Axisymmetric Loading of a Plate on a Local Support in the Framework of the Rodionova-Titayev-Chernykh Theory

UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES ◽

10.18522/1026-2237-2021-1-4-14 ◽

2021 ◽

pp. 4-14

Author(s):

Viktor E. Velichko ◽

Pavel S. Mostovykh

Keyword(s):

Plate Thickness ◽

Three Dimensional ◽

Theory Of Elasticity ◽

Optical Quality ◽

Half Width ◽

Constant Thickness ◽

Mirror Surface ◽

Optical Telescopes ◽

Peak Value

The problem of an axisymmetric bending of a plate with constant thickness under its own weight is considered. The plate has a circular support. The problem is solved in an axisymmetric statement using a nonclassical shell theory of Rodionova-Titaev-Chernykh (RTCh), which takes plate compression in thickness into account. The solution for this theory is obtained using Godunov's orthogonal sweep method. This solution is compared with the solution obtained using the general three-dimensional theory of elasticity, implemented in an open-source package Code_Aster using axisymmetric finite elements. The motivation for this study is the description of a stress-strain state of some variants of primary mirrors of large optical telescopes under the action of gravity. The obtained results characterizing the optical quality of the mirror surface are: the peak value (PV) and the root-mean square (RMS) of its displacement. A parametric study was carried out, i.e., the thickness of the plate and the half-width of the support were varied. The two methods were compared. It is shown that, as the plate thickness increases or the half-width of the support decreases, non-physical behaviour of the mirror surface takes place within the limits of the nonclassical theory of RTCh. A criterion of its applicability is therefore proposed.

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FREE VIBRATION OF AN ORTHOTROPIC HOLLOW BODY OF REVOLUTION

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455405001532 ◽

2005 ◽

Vol 05 (02) ◽

pp. 299-312

Author(s):

D. REDEKOP

Keyword(s):

Natural Frequencies ◽

Differential Quadrature Method ◽

Three Dimensional ◽

Theory Of Elasticity ◽

Constant Thickness ◽

Body Of Revolution ◽

Orthotropic Cylinder ◽

Dimensional Theory ◽

Hollow Cylinders ◽

Definition Of

A method is developed to determine the natural frequencies of vibration of an orthotropic hollow body of revolution of constant thickness but of arbitrary smooth meridian. Equations are derived using the linear three-dimensional theory of elasticity, and a numerical solution is obtained using the differential quadrature method. The geometric generality of the solution is attained by delaying definition of local geometric parameters until the solution stage. Validation is by comparison with previously published results, including results for a hollow orthotropic cylinder. Sample results are given for orthotropic hollow cylinders and spherical segments, and conclusions are drawn.

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Alumina-Doped Silica Gradient-Index (GRIN) Lenses by Slurry-Based Three-Dimensional Printing (S-3DP™)

MRS Proceedings ◽

10.1557/proc-758-ll1.5 ◽

2002 ◽

Vol 758 ◽

Cited By ~ 1

Author(s):

Hong-Ren Wang ◽

Michael J. Cima ◽

Emanuel M. Sachs

Keyword(s):

Complex Structure ◽

Three Dimensional ◽

Optical Quality ◽

Vacuum Furnace ◽

Gradient Index ◽

Alumina Concentration ◽

Graded Materials ◽

Three Dimensional Printing ◽

Dimensional Printing

ABSTRACTThe traditional Slurry-based Three-dimensional Printing (S-3DP™) process has been used to fabricate complex structure materials by printing organic binders in selected positions on each printing layer. This process is modified to fabricate functional graded materials, such as gradient index (GRIN) lenses, by depositing different concentrations of dopant at different positions. The modified S-3DP™ process offers advantages over conventional GRIN lens processes, including reduced processing time, improved compositional flexibility, and increased index profile dimensionality. Two different approximately parabolic dopant concentration profiles, which have maximum alumina concentrations of 1.63 mol% and 2.50 mol%, are printed into silica powder beds using S-3DP™. The samples with maximum alumina concentration of 1.63 mol% have been sintered into optical transparency at 1650 0C for 30 minutes in a vacuum furnace (5×10-6 torr) while an additional dehydration process before sintering was required for the samples with maximum alumina concentration of 2.50 mol%. The magnifying effects of GRIN lenses with profiles of 1.63 mol% and 2.50 mol% alumina were observed, yielding effective focal lengths of 10 cm and 6.1 cm, respectively. Light diffraction, which results from the locally inhomogeneous dopant distribution and reduces the optical quality of GRIN lenses, was also observed.

