On the effect of the residual stresses on the crack opening displacement in a cracked sheet

The residual stress and displacement fields caused by localized plastic flow near a mode I crack tip in a sheet under plane stress conditions are investigated. The present study founds on the classical Dugdale scheme of the plastic flow localization. The residual stress field is considered to be induced by reversed plastic flow near the crack tip caused by an unloading. As it is found the residual stresses around the crack compress the crack tip, while the residual tensile stresses in a distant from the crack tip zone occur. It is shown the maximum residual tensile stresses can reach the significant value of the one third of the yield limit. The length of the compressed plastic zone and the residual displacement distributions are obtained. The exact formula for the residual crack opening displacement to estimate the crack closure is found. Then the next loading of the cracked plate is considered. It is shown that the second loading causes the origin of two plastic zones localized near the crack tip and at the point, where the maximum residual tensile stresses are concentrated. Again, according to the Dugdale scheme of the plastic localization, both the plastic flow zones are modelled as narrow stripes on the line extending the crack. To determine three non-dimensional parameters, characterizing the position of the segment-like plastic flow zones, a non-linear system of equations is obtained and analyzed. The exact formula for the crack opening displacement after a loading–unloading cycle is obtained. An asymptotic analysis (as the linear dimension of the distant plastic flow zone compared with the actual crack length is small) is given. It shows that the effect of the distant plastic flow zone appears as some complementary crack closure.     

Elastic and plastic stress analysis of an interface crack between two dissimilar layers

In the present paper, the mixed-mode Dugdale model is applied to investigate the plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers. In the analysis, both normal and shear stresses are assumed to exist in the plastic zones and satisfy the Von Mises yield criterion. The plastic zone sizes can be determined on condition that the stress intensity factors caused by the stresses in the plastic zones and applied loading are zero. Then, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, the plane stress condition is considered. The plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers under a uniform load are examined. The effects of Dundurs’ parameters and the thickness of materials on the plastic zone size and the crack tip opening displacement are investigated in detail. Numerical results show that in the case of small thickness, the values of the normalized plastic zone size and the normalized crack tip opening displacement decrease with increasing Dundurs’ parameters, α and β, while, in the case of infinite thickness, the value of the normalized plastic zone size is independent of α, and the value is symmetric about the axis on which β = 0.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

Elastic–plastic fracture behavior of a radial matrix crack interacting with a circle inclusion with generalized Irwin corrections

A generalized Irwin plastic zone model is proposed to investigate the interaction problem for a circular inclusion embedded in an elastic–plastic matrix that contains a radial crack, oriented at an arbitrary angle from a remote load. The distributed dislocation technology is applied to formulate the current problem. The effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are evaluated by solving the formulated singular integral equations. In the numerical examples, the inclusion is taken as a void and a rigid body. The effects of the crack angle and the inclusion–crack distance (the distance from the inclusion center to the crack center) on the effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are discussed in detail. Numerical results show that if the crack angle is not large, the values of the plastic zone size and the crack tip opening displacement are less than the corresponding values in the homogenous case when the inclusion is a rigid body; when the inclusion is a void, these values are larger than the corresponding values in the homogenous case.     

Film thickness effects on craze micromechanics

Air crazes have been grown from indentor “crack” tips in polystyrene films of thicknesses 0.11, 0.57 and 1.2μm. Quantitative transmission electron microscopy is used to measure craze thickness and fibril volume fraction profiles. From these, profiles of craze fibril extension ratio, λ(x), craze surface displacement,w(x), and craze surface stress,S(x), have been computed. For all thicknesses of film, the λ(x) profiles prove that the craze thickens by drawing more material into the fibrils from the craze-matrix interface, rather than by fibril creep. The form ofS(x) is also similar for all thicknesses of film, with a maximum at the craze tip and a minimum approximately half way along the craze. The extension ratio profiles also show a maximum at the craze tip. The midrib, which develops in the high stress region behind the craze tip as the craze propagates, has a value of λ comparable to that found at the stationary craze tip. When an isolated craze grows in an initially homogeneous stress field the midrib is observed to be of constant thickness. In contrast the midribs of crazes grown from crack tips decrease in thickness to a constant value with distance from the crack tip. All these observations are explained by the surface drawing mechanism of craze thickening. The computed values of λ and the fibril stress,σ _t, for the thinnest film, are significantly lower than for the thicker two films. These changes are attributed to the absence of plastic constraint in the thinnest film, which decreases the fibril true stress necessary for surface drawing.     

