Distributed Dislocation Dipole Research Articles

Development of the distributed dislocation dipole technique for the analysis of closure of complex fractures involving kinks and branches

Abstract This paper presents the development of the distributed dislocation dipole technique (DDDT) for the analysis of straight, kinked and branched cracks where parts of the cracks may close during loading. The method has been developed for plane problems. Crack cases in which closure occurs are analyzed by reformulating the Bueckner's principle, taking into account the contact stresses at the contacting portions of the crack surfaces. Stress intensity factors corresponding to opening and the in-plane sliding mode of deformation at the crack tips are computed. Several test cases involving straight, kinked and/or branched cracks where parts of the cracks undergoes crack surface closure when subjected to the outer loading are analyzed. The results obtained from the DDDT are compared to those obtained from a Finite Element Method (FEM) analysis of the same crack cases. This comparison shows that the computation of stress intensity factors for the cases involving crack surface closure are less accurate than those for fully open crack cases. However, for the cases under consideration, the stress intensity factors were still computed with a maximum difference of approximately 2 per cent compared to the FEM calculations if Jacobi polynomial expansions of at least the twelfth order were used to represent the crack surface opening and sliding displacements. In most cases under consideration, sixth order Jacobi polynomial expansions were sufficient to obtain results within that margin of deviation.

Development of a distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half-plane

A distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane has been developed. The dipole density distribution is represented with a weighted Jacobi polynomial expansion where the weight function captures the asymptotic behaviour at each end of the crack. To allow for opening and sliding at crack kinking and branching the dipole density representation contains conditional extra terms which fulfills the asymptotic behaviour at each endpoint. Several test cases involving straight, kinked and branched cracks have been analysed, and the results suggest that the accuracy of the method is within 1% provided that Jacobi polynomial expansions up to at least the sixth order are used. Adopting even higher order Jacobi polynomials yields improved accuracy. The method is compared to a simplified procedure suggested in the literature where stress singularities associated with corners at kinking or branching are neglected in the representation for the dipole density distribution. The comparison suggests that both procedures work, but that the current procedure is superior, in as much as the same accuracy is reached using substantially lower order polynomial expansions.

A Natural Transition Between Equilibrium Patterns of Dislocation Dipoles

The equilibrium configurations of a row of uniformly distributed dislocation dipoles are first studied. The analysis is then generalised to study dipoles in two-dimensional rectangular periodic lattices. By examining the stability of the equilibrium configurations we find that the system may undergo a natural transition from the Taylor lattice to a row of dipole walls. This bifurcation may be involved in the transition from channel-vein to persistent slip band (PSB) structures in the early stage of metal fatigue.

Numerical Simulation of Crack Tip Behavior under Fatigue Loading

Fatigue damage is a localized phenomenon controlled by the near-tip crack behavior. This paper presents an application of a dislocation distribution technique to the simulation of crack tip behavior under fatigue loading. A centre-cracked tension specimen under uni-axial fatigue loading is used in the study. Crack opening and plastic deformation around the crack tip are simulated by distributions of dislocation dipoles in crack plane and four inclined planes ahead of the crack tip. Climb dislocation dipole is used to model the opening and closing of the crack while glide dislocation dipole is used to simulate the backward and forward slip in the inclined planes during loading and unloading of the fatigue cycle. Stress field around the crack tip is obtained by the superposition of the contributions of the applied external load and the distributed dislocation dipoles. Correct boundary conditions of the model are achieved by employing a quadratic programming technique to minimize a properly constructed non-negative object function. It is found that the simulated crack closure variations under the constant amplitude fatigue load agree well with the result of a previously developed modified strip yield model with an appropriate constraint factor.

Characteristics of short fatigue crack growth in the vicinity of a low angle grain boundary

The influence on the crack growth behaviour of a short edge crack under fatigue due to different configurations of a nearby located low angle grain boundary is investigated under quasi-static and plane strain conditions. The geometry is modelled by distributed dislocation dipole elements in a boundary element method approach, and the plasticity is described by discrete dislocations, located along specific slip planes in the material. The crack is assumed to grow in a single shear mechanism due to nucleation, glide and annihilation of dislocations in the material. It was found that both the nature of the dislocations in the grain boundary, the distances between them and the placement of the dislocations in the grain boundary with respect to preferred slip plane directions strongly influenced the growth behaviour. It was found that, a boundary consisting of dislocations with Burgers vector pointing inwards into the material attracted the dislocations in the plastic zone. This resulted in lower growth rates as compared with a boundary consisting of dislocations having a Burgers vector pointing outwards of the material.