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Refined hyperbolic shear deformation plate theory

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406214563739 ◽

2014 ◽

Vol 229 (15) ◽

pp. 2675-2686 ◽

Cited By ~ 2

Author(s):

K Nareen ◽

RP Shimpi

Keyword(s):

Shear Deformation ◽

Plate Theory ◽

Plate Thickness ◽

Three Dimensional ◽

Theory Of Elasticity ◽

Governing Equations ◽

Dimensional Theory ◽

Classical Plate Theory ◽

Shear Deformation Plate Theory ◽

Elasticity Solutions

The paper presents a novel shear deformation plate theory involving only two variables. Taking a cue from exact three-dimensional theory of elasticity solutions for a plate, hyperbolic functions are used for describing displacement variation across plate thickness. The theory involves only two governing equations, which are uncoupled for statics and are only inertially coupled for dynamics. The shear stress free surface conditions are satisfied. No shear correction factor is required. The theory is variationally consistent, has a strong similarity with classical plate theory, and is simple, yet accurate. Illustrative examples for free vibration and for static flexure demonstrate the effectiveness of the theory.

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Scanning Electron Microscopy in Hybrid Microelectronics

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100092244 ◽

1976 ◽

Vol 34 ◽

pp. 456-457

Author(s):

S. Khadpe ◽

R. Faryniak

Keyword(s):

Integrated Circuits ◽

Thick Film ◽

Integrated Circuit ◽

Failure Modes ◽

Three Dimensional ◽

Circuit Components ◽

Hybrid Microcircuit ◽

Electrical Connections ◽

Scanning Electron

The Scanning Electron Microscope (SEM) is an important tool in Thick Film Hybrid Microcircuits Manufacturing because of its large depth of focus and three dimensional capability. This paper discusses some of the important areas in which the SEM is used to monitor process control and component failure modes during the various stages of manufacture of a typical hybrid microcircuit.Figure 1 shows a thick film hybrid microcircuit used in a Motorola Paging Receiver. The circuit consists of thick film resistors and conductors screened and fired on a ceramic (aluminum oxide) substrate. Two integrated circuit dice are bonded to the conductors by means of conductive epoxy and electrical connections from each integrated circuit to the substrate are made by ultrasonically bonding 1 mil aluminum wires from the die pads to appropriate conductor pads on the substrate. In addition to the integrated circuits and the resistors, the circuit includes seven chip capacitors soldered onto the substrate. Some of the important considerations involved in the selection and reliability aspects of the hybrid circuit components are: (a) the quality of the substrate; (b) the surface structure of the thick film conductors; (c) the metallization characteristics of the integrated circuit; and (d) the quality of the wire bond interconnections.

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Semi-automated methods for 3-D reconstruction of macromolecular complexes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136581 ◽

1995 ◽

Vol 53 ◽

pp. 42-43

Author(s):

B. Carragher ◽

M. Whittaker

Keyword(s):

Three Dimensional ◽

Time Saving ◽

Macromolecular Complexes ◽

Symmetry Properties ◽

Dimensional Reconstruction ◽

Experienced Operator ◽

Projection Images ◽

The Individual ◽

Natural Symmetry

Techniques for three-dimensional reconstruction of macromolecular complexes from electron micrographs have been successfully used for many years. These include methods which take advantage of the natural symmetry properties of the structure (for example helical or icosahedral) as well as those that use single axis or other tilting geometries to reconstruct from a set of projection images. These techniques have traditionally relied on a very experienced operator to manually perform the often numerous and time consuming steps required to obtain the final reconstruction. While the guidance and oversight of an experienced and critical operator will always be an essential component of these techniques, recent advances in computer technology, microprocessor controlled microscopes and the availability of high quality CCD cameras have provided the means to automate many of the individual steps.During the acquisition of data automation provides benefits not only in terms of convenience and time saving but also in circumstances where manual procedures limit the quality of the final reconstruction.

