Rounded Vertices Research Articles

Stresses in an elastic plane with periodic system of closely located holes

By the method of singular integral equations, we study a two-dimensional periodic problem of the theory of elasticity for a plane with infinite collection of closely located curvilinear holes. Special attention is given to the unified approach to the solution of the problems of stress concentration near holes with sharp or rounded vertices. In this way, we obtain the solutions of the problems of elastic interaction of elliptic, oval, and rhombic holes and physical flaws for arbitrary distances between the holes. By passing to the limit as the distance between the boundary contours tends to zero, we determine the stress concentration and stress intensity factors at the rounded and sharp tips of the corresponding bilateral notches in the elastic plane.

Diffusion Tensors for Processing Sheared and Rotated Rectangles

Image restoration and simplification methods that respect important features such as edges play a fundamental role in digital image processing. However, known edge-preserving methods like common nonlinear diffusion methods tend to round vertices for large diffusion times. In this paper, we adapt the diffusion tensor for anisotropic diffusion to avoid this effects in images containing rotated and sheared rectangles, respectively. In this context, we propose a new method for estimating rotation angles and shear parameters based on the so-called structure tensor. Further, we show how the knowledge of appropriate diffusion tensors can be used in variational models. Numerical examples including orientation estimation, denoising and segmentation demonstrate the good performance of our methods.

Effect of chip geometry on breakdown voltage of GaInN light-emitting diodes

Reverse leakage current characteristics of GaInN/GaN multiple quantum well light-emitting diodes (LEDs) with various chip geometries are examined. The effect of chip geometry on the reverse leakage current is negligible at a low voltage, but becomes apparent at a high voltage. The reverse breakdown voltage of LEDs decreases as the angle of vertex in the chip geometry decreases presumably because of a highly localised electric field strength near the vertex. This suggests that a chip geometry with a rounded vertex is suitable for reliable high-power LEDs.

Most Hydraulically Efficient Standard Lined Canal Sections

Cross-sectional dimensions of the most hydraulically efficient lined canals are evaluated based on an analysis of a generalized trapezoidal shape that reduces to two standard sections with rounded bottom vertices used in India, as well as to the commonly used trapezoidal section with sharp bottom vertices. The method of Lagrange multipliers is applied to find the dimensions of optimal sections when the only constraint imposed is that of uniform flow and normal depth and, in addition, when values of either channel side slope, bottom width, top width, or supply depth are specified as well. The analytic solutions obtained for the generalized trapezoidal section are shown to match known solutions for limiting cases including those for sections in the shape of sharp-cornered trapezoids, rectangles, triangles, and semicircles. Solutions presented will be useful for evaluating standard cross-sectional shapes used for lined canals in India, as well as other sections that can be obtained from the generalized trapezoid with rounded bottom vertices.

A model of the motion of the Nneuschwanstein bolide in the atmosphere

A new model of the atmospheric entry of the Neuschwanstein fireball has been developed. The fireball was photographed in Germany on April 6, 2002, and three fragments of it were found during a subsequent search in the territory predicted according to observations. In this study, the form of the meteoric body is assumed to be a cube with rounded vertices and edges. The estimated mass of the meteoric body at its entry into the atmosphere is close to the literature data obtained using seismic, acoustic, and infrasonic analysis. It is noted that the analysis of this fireball was for the first time made without using the photometric approach.

An approximate expression for the calculation of the electric field in vicinity of the cube electrode with rounded wedges

In this paper, an approximate analytical expression for calculation electric field in the centre of the rounded vertex of the metallic cube will be proposed. This expression is determined according to the numerical results obtained using different numerical methods. .

Yield criteria for porous ductile sheet metals with planar anisotropy under plane stress conditions

Modified yield criteria for porous ductile sheet metals with planar anisotropic matrices are investigated by a finite element analysis. A sheet model with a centric through-thickness hole is adopted for an analysis model. A new proposed plane stress yield function accounting for rounded vertices of a yield surface, in particular, near the uniaxial and equal biaxial loading directions based on experiments is employed for the matrix surrounding the hole. Strain hardening behavior of the matrix is considered in the present study. Two different definitions of the macroscopic yield stress are implemented here. Macroscopic yield loci and macroscopic stress–strain relationships under plane stress conditions based on the finite element computational results and those based on the modified anisotropic yield criteria are then compared and found to be in reasonable agreement.

