Circular Dislocation Loops Research Articles

Anisotropic Elasticity of a Prismatic Dislocation Loop in a Hexagonal Crystal

Elasticity solutions for the stress and displacement fields around a finite circular prismatic dislocation loop in the basal plane of a hexagonal crystal have been obtained by taking into account crystal anisotropy. Following the method proposed by Elliott for the problems of axial symmetry in hexagonal crystals, the Hankel transforms of two stress functions were determined from the boundary conditions appropriate for a circular prismatic loop. From these stress functions, the components of stress and displacement have been derived in terms of a series of definite integrals which could in turn be expressed in terms of elliptic integrals. The expressions of anisotropic elasticity approach those of isotropic theory derived by Kroupa when the conditions of elastic isotropy are imposed on the elastic constants. Numerical calculations have been carried out specifically for the case of graphite. The over-all effect of crystal anisotropy is to reduce the magnitude of the stresses and the displacements around the loop, except for the component of displacement normal to the loop plane which is increased by about a factor of 2.

Circular prismatic dislocation loop in a two‐phase material

AbstractThe elastic fields of a circular prismatic dislocation loop in a two‐phase material are evaluated to arrive at the elastic energy and Peach‐Koehler force. The results can be expressed in terms of complete elliptic integrals. The asymptotic forms reveal that the interaction between the prismatic loop and the second phase is not necessarily restricted to simple repulsion or attraction, and that for certain combinations of the elastic constants stable or unstable equilibrium positions are possible.

Circular edge dislocation loop in sandy soil

Stresses and displacements in semi-infinite sandy soil due to a circular dislocation loop have been obtained. Some numerical results are also presented.

Electron Microscope Image Contrast from Dislocation Loops

AbstractExperimental electron microscope images of dislocation loops in molybdenum have been compared to theoretical loop images displayed in the form of computer simulated micrographs. The exact displacement field around a circular edge dislocation loop has been calculated using isotropic elasticity theory; this displacement field together with the two‐beam dynamical theory of electron diffraction contrast have been used to compute image intensities. Specific results are given for loops of radius ≈ 250 Å and ≈ 25 Å. Their contrast has been studied over a narrow range of the numerous variables which are known to influence image features. It has been found that the general topographical character of certain prominent experimental images is qualitatively accounted for by the simple “elastic” defect model with the two‐beam theory of electron diffraction.

On the annealing of damage produced by boron ion implantation of silicon single crystals

Abstract Using the techniques of Rutherford scattering, electron microscopy and infra-red absorption, the damage produced by implantation of 100 to 400 keV boron (11B) ions in silicon single crystals, held at room and liquid nitrogen temperatures, has been examined, and the annealing behaviour investigated. The rapid annealing of disorder in the range between room temperature and 325°C, as observed by the Rutherford scattering of channelled analysing particle beams, correlates with a shrinkage of the 1. 8μ divacancy infra-red absorption band, and with a coarsening of some of the defects observed in the electron microscope. Amorphous zones observed in samples implanted with 1015 boron ions/sq. cm. at liquid nitrogen temperature do not anneal until temperatures in the vicinity of 450 °C are attained. The consequences of aggregation of point defects, first as small crystallographic clusters, for temperatures in the vicinity of 300 °C, and then as large circular dislocation loops following isochronal anneals ...

Elastic Fields and Energies of a Circular Edge Disclination and a Straight Screw Disclination

The displacement and stress fields for a circular edge disclination are derived from Green's tensor function. The elastic strain energy of this circular edge disclination is calculated and compared with the energy of two different kinds of circular dislocation loops. Finally, the stress field and elastic strain energy of a straight screw disclination is computed and is compared to the case of a straight screw dislocation.

Numerical Analysis of Deformation Twin Behavior. Small Static Twin Lamellas

A detailed numerical analysis has been made of the energies and equilibrium configurations of deformation twin lamellae under various conditions of applied stress and twin boundary energies. Exact expressions have been used for the self-energies and interaction energies associated with the various partial dislocations comprising the twin lamellae using isotropic elasticity theory developed for circular glide dislocation loops. It has been found that as the applied stress and/or twin boundary energy is increased, the size of the stable twin lamella as well as the number of dislocation loops comprising it decreases. Accompanying this size decrease is a correspondingly smaller energy for activation. In this respect it is found that the critical step in the formation of a twin lamella is the activation of the first loop which requires the formation of a stacking fault. Subsequent loop nucleation on adjacent planes, which increases the twin thickness, occurs spontaneously.

Junction reaction hardening by dislocation loops

Abstract The movement of a flexible dislocation line through a random array of circular dislocation loops has been studied, using a digital computer, on the basis of the line tension approximation. The dislocation moves on a {111} plane of a face-centred cubic crystal and interacts with {110} prismatic loops by the formation of short junction reactions at the points of intersection. The critical stress required to propagate the dislocation through the loops is μb/1·75L, where L is the average spacing of the reactive loops cutting the glide plane (one-third of all the loops are reactive). This result is independent of the radius and sense of the loops and the edge-screw character of the glide dislocation. It is slightly increased when the loops are distributed more uniformly, in disagreement with the factor two reduction predicted by Westmacott et al: (1966). The present estimate of loop hardening is comparable with the tetragonal distortion model of Fleischer (1962), and is in agreement with the measured hardening of quenched aluminium. It is almost identical with the value for a random array of simple obstacles with a flat spectrum of strengths. The present result is an order of magnitude larger than the previous estimate by Friedel (1963), which was based on the zig-zag model. This model is applicable only at zero stress, since it neglects the additional pinning points that are encountered by a dislocation when it bows under stress. The critical stress may be increased if the mobility of the loops allows the formation of longer junctions.