Crack arrest model for a piezoelectric plate — A generalised Dugdale model

The constant search for new materials has provided impetus to research in piezoelectric materials. An anti-plane problem for a cracked unbounded two-dimensional poled piezoelectric plate has been investigated. The crack rims open on account of shear mechanical forces applied at the remote boundary and in-plane electric displacement field prescribed at the infinite boundary. Thus the crack yields both mechanically and electrically. Consequently, a plastic zone and a saturation zone protrude ahead of each tip of the crack. These developed zones are in turn closed by applying yield point shear stress at the rims of plastic zone and normal closing saturation limit displacement on the rims of saturation zones. Two cases are investigated when (i) the developed saturation zone length exceeds that of the developed plastic zone, and (ii) saturation zone length is smaller than that of the plastic zone. Fourier integral transform method is used in each case to obtain the length of plastic zone and saturation zone. Closed form analytic expressions are obtained in each case. Crack opening displacement and potential drop across the rims of the crack are also obtained. The effect of mechanical loads on crack closure in the presence of electric field is investigated and vice-versa. Also energy release rate expressions are obtained for both the cases.     

Distribution of dislocations at a mode I crack tip and their shielding effect

Distribution of dislocations at a finite mode I crack tip is formulated. Closed form solutions for the dislocation distribution function, the dislocation-free zone (DFZ), the local stress intensity factor and the crack tip stress field are obtained. The dislocation distribution has similar features to a mode III crack model. Under a given applied stress, there may exist different configurations of plastic zone and DFZ. Crack tip shielding by dislocations depends on both applied stresses and the configuration.     

Plastic intensity factors for cracked plates

An elastic-plastic analysis is performed for two problems relevant to fracture mechanics: a semiinfinite body with an edge crack in a far out-of-plane shearing field and an infinite plate under plane stress conditions containing a finite line crack in a remote tensile field. Amplitudes of the dominant singularity in the plastic region at the crack tip, the plastic stress and strain intensity factors, are calculated for applied stress levels approaching the yield stress. A technique is developed for using the dominant singular solution in conjunction with the finite element method to make accurate calculations for the near-tip fields. Additionally, a comparative study of deformation theory with flow theory is performed for cracks in an anti-plane shear field. Elastic fracture mechanics is extended to high levels of applied stress for which the plastic zone is no longer small compared to the crack length by relating the critical stress for fracture initiation to the plastic intensity factors.     

In-situ Observation of Initiation and Propagation of Cleavage Microcrack in TiAl

Nucleating and propagating of nancrack formed in dislocation free zone (DFZ) for the brittle TiAl alloy has been studied through in-situ tensile test in TEM and analyzed using microfracture mechanics. The resufts show that a lot of dislocations can be emitted from a crack tip when the applied stress intensity Kla i5 larger than the stress intensity for dislocatin emission Kle=1.4 M Pa·m1/2 and a dislocation free zone, which smetimes is a close zone, can form after reaching equilibrium. The DFZ is a elastic zone with large strain and then the stress in the DFZmight equal to the cohesive strength σth because the crack tip is still sharp. When Kla is larger than the stress intensity for nanocrack nucleation Kli =2.4 M Pa·m1/2, the stress within a certain range in the DFZ would equal to σth and then a nanocrack initiates in the DFZ or sometimes at the notch tip. The nanocrack formed in the DFZ is stable and can propagate a small distance in cleavage mode through multiplication and movement of dislocation in the plastic zone, during keeping constant displacement. Increasing Kla can make the crack stably propagate continuously or discontinuously and it means that the stre5s intensity for crack propagation, Klp, is larger than Kli. Therefre, Kle 相似文献   

