Transient analyses of dislocation emission in the three modes of fracture

Exact transient solutions for a micromechanical process of dislocation emission from stationary cracks in each of the three Modes of fracture are assembled. Each process involves a possible slip mechanism for the Mode, and the loading in each case is due to plane wave diffraction.

The solutions are examined in view of an emissions criterion similar to one developed for quasi-static studies on the basis of standard dislocation force concepts. Among the results are formulas for three parameters: emission time, distance traveled by an emitted dislocation prior to arrest, and the time of arrest. The formulas involve material properties such as yield stress and surface energy, as well as a dimensionless variable parameter governing the size of a zone of dislocation attraction at the crack edge.

The parameters themselves exhibit dislocation speed-dependent dynamic effects, but, on the other hand, the Mode I and Mode III cases achieve minimum emission times for the zero-speed limit. The effect of crack blunting serves to raise the emission time required. Order-of-magnitude estimates of the parameters indicate that they are definitely on a micromechanical scale, suggesting that emission may depend only on the region very near the crack edge.     

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.     

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.     

Comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane

A comparison of elastic interaction of a dislocation and a crack for four bonding conditions of the crack plane was made. Four cases of single crystalline material, sliding grain boundary, perfectly bonded interface, and sliding interface were considered. The stress intensity factors arising from edge and screw dislocations and their image forces for the above four cases were compared. The stress intensity factor at a crack tip along the perfectly bonded interface arising from screw dislocation can be obtained from that in a single crystalline material if the shear modulus in the single crystalline material is replaced by the harmonic mean of both shear moduli in the bimaterial. The stress intensity factor at a crack tip along the sliding interface arising from edge dislocation in the bimaterial can be obtained from that along the sliding grain boundary in the single material if the μ/(1−ν) in the single material is substituted by the harmonic mean of μ/(1− ν) in the bimaterial where μ and ν are the shear modulus and Poisson's ratio, respectively. The solutions of screw dislocation near a crack along the sliding grain boundary and sliding interface are the same as that of screw dislocation and its mirror image. Generally, the effect of edge dislocation for perfectly bonded interface on the crack propagation is more pronounced than that for the sliding interface. The effect of edge dislocation on the crack propagation is mixed mode for the cases of perfectly bonded interface and single crystalline material, but mode I fracture for the cases of sliding interface and sliding grain boundary. All curves of Fx versus distance r from the dislocation at interface to the right-hand crack tip are similar to one another regardless of dislocation source for both sliding interface and perfectly bonded interface. The level of Fx for m=0 is larger than that for m=−1. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Elastic analysis of an edge dislocation and two collinear cracks of different length

The elastic interaction between an edge dislocation and two collinear internal cracks of different length has been investigated. The effect of the distance between two collinear cracks on the crack shielding and image force on the edge dislocation were examined. The effect of the length of the right-hand-side crack on the shielding of the left-hand-side crack and the image force of the dislocation were also considered. The dislocations in the crack play an important role in fracture. Three conditions consisting of an edge dislocation emitting from the right-hand-side crack, originating elsewhere, and emitting from the left-hand-side crack are discussed. We compared the mechanical behavior between edge and screw dislocations near two collinear cracks. Newton's third law is satisfied in this system. Three special cases are discussed.     

Shielding and amplification of a penny-shape crack due to the presence of dislocations

The interaction between a penny-shape crack and a dislocation in crystalline materials is investigated within the framework of dislocation dynamics. The long-range and singular stress field resulting from the crack is determined by modeling the crack as continuous distribution of dislocation loops. This distribution is determined by satisfying the traction boundary condition at the crack face, resulting into a singular integral equation of the first kind that is solved numerically. This crack model is integrated with the dislocation dynamics simulation technique to yield the stress field of the combine system of crack and different types of dislocations situated at different positions in a three dimensional space. The integrated system is then used to investigate the dislocation behavior and its influence on the crack opening displacement and the characteristic of the stress field near the crack tip. It is shown that, depending on the relative position of the dislocation and its character, the dislocation may result in reduction in the stress amplitude at the crack tip and in some cases in closure of the crack tip. These analyses yield shielding and amplification zones near the crack providing an insight of the dislocation influence on the crack. The full dislocation dynamic analysis reveals the nature of the crack dislocation interaction and the manner in which the dislocation morphology changes as it is attracted to the crack surfaces, as well as the changes it causes to the crack profile.     

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 θ_c=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.     

Shielding effects and residual stresses at cleavage due to pre-existing dislocations

The motion of pre-existing edge dislocations in an infinite linear elastic body is studied at initiation of crack growth and at quasi-static steady-state crack growth. Dislocation nucleation is assumed not to occur. Thus, the study concerns only dislocations that are present in the virgin material. A dislocation is assumed to glide if its driving force exceeds a critical value. Changes in dislocation density, crack tip shielding and residual stresses are obtained. The shielding of a stationary crack tip is found to be small compared with the shielding of a growing crack tip. At steady-state the residual stresses far behind the crack tip are tensile near the crack, decreasing to zero at a certain distance from the crack plane. It is shown that the shielding due to pre-existing dislocations, e.g., for cleavage in α-iron crystals may be considerable. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Stress Intensity Factor of an Edge Dislocation Near an Elliptically Blunted Crack Tip

The stress field around the tip of an elliptically blunted crack induced by an edge dislocation has been obtained in closed form, from which the mode I and mode II stress intensity factors induced by the edge dislocation are obtained. The solutions apply to the edge dislocation either emitted from crack-tip surface or originated elsewhere, and for the dislocation located anywhere around the crack tip. The effects of the crack length, the crack-tip bluntness, the origination and position of the dislocation on the stress intensity factors are examined.