Dislocation Configurations in Cylindrical Coordinates

General expressions are derived for the stress components due to an arbitrary continuous distribution of dislocations prescribed in a cylindrical coordinate system. Results are obtained for the stress field of a helical dislocation of uniform shape with the Burgers vector along its axis. The additional energy arising from the interaction of a point defect with the helix is obtained and the behavior of the defect examined. It is also shown how results for circular dislocation loops are readily obtained from the general expressions and in particular the problem of a rotational loop, which is formed by twisting the two faces of the slip surface relative to each other, is solved.

Circular dislocation pile-ups

Abstract A model of concentric circular dislocation loops is used to examine the Hall-Petch relation. For a given critical stress concentration at the grain boundary, the Hall-Petch slope produced by this model is higher than that from either single or double-ended pile-ups of infinite straight dislocations. The minimum stress required to maintain a certain number of circular dislocations is compared with the stress necessary for the continuous distribution of these dislocations in a circular slip zone.

Energy of polygonal and elliptical dislocation loops

AbstractA general expression for the elastic strain energy of an arbitrary N‐sided planar polygonal dislocation loop is obtained from the stress field of dislocation segments using isotropic elasticity. This expression can be used to approximate the energy of any curved dislocation loop to any degree of accuracy. For example, the energies of regular polygonal dislocation loops and irregular polygonal dislocation loops inscribed in several ellipses are calculated and, by extrapolating to N = ∞, are shown to agree with the energies of circular and elliptical dislocation loops for which simple analytical expressions are available. The equilibrium shape of elliptical loops and the possibility of reorientation of these loops along a glide cylinder are consistent with experimental results in h.c.p., b.c.c., and f.c.c. crystals and in α‐uranium. The effect of anisotropy in h.c.p. metals is calculated for large elliptical loops in the basal plane.

Elastic interaction of dislocation loops and point defects

The elastic interaction energy of a circular dislocation loop with interstitial atoms and vacancies characterized as dilatation or relaxation centres is calculated. Further, the forces which the dislocation loops exert on point defects through elastic interaction are discussed.

Interaction of dislocation loop with free surface

The stress due to a circular dislocation loop with Burgers vector normal to the plane of the loop in half-space is calculated. The forces by which the surface attracts the dislocation loop are calculated and finally the possible applicability of the given solution to foils is discussed.

The energy of circular dislocation loops in anisotropic hexagonal plates

Analysis of the energy of circular dislocation loops in thin plates is extended to anisotropic hexagonal materials. The derived expressions are used in computations for a group of common hexagonal crystals. Modification due to anisotropy of the thinning effect on loop energy is small for most common hexagonal metals. For Cd and Zn the modification is slightly greater. Two distinct types of high anisotropy produce large modifications, depending on the algebraic values of three anisotropy parameters, λ, C and η. The first may be demonstrated with materials having large values of λ, C and η. In such instances the ratio of loop energies W W inf ., where W inf. is the loop energy in an infinite medium, is greater than that in isotropic materials. Thus, the effect of high anisotropy opposes the effect of thinning on loop energy. Graphite is an example. The second type may be demonstrated in materials with negative C and with λ and η less than unity. The calculated values are less than those obtained on the assumption of isotropy. The effect of high anisotropy therefore enhances the effect of thinning.

The energy of circular dislocation loops in thin plates

Expressions for the elastic energy of a circular dislocation loop lying in the central plane of an isotropic thin plate are formulated. Computed results are given. These results show that, as might be expected, the thickness effect becomes sensible when the plate thickness approaches the dimension of the loop. The effect is greater for loops with Burgers vector normal to the plane of the loop than those with Burgers vector parallel to it. An example which is presented for loops with Burgers vector normal to the plane of the loop shows that for a plate of half-thickness h = 1000 r 0 (where r 0 is the inner cut-off length) the loop energy is overestimated by 100, 60, 17, 0.07 and ~0 per cent for loop radii of 10 5, 10 4, 10 3, 10 2 and 10 r 0, respectively. For loops with Burgers vector parallel to the plane of the loop and Poisson's ratio equal to 1 3 , the corresponding percentages are 74, 38, 6, 0.03 and ~0.

The interaction between prismatic dislocation loops and straight dislocations. part I

Abstract The elastic interaction between a prismatic circular dislocation loop and a straight dislocation moving in its glide plane in an infinite isotropic medium is studied. The interaction is discussed in terms of the total interaction energy, the total force on the dislocation and on the loop and of forces on line elements of the dislocation and the loop.

The spacing of prismatic dislocation loops

Abstract The stresses round a circular prismatic dislocation loop are determined and used to calculate the spacings between such loops in a row generated from a source of internal stress. The results are consistent with the available experimental observations.