Dislocation-free Zone Model Research Articles

The effect of specimen thickness on the dislocation-free zone model of fracture

We use the method of continuously distributed dislocations to study the dislocation-free zone (DFZ) model of fracture in an infinite plate of finite thickness under externally applied uniform anti-plane shear stress. We formulate the equilibrium condition in terms of a set of two coupled singular integral equations using the representation of the total interaction force between two screw dislocations in a thin plate obtained by Eshelby and Stroh (1951). This set of singular integral equations is then solved numerically using the Gauss-Chebyshev integration formula resulting in the dislocation distribution functions in the crack and in the two plastic zones, the total number of screw dislocations in the plastic zone, the DFZ condition for a crack in a thin plate and the local mode III stress intensity factor at the crack tip.

How to Measure a Dislocation’s Breakthrough Stress to Estimate the Grain Boundary Resistance against Slip Transfer Based on the DFZ-Model of Fracture

Stage-I-fatigue-cracks are used as highly localized dislocation sources with well-known Burger’s vectors to study the interaction between dislocations and grain boundaries. This interaction in the plastic zone is of particular interest to understand the fluctuating crack growth in the very short crack regime. In the case of a blocked slip band the dislocations pile up at the grain boundary causing a local stress concentration. The resulting local stress distribution is calculated based on measurements of the dislocation density distribution in the plastic zone. For this purpose the slip line profiles were measured by AFM, the dislocation density distribution was determined and the dislocation-free zone model of fracture (DFZ) was validated. With this it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer.

Quantifying the grain boundary resistance against slip transfer by experimental combination of geometric and stress approach using stage-I-fatigue cracks

In recent studies, many groups have investigated the interaction of dislocations and grain boundaries by bi-crystals and micro-specimen experiments. Partially, these experiments were combined with supplementary simulations by discrete dislocation dynamics, but quantitative data for the grain boundary resistance against slip transfer is still missing. In this feasibility study with first results, we use stage-I-fatigue cracks as highly localised sources for dislocations with well-known Burgers vectors to study the interaction between dislocations in the plastic zone in front of the crack tip and selected grain boundaries. The stress concentration at the grain boundary is calculated with the dislocation-free zone model of fracture using the dislocation density distribution in the plastic zone from slip trace height profile measurements by atomic force microscopy. The grain boundary resistance values calculated from common geometric models are compared to the local stress distribution at the grain boundaries. Hence, it is possible to quantify the grain boundary resistance and to combine geometric and stress approach for grain boundary resistance against slip transfer to a self-contained concept. As a result, the prediction of the grain boundary resistance effect based on a critical stress concept is possible with knowledge of the geometric parameters of the grain boundary only, namely the orientations of both participating grains and the orientation of the grain boundary plane.

Fatigue crack growth based on the dislocation-free zone (DFZ) model

In the present work, a microscopic mechanism model is presented to predict fatigue crack growth, which is developed based on the combination of fracture physics and cyclic plasticity effect. The gradual degradation of materials near crack tip is regarded as a result of cyclic plastic deformation. The J-integral within cohesive zone under cyclic loading is selected as the crack growth criterion, and determined based on dislocation-free zone and cohesive theory. The dislocation-free zone model is validated with experimental data from literature. It has been shown that the results predicted by the proposed model agree well with the experiment.

Stage-I fatigue crack studies in order to validate the dislocation-free zone model of fracture for bulk materials

Stage-I fatigue cracks are commonly described by the model of Bilby, Cottrell and Swinden (BCS model). However, since several experimental investigations have shown a dislocation-free zone (DFZ) in front of crack-tips, it is necessary to validate the new DFZ model and to examine the deviations to the BCS model. Therefore, the dislocation density distribution is derived from height profiles of slip lines in front of stage-I fatigue cracks in CMSX4® single crystals measured by contact-mode atomic force microscopy. This is possible, because the cracks are initiated at notches milled by focused ion beam technique directly on slip planes with a high Schmid factor. Consequently, the directions of the Burgers vectors are well known; it is possible to calculate the dislocation density distributions from the height profiles. The measured distributions are compared to the calculated distribution function of the DFZ model proposed by Chang et al. The additionally measured microscopic friction stress of the dislocations is then used to calculate the influence of grain boundaries on the dislocation density distribution in front of stage-I cracks. The calculation is done by the extended DFZ model of Shiue et al. and compared with the measured distribution function in polycrystalline specimens. Finally, the crack-tip sliding displacement as a measure for the crack propagation rate is compared for the DFZ model and the BCS model with the experimentally revealed values. The important result: the often used BCS model does not reflect the experimental measurements. On the contrary, the DFZ model reflects the measurements at stage-I cracks qualitatively and quantitatively.

