Thermal stresses near tips of an insulated line crack in a semi-infinite medium under uniform heat flow

On the basis of the two-dimensional theory of thermoelasticity, the thermal stress field near the tips of a thermally insulated line crack in a semi-infinite medium under steady-state uniform heat flow is discussed. The crack is replaced by continuous distributions of sources of temperature discontinuity and edge dislocations. Then we obtain a set of simultaneous singular integral equations for density functions. The solution is given in the form of the product of the series of Tchebycheff polynomials of the first kind and their weight function. 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. The effects of the distance from the crack tip to the bounding plane surface of the semi-infinite medium and the angle of inclination of the line crack on the stress intensity factors are shown graphically.     

Edge dislocations near a cracked sliding interface

The edge dislocations near a cracked sliding interface were investigated. A continuous distribution of edge dislocations with Burgers vector along the y direction was used to simulate a crack of finite length along the sliding interface. From the dislocation distribution the stress field in the entire space was obtained. The stress intensity factors at both crack tips and image force on the edge dislocation were derived. The effects of the dislocation source and shear modulus ratio on both stress intensity factors and image force were also studied. Only mode I stress intensity factors at both tips were found in the composite materials with a sliding interface. The edge dislocations with Burgers vector along the y direction emitted from the crack always shield it to prevent propagation. The above results may reduce to an edge dislocation near a semi-infinite crack along a sliding interface including a sliding grain boundary. This revised version was published online in August 2006 with corrections to the Cover Date.     

An edge dislocation of constant velocity near a static internal crack

An edge dislocation of constant velocity near a static internal crack was investigated. The dislocation slip and climb and dislocation source were considered. The crack surface was simulated with static continuous dislocations. After obtaining the distribution of static dislocations in the crack, we calculated the stress field in the entire space. Using the stress distribution, we then computed the stress intensity factors at both crack tips and the image force on the edge dislocation. Numerical results are provided to describe in detail the effect of velocity and crack length on toughness and image force.     

Interaction between a cracked hole and a line crack under uniform heat flux

This article deals with the interaction between a cracked hole and a line crack under uniform heat flux. Using the principle of superposition, the original problem is converted into three particular cracked hole problems: the first one is the problem of the hole with an edge crack under uniform heat flux, the second and third ones are the problems of the hole under distributed temperature and edge dislocations, respectively, along the line crack surface. Singular integral equations satisfying adiabatic and traction free conditions on the crack surface are obtained for the solution of the second and third problems. The solution of the first problem, as well as the fundamental solutions of the second and third, is obtained by the complex variable method along with the rational mapping function approach. Stress intensity factors (SIFs) at all three crack tips are calculated. Interestingly, the results show that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip. The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension.     

Influence of an insulated circular hole on thermal stress singularities at tips of a crack

On the basis of the steady-state two-dimensional theory of thermoelasticity, the influence of an insulated circular hole on the thermal stress singularities at the tips of a line crack, whose surface is cooled, is discussed. The crack is located arbitrarily in an infinite matrix. The method of continuous distribution of dislocations is extended by newly introducing a continuous distribution of quasi-Volterra dislocations corresponding to line heat sources. Replacing the crack by the continuous distributions of quasi-Volterra dislocations and edge dislocations, we obtain a set of simultaneous singular integral equations for dislocation density functions. 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. The variations of the stress intensity factors and the initial direction of crack extension with the distance of the crack from the insulated circular hole and the angle of inclination of the crack are shown graphically.

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Résumé Sur la base d'une théorie bidimensionnelle en état statique de la thermoélasticité, on discute l'influence d'un trou circulaire sur les singularités des contraintes thermiques aux extrémités d'une fissure linéaire dont la surface est refroidie. La fissure est localisée arbitrairement dans une matrice infinie. La méthode de distribution continue des dislocations est généralisée par l'introduction d'une distribution continue de dislocations du type quasi Volterra correspondant à des sources de chaleur en ligne. En remplaçant la fissure par des distributions continues de dislocations du type quasi Volterra et par des dislocations-coin, on obtient une série d'intégrales singulières simultanées correspondant aux fonctions de densité de dislocations. Par cette méthode, les singularités de contraintes thermiques aux extrémités d'une fissure sont estimées de façon exacte et les facteurs d'intensité de contraintes peuvent être évaluées aisément. Les variations des facteurs d'intensité des contraintes ainsi que la direction initiale de l'extension de la fissure suivant la distance de la fissure à partir d'un trou circulaire isolé et l'angle d'inclinaison de la fissure sont exprimés graphiquement.

