On the fracture behavior of a Zener–Stroh crack with plastic zone correction in three-phase cylindrical composite material

With crack tip plastic zone correction, stress investigation on the fracture behavior of a Zener–Stroh crack in three-phase composite was carried out. A Zener–Stroh crack (in the matrix phase) is near a circular inclusion, with the three-phase cylindrical composite model used to represent the composite material. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. The Dugdale model of small scale yielding is used to introduce a thin strip of yielded plastic zone each crack tip. The physical problem is formulated into a set of singular integral equations, using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function. The singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements using Erdogan and Gupta’s method with some iterative numerical procedures.     

Stress analysis on a Zener crack nucleation from an eccentric wedge disclination in a cylinder

The stress relaxation of an eccentric (off-center) negative wedge disclination in an isotropic homogeneous cylinder by nucleation of a Zener crack has been investigated with a continuum model. The nucleated Zener crack is simulated with distributed edge dislocations. The stress intensity factor (SIF) at the sharp tip of the Zener crack is computed through solving the singular integral equations formulated. By enforcing the fracture criterion at the sharp tip, the critical disclination power to nucleate a Zener crack is determined. The equilibrium crack lengths of the crack are then calculated when the disclination power is above the critical value. It is found that there is a special position at which the critical disclination power reaches the minimum value. As the disclination deviates from this position, the critical disclination power increases. Two or four equilibrium crack lengths could be found for the Zener crack, dependent upon the power and off-center position of the disclination. The influence of the off-center distance on the equilibrium crack lengths and the dependence of the critical disclination power and stable equilibrium crack lengths on cylinder radius are also discussed.     

On the plastic zone size and CTOD study for a Zener�CStroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener?CStroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green??s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan??s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson??s ratio, and loading condition are studied.     

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.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

On elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.     

A Zener–Stroh crack at the interface of a thin film bonded to a substrate

A Zener–Stroh crack can nucleate at the interface of a multi-layered structure when a dislocation pileup is stopped by the interface which works as an obstacle. During the entire fracture procedure of a crack, Zener–Stroh crack mechanism controls the initial stage, or the first phase of crack initiation and propagation. In our current research, stress investigation on a Zener–Stroh crack initiated at the interface of a thin film bonded to a half plane substrate has been carried out. With the application of dislocation-based fracture mechanics, the micro crack is simulated by the distributed dislocations along the crack line. To eliminate the contradictory oscillation phenomenon for the stress field near the interfacial crack tip, a contact zone behind the crack tip is introduced. The physical problem is thus formulated into a set of non-linear singular integral equations. Through careful examination of the crack singularities at the crack tips for different configurations, the formulated integral equations are solved with numerical methods developed in our research. The contact zone length, the stress fields near the crack tip and the stress intensity factors of the crack are evaluated accordingly. Numerical examples based on practical engineering structures are provided to discuss the influence of the key parameters, such as the thickness of the film, and the Dundurs constants, on the fracture behaviour of the crack.     

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.     

Plastic CTOD as fatigue crack growth characterising parameter in 2024‐T3 and 7050‐T6 aluminium alloys using DIC

The plastic range of crack tip opening displacement (CTOD) has been used for the experimental characterisation of fatigue crack growth for 2024‐T3 and 7050‐T6 aluminium alloys using digital image correlation (DIC). Analysis of a complete loading cycle allowed resolving the CTOD into elastic and plastic components. Fatigue tests were conducted on compact tension specimens with a thickness of 1 mm and a width of 20 mm at stress ratios of 0.1, 0.3 and 0.5. The range of plastic CTOD could be related linearly to da/dN independent of stress ratio for both alloys. To facilitate accurate measurements of CTOD, a method was developed for correctly locating the crack tip and a sensitivity analysis was performed to explore the effect of measurement position behind the crack tip on the CTOD. The plastic range of CTOD was demonstrated to be a suitable alternate parameter to the stress intensity factor range for characterising fatigue crack propagation. A particularly innovative aspect of the work is that the paper describes a DIC‐based technique that the authors believe gives a reliable way to determine the appropriate position to measure CTOD.     