Effect of grain boundary ledge on the dislocation-free zone model of fracture: Transgranular microcrack nucleated from a grain boundary ledge

The effect of grain boundary ledge on the dislocation-free zone model of fracture is investigated in the case of a transgranular microcrack nucleated from a grain boundary ledge. The dislocation distribution functions to simulate the crack and plastic zone, the numbers of dislocations in the crack and plastic zone, the stress field, and the stress intensity factor at the crack tip (or the energy barrier for the dislocation emission from the crack tip) are obtained. If the Burgers vector of dislocations comprising the grain boundary ledge has the same sign as that of plastic zone dislocations, the dislocation distributions to simulate the plastic zone, the number of dislocations in the plastic zone, the stress field in the dislocation-free zone, and the stress intensity factor at the crack tip increase with increasing number of grain boundary ledge dislocations, but increase with decreasing grain size. Both sides of the grain boundary ledge dislocations have different effects on the dislocation-free zone size, plastic zone size, and the applied stress to accumulate the plastic zone dislocations. The stress intensity factor at the crack tip increases with increasing dislocation-free zone size. When there is no dislocation-free zone in front of the crack tip, the stress intensity factor is zero regardless of whether the grain boundary ledge dislocations exist.

Dislocation-free zone model of mode III fracture: the effect of crack bluntness

The effect of crack blunting on mode III fracture is investigated using a discrete dislocation approach. The blunt crack is simulated by a small sharp crack emanating from a semi-elliptic hole. It is found that the applied stress intensity factor for dislocation emission increases with the number of emitted dislocations, but decreases with the increasing extent of bluntness. For a given number of emitted dislocations, the region between the crack tip and furthest dislocation increases with the bluntness of crack. However, the distance between the first dislocation and the crack tip becomes larger as the crack is more blunt. It is implied that the plastic zone and dislocation-free zone sizes increases with the bluntness of the crack. In addition, the dislocation-free zone size also increases with decreasing friction stress. The ductile vs. brittle behavior of the material is also studied. A material with a blunt crack has a lower applied stress intensity factor for dislocation emission, so it behaves more ductile.

The DFZ model of fracture at crack tip and crack initiation criterion

Direct observation by transmission electron microscopy (TEM) has been made on the distribution of dislocations in front of the crack tip during tensile deformation of aluminum. A microfracture model has been established to describe the equilibrium configuration of the dislocations in the presence of a dislocation-free zone (DFZ). The site of void nucleation observed from TEM experiments was found to be at about the place of maximum dislocation density predicted from the model. The relationship between the size of crack, DFZ and crack opening displacement (COD) was obtained as a function for a crack initiation criterion.

The dislocation-free zone model of mode-Ill fracture: A discrete dislocation approach

Abstract The dislocation-free zone model of mode-Ill fracture using the discrete dislocation model is investigated. It is found that the stress intensity factor KD for dislocation emission is dependent on the crack length and the dislocations in the system but is not a materials constant as reported in the literature. On the basis of this argument, we can explain the mechanical properties of materials, such as toughness and the ductile-to-brittle transition temperature. When the crack length is shorter than the critical value, no equilibrium dislocation exists in the finite space. In addition, we also found that the crack length increases with a decrease in the size of the dislocation-free zone. Finally, a comparison between the discrete and continuous dislocation approaches is made.

