The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

The method of caustics for the determination of normal loads acting on surfaces of elastic bodies

1980 ◽  
Vol 15 (1) ◽  
pp. 37-41 ◽  
Author(s):  
P S Theocaris ◽  
N I Ioakimidis
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Elastic Bodies ◽  
Plate And Shell ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Quantities Of Interest

The optical method of caustics constitutes an efficient experimental technique for the determination of quantities of interest in elasticity problems. Up to now, this method has been applied only to two-dimensional elasticity problems (including plate and shell problems). In this paper, the method of caustics is extended to the case of three-dimensional elasticity problems. The particular problems of a concentrated force and a uniformly distributed loading acting normally on a half-space (on a circular region) are treated in detail. Experimentally obtained caustics for the first of these problems were seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of various, more complicated, three-dimensional elasticity problems, including contact problems, by the method of caustics is also possible.

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

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.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

Shear-Driven Flow in a Toroid of Square Cross Section

2003 ◽  
Vol 125 (1) ◽  
pp. 130-137 ◽  
Author(s):  
J. A. C. Humphrey ◽  
J. Cushner ◽  
M. Al-Shannag ◽  
J. Herrero ◽  
F. Giralt
Keyword(s):  
Cross Section ◽  
Finite Length ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Transition To Turbulence ◽  
Radius Of Curvature ◽  
Infinite Length ◽  
Core Flow ◽  
Reynolds Number Flow ◽  
End Wall

The two-dimensional wall-driven flow in a plane rectangular enclosure and the three-dimensional wall-driven flow in a parallelepiped of infinite length are limiting cases of the more general shear-driven flow that can be realized experimentally and modeled numerically in a toroid of rectangular cross section. Present visualization observations and numerical calculations of the shear-driven flow in a toroid of square cross section of characteristic side length D and radius of curvature Rc reveal many of the features displayed by sheared fluids in plane enclosures and in parallelepipeds of infinite as well as finite length. These include: the recirculating core flow and its associated counterrotating corner eddies; above a critical value of the Reynolds (or corresponding Goertler) number, the appearance of Goertler vortices aligned with the recirculating core flow; at higher values of the Reynolds number, flow unsteadiness, and vortex meandering as precursors to more disorganized forms of motion and eventual transition to turbulence. Present calculations also show that, for any fixed location in a toroid, the Goertler vortex passing through that location can alternate its sense of rotation periodically as a function of time, and that this alternation in sign of rotation occurs simultaneously for all the vortices in a toroid. This phenomenon has not been previously reported and, apparently, has not been observed for the wall-driven flow in a finite-length parallelepiped where the sense of rotation of the Goertler vortices is determined and stabilized by the end wall vortices. Unlike the wall-driven flow in a finite-length parallelepiped, the shear-driven flow in a toroid is devoid of contaminating end wall effects. For this reason, and because the toroid geometry allows a continuous variation of the curvature parameter, δ=D/Rc, this flow configuration represents a more general paradigm for fluid mechanics research.

Heat Transfer for Laminar Flow in Spiral Ducts of Rectangular Cross Section

2005 ◽  
Vol 127 (3) ◽  
pp. 352-356 ◽  
Author(s):  
Michael W. Egner ◽  
Louis C. Burmeister
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Laminar Flow ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Radius Of Curvature ◽  
Dean Number ◽  
Aspect Ratios ◽  
Small Pitch

Laminar flow and heat transfer in three-dimensional spiral ducts of rectangular cross section with aspect ratios of 1, 4, and 8 were determined by making use of the FLUENT computational fluid dynamics program. The peripherally averaged Nusselt number is presented as a function of distance from the inlet and of the Dean number. Fully developed values of the Nusselt number for a constant-radius-of-curvature duct, either toroidal or helical with small pitch, can be used to predict those quantities for the spiral duct in postentry regions. These results are applicable to spiral-plate heat exchangers.

The Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
General Solution ◽  
Elastic Body ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
Two Dimensional ◽  
Stroh's Formalism ◽  
Elasticity Problems ◽  
Anisotropic Elastic ◽  
Stroh’S Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

Geometry Effects in Eulerian/Granular Simulation of a Turbulent FCC Riser with a (kg-?g)-KTGF Model

2010 ◽  
Vol 8 (1) ◽  
Author(s):  
Hamid Reza Nazif ◽  
Hassan Basirat Tabrizi ◽  
Farhad A Farhadpour
Keyword(s):  
Cross Section ◽  
Large Scale ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Good Prediction ◽  
Model Predictions ◽  
Secondary Circulations ◽  
Sharp Corners ◽  
Granular Simulation

Three-dimensional, transient turbulent particulate flow in an FCC riser is modeled using an Eulerian/Granular approach. The turbulence in the gas phase is described by a modified realizable (kg-?g) closure model and the kinetic theory of granular flow (KTGF) is employed for the particulate phase. Separate simulations are conducted for a rectangular and a cylindrical riser with similar dimensions. The model predictions are validated against experimental data of Sommerfeld et al (2002) and also compared with the previously reported LES-KTGF simulations of Hansen et al (2003) for the rectangular riser. The (kg-?g)-KTGF model does not perform as well as the LES-KTGF model for the riser with a rectangular cross section. This is because, unlike the more elaborate LES-KTGF model, the simpler (kg-?g)-KTGF model cannot capture the large scale secondary circulations induced by anisotropic turbulence at the corners of the rectangular riser. In the cylindrical geometry, however, the (kg-?g)-KTGF model gives good prediction of the data and is a viable alternative to the more complex LES-KTGF model. This is not surprising as the circulations in the riser with a circular cross section are due to the curvature of the walls and not due to the presence of sharp corners.

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