Analysis of the wedge-shaped damage zone in edge-notched polypropylene

The irreversible deformation behaviour of polypropylene during sharp single-edge-notched tension testing has been studied as a function of temperature. Specimens were tested at room temperature, –20, –40, and –60 °C with photographs taken of the notch tip area during testing. BelowT _g, a narrow wedge-shaped damage zone grew from the notch tip with increased stress. The damage zone length correlated with the ratio of applied stress to yield stress in agreement with the Dugdale model. The crack tip opening displacement (CTOD) was found to follow the predicted Dugdale CTOD when modified by using the secant modulus to account for viscoelasticity. The shape of the damage zone did not agree with the Dugdale model near the notch tip, but instead was found to follow a path of the minor principal stress trajectory. AboutT _g, the damage zone had a lower length-to-width ratio which no longer resembled the Dugdale model.     

Application of local approach to inhomogeneous welds. Influence of crack position and strength mismatch

In steel welds there is often a large variation in fracture toughness and mechanical properties between the weld metal, base material and the various heat affected zone (HAZ) microstructures. The stress field in front of a crack in a weldment can be noticeably affected by the strength mismatch between the weld metal, HAZ and the base material. The crack position relative to the various microstructures will clearly influence the strength mismatch effect. In this paper the influence of crack tip positioning on the fracture performance of strength mismatched steel welds has been studied both experimentally and by FEM analysis. For a mismatched weld with local brittle zones small changes in crack tip location can give considerable changes in the fracture performance of a CTOD specimen. A high degree of strength mismatch increases the effect of crack positioning. Weld metal overmatch increases the stress level in the heat affected zone due to material constraint and thereby reduces the cleavage fracture resistance of the weldment when the coarse grained HAZ (CGHAZ) controls the fracture. The detrimental effect of high overmatch is most pronounced for specimens with notch position at fusion line and a short distance into the brittle CGHAZ. The Weibull stress has been shown to be a suitable fracture parameter in the case where one microstructure clearly controls the cleavage fracture and the calculation of the Weibull stress therefore can be limited to this zone.     

On the fracture of constrained layers

The problem of a crack embedded in a layer sandwiched between two elastic adherends is analysed accounting for the influence material property mismatch on the crack tip plastic deformation, which is contained in the layer. The cohesive crack model developed by Dugdale and Barrenblatt is adopted to model the strip yielding behaviour in a constrained layer. It is found that, due to the constraint imparted by elastic adherends with higher moduli, the near tip plastic deformation exhibits a sharp transition (plastic zone grows faster than the square of stress intensity factor) from small scale to large scale yielding. Because the region of singularity dominance for a crack embedded in a layer is generally much smaller than the layer thickness when the layer has a modulus much lower than the adherends, the prevailing failure mode of most bonded joints should be under large scale yielding conditions. A model based on energy balance is proposed to determine the fracture energy of bonded joints under such condition, taking into account of the plastic dissipation in the constrained layer. Comparison with experimental results demonstrates that the theory correctly predicts the dependence of fracture toughness on layer thickness as observed in experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Elastic–plastic analysis of interaction between an interfacial crack and a subinterfacial microcrack in bi-materials

In the present work the finite element method is used to analyze the effect of interaction between an interfacial crack and a microcrack in ceramic/aluminum bi-materials. The behaviour is analyzed by the determination of the J integral, the plastic zone at the tips of the interfacial crack and the microcrack. The effects of longitudinal and transversal distance between the tips of the two cracks and the rotation of the microcrack are examined. The obtained results allow us to deduce a correlation between the position of the microcrack and the J integral and the plastic zone.The obtained results shows that the J integral at the interfacial crack tip reaches a maximum value when the microcrack is moved in the vicinity of the interfacial crack. With this distance the effect of interaction is marked more; the stress field at the microcrack tip and that of the interfacial crack generates only one plastic zone at the interfacial crack tip. The maximum size of the plastic zone is localised at the interfacial crack tip. Those of the two tips of the microcrack are very weak and even negligible in front of the zone plasticized at the interfacial crack tip.     

