g., due to intersections of dislocation lines). In this paper the influence of such point defects on metric properties of the continuized dislocated Bravais crystalline structure is assumed to be isotropic. The influence of the point defects on the distribution of dislocations is then modeled by treating dislocations as those located in a conformally flat space. This approach leads (among others) to new results concerning the geometry of glide surfaces.