scholarly journals The Generalized Twist for the Torsion of Piezoelectric Cylinders

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
István Ecsedi ◽  
Attila Baksa
Keyword(s):  
Elastic Properties ◽  
Cross Section ◽  
Classical Theory ◽  
Elastic Cylinder ◽  
Theory Of Elasticity ◽  
Torsion Problem ◽  
Displacement Fields ◽  
The Cross

In the classical theory of elasticity, Truesdell proposed the following problem: for an isotropic linearly elastic cylinder subject to end tractions equipollent to a torque T, define a functional τ(u) on Q such that T=Kτ(u), for each u∈Q, where Q is the set of all displacement fields that correspond to the solutions of the torsion problem and K depends only on the cross-section and the elastic properties of the considered cylinder. This problem has been solved by Day. In the present paper Truesdell’s problem is extended to the case of piezoelastic, monoclinic, and nonhomogeneous right cylinders.

The Elastic Plane With a Circular Insert, Loaded by a Tangentially Directed Force

1962 ◽  
Vol 29 (2) ◽  
pp. 362-368 ◽  
Author(s):  
M. Hete´nyi ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Elastic Plane ◽  
Elementary Functions ◽  
Explicit Formulas ◽  
Concentrated Force

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated force in the plane of the plate and in a direction tangential to the circle. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

scholarly journals The torsion and flexure of shafting with keyways or cracks

1932 ◽  
Vol 138 (836) ◽  
pp. 607-634 ◽  
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Torsional Rigidity ◽  
Circular Cross Section ◽  
Transverse Load ◽  
Torsion Problem ◽  
Circular Section ◽  
Physical Conditions ◽  
The Cross ◽  
Straight Lines

The object of the paper is to investigate the properties of shafts of circular cross-section into which keyways or slits have been cut, first when subjected to torsion, and second when bent by a transverse load at one end. The torsion problem for similar cases has been treated by several writers. Filon has worked out an approximation to the case of a circular section with one or two keyways ; in his method the boundary of the cross-section was a nearly circular ellipse and the boundaries of the keyways were confocal hyperbolas. In particular he considered the case when the hyperbola degenerated into straight lines starting from the foci. The solution for a circular section with one keyway in the form of an orthogonal circle has been obtained by Gronwall. In each case the solution has been obtained by the use of a conformal trans­formation and this method is again used in this paper, the transformations used being ρ = k sn 2 t . ρ = k 1/2 sn t , ρ = k 1/2 sn 1/2 t where ρ = x + iy , t = ξ + i η. No work appears to have been done on the flexure problem which is here worked out for several cases of shafts with slits. 2. Summary of the Problems Treated . We first consider the torsional properties of shafts with one and with two indentations. In particular cases numerical results have been obtained for the stresses at particular points and for the torsional rigidity. The results for one indentation and for two indentations of the same width and approximately the same depth have been compared. We next consider the solution of the torsion problem for one, two or four equal slits of any depth from the surface towards the axis. The values of the stresses have not been worked out in these cases since the stress is infinite at the bottom of the slits. This in stress occurs because the physical conditions are not satisfied at the bottom of the slits, but as had been pointed out by Filon this does not affect the validity of the values of the torsional rigidity. We compare the effect on the torsional rigidity of the shaft of one, two and four slits of the same depth in particular cases. We also compare the results for one slit with those obtained by Filon by another method, and find very good agreement which is illustrated by a graph. The reduction in torsional rigidity due to a semicircular keyway is compared with that due to a slit of approximately the same depth. Finally the distortion of the cross-sections at right angles to the planes is investigated, and in this, several interesting and perhaps unexpected features appear. The relative shift of the two sides of the slits is calculated in several cases.

