The condition for the cyclic plastic deformation of the crack tip: the influence of dislocation obstacles

The stress intensity range below which no cyclic plastic deformation at the crack tip and, hence, no fatigue crack propagation occurs is investigated. The emission of dislocations from the crack tip is assumed as mechanism for the dislocation generation. For a mode III crack, a computer simulation is carried out to study the influence of dislocation obstacles. Both the distance between the crack tip and the obstacle and the strength of the obstacle are varied and the characteristic dislocation arrangements are shown.The stress intensity range necessary to return one dislocation to the crack tip is mainly controlled by the critical stress intensity factor sufficient to emit a dislocation. The influence of the obstacles is not very significant.     

The elastic interaction of screw dislocations and a star crack with a central hole

The elastic interaction of screw dislocations and a star crack with a central hole was investigated. The complex potential of the present problem was obtained from that of an internal crack in an infinite medium using the conformal mapping technique. The stress field, image force and strain energy of dislocation, and stress intensity factor at the crack tip were derived. The critical stress intensity factor for dislocation emission was calculated based on the spontaneous dislocation emission criterion. The influence of the ratio of crack length to hole radius, crack number, and dislocation source on the above mechanical variables were studied. The present solution was reduced to several special cases previously reported in the literature.     

Screw dislocation in the two-phase isotropic thin film of an interfacial crack

The screw dislocation in the two-phase isotropic thin film of an interfacial crack has been investigated. The stress field, stress intensity factors at the crack tip and for dislocation emission, crack extension force, strain energy and the image force on the dislocation are obtained and found to be related to the thickness and effective shear modulus. The effect of size on fracture is pronounced when the thickness is smaller than the distance between dislocation and crack tip by a factor of 1000. The effect of the second phase on fracture is pronounced when μ⁽²⁾/μ⁽¹⁾ is in the range from 0.01 to 100. Newton's third law is proved to be valid for any thickness and shear modulus ratio. This result can be reduced to three special cases.     

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.     

Interfacial crack problems in magneto-electroelastic solids

In this paper, we present an explicitly analytic solution for an electrically permeable interface crack between two dissimilar magneto-electroelastic solids. Using the Stroh formalism, we first derive the general solution under arbitrary loads by reducing the generalized 2D problem to an equivalent interface crack problem in two-elastic anisotropic media, which can be solved with well-established methods. With the analytic solution, we then study the interface crack loaded by remote uniform loads, by a generalized line force or/and by a generalized line dislocation. The results show the singular and oscillatory fields ahead of the crack tips. The electric-magnetic field within the crack has a similar singularity at the crack tip as the electric-magnetic field within the solids. In particular, when uniform mechanical-electric-magnetic loads are applied at infinity, the singular intensity depends on the material properties and the mechanical loads, but not on the electric-magnetic loads.     

Micro-mechanical analysis and TEM study of crack initiation in dislocation free zone

Dislocation emission, dislocation-free zone (DFZ) formation and crack initiation in the DFZ and/or at the crack tip were analyzed by micromechanics. The results show that a DFZ is formed after dislocation emission under constant load. The DFZ size decreases with increase in the applied stress intensity factor K _Ia or lattice friction stress _f. There are two stress peaks ahead of the crack tip. The first one is located at the blunt crack tip and the second one in the DFZ. With increasing in the applied stress intensity factor K _Ia, the peak stress at the crack tip may decrease while the peak stress in the DFZ increases monotonically. Microcrack will initiate when the peak stress is equal to the cohesive strength. In situ tensile tests in a transmission electron microscope (TEM) show that microcrack initiates in DFZ or/and at a blunt crack tip after dislocation emitting and DFZ formation.     

Effect of cooperative grain boundary sliding and migration on dislocation emitting from a semi-elliptical blunt crack tip in nanocrystalline solids

A theoretical model is established to investigate the interaction between the cooperative grain boundary (GB) sliding and migration and a semi-elliptical blunt crack in deformed nanocrystalline materials. By using the complex variable method, the effect of two disclination dipoles produced by the cooperative GB sliding and migration process on the emission of lattice dislocations from a semi-elliptical blunt crack tip is explored. Closed-form solutions for the stress field and the force acting on the dislocation are obtained in complex form, and the critical stress intensity factors for the first dislocation emission from a blunt crack under mode I and mode II loadings are calculated. Then, the influence of disclination strength, curvature radius of blunt crack tip, crack length, locations and geometry of disclination dipoles, and grain size on the critical stress intensity factors is presented detailedly. It is shown that the cooperative GB sliding and migration and the grain size have significant influence on the dislocation emission from a blunt crack tip.     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

Interaction between a surface crack and a subsurface inclusion

A numerical method for the integration of the singular integral equation resulting from the interaction of a surface crack with a subsurface inclusion is presented. The crack is modelled as a pile-up of dislocations, and the dislocation density function is partitioned into three parts: A singular term due to the load discontinuity imposed by the inclusion, a square root singular term from the crack tip, and a bounded and continuous residual term. By integrating the singular terms explicitly the well behaved residual dislocation density function only has to be determined numerically, together with the intensity of the square root singular term. The method is applied to the determination of the stress intensity factor for a surface crack growing towards and through a circular inclusion whose diameter is equal to the distance from the free surface, and to the determination of the characteristic stress intensity factors when the crack enters the inclusion and leaves it for arbitrary ratios between the inclusion diameter and the distance from the surface.     

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.     

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.     

