A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

The impulsive Reissner-Sagoci problem in a composite medium

The small-time Reissner-Sagoci problem in a bi-material elastic half-space under an impulsive twist is investigated. The problem is reduced to an integral equation, which, by asymptotic analysis and application of the Wiener-Hopf technique, is further reduced to a Fredholm integral equation of the second kind. Approximate analytical results, whose accuracy can be improved by retaining higher order terms in the approximations, are obtained for the free surface displacement outside the disc and the reactive couple necessary to keep the disc in the twisted position, for a bi-material medium with small shear wave speed disparity between the two media.     

Energy bounds for a mixture of two linear elastic solids occupying a semi-infinite cylinder

Summary The aim of this paper is to establish some forms of the Saint-Venant principle for a mixture of two linear elastic solids occupying a semi-infinite prismatic cylinder. We examine the behaviour of the energy for both static and dynamical problems. Under mild assumptions on the asymptotic behaviour of the unknown fields at infinity, we show that in the static case the elastic energy of the portion of the cylinder beyond a distancex ₃ from the loaded region decays exponentially withx ₃. For the dynamical problem we estimate through the data the total energy stored in that part of the cylinder whose minimum distance from the loaded end isx ₃; these estimates, which are based on the assumption that the initial total energy is finite, depend uponx ₃ but do not depend upon the timet.     

Axisymmetric solution for a semi-infinite elastic circular cylinder of one material indenting an elastic half-space of another material

The axisymmetric bonded contact problem of a semi-infinite right circular cylinder of one elastic material indenting a half-space of a different elastic material is reduced to a system of singular integral equations of the second kind. The kernels of the integral equations are found to contain Cauchy and generalized Cauchy-type singularities. The index of the singularity for various material parameters combinations is determined by solving a characteristic determinant, which is obtained by considering the dominant part of the kernels. Using a modification of the method employed in [1], the system of singular integral equations is reduced to a system of simultaneous algebraic equations. The latter may then be solved numerically as in [1].     

Reissner-Sagoci problem for a non-homogeneous half-space with surface constraint

This paper deals with the problem of twisting of a non-homogeneous, isotropic, half-space by rotating a circular part of its boundary surface (0ra, z = 0) through a given angle. A ring (a<r<b, z = 0) outside this circle is stress-free and the remaining part (r>b, z = 0) is rigidly clamped. The shear modulus is assumed to vary with the cylindrical coordinates r, z by the power law ( = _, r z ). Such a dependence is of practical interest in the context of Soil Mechanics. The problem leads to a Fredholm integral equation of the second kind which is solved numerically, giving an evaluation of the influence of non-homogeneity on the torque at the surface and the stress intensity factor. The homogeneous case studied in [16] is recovered. Expressions for some quantities of physical importance such as the torque applied at the surface and stress intensity factor are obtained. It appears from our investigation that the influence of clamping dies out with increasing and . Quantitative evaluations are given and some curves for the relative increase, due to clamping, in the torque and in the stress intensity factor are presented.     

Axisymmetric contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the elastostatic contact problem of a semi-infinite cylinder compressed against a layer lying on a rigid foundation. It is assumed that all the contacting surfaces are frictionless and that only compressive normal tractions can be transmitted through the interfaces. Upon loading the contact along the layer-foundation interface shrinks to a circular area whose radius is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of the contact area between the layer and the foundation, and the stress intensity factor round the edge of the cylinder are calculated for various material pairs.     

The Reissner-Sagoci problem for a non-homogeneous half-space with a penny-shaped crack

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular disc bonded to a non-homogeneous half-space containing a penny-shaped crack. The shear modulus of the half-space is assumed to vary with depth according to the relation (z) = ₁(z + c), c > 0 and ₁, are constants. Using Hankel transforms, the solution of the problem is reduced to integral equations and finally to simultaneous Fredholm integral equations of the second kind. By solving numerically the simultaneous Fredholm integral equations, results are obtained which are used to estimate the stress intensity factor at the crack tip and the torque required to rotate the disc through an angle ₀.     

Saint-Venant's problem for a micropolar elastic circular cylinder

In the present paper Saint-Venant's problem in the linear theory of micropolar elasticity is solved for a circular cylinder. Expressions for the macrodisplacement vector and microrotation vector are obtained.     