Axial Loading of Annular Bonded Rubber Blocks

2003 ◽  
Vol 76 (5) ◽  
pp. 1194-1211 ◽  
Author(s):  
J. M. Horton ◽  
G. E. Tupholme ◽  
M. J. C. Gover
Keyword(s):  
Experimental Data ◽  
Stress Distribution ◽  
Closed Form ◽  
Cross Section ◽  
Classical Theory ◽  
Cross Sections ◽  
Axial Loading ◽  
Theory Of Elasticity ◽  
Governing Equations ◽  
Annular Cross Section

Abstract Closed-form expressions are derived using a superposition approach for the axial deflection and stress distribution of axially loaded rubber blocks of annular cross-section, whose ends are bonded to rigid plates. These satisfy exactly the governing equations and conditions based upon the classical theory of elasticity. Readily calculable relationships are derived for the corresponding apparent Young's modulus, Ea, and the modified modulus, Ea′, and their numerical values are compared with the available experimental data. Elementary expressions for evaluating Ea and Ea′ approximately are deduced from these, in forms which are closely analogous to those given previously for blocks of circular and long, thin rectangular cross-sections. The profiles of the deformed lateral surfaces of the block are discussed and it is confirmed that the assumption of parabolic lateral profiles is not valid generally.

The General Theory of Cylindrical and Conical Tubes under Torsion and Bending Loads

1947 ◽  
Vol 51 (443) ◽  
pp. 884-930 ◽  
Author(s):  
J. Hadji-Argyris ◽  
P. C. Dunne
Keyword(s):  
General Theory ◽  
Elastic Properties ◽  
Cross Section ◽  
Secular Equation ◽  
Bending Loads ◽  
The Cross

In 3.2.1 it was indicated that ξi, ηi may be found from the appropriate co-factors of the secular equation and appear directly as functions of the geometric and elastic properties of the root cross-section. The method will be applied in 5.6.2 for the four-boom tube, in 5.7.3 for the n-boom tube, and in 5.8 for the trapezoidal tube with direct stress carrying top and bottom panels. This procedure is advantageous when it is desired to study the effect on ξ, η of varying particular parameters of the cross-section.However, when in any case the h-functions have to be obtained, it is often more expedient to determine ξ, η by one of the following procedures.

Elastic Properties of Normal and Binormal Helical Nanowires

2006 ◽  
Vol 963 ◽  
Author(s):  
Alexandre Fontes da Fonseca ◽  
C P Malta ◽  
Douglas S Galvão
Keyword(s):  
Simple Model ◽  
Young’S Modulus ◽  
Elastic Properties ◽  
Cross Section ◽  
Young's Modulus ◽  
Helical Axis ◽  
Geometric Features ◽  
Kirchhoff Rod ◽  
Rod Model ◽  
The Cross

ABSTRACTA helical nanowire can be defined as being a nanoscopic rod whose axis follows a helical curve in space. In the case of a nanowire with asymmetric cross section, the helical nanostructure can be classified as normal or binormal helix, according to the orientation of the cross section with respect to the helical axis of the structure. In this work, we present a simple model to study the elastic properties of a helical nanowire with asymmetric cross section. We use the framework of the Kirchhoff rod model to obtain an expression relating the Hooke's constant, h, of normal and binormal nanohelices to their geometric features. We also obtain the Young's modulus values. These relations can be used by experimentalists to evaluate the elastic properties of helical nanostructures. We showed that the Hooke's constant of a normal nanohelix is higher than that of a binormal one. We illustrate our results using experimentally obtained nanohelices reported in the literature.

A Torsion Problem of a Prismatic Rod Composed of Two Orthotropic Materials

2014 ◽  
Vol 1020 ◽  
pp. 280-285
Author(s):  
Vardges Yedoyan
Keyword(s):  
Cross Section ◽  
Orthotropic Materials ◽  
Anisotropy Axis ◽  
Section Plane ◽  
Cartesian Coordinate System ◽  
Torsion Problem ◽  
Boundary Problems ◽  
Composite Rod ◽  
The Cross ◽  
Cartesian Coordinate

A rod composed of two different prismatic rods built-up with rectilinear orthotropic materials is studied. Composite parts of the prismatic rods are connected entirely by their common surface. The anisotropy axis is perpendicular to the cross-section plane. The problem has been solved in Cartesian coordinate system. The functions of stresses are presented by the sums of solutions corresponding to positive eigenvalues of homogeneous boundary problems and partial solutions of inhomogeneous boundary problems. From condition of existence of the non-trivial solution of the homogeneous boundary problem an equation in respect to eigenvalues is derived of which roots are real and different. If there are roots in (0; 1) interval, then the stresses tend to infinity at the vertex of the cross-section of the composite rod have a feature with the order equal to where - is the smallest root in the interval (0, 1).

