Rectilinear Boundaries Research Articles

Boundary correlation functions and interface formation in two-dimensional solids

Two-point correlation functions at the rectilinear boundary of a two-dimensional crystal half-space on a reference lattice are studied for the case of free boundary conditions. We show that the power of the decay law of the positional correlation function at the surface satisfies η surf = 2η bulk, within the approximation that dislocation pair correlations are taken into account. Dislocations at the boundary form a quasi-one-dimensional plasma for T ≠ 0 and this may lead to a dynamic roughening of the surface. Formation of dynamic interfaces in grain boundaries and near free boundaries is discussed in terms of 1-dimensional charged gas models with long-range interaction. Melting of boundary layers at T surf c< T bulk c is suggested to be possible only if topological effects of moving dislocations on the reference lattice are taken into account.

Qualitative study of the boundary correlation function in the two-dimensional XY-model

Abstract The two-point correlation function at the rectilinear boundary of the two-dimensional half-space is studied for the XY-model for the case of free boundary conditions. It is shown that defects of vortex-type at the boundary form a one-dimensional plasma for T ≠ 0, but that this does not lead to exponential decay of the boundary correlation function for T

Motion of a Rigid Cylinder Between Parallel Plates in Stokes Flow: Part 1: Motion in A Quiescent Fluid and Sedimentation.

A general numerical scheme for solution of two-dimensional Stokes equations in a multiconnected domain of arbitrary shape [1,2] is applied to the motion of a rigid circular cylinder between plane parallel boundaries. Numerically generated boundary-conforming coordinates are used to transform the flow domain into a domain with rectilinear boundaries. The transformed Stokes equations in vorticity-stream function form are then solved on a uniform grid using an iterative algorithm. In Part I coefficients of the resistance matrix representing the forces and torque on the cylinder due to its translational motion parallel or perpendicular to the boundaries or due to rotation about its axis are calculated. The solutions are obtained for a wide range of particle radii and positions across the channel. It is found that the force on a particle translating parallel to the boundaries without rotation exhibits a minimum at a position between the channel centerline and the wall and a local maximum on the centerline. The resistance matrix is utilized to calculate translational and angular velocities of a free particle settling under gravity in a vertical channel. It is found that the translational velocity has a maximum at some lateral position and a minimum on the centerline; the particle angular velocity is opposite in sign to that of a particle rolling along the nearer channel wall except when the gap between the particle and the wall is very small. These results are compared with existing analytical solutions for a small cylindrical particle situated on the channel centerline, and with solutions of related 3-D problems for a spherical particle in a circular tube and in a plane channel. It is shown that the behavior of cylindrical and spherical particles in a channel in many cases is qualitatively different. This is attributed to different flow patterns in these two cases. The motion of a spherical particle in a circular tube has qualitative and quantitative features similar to those for a cylindrical particle in a plane channel.

Criteria for the differentiation of en echelons and hackles in fractured rocks

Straight en echelons and hackles are two morphological aspects of cracked surfaces in rocks. Although they often resemble each other, they result from different modes of fracture. Eight criteria enabling their differentiation are presented. En echelons commonly maintain consistent orientations of smooth surfaces with respect to the parent fracture, and uniform azimuths in individual outcrops along the rectilinear boundaries of the parent fractures. Hackles, generally appear as rough surfaces that do not follow such constant orientations and uniform azimuths, and they propagate from circular perimeters. Whereas en echelons characteristically alternate intermittently with steps, may have plumose markings on their surfaces, and have consistent length/width ratios, hackles do not show any of these features. Hackles are formed on a planar surface inclined to the parent fracture; such an angular relationship does not occur between en echelons and their parent joints. Fracture surface morphology also allows for differentiating between natural joints and man-made artificial fractures, since in natural joints the ratio mirror radius/average length of hackles is high, whilst it decreases considerably in man-made artificial fractures.

