End displacements of semi-infinite cylinders due to annular loadings

Surface displacements at the end of a semi-infinite, circular cylinder due to an axisymmetric ring of forces on the end are examined. The solution which has been found may then be used to find surface displacements for general axisymmetric loadings by convolution. The solution, in tabular form, is given as corrections to the counter-part half-space solution. The method of solution involves a three step superposition process. First, the displacement due to a ring of forces on a half-space is found by using the Boussinesq solution. Then, the excess tractions on the half-space, over that of the cylinder, are removed. This is done in two parts. The problem of an infinite cylinder with linearly varying pressure and shear over a short length of the lateral surface is solved by using Fourier integrals. This is used for the removal of the pressure and shear on the lateral surface of the cylinder by convolution. Next, the stresses at the mid-section of the infinite cylinder are removed. This is done by finding a set of boundary conditions for the end which yields zero tractions on the lateral surface. Then a series of these boundary conditions is used to approximate the tractions which must be removed. With the solution thus obtained, two sample problems are shown: 1. an elastic cylinder in contact with a half-space; 2. a rigid punch in contact with an elastic cylinder.     

Torsional impact of a layer or a cylinder bonded to an elastic half-space

In this paper two torsional impact problems are considered. The first problem deals with the solution of a layer bonded to an elastic half-space when the layer is driven by the torsional impact over a bonded rigid circular disc. In the second problem sudden torsion by a rigid disc attached over the plane face of a circular cylinder is considered and the rest of the plane surface of the cylinder is stress free. The cylinder is bonded to the half-space, making use of Laplace and Hankel transforms the solution of each problem is reduced into Fredholm integral equations of the second kind. A numerical Laplace inversion technique is then used to recover the time depencence of the solution. The numerical values for the applied torque at the surface of rigid disc are calculated for each problem and then are displayed graphically.     

Axisymmetric contact of an elastic layer underlain by rigid base

Axisymmetric contact problems of an elastic layer are studied in this paper. The lower surface of the layer is either bonded to or in smooth contact with a rigid base. The upper surface is in tensionless smooth contact with a rigid flat cylinder, a rigid sphere, an elastic sphere or a circular plate, respectively. A general method is developed so that all these related problems are treated in a straightforward and like manner. In comparison to the other methods used in previous papers, the simplicity and high numerical accuracy are the major advantages of this method. The convergence is proved by comparing the numerical values to some available analytical solutions. Extensive numerical results for contact radii, displacements and contact pressures are provided.     

Transient thermal stresses in a transversely isotropic finite circular cylinder due to an arbitrary internal heat-generation

The transient thermal stresses in a transversely isotropic, finite, solid, circular cylinder resulting from a unsteady-state, axisymmetrical temperature field are examined. A general method of solution based upon a set of stress functions is presented for the case of an arbitrary internal heat-generation. The temperature field is exactly determined by combined use of finite Fourier-cosine and finite Hankel transforms. Numerical calculations are carried out for various cases of the thermal and elastic anisotropies of the materials.     

The burial of a slender elastic vertical cylinder

Formulas are presented for the forces and stresses induced by the burial in a vertical hole of a long slender elastic cylinder, such as a pipe or a cable. These formulas are developed for both the elastic cylinder and the stemming or burial material during the process of backfilling the hole in which the cylinder is suspended. They also account for the load and stress reduction achieved by controlled displacement of the ground surface end of the elastic cylinder downhole.     

Oblique wave scattering by a circular cylinder submerged beneath an ice-cover

When a train of small-amplitude surface water waves is normally incident on a very long horizontal circular cylinder fully submerged in deep water with a free surface, it is well known that it passes over and below the cylinder with a change of phase without experiencing any reflection. However the cylinder does experience reflection for oblique incidence of the surface wave train. It is shown here that the same phenomenon also holds good when the deep water has an ice-cover instead of a free surface, the ice-cover being modelled as a thin elastic plate. Here, for oblique incidence, the reflection and transmission coefficients are obtained approximately and depicted graphically against the wave number in a number of figures.     