An electron microscope study of crack tip deformation and its impact on the dislocation theory of fracture

Dislocation behavior has been observed near the crack tip of metals during in situ deformation in an electron microscope. All three modes of crack propagation and deformation were observed depending on the specimen geometry, although shear cracks of mode III type with screw dislocations in the plastic zone were most frequently observed. Dislocations were emitted from the crack tip during early stages of crack propagation and were driven out of the crack tip area, leaving behind a dislocation-free zone. The dislocations were piled up in the form of a linear array on a slip plane which was coplanar with the crack in metals of low stacking fault energy. In metals of high stacking fault energy the dislocations cross slipped out of the original slip plane and formed a broad plastic zone. The cracks propagated by a combination of plastic and elastic processes in which the plastic portion of the crack opening was created by the dislocations that were emitted from the crack tip. The elastic process occurred as a result of brittle fracture of the dislocation-free zone. As the cracks moved into the thicker part of the specimen, they often propagated in a zigzag manner by emitting dislocations on two alternating slip planes. In order to understand the observed behavior of dislocations near the crack tip, the elastic interaction between a crack and a dislocation must be examined and the force on the dislocation close to the crack tip estimated. It was shown that a dislocation was generated if the applied stress was sufficiently large that the force on the dislocation was repulsive down to the core distance from the crack tip. The emission condition was expressed in terms of a critical stress intensity factor for dislocation generation. The magnitude of the critical stress intensity factor for dislocation generation relative to the critical stress intensity factor for brittle fracture could be used as a criterion for determining the ductile versus brittle behavior of a metal. Once generated, the dislocations moved out of the crack tip area because of a repulsive force. The dislocations came to rest at a point where the repulsive force was balanced by the lattice friction. The dislocation-free zone thus created would restore the elasticity to the crack tip region and it was possible to define a local stress intensity factor. Without the dislocation-free zone, the local stress intensity factor approached zero and the crack did not propagate because there was no elastic energy release associated with the propagation. The dislocation-free zone model was also applied to describe the mechanisms of ductile fracture as well as the ductile-to-brittle transition. A mechanism of brittle fracture in the presence of crack tip deformation was also proposed from the model. The shielding of a crack tip by the dislocations in the plastic zone was discussed in terms of the local stress intensity factor. Finally, the local stress intensity factors were determined experimentally from the observed crack tip geometry and were compared with the values predicted by theory.

Distribution function of dislocations and condition of finite stress for the dislocation-free zone model of fracture

Distribution function of dislocations and condition of finite stress for the dislocation-free zone model of fracture

Dislocation-free zone model of fracture comparison with experiments

The dislocation-free zone (DFZ) model of fracture has been extended to study the relationship between the stress intensity factor, extent of plastic deformation, and crack tip geometry of an elastic-plastic crack as a function of applied stress. The results show that the stress intensity factor K decreases from the elastic value at first slowly, then goes rapidly to zero as the number of dislocations in the plastic zone increases. The crack with a zero stress intensity factor has its crack tip stress field completely relaxed by plastic deformation and hence is called a plastic crack. Between the elastic and plastic cracks, a wide range of elastic-plastic cracks having both a stress singularity and a plastic zone are possible. These elastic-plastic cracks with a DFZ are predicted if there is a critical stress intensity factor Kg required for the generation of dislocations at the crack tip. The expression for Kg is obtained from the crack tip dislocation nucleation model of Rice and Thomson. In most metals, the magnitude of Kg is less than the critical stress intensity factor for brittle fracture Kc. The values of K are determined from electron microscope fracture experiments for various metals and they are found to be in good agreement with the Kg predicted from the model. It is concluded that for most ductile and semibrittle metals, the mechanism of dislocation generation is more important than the fracture surface energy in determining the stress intensity factor at the crack tip.

Dislocation-free zone model of fracture

Recent observations by electron microscopy have shown that a dislocation-free zone (DFZ) is present between the crack tip and the linear pileup of dislocations in the plastic zone. A singular integral equation is formulated to describe the equilibrium configuration of the dislocations. The distribution function of the dislocations is obtained in terms of elliptic integrals. The condition of compatibility and the elastic stress intensity factor at the crack tip are also derived. At the crack tip the stress varies as 1/√r. It is found that the stress intensity factor K is approximately a function of the length of the DFZ, whereas the externally applied stress approximately determines the length of the plastic zone. Based on the mechanism of dislocation generation at the crack tip proposed by Rice and Thomson, it is shown that the formation of the DFZ is anticipated when the stress intensity factor K is less than a critical stress intensity factor Kg defined for a spontaneous generation of dislocations. The magnitude of Kg relative to the critical stress intensity factor for brittle fracture Kc determines the brittle-ductile nature of a material.