Elastic interaction between screw dislocations and the internal crack near a free surface

The elastic interaction between screw dislocation and the internal crack near a free surface has been investigated. The stress intensity factor at the crack tip, crack extension force, the image force on the dislocation are affected by the free surface. The number and nature of dislocations, m, inside the crack also play an important role in fracture. In order to understand the plastic zone, the zero-force points of dislocation along the x-axis are involved. The dislocation emitted from the right-hand crack tip is enhanced by positive m and reduced by negative m. On the other hand, if the internal crack is closer to the free surface, a dislocation generated from the right-hand crack tip is easier for negative m and more difficult for positive m. However, the role of m on the dislocation emission for the left-hand crack tip is opposite to that for the right-hand crack tip. Finally, three special cases can be obtained from our results. (1) The interaction between a dislocation and a surface crack; (2) the interaction between a dislocation and an internal crack; (3) the interaction between two dislocations.     

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.     

A Griffith crack shielded by a dislocation pile-up

The dislocation free zone at the tip of a mode III shear crack is analyzed. A pile-up of screw dislocations parallel to the crack front, in anti-plane shear, in the stress field of a crack has been solved using a continuous distribution of dislocations. The crack tip remains sharp and is assumed to satisfy Griffith's fracture criteria using the local crack tip stress intensity factor. The dislocation pile-up shield the sharp crack tip from the applied stress intensity factor by simple addition of each dislocation's negative contribution to the applied stress intensity value. The analysis differs substantially from the well known BCS theory in that the local crack tip fracture criteria enters into the dislocation distributions found.     

A model for fatigue crack growth with a variable stress intensity factor

A model for fatigue crack growth, similar to that of Majumdar and Morrow, is proposed where the crack growth rate is determined from the low cycle fatigue and cyclic stress-strain response of the material. The model is for a constant stress range at infinity, but does allow for a variable stress intensity factor due to the changing crack length. The study also includes an analysis of the strain range in the neighborhood of the crack tip. Further it is shown that the model predicts the critical stress intensity factor. A prediction of the crack growth rate is made for 2024-T351 aluminium, copper and CU-6.3 AL alloy and is compared to the experimental observations.     

Dielectric breakdown model for a conductive crack and electrode in piezoelectric materials

The strip dielectric breakdown (DB) model introduced by Zhang and Gao [T.Y. Zhang, C.F. Gao, Fracture behavior of piezoelectric materials, Thero. Appl. Fract. Mech. 41 (2004) 339–379] is used to study the generalized 2D problem of a conductive crack and an electrode in an infinite piezoelectric material. The energy release rate and stress intensity factors are derived based on the Stroh formalism, and then they are applied as failure criteria to predict the critical fracture loads. It is found that the DB strip may take the shielding effect on a conductive crack or electrode. For the case of an electrode, the local energy release rate and stress intensity factor become zero when DB happens ahead of the electrode tip. For the case of a mode-I conductive crack in a transversely isotropic piezoelectric solid, the results based on the DB model show that the critical stress intensity factor linearly increases as the applied electric field parallel to the poling direction increases, while it linearly decreases as the applied electric field anti-parallel to the poling direction increases. Finally, the upper and lower bounds of the actual critical fracture loads are proposed for a conductive crack in a piezoelectric material under combined mechanical–electrical loads.     

Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D combined mode loading

The dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body are analyzed. The crack is subjected to a traction distribution consisting of two pairs of suddenly-applied shear point loads, at a distance L away from the crack tip. The exact expression for the combined mode stress intensity factors as the function of time and position along the crack edge is obtained. The method of solution is based on the direct application of integral transforms together with the Wiener-Hopf technique and the Cagniard-de Hoop method, which were previously believed to be inappropriate. Some features of solutions are discussed and the results are displayed in several figures.     

Hold-time effects on high temperature fatigue crack growth in Udimet 700

Crack growth behaviour under creep-fatigue conditions in Udimet 700 has been studied, and the crack growth data were analysed in terms of the stress intensity factor as well as theJ-integral parameter. Crack growth behaviour is shown to depend on the initial stress intensity level and the duration of hold-time at the peak load. For stress intensities that are lower than the threshold stress intensity for creep crack growth, the crack growth rate decreases with increase in hold time even on a cycle basis, da/dN, to the extent that complete crack arrest could occur at prolonged hold times. This beneficial creep-fatigue interaction is attributed to the stress relaxation due to creep. For stress intensities greater than the threshold stress intensity for creep crack growth, the growth rate on a cycle basis increases with increase in hold time. For the conditions where there is no crack arrest, the crack growth appears to be essentially cycle-dependent in the low stress intensity range and time-dependent in the high stress intensity range. Both the stress intensity factor and theJ-integral are shown to be valid only in a limited range of loads and hold-times where crack growth rate increases continuously.     

Dynamic interaction of a propagating crack with a hole boundary

Summary The optical method of caustics was used along with high speed photography to study crack propagation and crack-hole interaction in plane PMMA specimens containing a transverse edge crack and a hole lying eccentrically to its axis. The specimens were fractured under different dynamic loads.Crack-hole interaction is characterized (for a limiting vertical distance of the crack axis from the center of the hole) by a process of attraction — repulsion of the crack towards the hole, interrupted by a momentary crack-arrest at the hole boundary. Increased values of crack propagation velocity and of the stress intensity factor at the tip of the propagating crack are detected during crack-hole interaction.With 7 Figures