The elastic contact problem for dissimilar orthotropic semi-infinite and infinite strips

This article examines the stress state which develops when the end of a symmetrically loaded orthotropic semi-infinite strip is supported on an orthotropic infinite strip. Both smooth and bonded contact between the strips are considered. By using Fourier transforms the problem is reduced to a system of singular integral equations. The dependence of the order of the stress singularity on the orthotropic properties of the materials is investigated. Numerical solutions are presented for the contact stresses corresponding to different loadings and material properties.     

The axisymmetric double contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the smooth receding contact between an elastic layer and a half space when the layer is compressed by a frictionless semi-infinite elastic cylinder. Upon loading, the contact along the layer-subspace interface shrinks to a circular area, radius of which is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of contact and the stress intensity factor round the edge of the cylinder are calculated for various material combinations.     

Two potential theory problems for a semi-infinite cylinder

Two potential theory problems for a semi-infinite cylinder are solved when mixed boundary conditions are prescribed at its end. Using integral transform techniques the problems are reduced to integral equations which can be solved by standard numerical techniques. Although the problems considered here usually lead to integral equations, whose kernel is an infinite integral with divergent integrand, the technique employed circumvents this difficulty.     

A mixed heat conduction boundary problem for a semi-infinite plate

An examination is made of a plane stationary heat conduction problem with mixed conditions assigned on the edge of a semi-infinite plate. There is a temperature field in the plate due to the action of a point source of heat located inside the plate.     

Axial eigenfunctions for the axisymmetric problem of an elastic circular cylinder

Constructing a biorthogonality relation it is shown how solutions can be obtained for the stress, displacement and mixed problems associated with the axisymmetric deformations in a cylinder whose plane faces are either stress-free or displacement-free. Numerical results are presented for the reproduction of the boundary stresses when a self-equilibrating load is prescribed on the circular boundary of the cylinder whose plane faces are stress-free.     

A Dirichlet problem for the biharmonic equation in a semi-infinite strip

This paper addresses the two-dimensional biharmonic problem for a semi-infinite strip with Dirichlet boundary conditions. The method of superposition is used to solve the problem. The object of this paper is to clarify mathematical questions connected with the solution of a special integral equation and to provide a rigorous justification of the applicability of the method of superposition. Mellin's transform technique of investigating the asymptotic behaviour of unknown density when the argument tends to infinity is used.     

A plane dynamic axisymmetric problem for a hollow cylinder

A dynamic problem for an elastic hollow cylinder whose surfaces are subjected to the action of arbitrary forces is solved by a new method based on the use of finite differences only with respect to time [12]. The numerical analyses of the time dependences of stress concentration are carried out for hollow cylinders of different thickness under impact loading. The proposed method is tested by solving a problem of circular hole in an infinite body subjected to the action of impact loads over the entire surface. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 7–13, January–February, 2008.     

A mixed two-dimensional stationary heat-conduction problem for a cylinder

We investigate the exact solution to the problem of calculating the stationary thermal field external to a cylinder on a portion of the surface of which the thermal flow density is constant and on the remaining portion of which the temperature is constant.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 3, pp. 476–479, September, 1971.     

Equilibrium of an elastic finite cylinder: Filon's problem revisited

This paper deals with an analytical solution of the axisymmetric boundary-value problem of the theory of elasticity for a finite circular cylinder with free ends and arbitrary loaded curved surface. The object of this paper is to employ the method of superposition to obtain accurate values of the stress field near the boundaries. The classical Filon (1902) problem of uniformly distributed tangential load applied along two rings at the curved surface is addressed in full detail. The distribution of stresses along some typical sections of the cylinder are shown graphically.     

The statical Reissner-Sagoci problem for an internally loaded transversely isotropic halfspace

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular punch which is bonded to the surface of a transversely isotropic elastic halfspace region. The rotation of the punch is induced by a concentrated couple which acts at a finite distance from the punch along the axis of symmetry. The resulting rigid rotation of the punch is evaluated in exact closed form.     

Axisymmetric solution of the end-problem for a semi-infinite elastic circular cylinder and its application to joined dissimilar cylinders under uniform tension

Using transform methods, axisymmetric end-problem for a semi-infinite elastic circular cylinder is reduced to a system of singular integral equations. The kernels of the integral equations are found to contain Cauchy as well as generalized Cauchy-type singularities. The dominant part of the equations is separated and analyzed to determine the index of the singularity for differing boundary conditions at the end. An approximate method is used to obtain a system of simultaneous algebraic equations from the system of singular integral equations. As an application, axisymmetric solution for joined dissimilar elastic semi-infinite cylinders under uniform tension is solved and various physical quantities of interest, such as normal and shear stresses at the interface, are obtained.