Influence from geometrical features on the growth rate of a micro-structurally short fatigue crack

The influence on the crack growth rate on a micro-structurally short edge crack subjected to fatigue loading from changes in crack length, distance to grain boundaries and applied load has been investigated. The crack is assumed to grow in a single shear mechanism due to nucleation, glide and annihilation of dislocations along preferred slip planes in the material. The external geometry is modelled by distributed dislocation dipole elements in a boundary element approach under quasi-static and plane strain conditions. The evolving plasticity is described by individual discrete dislocations along a slip plane emanating from the crack in the crack direction. The crack growth rate is shown to be controlled by the plasticity, which in turn is controlled by geometrical parameters in combination with the external load.

Crack growth rates for short fatigue cracks simulated using a discrete dislocation technique

The influence on the crack growth rate for a short edge crack under fatigue loading due to changes in crack length, grain size, load range and grain boundary configuration, is investigated under quasi-static and plane strain conditions. The geometry is modelled by distributed dislocation dipole elements in a boundary element method approach and the plasticity is described by discrete dislocations. The crack is assumed to grow due to nucleation, glide and annihilation of dislocations along slip planes in the material in a single shear mechanism. The results of the investigation are compared to typical growth rates for long cracks and it is found that the increase in growth rate due to a prescribed stress intensity factor range was much less pronounced as compared to what holds for long cracks.

Influence of fatigue load range on the growth of a microstructurally short edge crack simulated by a discrete dislocation formulation

Quasistatic propagation of a short edge crack, located within one grain of a bcc material has been studied using a discrete dislocation technique. The geometry is modelled by distributed dislocation dipole elements and the plasticity by discrete dislocations. The crack is assumed to grow through a single shear mechanism due to nucleation and annihilation of dislocations along preferred slip planes in the material. The change in growth rates due to different load cycles and due to a small single overload was investigated. It was found that the growth rates were strongly influenced by the applied load cycles, and that a single overload affects the growth differently depending on the number of cycles prior to the overload.

A study of the influence of grain boundaries on short crack growth during varying load using a dislocation technique

The propagation of short cracks in the neighbourhood of grain boundaries have been investigated using a technique were the crack is modelled by distributed dislocation dipoles and the plastic deformation is represented by discrete dislocations. Discrete dislocations are emitted from the crack tip as the crack grows. Dislocations can also nucleate at the grain boundaries. The influence on crack growth characteristics of the distance between the initial crack tip and the grain boundary has been studied. It was found that crack growth rate is strongly correlated to the dislocation pile-ups at the grain boundaries.

A tool to model short crack fatigue growth using a discrete dislocation formulation

A method is presented that combines the modelling of cracks by distributed dislocation dipoles with developing plasticity represented by discrete dislocations moving along slip bands. Crack growth is due to the emission of dislocations from the crack tip along preferred slip planes. Eventual annihilation of dislocations occurs by reunion with the corresponding displacement steps of the crack surface. Crack surface overlap is not allowed. The equilibrium state for each load increment is solved iteratively, allowing various crack geometries. The method is applied to the problem of a short edge crack growing in mode I due to fatigue loading. It is shown that the development of a local plastic zone and the propagation of the crack can be monitored in detail.

Modelling cracks in finite bodies by distributed dislocation dipoles

The method of continuous distribution of dislocations is extended here to model cracks in finite geometries. The cracks themselves are still modelled by distributed dislocations, whereas the finite boundaries are represented by a continuous distribution of dislocation dipoles. The use of dislocation dipoles, instead of dislocations, provides a unified formulation to treat both simple and arbitrary boundaries in a numerical solution. The method gives a set of singular integral equations with Cauchy kernels, which can be readily solved using Gauss–Chebyshev quadratures for finite bodies of simple shapes. When applied to arbitrary geometries, the continuous distribution of infinitesimal dislocation dipoles is approximated by a discrete distribution of finite dislocation dipoles. Both the stress intensity factor and the T‐stress are evaluated for some well‐known crack problems, in an attempt to assess the performance of the methods and to provide some new engineering data.

Computer Modelling of Dynamically-Induced Dislocation Patterning

ABSTRACTThe evolution of a random and initially homogeneous distribution of parallel and infinitely extended edge dislocations is studied by using elastic energy minimization without and in presence of a periodic external stress, τa. During the energy minimization without external stress (relaxation), randomly distributed dislocation dipoles are formed whereas, when the external stress is acting, the dislocations condense in walls. We investigated the spatial periodicity of this microstructure, λ, as a function of, τa, and of the total dislocation density. The elastic energy of the stress-induced microstructure is found to be comparable to the value obtained by relaxation. Thereby, emphasis is given to the dynamical character of patterning. A phenomenological model has been developed, explaining the correlation between λ and τa found in the simulations and comparing favorably with existing experimental data.