Spinodal decomposition in InGaAsP epitaxial layers

In GaAsP epitaxial layers emitting in the 1.25−1.37 μm wavelength region have been evaluated using transmission electron microscopy, X-ray diffraction and photoluminescence. Electron diffraction patterns show elongation of the 400 spots in the <> direction and diffuse intensity on the low angle as well as on the high angle side. This observation is consistent with the extra scattering observed close to the 400 Bragg peaks by X-ray diffraction. Furthermore, microstructures exhibit quasi periodic fine scale structure having a wavelength of ≈ 150 Å, rectilinear boundaries and a weakly developed basket-weave pattern. These observations are compatible with the occurrence of spinodal decomposition in the layers. In addition, photoluminescence studies indicate that the presence of the fine scale composition modulations does not adversely affect the luminescence properties of these materials.

Finite element formulation of radioisotope diffusion in metal grain textures

The problem of grain boundary diffusion of radiosotopes is analysed by the finite element method for a plane semiinfinite situation. Mathematically this problem consists of solving the time-dependent linear diffusion equation with a fixed Dirichlet condition at the finite rectilinear boundary. But peculiarities arise because of the existence of a thin straight region (the grain boundary) of about 10 −8 cm width separating two grains, whose conductivity is five orders of magnitude greater than that of the metal grains. Reasonable solutions were obtained by using quadrilateral Lagrange elements for the grains on the central finite region and one-dimensional linear elements for the grain boundary. The main purpose of the paper is to establish the underlying ideas for a natural finite element formulation of this kind of problem, based on a variational approach.

On the domain structure of (Au,Cu)3Zn alloys

In order to investigate the high-temperature phase H, two methods were applied: addition of copper and deviation of the composition from stoichiometric Au3Zn. A model for the two different types of interface between periodically ordered domains is presented and a trace analysis of rectilinear boundaries is carried out. Extra spots in the diffraction pattern are predicted by treating the formation of periodic domains as a twinning in the tetragonal lattice.

Partially loaded elastic plates

A general procedure for the analysis of elastic plates under partial loading is described. The method applies to any elastic plate with rectilinear boundaries. Loads distributed over a circular region and a rectangular region are investigated.

On the two-dimensional groundwater movement

A simple technique to solve the linearized two-dimensional Boussinesq equation in a nonsteady flow field with arbitrary shape is presented. According to this technique the reservoir boundaries are replaced by small sources emitting attenuating waves. The characteristic content of each source depends on the shape of the flow field. The variation of the free surface at any point of the flow field results from the interaction of all the waves arriving at that point. This technique is applied to a flow field which is confined by rectilinear and curvilinear boundaries. The results indicate satisfactory proximity to those obtained by numerical analysis.

Potential flow of a nonviscous incompressible liquid around a profile, in the presence of rectilinear boundaries

Potential flow of a nonviscous incompressible liquid around a profile, in the presence of rectilinear boundaries

Nonlinear filtration of a liquid subject to a power resistance law

A method of solving the plane problem for the nonlinear filtration of an incompressible liquid is proposed, assuming a power-type law of resistance, with rectilinear boundaries of the region of motion. The method is demonstrated for the case of filtration through a uniformlyporous wedge.

Symmetrisierung von schwingenden Membranen durch Identifikation

A method of domain symmetrization, based on identification of rectilinear boundary segments, is sketched in some examples. It allows to prove some simle extremal properties of the first eigenvalue of a symmetric membrane.

