Solution of mixed problems on the torsion of an infinite elastic cylinder by the method of dual integral equations and by differentiation of the boundary conditions

1965 ◽  
Vol 29 (3) ◽  
pp. 628-643
Author(s):  
G.M. Valov
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Elastic Cylinder ◽  
Dual Integral Equations ◽  
Mixed Problems

scholarly journals Simple and accurate correlations for some problems of heat conduction with nonhomogeneous boundary conditions

2017 ◽  
Vol 21 (1 Part A) ◽  
pp. 125-132
Author(s):  
Najib Laraqi ◽  
El-Khansaa Chahour ◽  
Eric Monier-Vinard ◽  
Nouhaila Fahdi ◽  
Clémence Zerbini ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Thermal Resistance ◽  
Integral Equations ◽  
Separation Of Variables ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Uniform Temperature ◽  
Dual Integral Equations

Heat conduction in solids subjected to non-homogenous boundary conditions leads to singularities in terms of heat flux density. That kind of issues can be also encountered in various scientists? fields as electromagnetism, electrostatic, electrochemistry and mechanics. These problems are difficult to solve by using the classical methods such as integral transforms or separation of variables. These methods lead to solving of dual integral equations or Fredholm integral equations, which are not easy to use. The present work addresses the calculation of thermal resistance of a finite medium submitted to conjugate surface Neumann and Dirichlet conditions, which are defined by a band-shape heat source and a uniform temperature. The opposite surface is subjected to a homogeneous boundary condition such uniform temperature, or insulation. The proposed solving process is based on simple and accurate correlations that provide the thermal resistance as a function of the ratio of the size of heat source and the depth of the medium. A judicious scale analysis is performed in order to fix the asymptotic behaviour at the limits of the value of the geometric parameter. The developed correlations are very simple to use and are valid regardless of the values of the defined geometrical parameter. The performed validations by comparison with numerical modelling demonstrate the relevant agreement of the solutions to address singularity calculation issues.

scholarly journals A Pair of Dual Integral Equations Involving Bessel Functions of the First and the Second Kind

1964 ◽  
Vol 14 (2) ◽  
pp. 149-158 ◽  
Author(s):  
R. P. Srivastav
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Function ◽  
Cylindrical Cavity ◽  
Bessel Functions ◽  
Boundary Value ◽  
Regularity Conditions ◽  
Dual Integral Equations ◽  
Image Position

In this paper we give a method for the solution of the dual integral equationswhere Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation forsubject to the usual regularity conditions and the following boundary conditions:

Periodic Contact and Mixed Problems of the Elasticity Theory (Review)

2021 ◽  
pp. 22-33
Author(s):  
Dmitrii A. Pozharskii
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Half Space ◽  
Three Dimensional ◽  
Contact Problems ◽  
Mixed Problems ◽  
Periodic Problems ◽  
Straight Line ◽  
Periodic Contact ◽  
Space Boundary

Results are reviewed collected in the investigations of periodic contact and mixed problems of the plane, axially symmetric and spatial elasticity theory. Among mixed problems, cut (crack) problems are focused integral equations of which are connected with those for contact problems. The periodic contact problems stimulate research of the discrete contact of rough (wavy) surfaces. Together with classical elastic domains (half-plane, half-space, plane and full space), we consider periodic problems for cylinder, layer, cone and spatial wedge. Most publications including fun-damental ones by Westergaard and Shtaerman deals with plane periodic problems of the elasticity theory. Here, one can mention approaches based on complex variable functions, Fourier series, Green’s functions and potential func-tions. A fracture mechanics approach to the plane periodic contact problem was developed. Methods and approaches are considered which allow us to take friction forces, adhesion and wear into account in the periodic contact. For spatial periodic and doubly periodic contact and properly mixed problems, we describe such methods as the localiza-tion method, the asymptotic methods, the method of nonlinear boundary integral equations, the fast Fourier trans-form. The half-space is the simplest model for elastic solids. But for the simplest straight-line periodic punch system, some three-dimensional contact problems (normal contact or tangential contact for shifted cohesive coatings) turn out to be incorrect because their integral equations contain divergent series. Considering three-dimensional periodic problems, I.G. Goryacheva disposes circular punches in special way (circular orbits, polar coordinated are used for centers of the punches), in this case one can prove convergence of the series in the integral equation (it is important that the punches are circular). For the periodic problems for an elastic layer, V.M. Aleksandrov has shown that the series in integral equations converge but the kernels become more complicated. In the present paper, we demonstrate that for the straight-line periodic punch system of arbitrary form the contact problem for a half-space turns out to be correct in case of more complicated boundary conditions. Namely, it can be sliding support or rigid fixation of a half-plane on the half-space boundary, the half-plane boundary should be parallel to the straight-line (the punch system axis) for arbitrary finite distance between the parallel lines. On this way, for sliding support, the kernel of the period-ic problem integral equation kernel is free of integrals, it consists of single convergent series (normal contact, the kernel is given in two equivalent forms). We consider classical percolation (how neighboring contact domains pene-trate one to another, investigated by K.L. Johnson, V.A. Yastrebov with co-authors) for the three-dimensional periodic contact amplification as well as percolation for the straight-line punch system. A similar approach is suggested for the case of periodic tangential contact (coatings system cohesive with a half-space boundary shifted along its axis or perpendicular to it). Here, one can separate out unique solutions of auxiliary problems because the line of changing boundary conditions on the half-space boundary can provoke non-uniqueness. The method proposed opens possibility to consider more complicated three-dimensional periodic contact problems for straight-line punch systems with changing boundary conditions inside the period.

Torsion of a Cylindrical Rod Welded to an Elastic Half Space

1967 ◽  
Vol 34 (3) ◽  
pp. 687-692 ◽  
Author(s):  
N. J. Freeman ◽  
L. M. Keer
Keyword(s):  
Integral Equation ◽  
Stress Distribution ◽  
Integral Equations ◽  
Half Space ◽  
Elastic Cylinder ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Dual Integral Equations ◽  
Single Integral

A solution is given for the problem of the torsion of an elastic cylinder welded to an elastic half space. The problem was formulated so as to involve coupling between dual-integral equations and Dini series. These equations were reduced to a single integral equation. Numerical results are given and functions found that approximate the stress distribution.

scholarly journals Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 330
Author(s):  
Gennaro Infante
Keyword(s):  
Fixed Point ◽  
Boundary Conditions ◽  
Integral Equations ◽  
General Class ◽  
Fixed Point Index ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Hammerstein Integral Equations ◽  
Functional Boundary ◽  
Functional Boundary Conditions

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

scholarly journals DUAL INTEGRAL EQUATIONS TECHNIQUE IN ELECTROMAGNETIC WAVE SCATTERING BY A THIN DISK

2009 ◽  
Vol 16 ◽  
pp. 107-126 ◽  
Author(s):  
Mikhail V. Balaban ◽  
Ronan Sauleau ◽  
Trevor Mark Benson ◽  
Alexander I. Nosich
Keyword(s):  
Electromagnetic Wave ◽  
Integral Equations ◽  
Wave Scattering ◽  
Thin Disk ◽  
Dual Integral Equations ◽  
Electromagnetic Wave Scattering
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