scholarly journals Two-Dimensional Electrostatic Problem in a Plane with Earthed Elliptic Cavity due to One or Two Collinear Charged Electrostatic Strips

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
B. M. Singh ◽  
J. G. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Charge Density ◽  
Closed Form ◽  
Integral Equations ◽  
Integral Transform ◽  
Weight Functions ◽  
Two Dimensional ◽  
Electrostatic Problem ◽  
Closed Form Solutions ◽  
Triple Integral Equations ◽  
Elliptic Cavity

A two-dimensional electrostatic problem in a plane with earthed elliptic cavity due to one or two charged electrostatic strips is considered. Using the integral transform technique, each problem is reduced to the solution of triple integral equations with sine kernels and weight functions. Closed-form solutions of the set of triple integral equations are obtained. Also closed-form expressions are obtained for charge density of the strips. Finally, the numerical results for the charge density are given in the form of tables.

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

2018 ◽  
Vol 24 (6) ◽  
pp. 1821-1848 ◽  
Author(s):  
Yuan Li ◽  
CuiYing Fan ◽  
Qing-Hua Qin ◽  
MingHao Zhao
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Elliptical Crack ◽  
Two Dimensional ◽  
Crack Face ◽  
Penny Shaped Crack ◽  
Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

scholarly journals Closed-form solution for piezoelectric layer with two collinear cracks parallel to the boundaries

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
B. M. Singh ◽  
J. Rokne ◽  
R. S. Dhaliwal
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Piezoelectric Layer ◽  
Electric Displacement ◽  
Closed Form Solution ◽  
Finite Width ◽  
Intensity Factors ◽  
Collinear Cracks ◽  
Triple Integral Equations ◽  
Electromechanical Loading

We consider the problem of determining the stress distribution in an infinitely long piezoelectric layer of finite width, with two collinear cracks of equal length and parallel to the layer boundaries. Within the framework of reigning piezoelectric theory under mode III, the cracked piezoelectric layer subjected to combined electromechanical loading is analyzed. The faces of the layers are subjected to electromechanical loading. The collinear cracks are located at the middle plane of the layer parallel to its face. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions for stress intensity factors, electric displacement intensity factors, and shape of crack and energy release rate are derived. As the limiting case, the solution of the problem with one crack in the layer is derived. Some numerical results for the physical quantities are obtained and displayed graphically.

Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

1990 ◽  
Vol 112 (3) ◽  
pp. 460-468 ◽  
Author(s):  
Tsung-Ju Gwo ◽  
Thomas J. Lardner
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Approximate Analytical Solution ◽  
Experimental Studies ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Rigid Substrate ◽  
Closed Form Solutions ◽  
Finite Element Calculations ◽  
Cylindrical Roller

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

On the Approximate Solution of Viscous-Flow Problems

1963 ◽  
Vol 30 (2) ◽  
pp. 263-268 ◽  
Author(s):  
J. A. Schetz
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
New Method ◽  
Two Dimensional ◽  
General Technique ◽  
Closed Form Solutions ◽  
Flow Problems

The need for a general technique for the approximate solution of viscous-flow problems is discussed. Existing methods are considered and a new method is presented which results in simple closed-form solutions. The accuracy of the method is demonstrated by comparisons with the results of known exact solutions, and finally the general technique is employed to determine a new solution for the fully expanded two-dimensional laminar nozzle problem.

Closed form solutions for the geometrical spreading in inhomogeneous media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 1189-1197 ◽  
Author(s):  
Amir‐Homayoon Najmi
Keyword(s):  
Closed Form ◽  
P Wave ◽  
Weight Functions ◽  
Reflection Coefficients ◽  
Geometrical Spreading ◽  
Main Subject ◽  
Closed Form Solutions ◽  
The Earth ◽  
Symmetry Properties ◽  
Kirchhoff Integral

True‐amplitude migration is a subject of great interest to exploration geophysicists. The procedure should provide a means of computing angle‐dependent reflection coefficients of reflectors within the Earth and is therefore essential in any AVO analysis. The migration weighting functions in the Kirchhoff integral include geometrical spreading factors whose determination in terms of traveltime functions and their end point derivatives are the main subject of this paper. Such “closed form” solutions for the geometrical spreading of an acoustic P‐wave in an isotropic and inhomogeneous medium are presented, and their symmetry properties are used to simplify the Kirchhoff integral migration weight functions. Emphasis is put on derivation of the equations based on simple physical and mathematical requirements. The result of applying the derived forms to a synthetic example comprised of a velocity field that varies linearly with depth and dipping reflectors is also included. It is suggested that the migration weight functions could be simplified substantially for smooth velocity backgrounds.

scholarly journals On triple trigonometrical equations

1973 ◽  
Vol 14 (2) ◽  
pp. 174-178 ◽  
Author(s):  
B. M. Singh
Keyword(s):  
Exact Solution ◽  
Charge Density ◽  
Surface Charge ◽  
Electrostatic Potential ◽  
Surface Charge Density ◽  
Equal Length ◽  
Two Dimensional ◽  
Electrostatic Problem ◽  
Unit Depth

An exact solution of triple trigonometrical equations is obtained by using the finiteHilbert transform. The solution of these equations is used to solve a two-dimensional electrostatic problem. The problem of determining the electrostatic potential due to two parallel coplanar strips of equal length, charged to equal and opposite potentials, each parallel to and equidistant from an earthed strip, is considered. Both the charged strips lie along the x-axis and they are equally spaced with respect to the y-axis. Finally the expression for the surface charge density (per unit depth) of the strip is derived

Two-dimensional planar plumes: non-Boussinesq effects

2014 ◽  
Vol 750 ◽  
pp. 245-258 ◽  
Author(s):  
T. S. van den Bremer ◽  
G. R. Hunt
Keyword(s):  
Closed Form ◽  
Uniform Density ◽  
Two Dimensional ◽  
Boussinesq Model ◽  
Axisymmetric Case ◽  
Closed Form Solutions ◽  
Plume Model ◽  
Entrainment Velocity ◽  
Planar Area ◽  
Pure Plume

AbstractIn an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

A Fourier generalized convolution transform and applications to integral equations

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Nguyen Thao ◽  
Vu Tuan ◽  
Nguyen Hong
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Integral Transform ◽  
Type Theorem ◽  
Generalized Convolution ◽  
Fourier Cosine ◽  
Fourier Sine

AbstractWe introduce an integral transform related to a Fourier sine-Fourier - Fourier cosine generalized convolution and prove a Watson type theorem for the transform. As applications we obtain solutions of some integral equations in closed form.

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