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ANALYSIS OF THE GENERAL EQUATIONS OF THE TRANSVERSE VIBRATION OF A PIECEWISE UNIFORM VISCOELASTIC PLATE

Computational nanotechnology ◽

10.33693/2313-223x-2020-7-3-52-56 ◽

2020 ◽

Vol 7 (3) ◽

pp. 52-56

Author(s):

MMATMATISA JALILOV ◽

◽

RUSTAM RAKHIMOV ◽

Keyword(s):

Cross Section ◽

Transverse Vibration ◽

Three Dimensional ◽

Viscoelastic Plate ◽

Constant Thickness ◽

Regional Problem ◽

Rigid Contact ◽

Under Construction ◽

Derivatives Of ◽

Theoretical Results

This article discusses the analysis of the general equations of the transverse vibration of a piecewise homogeneous viscoelastic plate obtained in the “Oscillation of inlayer plates of constant thickness” [1]. In the present work on the basis of a mathematical method, the approached theory of fluctuation of the two-layer plates, based on plate consideration as three dimensional body, on exact statement of a three dimensional mathematical regional problem of fluctuation is stood at the external efforts causing cross-section fluctuations. The general equations of fluctuations of piecewise homogeneous viscoelastic plates of the constant thickness, described in work [1], are difficult on structure and contain derivatives of any order on coordinates x, y and time t and consequently are not suitable for the decision of applied problems and carrying out of engineering calculations. For the decision of applied problems instead of the general equations it is expedient to use confidants who include this or that final order on derivatives. The classical equations of cross-section fluctuation of a plate contain derivatives not above 4th order, and for piecewise homogeneous or two-layer plates the elementary approached equation of fluctuation is the equation of the sixth order. On the basis of the analytical decision of a problem the general and approached decisions of a problem are under construction, are deduced the equation of fluctuation of piecewise homogeneous two-layer plates taking into account rigid contact on border between layers, and also taking into account mechanical and rheological properties of a material of a plate. The received theoretical results for the decision of dynamic problems of cross-section fluctuation of piecewise homogeneous two-layer plates of a constant thickness taking into account viscous properties of their material allow to count more precisely the is intense-deformed status of plates at non-stationary external loadings.

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Towards Predicting Relative Belt Edge Endurance With the Finite Element Method

Tire Science and Technology ◽

10.2346/1.2141701 ◽

1990 ◽

Vol 18 (4) ◽

pp. 216-235 ◽

Cited By ~ 19

Author(s):

J. De Eskinazi ◽

K. Ishihara ◽

H. Volk ◽

T. C. Warholic

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Three Dimensional ◽

The Finite Element Method ◽

Shear Deformations ◽

Element Analysis ◽

Vertical Loading ◽

Predicted Values ◽

Element Method

Abstract The paper describes the intention of the authors to determine whether it is possible to predict relative belt edge endurance for radial passenger car tires using the finite element method. Three groups of tires with different belt edge configurations were tested on a fleet test in an attempt to validate predictions from the finite element results. A two-dimensional, axisymmetric finite element analysis was first used to determine if the results from such an analysis, with emphasis on the shear deformations between the belts, could be used to predict a relative ranking for belt edge endurance. It is shown that such an analysis can lead to erroneous conclusions. A three-dimensional analysis in which tires are modeled under free rotation and static vertical loading was performed next. This approach resulted in an improvement in the quality of the correlations. The differences in the predicted values of various stress analysis parameters for the three belt edge configurations are studied and their implication on predicting belt edge endurance is discussed.

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