Asymptotic Analysis Of Extensive Crack Growth Parallel To Free Boundary

Asymptotic techniques are introduced for modelling interaction between a crack and a parallel free boundary. For long cracks, the part of the material between the crack and the boundary is represented as a beam (plate). The area of the crack opening is calculated directly. The stress intensity factors are calculated by matching the beam calculations with Zlatin and Khrapkov's solution for a semi-infinite crack parallel to the boundary. This allows the calculation of two main asymptotic terms. Small cracks are treated as being subjected to the external load and an additional one reflected from the boundary. The additional load is assumed to be uniform over the crack size. For intermediate distances between the crack and the boundary, interpolation formulae are suggested. INTRODUCTION Crack growth in the presence of a parallel rectilinear boundary is a situation important for various applications. The full solution for such problems can be obtained only numerically. However, if the crack is situated too close to the boundary, numerical solutions can lose accuracy. For example, Murakami [1] presents two numerical solutions for the case of uniformly loaded crack parallel to the free boundary. For long cracks, these solutions differ more than 40% (see Figure 2 in this paper). Therefore, there is a strong requirement for asymptotic solutions that are free of such shortcomings and give relatively simple analytical expressions, which are very useful when the behaviour of many cracks is modelled. The classical beam approach consists in considering the material between the crack and the boundary as a beam under the given load (eg, Rice [2]). This representation makes it possible to calculate the change of the elastic energy due to an infinitesimal step of the crack propagation. However, this energy approach does not allow separating the stress intensity factors (SIFs). Such solutions can be found by matching the beam considerations with the problem for a semi-infinite crack playing the role of inner asymptotics. Transactions on Engineering Sciences vol 6, © 1994 WIT Press, www.witpress.com, ISSN 1743-3533 624 Localized Damage BEAM ASYMPTOTICS Consider the plane problem for a crack parallel to a boundary of an isotropic elastic half-plane (Figure la). Let the distance, /?, to the boundary be much less than the crack length, 21. From the large-scale point of view, the material between the crack and the boundary can be treated as a beam (plate for the plane strain case) subjected to the given external load (Figure Ib). The expression for the beam deflection gives an asymptotic approximation (with respect to the small parameter s=h/l) for displacements of the crack surfaces. This is valid for all the points of the crack situated far from its ends compared to h. Therefore this is the solution for the outer region of the considered problem. The solution for the inner region can be obtained by considering the crack tip as a semi-infinite crack (Figure Ic), since /»/?. This representation is valid for the points situated far enough from the other tip of the crack. The matching region for these two asymptotics consists of the points, x, such that h«x«l.

New points of view in the design of electron guns for cylindrical beams of high space charge

The paper outlines the influences of the anode aperture, disregarded in the theory of the Pierce gun, which lead to an unsatisfactory functioning of such electron guns, particularly in the case of a high perveance. These influences are taken into consideration by imposing a rectilinear boundary on the beam, so that the space charge potential can be simply determined in the electrolytic tank. For this purpose, the anode is made tubular and sharp edged, and besides the Wehnelt electrode, an additional electrode of substantially the same potential is introduced, being mounted in the immediate vicinity of the anode aperture. Simple electrode forms and their favourable positions are given for a wide range of dimensioning. By the introduction of corrections, the dimensioning rules are extended to high perveance values. The magnitude of the current leaving the assumed boundary of the beam because of its initial thermal velocity is determined.

Book Review: Two-Dimensional Potential Problems Connected with Rectilinear Boundaries

Book Review: Two-Dimensional Potential Problems Connected with Rectilinear Boundaries

Two Dimensional Potential Problems connected with Rectilinear Boundaries. By B. R. Seth. Pp. vi, 124. Rs. 2.12. 1939. Lucknow University Studies, 13. (University of Lucknow)

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Two-Dimensional Potential Problems

The solution of problems in two-dimensional potential theory depends, generally speaking, on the discovery of a conformal transformation suitable for the particular region involved, and the degree of success in the applicability of the method seems to be seriously limited by the complexity of the transformation. Only one serious attempt seems to have been made, by W. G. Bickley*, to formulate a general theory coordinating the different types of boundary value problem for general and inclusive classes of curves, but Bickley's analysis is somewhat involved and not very suitable for certain types of problem. In a re-examination of the subject on the basis of certain suggestions made to me by Prof. Livens, I have discovered what appears to be a much simpler and equally general method of formulating directly the solution for a comprehensive class of problem dealing with simple closed cylinders with curved or rectilinear boundaries, and including all those cases for which solutions are known. In its theoretical aspects the method seems to be perfectly general, but to put it into practice it is necessary to determine explicitly the particular form for the conformal transformation of the space outside the cross-section of the cylinder to the space outside a circle with the points at infinity coinciding.