Material tailoring for functionally graded hollow cylinders and spheres

We present a technique to tailor materials for functionally graded (FG) linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant hoop (or circumferential) stress or a constant in-plane shear stress. The volume fractions of two phases of a FG material (FGM) are assumed to vary only with the radius and the effective material properties are estimated by using either the rule of mixtures or the Mori-Tanaka scheme; the analysis is applicable to other homogenization methods. For a FG cylinder we find the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the hoop stresses uniform throughout the thickness. The through-the-thickness uniformity of the hoop stress automatically eliminates the stress concentration near the inner surface of a very thick cylinder. The through-the-thickness variations of Young’s moduli obtained with and without considering the variation of Poisson’s ratio are very close to each other for a moderately thick hollow cylinder but are quite different in a very thick hollow cylinder. For an FG sphere the required radial variation of the volume fractions of the two phases to get a constant circumferential stress is similar to that in an FG cylinder. The material tailoring results presented here should help structural engineers and material scientists optimally design hollow cylinders and spheres comprised of inhomogeneous materials.     

Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a pre-assigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r². The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r². For a hollow cylinder fixed on the inner surface and subjected to tangential tractions on the outer surface (or vice versa) the through-the-thickness in-plane shear stress distribution is also universal and is determined by surface tractions and the outer radius of the cylinder; it is independent of the spatial variation of the shear modulus.     

Incremental elastic compliance and electric resistance of a cylinder with partial loss in the cross-sectional area

Motivated by material science applications, the paper focuses on quantitative characterization and comparison of two microstructural elements typical for lamellar materials - crack and contacting area - in the context of their effect on macroscopic elastic and conductive (thermal or electrical) properties of a body of finite size. The problem is solved in axisymmetric formulation - axial load or axial heat flux is applied to a circular cylinder containing a centered crack, either internal or external. The latter case corresponds to the welding of two halves of the cylinder at the center. The changes in the elastic and conductive properties of the cylinder due to these types of cracks are obtained in explicit analytical form. It is shown that the contributions of internal and external cracks into elastic and conductive properties are similar if the relative loss in the cross-sectional between two parts of the cylinder is up to 70% for elasticity problem and up to 85% for conductivity problem. We also show that the changes in elastic compliance and conductive properties generated by both microstructural elements are interrelated by cross-property connection identical to one obtained for an unbounded material.     

Surface waves in fibre-reinforced anisotropic elastic media

The aim of this paper is to investigate surface waves in anisotropic fibre-reinforced solid elastic media. First, the theory of general surface waves has been derived and applied to study the particular cases of surface waves — Rayleigh, Love and Stoneley types. The wave velocity equations are found to be in agreement with the corresponding classical result when the anisotropic elastic parameters tends to zero. It is important to note that the Rayleigh type of wave velocity in the fibre-reinforced elastic medium increases to a considerable amount in comparison with the Rayleigh wave velocity in isotropic materials.     

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.     

Frictionless elastic contact problem for a curved rigid punch of arbitrary shape

Summary A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.With 3 Figures     

Generalized debye series for acoustic scattering from objects of separable geometric shape

Summary In a classical paper of 1908, Debye has resolved the electromagnetic field scattered by a dielectric cylinder into a series of waves multiply internally reflected in the cylinder. For acoustic scattering by elastic cylinders, a corresponding series was derived from the conventional solution (obtained by satisfying the overall or global continuity conditions) by Brill and Überall, taking into account mode conversions of longitudinal (L) into transverse (T, shear) waves, or vice versa, upon internal scattering in a some-what involved fashion. In a series of papers, Gérard has shown that this approach could be greatly simplified by introducing local reflection and transmission coefficients at each interface, which is suitable for generalizing the Debye series to the case of elastic waves coupled by the continuity conditions at the external and each of any possible (multiple) internal interfaces of the scattering object. The approach is then applicable to all elastic objects for which surface and interfaces form coordinate surfaces of any separable geometry; the corresponding derivation is given here in the most general fashion, and is concretely illustrated by the examples of an elastic plate, infinite cylinder and sphere.     

Steady circulatory flow of a second grade fluid about a rotating porous cylinder

An exact solution for the circulatory flow of an incompressible second grade fluid about a rotating porous cylinder is given. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number, the elastic number and the ratio of the circulation at infinity to that on the surface of the cylinder. The velocity, the vorticity and the torque exerted by the fluid on the cylinder are calculated. It is shown that there are some discrepancies between the results obtained by the exact solution and those obtained by the perturbation solution which is valid for small values of the elastic number.     