3D Analysis of cold rolling using a constitutive model for interface friction

Full three-dimensional numerical analyses are carried out for the cold rolling of plates of finite width, to study the effect of the width spread during rolling. The contact and friction between roll and plate is modeled in terms of an interface constitutive model that accounts for the friction forces in the rolling direction as well as those in the transverse direction that give resistance to the width spread. At low normal pressures Coulomb friction is represented while at high normal pressure a yield stress limitation of the maximum tangential stress is incorporated, and slip as well as no slip is accounted for. Finite strain elasto-plasticity is applied for the plate material, using mostly isotropic hardening or in a few cases kinematic hardening to represent the effect of a rounded vertex on the yield surface. In addition, for a given plate thickness and degree of reduction the effect of different values of the roll radius and the effect of different values of the plate width are analysed.

Analysis of mode characteristics for equilateral triangle semiconductor microlasers with imperfect boundaries

The influence of imperfect boundaries on the mode quality factor is investigated for equilateral-triangle-resonator (ETR) semiconductor microlasers by the finite difference time domain technique and the Padé approximation with Baker's algorithm. For 2-D ETR with a refractive index of 3.2 and side length of 5 μm, the confined modes can still have a quality factor of about 1000 as small triangles with side length of 1 μm are cut from the vertices of the ETR. For a deformed 5 μm ETR with round vertices and curve sides, the simulated mode quality factors are comparable to the measured results.

Characterization and modelling of the hollow beam produced by a real conical lens

The properties of the hollow beam produced by a conical lens are studied in detail. In particular, the impact of a rounded vertex is examined. It is shown that it could lead to drastic changes in the transverse distribution of the hollow beam, determined by the ratio between the transverse size of the incident beam and the size of the blunt area. An adequate choice for this ratio allows us to either minimize the losses or optimize the distribution symmetry.

Rounding Voronoi diagram

Computational geometry classically assumes real-number arithmetic which does not exist in actual computers. A solution consists in using integer coordinates for data and exact arithmetic for computations. This approach implies that if the results of an algorithm are the input of another, these results must be rounded to match this hypothesis of integer coordinates. In this paper, we treat the case of two-dimensional Voronoi diagrams and are interested in rounding the Voronoi vertices to grid points while interesting properties of the Voronoi diagram are preserved. These properties are the planarity of the embedding and the convexity of the cells. We give a condition on the grid size to ensure that rounding to the nearest grid point preserves the properties. We also present heuristics to round vertices (not to the nearest grid point) and preserve these properties.

Shortest Path Geometric Rounding

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. In particular, coordinate bit-complexity can grow exponentially when algorithms are cascaded : the output of one algorithm becomes the input to the next. Cascading is a significant problem in practice. We propose a geometric rounding technique: shortest path rounding . Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to all algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact floating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on floating point input coordinates; the exact output coordinates can be rounded to accurate floating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware floating point operation.

Effects of ductile matrix failure in three dimensional analysis of metal matrix composites

Full three dimensional numerical cell model analyses are carried out for a metal reinforced by short fibers, to study the development of ductile matrix failure. A porous ductile material model is used to describe the effect of the nucleation and growth of voids to coalescence. In each case studied the computations are continued through the mechanically unstable regime, where an open crack forms near the ends of the fibers by the coalescence of voids in the matrix. Comparison of predictions for an isotropic hardening model and a kinematic hardening model are used to evaluate the effect of a metal that forms a rounded vertex on the yield surface. The full three dimensional model is used to study effects of deviations from equal transverse tension in directions perpendicular to the fibers.

A simple technique for the evaluation of a vertex using interference phenomena in the shadow region

We have studied the dependence of the finish of the vertex of a wedge on the diffraction pattern experimentally. It is found that the three ray interference pattern in the shadow region of the samples arises from the rounded vertex of the wedge. The method reported in this article is shown to be a valuable technique to evaluate the finish of a vertex of a small wedge with an obtuse angle.

Shear banding in strain hardening polycrystals during rolling

Initiation and development of shear band (SB) in f.c.c. strain hardening polycrystals during rolling are modelled in terms of crystallographic texture. The constitutive law of the material is expressed in terms of the texture-dependent normalized yield surface and the critical shear stress which evolves with strain. The normalized yield surface is predicted by the Taylor model as a function of rolling texture. It is shown that a rounded vertex (RV) develops at the loading point as the rolling texture becomes more and more marked. A detailed characterization of the RV is carried out. It is found that the normalized curvature radius of the RV decreases from unity towards zero at very large strain. This allows for a small stress perturbation to induce a shear strain perturbation with a large orientation deviation of deformation. By linearized stability analysis, the condition for initiation of SB from the shear strain perturbation is obtained. Development of SB is analysed by solving the established governing equations of shear banding. It is shown that the conditions for SB initiation and saturation of shear localisation depend strongly on the texture. Based on this model problem, a long discussion is carried out which allows a better understanding of the basic physical origin and saturation of SB in strain-hardening polycrystals, as well as the effects of yield surface curvature and yield surface rotation whose general form is derived.