Energy partitioning for a crack under remote shear and compression

The true nature and characteristics of crack growth mechanisms in geologic materials have not been adequately described and are poorly understood. The process by which deformation energy is converted to slipping and growing cracks under compressive stresses is complex and difficult to measure. A hybrid technique employing moiré interferometry as an experimental boundary condition to a finite element method (FEM) was employed for through-cracked polycarbonate plates under remote shear and compression. Cohesive end zone and dislocation slip models are used to approximate experimentally observed displacement characteristics. Shear-driven linear elastic fracture mechanics displacement predictions are shown to be inadequate for initial displacement progression. Moiré displacement fields of relative crack face slip reveal a near tip cohesive zone. The pre-slip moiré-FEM stress fields reveal that the maximum crack tip tensile stress occurs at approximately 45 degrees and further infers cohesive zone presence. A J integral formulation uses moiré displacement data and accounts for stored energy along the crack before and after shear driven crack face slip. These energy-partitioning results track the transfer of stored energy along the crack face to the crack tip until the entire crack is actively slipping. These laboratory-scale experiments capture basic mechanical behavior and simulate thousands of years of large-scale geologic feature displacement history in just a few hours.     

SHORT FATIGUE CRACK GROWTH IN PLAIN AND NOTCHED SPECIMENS OF AN 8090 Al-Li ALLOY

Abstract— Fatigue tests on notched and plain specimens were carried out under a combined acoustic and optical microscope. Crack initiation in plain specimens occurs invariably at inclusions which act as sites of stress concentration. Short crack propagation behaviour shows the usual pattern of accelerations and retardations associated with microstructural variations. Growth can only be sustained if the applied stress is sufficiently high to overcome the microstructural barriers. For the notched specimens the extent of the notch tip and crack tip plastic zone sizes control crack propagation. A short crack will continue to propagate only if its own plastic zone can sustain growth as the crack tip extends beyond the notch zone, otherwise it becomes non-propagating. Notch plastic zones are calculated using both approximate and accurate solutions. Some microstructural effects are discussed.     

Dislocation Behaviour near the Crack Tip in Bulk Fe-3% Si Single Crystal

An investigation has been made of the disloca-tion distribution and dislocation free zone near thecrack tip in bulk Fe-3% Si single crystal duringdeformation in SEM.It has been found that anumber of dislocations were emitted from the cracktip during deformation.After that,the dislocationsmoved rapidly away from the crack tip,which indi-cated that they were strongly repelled by the stressfield at the crack tip.Between the crack tip and theplastic zone there is a region of dislocation-free,which is referred to as dislocation-free zone (DFZ).The length of DFZs is roughly estimated 100μm which is much longer than that found in thinfoil specimen.The variation of dislocation densityas a function of the distance from the crack tip wasmeasured,which showed that the dislocations areinversely piled up in the plastic zone.The length ofDFZs increased with both the length of pre-crackand the amplitude of applied stress.     

Analytical and numerical study of cohesive zones for a crack in an adhesive layer between identical isotropic materials

A crack in a thin adhesive elastic-perfectly plastic layer between two identical isotropic elastic half-spaces is considered. Uniformly distributed normal stress is applied to the substrates at infinity. First, stress distribution in the cohesive zones and the J-integral values are defined numerically by the finite element method (FEM). Further, a mathematical formulation of the problem is given and its analytical solution is proposed. It is assumed that, at the crack continuations, there exist cohesive zones. The interlayer thickness is neglected since it is much smaller than the crack length. The distribution of the normal stress, which was obtained by means of the FEM, is now approximated by a piecewise-constant function and assumed to be applied at the faces of the cohesive zones. The formulated problem is solved analytically and an equation for determination of the cohesive zone lengths is derived. Also, closed expressions for the crack tip opening displacement and for the J-integral are obtained in an analytical form. These parameters are found with respect to the values of the normal stress applied at infinity. Finally, a universal approximating function, which describes the stress distribution in the cohesive zones, is constructed. This function depends on the ratio between the interlayer thickness and the crack length and on the ratio between the normal stress applied at infinity and the yield limit of the interlayer’s material. Once again, the problem is solved analytically, but this time for the stress distribution prescribed by the universal approximating function. The cohesive zone lengths, the values of the crack tip opening displacement and of the J-integral are calculated. A comparative analysis of the obtained results is carried out. A good agreement of the J-integral values calculated by means of the developed analytical models and by the associated finite element analysis is demonstrated.     