Analysis of elastic wave propagation in a functionally graded thick hollow cylinder using a hybrid mesh-free method

In this paper, a hybrid mesh-free method based on generalized finite difference (GFD) and Newmark finite difference (NFD) methods is presented to calculate the velocity of elastic wave propagation in functionally graded materials (FGMs). The physical domain to be considered is a thick hollow cylinder made of functionally graded material in which mechanical properties are graded in the radial direction only. A power-law variation of the volume fractions of the two constituents is assumed for mechanical property variation. The cylinder is excited by shock loading to obtain the time history of the radial displacement. The velocity of elastic wave propagation in functionally graded cylinder is calculated from periodic behavior of the radial displacement in time domain. The effects of various grading patterns and various constitutive mechanical properties on the velocity of elastic wave propagation in functionally graded cylinders are studied in detail. Numerical results demonstrate the efficiency of the proposed method in simulating the wave propagation in FGMs.     

Dispersion of Axisymmetric Longitudinal Waves in A Bi-Material Compound Solid Cylinder Made of Viscoelastic Materials

The paper studies the dispersion of axisymmetric longitudinal waves in the bi-material compound circular cylinder made of linear viscoelastic materials. The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact equations of linear viscoelasto-dynamics. The corresponding dispersion equation is derived for an arbitrary type of hereditary operator and the algorithm is developed for its numerical solution. Concrete numerical results are obtained for the case where the relations of the constituents of the cylinder are described through fractional exponential operators. The influence of the viscosity of the materials of the compound cylinder on the wave dispersion is studied through the rheological parameters which indicate the characteristic creep time and long-term values of the elastic constants of these materials. Dispersion curves are presented for certain selected dispersive and non-dispersive attenuation cases under various values of the problem parameters and the influence of the aforementioned rheological parameters on these curves is discussed. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the components of the compound cylinder are the same, the viscosity of the layers' materials causes the axisymmetric wave propagation velocity to decrease.     

Die Abplattung im Kontaktproblem paralleler Zylinder

To calculate the deflection of cylinders with parallel axes contacting under load, i.e. the approach of the two axes, a simple equation of general validity is given. Such an equation was searched for since the fundamental works of Heinrich Hertz in 1881 and 1882 solving the contact problem of elastic bodies under load. The objective of the presented project was to close this gap on the base of theoretical, numerical and experimental investigations. The principal results can be summarized as follows: The deflection of parallel cylinders is directly proportional to the loading force, but reciprocal to the materials stiffness and to the length of the cylinder. Besides this it is independent of the curvature relations, therefore not influenced by the values of the cylinder's radius and the combinations of curvatures which are convex/convex, convex/plane and convex/concave. These statements are explained theoretically and verified experimentally as well as with FEM-calculations. In this form, they are valid for whole cylinders and for bodies with cylindrical surface in the contact region.     

Torsional buckling of elastic cylinders with hard coatings

The torsional buckling characteristic of an elastic cylinder with a hard surface coating layer is addressed in this paper. Deformations of the core and surface layer are obtained analytically through the Navier??s equation and thin shell model, respectively. Both infinitely and finitely long cylinders are studied and the effects of the surface layer??s stiffness, thickness, residual stresses, as well as the cylinder lengths on the critical torsional angle and buckling morphologies, are discussed. It is found that either the surface rippling or global buckling mode may occur when there exist residual stresses within the surface layer. The critical torsional angle increases when the surface layer becomes stiffer and thinner. In addition, higher-order rippling modes frequently occur for a finite-length cylinder with stiffer and thinner surface layer.     

Interaction of torsional waves with an annular crack in an infinitely long cylinder

The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infinite elastic cylinder, which is excited by normal torsional waves. The curved surface of the cylinder is assumed to be stress free. Solution of the problem is reduced to three simultaneous Fredholm integral equations. By finding the numerical solution of the simultaneous Fredholm integral equations the variations of the dynamic stress-intensity factors are obtained which are displayed graphically.     

Diffraction of torsional waves by a penny-shaped crack in an infinitely long cylinder bonded to an infinite medium

This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.