Gunther Schoeck

Dislocation emission from crack tips

Abstract The emission of a dislocation from a crack tip is described in two dimensions by the build-up of a continuous distribution of infinitesimal dislocations ahead of the crack tip. The shape of the incipient dislocation is actually determined by the atomic potential in the glide plane but is approximated here by a truncated arctan (x/w) function with ‘dislocation width’ w as adjustable parameter. The elastic self-stress in the glide plane balanced by the atomic interaction across the glide plane and the image stress due to the crack surface can be calculated and the energy of the configuration as a function of the stress intensity K caused by an externally applied stress can be determined. For small K there exists an energy minimum where the incipient dislocation is partially emitted, its complete emission occurring across an energy barrier. When K increases, the energy barrier decreases and vanishes at a critical K = K e. This situation corresponds to spontaneous emission in two dimensions. The real...

The generalized Peierls-Nabarro model

Abstract In the original Peierls-Nabarro model the core structure of a dislocation is determined as the solution of an integrodifferential equation. This equation describes the balance between the forces resulting from the deformation of two elastic half-spaces and from a one-dimensional periodic lattice potential acting across the glide plane. A method is described here which allows the core structure and core energy to be obtained for a straight dislocation with arbitrary Burgers vector in an arbitrary glide plane in a crystal of arbitrary anisotropy, for which the displacement potential is represented by a two-dimensional Fourier series. This is accomplished by describing the internal displacements by appropriate trial functions with a set of free parameters whose value is then determined by minimizing the total energy. The method is applied to obtain the core configuration of a screw dislocation dissociated in a {111} plane of a f.c.c. lattice.

The core structure, recombination energy and Peierls energy for dislocations in Al

Abstract In the framework of the Peierls model generalized to two dimensions the dissociation of a dislocation in the {111} plane of a fcc lattice into two Shockley partials is studied by a variational procedure. Each partial is made up by a distribution of infinitesimal dislocations with a density obtained by a superposition of three closely spaced Lorentz peaks of adjustable height, width and separation. The atomic misfit energy in the glide plane is obtained from the γ surface represented by a two-dimensional Fourier series showing the symmetry of the {111} plane. The procedure is applied to Al for which a set of γ values in the {111} plane has been obtained by Hartford et al. using ab-initio electron density functional theory. The dissociation width of the edge dislocation is found to be 0.74 nm, which almost agrees with the experimental value of about 0.8 nm. The screw dislocation is not split in the usual way but rather shows a widely extended core with some edge components. The energy to compress the core to a pure screw dislocation is ΔE c = 0.042 eV/b. The Peierls energy ΔE p can be evaluated by numerical summation of the energy at the atom positions. However, contrary to previous treatments the core configuration has been relaxed and hence changes in elastic energy contribute to ΔE p. The result is not quite unambiguous.

The formation of dislocation loops at crack tips in three dimensions

Abstract The formation of finite dislocation loops in three dimensions (3D) at a crack tip in the crack plane under mode II (or mode III) loading is considered. As suggested by Schoeck (1990) the formation of a dislocation in 2D is treated in the Peierls model where the incipient dislocation is described by a continuous distribution of parallel infinitesimal dislocations. The resulting atomic displacement ahead of the crack tip is determined by the interatomic potential and is of the general type arctg(x/w) where we treat the dislocation width w as adjustable parameter. The finite incipient loop in 3D is modelled by a distribution of infinitesimal dislocation loops (Kroupa 1962) but with varying Burgers vectors. The saddle-point configuration for the formation of rectangular incipient loops is obtained numerically as function of the stress intensity K by studying the energy contours in phase-space. The free enthalpy of activation to emission increases continuously when the stress intensity K e drops below...

Dissociated dislocations in the Peierls potential

Abstract Dislocations in close-packed lattices dissociated in two partial dislocations show a complex behaviour when moving over a periodic Peierls potential. Several equilibrium states are possible, and the energy barrier to motion of the combined system depends on the exact equilibrium separation of the partials, very sensitively so when they are strongly coupled. In this case the effective Peierls potential varies between zero and twice the potential for an individual partial. This mechanism may resolve an old discrepancy between low-temperature flow stress and the Peierls stress as deduced from internal friction measurements.

The Peierls energy revisited

The Peierls energy for dislocations, that is the variation in energy when the dislocation is moving is normally calculated by a procedure used by Peierls in 1940 and Nabarro in 1947. They sum the energy at the position of the atoms when the displacement profile of the dislocation is shifted rigidly. This procedure does not allow for any changes in shape and resulting changes on elastic energy. It is shown here that changes in shape do take place and lead to considerable changes in elastic and misfit energy. The two contributions are, however, of opposite sign and nearly cancel each other.

The line tension of dislocations in anisotropic media

Abstract The line tension of a dislocation determines its stability against small bow-outs from an equilibrium position. It is composed of a local term which includes a logarithmic singularity (avoided by some cut-off procedure), and a term which represents the interaction with the rest of the configuration. Whereas the prelogarithmic factor of the local term is known for elastic anisotropy, the interaction term has so far been determined only for isotropic media. In the present paper this interaction is calculated for arbitrary anisotropy for bow-outs of various shapes in straight dislocation lines, thereby establishing the final resolution of the elastic line-tension problem.

Dislocation core energies in the Peierls model

Abstract Within the framework of the Peierls-Nabarro model, dislocation core energies and core tensions are computed for Cu, Ni, Ag and Cu-Al alloys. Dissociation into partial dislocations and elastic anisotropy are taken into account. The splitting width and the core radius are less for screw than for edge dislocations. Core energies (tensions) are higher (lower) for screw than for edge orientation. Since this is opposite to the tendency of the elastic continuum contribution, the core terms have a remarkable influence on the shape of dislocation loops.

The Peierls dislocation: Line energy, line tension, dissociation and deviation

Abstract Atomic simulations of the dislocation core show that the atomic misfit is often concentrated in the glide plane. Instead of using a step function to describe the displacement as in a classical Volterra dislocation, a better description is obtained by a Peierls dislocation for which the displacement is assumed to have an arctg like shape. The slope in the center is determined by requiring that the total energy must be a minimum. The elastic energy can be expressed in closed form, and with the availability of high speed computing the atomic misfit energy in the glide plane can be calculated by standard numerical integration without any difficulties. When the Peierls model is extended to two dimensions the resulting line energy, line tension and resistance against bow-outs of straight dislocations can be obtained realistically without any adjustable parameters and the way that these quantities are influenced by the interplanar atomic potential can be studied. In addition to undergoing the well-known “dissociation”, a mixed dislocation may lower its energy by a “deviation” in which the displacement vector deviates from the direction of the crystallographic Burgers vector even when this runs along a path of lowest misfit energy.

The core structure of dislocations in Al: a critical assessment

Abstract The core structure of dislocations in Al has been the subject of various recent investigations. Whereas atomic simulations, which must rely on empirical potentials, are not very reliable, the generalized Peierls model using ab initio misfit energy values from electron density functional theory is able to give accurate results, when appropriate methods are used and in the calculations certain basic conditions are satisfied. It is shown that discrepancies in the results obtained in the literature result from inappropriate choice of trial functions in the variational problem for the dislocation energy and from incorrect treatments. Due to its inherent approximations the Peierls model can give only order of magnitude values for the Peierls energy.