     

TEM observations of dislocation emission at crack tips in aluminium

Direct observations were made of the propagation of ductile cracks and associated dislocation behaviour at crack tips in aluminium during tensile deformation in an electron microscope. In the electropolished area, the cracks propagated as a Mode III shear-type by emitting screw dislocations on a plane coplanar to the crack plane. A zone free of dislocations was observed between the crack tip and the plastic zone. As the cracks propagated into thicker areas, the fracture mode changed from Mode III to predominantly Mode I. The crack top of the Mode I cracks was blunted by emitting edge dislocations on planes inclined to the crack plane. The blunted cracks did not propagate until the area ahead of the crack tip was sufficiently thinned by plastic deformation. The cracks then propagated abruptly, apparently without emitting dislocations. The stress intensity factor was measured from the crack tip geometry of Mode III cracks and it was found to be in good agreement with the critical value of the stress intensity factor required for dislocation generation.     

Fundamental solutions for half plane with an oblique edge crack

An elastic half plane with an oblique edge crack is considered in this paper. A pair of concentrated forces or point dislocations is assumed to act at an arbitrary point in the half plane. The half plane with an edge crack is first mapped into a unit circle by a rational mapping function so that the following analysis can be carried out on the mapped plane analytically. Then the complex stress functions are derived by separating the whole problem into two parts; one is the principal part corresponding to the infinite plane acted on by concentrated forces or dislocations, the other is the holomorphic part, which can be determined by making use of the property of regularity of complex stress functions. The stress intensity factors of the crack can be calculated with different inclined angles of the crack, and the displacement and stress components at an arbitrary position in the half plane can be expressed explicitly.     

Fracture of coated plates and shells under thermal shock

The main interest in this study is in the subcritical crack propagation and fracture of coated materials, specifically of cylindrical shells under repeated thermal shock. First it is shown that the circumferential crack problem in a cylindrical shell may be approximated by a plate on an elastic foundation under plane strain conditions. The thermal shock problem for a layered plate supported by an elastic foundation containing a crack in each layer of arbitrary sizes and locations is then considered. An additional factor studied is the influence of the cooling rate of the plate surface on the stress intensity factors at the crack tips. The problem is formulated in terms of a pair of singular integral equations which are solved for a number of typical crack geometries such as an edge crack, a crack terminating at the interface, an undercoat crack, and a crack crossing the interface. The main results of this paper are the stress intensity factors.     

Dislocation modeling of quasi-static crack propagation in an elasto-plastic medium

A dislocation model for simulating two-dimensional quasi-static crack propagation is presented. The crack and plastic flow along slip planes are described using dislocation dipoles. A stationary crack can be modeled as well as a propagating crack along a straight line inclined at an arbitrary angle to a free surface of a semi-infinite medium. Cracks are also allowed to kink. A superdipole algorithm is introduced to save simulation time without loosing important information and necessary geometric details. It reduces the number of dislocation dipoles on slip planes in the plastic wake. The paper gives results on crack shapes for stationary and advancing cracks as well as it describes how the size of the plastic zone depends on crack inclination angles. Results on stress intensity factors (SIF) are given using two different approaches as well as kinking cracks are introduced and SIF at kinked crack tips are calculated.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

FRACTURE MECHANICS ASSESSMENT OF PLANAR BRANCHED CRACKS IN AN EQUIBIAXIAL STRESS FIELD

Abstract A simplified fracture mechanics assessment is presented of branched planar cracks in an equibiaxial stress state. In linear-elastic fracture mechanics the stress intensity factors which characterize the load at the crack tips depend, for a given external load, only on the crack geometry. The stress intensity factors of a large number of branched cracks were evaluated using the Boundary Element method, and correlations between the stress intensity factors and the crack geometry were investigated. Formulae are presented which assign an individual effective crack length to each crack tip of a branched crack and hence allow approximate stress intensity factors to be determined for very complicated crack geometries. An algorithm is used for the stochastic simulation of an irregular crack pattern formation in thermal fatigue.     

Automatic 3-D crack propagation calculations: a pure hexahedral element approach versus a combined element approach

This article presents an evaluation of two different crack prediction approaches based on a comparison of the stress intensity factor distribution for three example problems. A single edge notch specimen and a quarter circular corner crack specimen subjected to shear displacements and a three point bend specimen with a crack inclined to the mid-plane are examined. The stress intensity factors are determined from the singular stress field close to the crack front. Two different fracture criteria are adopted for the calculation of an equivalent stress intensity factor and crack deflection angle. The stress intensity factor distributions for both numerical methods agree well to available reference solutions. Deviations are recorded at crack front locations near the free surface probably due to global contraction effects and the twisting behaviour of the crack front. Crack propagation calculations for the three point bending specimen give results that satisfy intuitive expectations. The outcome of the study encourages further pursuit of a crack propagation tool based on a combination of elements.     

USE OF THE DISTRIBUTED DISLOCATIONS METHOD TO DETERMINE THE T-STRESS

Abstract— This paper demonstrates a method to determine the elastic T -stress for a semi-infinite half-plane containing a surface-breaking crack which is loaded by an arbitrarily distributed far-field tension. The method consists of representing the crack by a continuous distribution of edge dislocations and forming singular integral equations to determine the equilibrium dislocation distributions. By numerically solving the integral equations, stress intensity factors and T -stresses are obtained for the example case of a crack which is normal and inclined to the free-surface of a half-plane and loaded by a uniform far-field tension.     

