Interior Corner Research Articles

Vibrations of cantilevered skewed plates with corner stress singularities

AbstractAccurate natural frequencies of skewed cantilevered thin plates are determined using the Ritz method. The present work is the first known vibrational study of the effects of stress singularities at the reentrant corner of skewed plates. In conjunction with classical thin plate theory, the Ritz trial space of assumed transverse displacement functions consists of algebraic polynomials, which are mathematically complete, and corner functions, which account for the singular behaviour of bending stresses at the reentrant corner. The corner functions were derived approximately four decades ago to study the nature of the stress singularities which occur at sharp interior corners or crack tips of typical plane elasticity and plate bending problems. Accurate non‐dimensional frequencies are calculated for thin plates having various side ratios and arbitrary degrees of skewness. Detailed numerical studies show that augmenting the Ritz trial space of polynomials with corner functions, having a co‐ordinate origin at the reentrant corner, enhances the convergence of the upper‐bound frequencies. The importance of using corner functions in skew plate vibrations is found to increase as the skew angle increases and as the side ratio decreases. Results obtained using the present method are compared with those obtained by other investigators using both theoretical and experimental methods.

Archaeological Testing Within the Southeast Corner of the Plaza at Mission Espada, San Antonio, Bexar County, Texas

In October 1990, pursuant to a contract with the National Park Service, the Center for Archaeological Research, The University of Texas at San Antonio, initiated test investigations in the southeast interior corner of the Mission Espada compound. The study focused on examining structural foundations and determining the depth of Mexican and Spanish colonial ground surfaces in the area. Colonial period foundations were discovered under the south wall, as well as the probable original construction period surface and footing trench. 'IWentieth-century foundations were found in the unit excavated under the east wall. These likely continue south to the north wall of the room where the bastion is located.

Pseudoline electron beam recrystallization of silicon-on-insulator

A pseudoline electron beam (e-beam) which is produced by scanning a spot e-beam along a line faster than the thermal response time of the substrate, has an advantage for recrystallization of silicon-on-insulator (SOI) structures in that the temperature profile along the line can be precisely controlled by the waveform of the scanning signal. In this paper, the current status of the pseudoline e-beam recrystallization method is reviewed, being focused on two essential problems: void generation and subboundary generation in the recrystallized films. First, in order to predict the temperature distribution during heating, the heat equation is solved for both static and dynamic cases using the Kirchhoff transformation and the Green function analysis, respectively. Then, generation mechanisms of voids and subboundaries are experimentally studied and the respective generation models are set up. It is concluded that voids are generated from isolated molten spots in the SOI film and a perforation seed structure in which rectangular seed regions are separately arranged along a line, is effective to suppress them. It is also concluded that subboundaries are generated at the interior corners of the folded 11facets which are formed at the solid-liquid interface. Other topics included in this paper are oblique scanning of the pseudoline e-beam, optimization of the scanning direction and scanning waveform, optimization of the seed direction, generation of twin defects, relation between the subboundary direction and the scanning velocity of the beam, and so on. As a result, a single-crystal SOI area of 100 μm square has been obtained.

Optical printability of defects in two-dimensional patterns

Experiments, simulation, and algebraic analysis are used to characterize the printability of defects in various classes of features in optical projection printing. A special test mask was designed with programmed defects located adjacent to isolated lines, arrays of lines, corners, elbows, and nested elbows. These patterns were then projection printed in an exposure–focus matrix in a positive resist process on various substrates and examined in detail via scanning electron micrographs. For small defects of both polarities, the linewidth variation on isolated lines is shown to be K* (defect area) where K is 1 and 2.4 for transparent and opaque defects, respectively. For equal line and space arrays of minimum‐sized features [0.8* λ/numerical aperture (NA)], opaque defects as small as 0.3* λ/NA cause bridging while transparent defects as large as 0.5* λ/NA do not cause opens. Elbows and nested elbows are slightly more sensitive than arrays of lines due to intrafeature proximity effects. The interior corners are also sensitive to defects. Underexposure greatly increases the effects of opaque defects while defocus has little impact. The effects on aluminum substrates show similar trends to those on silicon while for thin resist on silicon, even much smaller defects print. The printability of transparent defects in positive resist is easy to model and a constant intensity threshold model is generally adequate. For opaque defects, nonvertical resist development effects typically occur and must be included in the simulation.

Characterization and entrainment of subboundaries and defect trails in zone-melting-recrystallized Si films

We have studied the morphology and crystallographic angular discontinuities of subboundaries and defect trails in zone-melting-recrystallized Si films. These subboundaries and defect trails, which originate at the interior corners of the faceted solidification front, are classified into seven types. Evidence is presented that in-plane stress due to temperature gradients plays a major role in causing such defects. Various schemes for entraining subboundaries and defect trails are described.