A unified model for dislocation nucleation, dislocation emission and dislocation free zone

In this paper, a unified model for dislocation nucleation, emission and dislocation free zone is proposed based on the Peierls framework. Three regions are identified ahead of the crack tip. The emitted dislocations, located away from the crack tip in the form of an inverse pileup, define the plastic zone. Between that zone and the cohesive zone immediately ahead of the crack tip, there is a dislocation free zone. With the stress field and the dislocation density field in the cohesive zone and plastic zone being, respectively, expressed in the first and second Chebyshev polynomial series, and the opening and slip displacements in trigonometric series, a set of nonlinear algebraic equations can be obtained and solved with the Newton-Raphson Method. The results of calculations for pure shearing and combined tension and shear loading after dislocation emission are given in detail. An approximate treatment of the dynamic effects of the dislocation emission is also developed in this paper, and the calculation results are in good agreement with those of molecular dynamics simulations.Presented at the Far East Fracture Group (FEFG) International Symposium on Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

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.     

Application of the distributed dislocation technique for calculating cyclic crack tip plasticity effects

This paper describes a method for modelling cyclic crack tip plasticity effects based on the distributed dislocation technique (DDT). A strip‐yield model is utilised to allow for the determination of the crack opening displacement, size of the plastic zones and in the case of a fatigue crack, the wake of plasticity. The DDT can be easily implemented for a wide range of cracked geometries with reliable control over the accuracy and convergence. Thickness effects can also be incorporated through a recently obtained solution for an edge dislocation in an infinite plate of finite thickness. Results for finite length cracks that have had limited growth, such that there is no plastic wake, are presented for a range of applied loads and R‐ratios. Further results are provided for a steady‐state fatigue crack in a plate of finite thickness. The present results are compared with analytical solutions and they show an excellent agreement.     

The Behavior of Screw Dislocations Emitted from a Star Crack with a Central Hole

The behavior of screw dislocations emitted from a star crack with a central hole was investigated using discrete dislocation modeling. Cracks are uniformly distributed along the circumference of a circular hole. Dislocations are assumed to be emitted one by one from the crack tips along the radial direction. Each emitted dislocation moves along the radial direction and its velocity is proportional to the third power of the effective shear stress. A dislocation-free zone exists based on the assumption that the crack tip must overcome an energy barrier to emit a dislocation. The effects of the central hole, slit crack, number of cracks and applied stress on the plastic zone, total number of dislocations emitted from all crack tips and the first crack tip, and the dislocation-free zone were studied for a given friction stress. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

The configuration of moving crack tip zones

The dynamic elasticity solution of a steady state crack is used for determination of the geometrical characteristics and of the displacement rates within the discrete crack zones which are formed as a result of the selective propagation of cleavage microcracks ahead of the tip of a running brittle crack in a mild steel plate. The zone length and the stress distribution in the zone are found to be strongly dependent on the assumed form of the stress-displacement relation for the progressively fracturing metal. On the other hand, the crack opening displacement rate is much less sensitive to the assumed form. The magnitudes of the nominal plastic strain rates, found by an approximate procedure, are of the order of 10⁵ to 10⁶ sec^?1. A comparison with the results of dynamic tension tests on similar steels indicates that the flow stress at these strain rates would exceed the twinning stress, and therefore, it is suggested in agreement with experimental observations, that twinning is the principal deformation mode in the crack zone. Accordingly, the crack tip boundary conditions considered in the solution of crack propagation problems can be assumed as independent of crack velocity. Also, their possible dependence on temperature would not be related to the deformation mode in the crack zone.