Parallel crack near the interface of magnetoelectroelastic bimaterials

A parallel crack near the interface of magnetoelectroelastic bimaterials is considered. The crack is modelled by using the continuously distributed edge dislocations, which are described by the density functions defined on the crack line. With the aid of the fundamental solution for the edge dislocation, the present problem is reduced to a system of singular integral equations, which can be numerically solved by using the Chebyshev numerical integration technique. Then, the stress intensity factor (SIF), the magnetic induction intensity factor (MIIF) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Using these fracture criteria, the cracking behaviour of magnetoelectroelastic bimaterials is investigated. Numerical examples demonstrate that the interface, mechanical load, magnetic load and material mismatch condition are all important factors affecting the fracture toughness of the magnetoelectroelastic bimaterials.     

Determination of thermal stress intensity factors for an interface crack under vertical uniform heat flow

In the case where an interface crack exists in an infinite two-dimensional elastic bimaterial, the crack surface is insulated under traction-free conditions and the uniform heat flow vertical to the crack from an infinite boundary is given, temperature and stress potentials are obtained by using the complex variable approach to solve Hubert problems, and the results are used to obtain thermal stress intensity factors. The mode II thermal stress intensity factor only occurs if both the shear moduli, as well as the Poisson's ratios in the upper and lower material, are the same. Otherwise, mode I and II thermal stress intensity factors exist but the value of the mode I thermal stress intensity factor is much smaller than that of mode II.     

Screw dislocations interacting with a partially debonded interface in cylindrically anisotropic composites

Interaction between screw dislocations and a partially debonded interface in cylindrically anisotropic composites subjected to uniform stress at infinity is investigated in this paper. Using Muskhelishvili’s complex variable method, the closed forms of complex potentials are obtained for a screw dislocation and a screw dislocation dipole located inside either matrix or inhomogeneity. Explicit expressions of stress intensity factors at the crack tips, image forces and image torques acting on dislocation or the center of dipole are provided. The results show that the crack and dislocation geometry combination plays an important role in the interaction between screw dislocations and interface crack. Furthermore, it is found that the anisotropy of solids may change the shielding and anti-shielding effects arising from screw dislocations and the equilibrium position of screw dislocations. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some novel or previously known results.     

Stress intensity factors in a cracked infinite elastic wedge loaded by a rigid punch

The paper considers splitting a plane elastic wedge-shaped solid through the application of a rigid punch. It is assumed that the coefficient of friction on the contact area is constant, the problem has a plane of symmetry with respect to loading and geometry, and the crack lies in the plane of symmetry. The problem is formulated in terms of a system of integral equations with the contact stress and the derivative of the crack surface displacement as the unknown functions. The solution is obtained for an internal crack and for an edge crack. The results include primarily the stress intensity factors at the crack tips, and the measure of the stress singularity at the wedge apex, and at the end points of the contact area.     

Calculation of the stress intensity factor for arbitrary finite cracked body by using the boundary weight function method

The state of thermal stresses for a periodic two-layered elastic space weakened by an interface thermally insulated Griffith crack and loaded by concentrated line heat sources is investigated. The analysis is performed within the framework of the homogenized model with microlocal parameters. The stress intensity factors at the crack tips are determined. Two examples of the application of the results obtained are detailed.     

Stress investigation on a Griffith crack initiated from an eccentric disclination in a cylinder

Disclinations are rotational line defects which may be introduced in metal wires during the manufacturing process. In this work, the relaxation of an eccentric (off-center) negative wedge disclination in a cylinder by nucleation of a Griffith crack is investigated. The nucleated crack is simulated with distributed edge dislocations. The stress intensity factors (SIFs) of the crack are evaluated by solving a set of singular integral equations. By enforcing the condition that the SIFs at the two crack tips should keep the same value in the nucleation process, the crack length growth on each side of the wedge disclination is determined. The critical disclination power and equilibrium crack lengths are then numerically determined. Some important characteristics of the Griffith crack nucleation are revealed. (1) The two tips of the nucleated Griffith crack grow asymmetrically when the disclination locates eccentrically. The tip closer to the cylinder edge travels a shorter length. This asymmetry is getting more severe as the normalized off-center distance increases. (2) The critical disclination power increases monotonically with the normalized off-center distance. (3) The normalized stable equilibrium crack length decreases as the normalized off-center distance increases while the normalized unstable equilibrium crack length shows an opposite dependence. The dependence of the critical disclination power and the equilibrium crack lengths on the disclination power and cylinder radius is also discussed in this work. It is believed that this work helps to predict the strength of disclinated metal wires at various length scales.