Microstructural effects on crack closure and propagation thresholds of small fatigue cracks

The crack closure behaviour of microstructurally small fatigue cracks was numerically simulated by combining the crack-tip slip band model with the plasticity-induced crack closure model. A Stage II crack started to propagate from an initiated Stage I crack. When the plastic zone was constrained by the grain boundary or the adjacent grain with higher yield stresses, the crack opening stress increased with crack extension, and the effective component of the stress range decreased. The crack-tip opening displacement range (Δ CTOD ), first decreased with crack extension due to the development of the residual stretch, then increased until the tip of the plastic zone reached the neighbouring grain boundary. When the plastic zone was blocked by the grain boundary, Δ CTOD began to decrease. The arrest condition of cracks was given by the threshold value of Δ CTOD . At the fatigue limit, the arrest of small cracks takes place just after the Stage II crack crosses the grain boundary when the grain boundary does not act as a barrier. Only when the grain boundary has a blocking strength and the yield stress of adjacent grains is not so high, the arrest of Stage II cracks takes place before the crack reaches the grain boundary. The fatigue limit decreases with the mean stress. The predicted relation between the fatigue limit and the mean stress is close to the modified Goodman relation.     

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags

An analytical investigation on the plastic zone size (PZS) of a crack near a circular inclusion has been carried out. Both the crack and the circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, two stripe plastic zones at both crack tips are introduced. Using the solution of a circular inclusion interacting with a single dislocation as the Green’s function, the physical problem is formulated as a set of singular integral equations. With the aid of Erdogan and Gupta’s method and iterative numerical procedures, the singular integral equations are solved numerically for the PZS and the crack tip opening displacement. The results obtained in the current work can be reduced to those simpler cases of the Dugdale model.     

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.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

Pile-up of screw dislocations at a circular inclusion

The behaviour of a pile-up of screw dislocations at a circular inclusion, with its tip away from the interface is analyzed using the method of continuously distributed dislocations. This leads for the first time, to a distribution function representing a shear crack at the inclusion. The stress required to extend the crack is derived and some new conclusions drawn on the deformation and fracture behaviour of the two-phase systems.     

Plastic extension of a shear crack inside a circular inclusion

The plastic relaxation of a shear crack situated at the interface inside a circular inclusion in an infinite matrix has been analyzed by treating it as a double pile-up of screw dislocations and the plastic zones at either tip of the crack as giant screw dislocations. The ratio of applied stress to yield stress and the magnitude of the Burgers vector of the giant screw dislocations which represent the relative displacement of the crack faces at the tips have been related to the crack parameters. Using a critical relative displacement of the crack faces at the tip of the crack as the criterion for brittle extension of the crack, the tendency of the shear crack to extend into the inclusion or into the matrix has been determined. The effect of shear modulus and size of the inclusion on the behaviour of the plastic zones at either tip of the crack has been discussed. Conclusions are made on the fracture behaviour of a circular inclusion in an infinite matrix.

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Résumé On a analysé la relaxation plastique d'une fissure de cisaillement située à l'interface d'une inclusion circulaire dans une matrice infinie en traitant cette fissure comme un double empilement de dislocation-vis, et les zônes plastiques à chacune de ses extrémités comme des dislocations-vis géantes. On a mis en relation avec les paramètres de la fissure le rapport de la contrainte appliquée à la limite d'écoulement, et la grandeur du vecteur de Burgers des dislocations-vis géantes, qui représente le déplacement relatif des faces de la fissure à ses extrémités.En utilisant comme critère d'extension fragile de la fissure un déplacement relatif critique des faces de la fissure à ses extrémités, on a pu déterminer si la fissure de cisaillement avait tendance à se développer dans l'inclusion ou dans la matrice.On discute l'influence du module de cisaillement et de la dimension de l'inclusion sur le comportement des zônes plastiques à chaque extrémité de fissure.On tire des conclusions sur le comportement à la rupture d'une inclusion circulaire dans une matrice infinie.