Analyses of Plastic Flow Localization in Metals

The continuum mechanics framework for analyzing plastic flow localization is reviewed. The prediction of the localization of deformation into shear bands is sensitive to the constitutive description. The classical isotropic hardening elastic-plastic solid with a smooth yield surface and normality is very resistant to localization, but deviations from these idealizations have a strong effect. Thus, a material that forms a sharp vertex on the yield surface, as predicted by crystal plasticity, shows flow localization at quite realistic levels of strain, and even the formation of a rounded vertex on the yield surface has an important influence. Also softening induced by material damage or by the heating due to plastic dissipation have significant influence in promoting the onset of flow localization. In a practical situation one effect, such as thermal softening under high deformation rates, may be the dominant cause of localization, but often the interaction of different effects appears to be the more realistic explanation of observed flow localization. Some relevant constitutive models are reviewed and the effect of the different material models on localization predictions is illustrated. Important information on localization behavior in uniformly strained solids is obtained by a relatively simple material stability analysis, but often failure by flow localization occurs in nonuniformly strained regions, where numerical solution procedures are necessary to obtain theoretical predictions. The numerical results reviewed cover localization under dynamic as well as quasi-static loading conditions.

The theory of stress concentrations with round vertices

Asymptotic formulas have been derived for the local elastic displacement patterns together with the corresponding strains and stresses around a curvilinear hole or crack or an absolutely rigid or elastic inclusion with allowance for the radius of curvature at the vertex of the stress concentrator. The pattern is described by means of generalized stress intensity coefficients, which are coefficients to the principal term (1/√gr+2r) in the expansion of the stresses σy and Txy along the longitudinal axis of the concentrator. In the limiting cases of an elastic inclusion, curvilinear holes, or rigid inclusion, the stress intensity coefficients in the classical sense (with 1/√2r dependence) are obtained by passing to the limit ρ→0 from the generalized coefficients. The general asymptotic representations are used to derive near and far asymptotes, which characterize the variations in those quantities correspondingly in the stress concentration and intensity zones around curved defects. The limits to the application of the asymptotes have been established, together with relationships of Irwin type for holes between the generalized intensity coefficients and the stress concentration coefficients (deformation coefficients) for the defect vertices. The limiting sharpness of a vertex has been identified starting with which a stress concentrator can be considered of crack type, i.e., having points of return on the edge.

3D-analysis of localization failure in a ductile material containing two size-scales of spherical particles

The failure process is analysed for a ductile material in which a population of relatively large spherical inclusions nucleate voids at an early stage, while smaller voids nucleate much later. A full three-dimensional numerical analysis is necessary to study the nucleation and growth of voids at the two populations of particles and the formation of a 3D shear band leading to void-sheet failure in the matrix material between the larger voids. The large inclusions as well as the small ones are represented in terms of a continuum model of a ductile porous material, in which the effect of the formation of a rounded vertex on the yield surface can be described in terms of kinematic hardening. Failure is analysed under macroscopic stress and strain conditions corresponding to plane strain, axisyrnmetry, or intermediate conditions; but locally the stress and strain fields do not satisfy such conditions. Results of the full 3D-analysis are compared with results of a plane strain analysis, in which the larger voids are represented as cylindrical holes.

Effect of rounded vertices in the yield surface on localized necking in biaxially stretched sheets

A plane stress constitutive relation for rigid-plastic sheets under biaxial stretching is presented, in which the yield surface expansion models the behaviour of a solid carrying a rounded vertex at the loading point. Following the approach of Marciniak and Kuczynski the rounded vertex model is used to investigate the process of strain localization in thin sheets undergoing balanced biaxial tension. Limit strains are found to depend strongly on the detailed description of the rounded vertex. The imperfection sensitivity of the limit strain is found to be realistic within an interval of relative strength imperfections from 0.01% to 1%. Based on the rounded vertex model an explanation is offered for the lack of correlation between strain hardening exponent and limit strain in balanced biaxial tension.

A simple nomogram for calculating the area under the absorption curve of an EPR signal

In the present work, nomograms were compiled using the integration of smooth curves by Gauss' method with rounded vertices, which makes it possible to decrease by a factor of approximately 2 the number of ordinates for calculating the area, with the same accuracy of integration. EPR signals A'(H) are usually recorded as the first derivative of the absorption curve A(H).