Growth Mechanisms During Thin Film Crystallization From the Melt

AbstractWe develop a model that appears to account for the existence of a new mode of crystallization recently discovered during zone-melt-recrystallization (ZMR) of silicon thin films on SiO 2 using a scanned strip heater or lamp. Transition to the new crystallization regime is induced by reducing the temperature of the scanned upper heater strip, thus reducing dT/dy, the thermal gradient along the direction of scan at the silicon solidification front. If dT/dy ≤:4 K/mm, the single crystal films have long non-branched subboundaries with tilt misalignments of 0.1 or less, a lateral separation in excess of 50 µm, and consist of rows of short dislocations threading through the film thickness and terminating at the two SiO2 layers. This is in marked contrast to material ZMR scanned at higher dT/dy which shows conventionally branched 1 to 3 subboundaries that consist of edge dislocations running in the plane of the film often for several hundred microns.Our model extends the {111} faceted-freezing-front picture we have developed previously to take into account the freezing profile with respect to film thickness, and in particular to such profiles at the intersections of pairs of {111} facets where subboundaries are known to form. We propose that the melt-freezing interface profiles at these interior corner intersections are aligned approximately normal to the scan in the high gradient case, but become tilted towards the plane of the SiO2 cap layer for the low gradient case. This tilting accounts in a natural way for the transition from in-plane to threading dislocations.

Child's Play

Consider the problem of finding the total number of ways each possible outcome (spot total) can be rolled using three standard dice. For the modem mathematician this problem is a fairly easy one. Nevertheless, it is scarcely child's play; the number of ways to get certain outcomes runs as high as twenty-seven. Dice are the oldest gaming instruments known to man, yet no solution to the problem was generally known until the time of Galileo. It took that great man himself to demonstrate one. So it is perhaps bold of me to assert, as I now do, that any reasonably intelligent child, working on his (or her) own, could find all the answers to the problem in ten or fifteen minutes! He could, that is, if briefly instructed in advance as to how to proceed. As you might suspect, he would have to employ a computational tool. But it wouldn't be what you'd think. Not a computer. Not even a calculator. Rather, an apparatus more primitive than an abacus: a supply of children's toy blocks! For the sake of a better explanation, it will be assumed in what follows that the blocks are available in a variety of colors. In addition, a cardboard box would be helpful (plus, of course, a pencil and paper to record results). The child is to stack the blocks in the simple and repetitive manner to be described, while counting the number of blocks required to complete each separate stage of construction. The numbers thus obtained represent the required solutions. That's all there is to it. Specifically, the child, having received instruction, starts by placing a single red block in an interior corner of the box. This constitutes the first stage of construction. The count of one block indicates the number of ways a 3 can be rolled with three dice and is so recorded. There will be three faces of this first block left visible. Next, using blocks of a second color, say, blue, the child places just enough blocks atop and beside the red one to cover these faces and to leave only the blue of the second course visible. The number of blocks (three) required for this second stage indicates the number of ways a 4 can be rolled. And, again, more blocks are laid, just sufficient to cover the blue, say, with a yellow this time. Now the number of blocks required indicates the number of ways a 5 can be rolled. And so on, color by color. It will be noted that like colors accumulate in diagonally sloping layers. Part of the instruction given will stipulate that the number of blocks in any coordinate direction is not to exceed six, i.e., no block is to be placed outside the dimensions of a 6 X 6 X 6 cube (which limits can be marked on the inside of the box). However, it is by no means necessary to set up the whole cube; half is sufficient. It will be obvious that beyond the midpoint of construction the block count will be only a duplication (in reverse order) of the figures already tabulated. When the child is through, he will have set up an interesting and ornamental structure, and will probably have had fun in the process. He will also have produced an accurate and legitimate solution to the original problem. But he won't be entitled to think he's smarter than Galileo! It's the blocks that are smart; they deserve the credit. To understand how they do the job, consider first the regularity of the structure as it grows. Just as the structure of a crystal or snowflake is said to be determined by a few molecules of seed, so here it is the initial block which establishes the final structure. Already by the time blocks of the second or third color have been added it is established that the blocks of each color will lie with their centers in a plane exclusive to that color; that all such planes will have identical slope and spacing from each other. Each of these color planes is normal to those interior diagonals of the blocks which extend in the general direction of added construction. The spacing between planes will be one-third of a block's diagonal (not the one-half one might intuitively expect). The most useful coordinates for the study of the resultant structure will be the usual