The Elastic Plane With a Circular Insert, Loaded by a Radial Force

1961 ◽  
Vol 28 (1) ◽  
pp. 103-111 ◽  
Author(s):  
J. Dundurs ◽  
M. Hete´nyi
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Classical Theory ◽  
Radial Force ◽  
Theory Of Elasticity ◽  
Elastic Plane ◽  
Elementary Functions ◽  
Explicit Formulas

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated radial force in the plane of the plate. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

Vibration of Thick Circular Disks and Shells of Revolution

2003 ◽  
Vol 70 (2) ◽  
pp. 292-298 ◽  
Author(s):  
A. V. Singh ◽  
L. Subramaniam
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Published Data ◽  
Axially Symmetric ◽  
Shells Of Revolution ◽  
Displacement Fields ◽  
Wide Range ◽  
Circular Disks

A fully numerical and consistent method using the three-dimensional theory of elasticity is presented in this paper to study the free vibrations of an axially symmetric solid. The solid is defined in the cylindrical coordinates r,θ,z by a quadrilateral cross section in the r-z plane bounded by four straight and/or curved edges. The cross section is then mapped using the natural coordinates (ξ,η) to simplify the mathematics of the problem. The displacement fields are expressed in terms of the product of two simple algebraic polynomials in ξ and η, respectively. Boundary conditions are enforced in the later part of the solution by simply controlling coefficients of the polynomials. The procedure setup in this paper is such that it was possible to investigate the free axisymmetric and asymmetric vibrations of a wide range of problems, namely; circular disks, cylinders, cones, and spheres with considerable success. The numerical cases include circular disks of uniform as well as varying thickness, conical/cylindrical shells and finally a spherical shell of uniform thickness. Convergence study is also done to examine the accuracy of the results rendered by the present method. The results are compared with the finite element method using the eight-node isoparametric element for the solids of revolution and published data by other researchers.

A New Approach to the Demonstration of Oxidase-Localization Using Tannic Acid Or Catechin

1973 ◽  
Vol 31 ◽  
pp. 698-699
Author(s):  
V. Mizuhira ◽  
Y. Futaesaku
Keyword(s):  
Cross Section ◽  
Tannic Acid ◽  
Elastic Fiber ◽  
The Other ◽  
New Approach ◽  
Tissue Block ◽  
Active Reaction ◽  
The Cross

Previously we reported that tannic acid is a very effective fixative for proteins including polypeptides. Especially, in the cross section of microtubules, thirteen submits in A-tubule and eleven in B-tubule could be observed very clearly. An elastic fiber could be demonstrated very clearly, as an electron opaque, homogeneous fiber. However, tannic acid did not penetrate into the deep portion of the tissue-block. So we tried Catechin. This shows almost the same chemical natures as that of proteins, as tannic acid. Moreover, we thought that catechin should have two active-reaction sites, one is phenol,and the other is catechole. Catechole site should react with osmium, to make Os- black. Phenol-site should react with peroxidase existing perhydroxide.

Energy Distribution in Electron Beams

1972 ◽  
Vol 30 ◽  
pp. 568-569
Author(s):  
Tamotsu Ohno
Keyword(s):  
Electron Beam ◽  
Energy Distribution ◽  
Cross Section ◽  
Integral Curve ◽  
Electron Beams ◽  
Electron Gun ◽  
Angular Divergence ◽  
Apparent Increase ◽  
Integral Width ◽  
The Cross

The energy distribution in an electron; beam from an electron gun provided with a biased Wehnelt cylinder was measured by a retarding potential analyser. All the measurements were carried out with a beam of small angular divergence (<3xl0-4 rad) to eliminate the apparent increase of energy width as pointed out by Ichinokawa.The cross section of the beam from a gun with a tungsten hairpin cathode varies as shown in Fig.1a with the bias voltage Vg. The central part of the beam was analysed. An example of the integral curve as well as the energy spectrum is shown in Fig.2. The integral width of the spectrum ΔEi varies with Vg as shown in Fig.1b The width ΔEi is smaller than the Maxwellian width near the cut-off. As |Vg| is decreased, ΔEi increases beyond the Maxwellian width, reaches a maximum and then decreases. Note that the cross section of the beam enlarges with decreasing |Vg|.

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