Mechanism for the Graphoepitaxy of Electrodeposited Tin

Tin islands electrodeposited from basic cyanide solutions were grown graphoepitaxially on a substrate into which a square wave relief pattern had been reactively ion etched. The [100] axis of each island was normal to the substrate and the [001] axis was parallel to the relief structure in the plane of the substrate. Tin has a body‐centered tetragonal crystal structure. At low cathodic polarizations , the islands nucleated at interior corners of the square wave pattern. Nucleation also occurred at the interior corners for all other electrodeposited metals studied (Cu, Pb, In, Zn). The tin islands grew with a highly faceted and linear morphology elongated in the [001] direction. Postdeposition and in situ observations suggested that the dominant mechanism for crystallographic orientation in the plane of the substrate was the physical confinement of the elongated islands in a single groove. The islands usually did not overgrow a step until they had grown as wide as the groove. This growth sequence suggests that the growing islands pushed away from the adjacent steps. Islands nucleated at random on the surface via a high cathodic polarization pulse and subsequently grown at a lower polarization also possessed a strong degree of graphoepitaxy, thus demonstrating that the graphoepitaxial orientation can be induced after nucleation. Nucleation at interior corners may be helpful in starting the orientation process at an earlier growth stage and in placing the islands on the lower level of the square wave pattern where confinement can take place. This demonstration of graphoepitaxy in an electrocrystallization process broadens the range of potential applications for graphoepitaxy, and new information on the orienting mechanism provides guidance in selecting systems for which graphoepitaxy might be successful.

Modeling ion milling

A string segment motion algorithm is presented and used in conjunction with SEM observation of line edge profiles to study phenomena in ion milling which result from the angular dependence of etch rates. The algorithm allows samples to be rotated and includes shadowing, automatic faceting,and special boundary conditions at interior and exterior corners. The accuracy of the algorithm is demonstrated by comparison with known mathematical results. Simulation is used to study how the shape of the angular etch rate curve and the modification of the local etch rate at corners affect the line edge profiles. Comparisons with experimental results are made to check the algorithm assumptions and applicability. The comparisons include preferentially etched Si, a soft Au mask on a hard NiFe substrate and a hard Ti mask on a soft Au substrate.

Numerical solutions for high Reynolds number separated flow past a semi-infinite compression corner

The present paper addresses the high Reynolds number, two-dimensional, steady laminar flow separation phenomenon near an interior (concave) corner. A very fast Alternating Direction Implicit finite difference approach is used to solve the Interacting Boundary Layer approximation to the Navier Stokes equations in a conformal plane. Solutions are presented for corner angles up to 18° for Reynolds numbers (based on forebody length) up to 10 8. Convergence properties and accuracy levels are identified in order to provide reliability estimates of the results. Limitations to the numerical algorithm for large separation regions at high Reynolds numbers are identified.

PLANFORM INFLUENCE ON FLUSHING AND CIRCULATION IN SMALL HARBORS

Laboratory data showing the influence of planform geometry on the tidal flushing characteristics of small harbors of simple surface shape. The tide ranges, water depths, and planform areas are typical of those encountered in small-boat marinas in Puget Sound, Washington. Each harbor investigated had a single, asymmetric entrance. Flushing and circulation patterns within such harbors depend strongly upon the characteristics of the angular momentum established within the basin and upon the effective penetration distance into the basin of the stream of ambient water entering the harbor on the flood tide. Experimental results confirm that best gross tidal flushing occurs when rectangular harbors have an aspect ratio L/B near unity, and that rounding interior corners of the basin has little effect on the gross tidal flushing but does improve local exchange. Aspect ratios L/B less than 3 lead to the creation of more than one circulation cell (gyre) within the basin.

A major advance in high-power electron-beam welding in air

A major extension of the capabilities of nonvacuum electron-beam welding follows from the development of a machine with higher beam power. Tests with a new atmospheric electron gun operating at 60 kW have shown that the advantages of nonvacuum electron-beam welding must be reevaluated. Gas heating produced by the beam itself becomes very pronounced, so that electron scattering is reduced causing the high-power density of the beam to be retained over larger working distances. Single-pass butt welds with a depth-to-width ratio of 4 : 1 can be made in 3.8-cm-thick steel at a speed of 0.77 cm/sec, while the ultimate welding depth exceeds 5 cm. Medium thick material, e.g., 1.3-cm steel, can be welded at a 5-cm work distance, with a depth-to-width ratio of 3 : 1. The large work distance permits access to more complex structures and interior corners such at T sections, which can now be fabricated from a plate, 1.3 cm thick, at a speed of 2–3 cm/sec, making a full through weld from one side. Also, seam welds of two 3-mm-thick hot-rolled steel sheets can be produced at speeds of up to 15 cm/sec. High-power machines need not be significantly larger or costlier than low-power guns, and the present 60 kW does not represent any technical limit. Welding efficiency improves with higher power; when welding steel of 2 cm thickness or more the energy efficiency of the process at 60 kW is better than 50%, while at 40 kW it is merely 30%. In addition, the high power permits greater welding speed. These developments translate directly into improved cost justification for electron-beam welding and a broad expansion of its possible applications.

Laminar heating in interior corners at Mach 19

Laminar heating in hypersonic vehicles interior corners, analyzing helium tunnel heat transfer data for various intersecting wedge corners