Finite Solid Cylinder Research Articles

On mean Green operator and Eshelby tensor of hollow cylinders with length from infinite tubes to flat rings in isotropic elastic media

Using the (RT-IRT) Radon transform and inversion formula method, the author has previously reported the exact formal expressions for the mean shape function (mSF) of a finite solid cylinder of general aspect ratio in any infinite medium and for the related mean Green operator (mGO) and mean Eshelby tensor (mET) in cases of isotropic elastic-like (including dielectric-like) linear properties. Nearly exact solutions were obtained provided an accurate analytical approximation of an elliptic integral. Using the same RT-IRT method, the present work addresses the case of hollow cylinders with finite length in same isotropic property contexts. From examining the general formal mSF expression (and comparing with the limits of infinitely long cylinders and infinitely flat platelets) a simple accurate approximation is provided for the mSF, mGO and mET of "thin" hollow cylinders, from tubes to rings what covers a wide range of practical situations. The same expression being shown to hold on the thick side limit, the conjecture that it may be relevant over the whole thickness range is questioned.

An asymptotic local solution for axi-symmetric heat conduction problems with temperature discontinuity on the boundary

An asymptotic local solution to the axisymmetric conduction problem with a discontinuous temperature boundary condition in the radial direction is presented. The Laplace's equation in cylindrical coordinates (r, z) is expanded near the discontinuity at r = 1, z = 0. The leading order term arises from the step-change in the boundary condition. The next term reflects the effect of curvature at the position of the temperature discontinuity and is a particular solution to the Poisson's equation. Subsequent higher-order terms involve general solutions to the Laplace's equation in Cartesian coordinates and depend on conditions on the far end boundaries. Three domain sizes are thus considered: (i) infinite solid cylinder (0 ≤ r < ∞, 0 ≤ z < ∞); (ii) semi-infinite solid cylinder with finite radius (0 ≤ r < a, 0 ≤ z < ∞); and (iii) finite solid cylinder (0 ≤ r ≤ a, 0 ≤ z ≤ a). Exact solutions for temperature in those three cases are used to determine the coefficients in the higher order terms of the asymptotic solution. As a result, a simple expression for local normal wall heat flux near the discontinuity is obtained and the accuracy is confirmed by comparing with the heat flux obtained through high order extrapolation of the exact temperature field near the wall.

Application of the Variational Method of Homogeneous Solutions for the Optimal Control of the Axisymmetric Thermoelastic State of a Cylinder

We propose a variational approach to the solution of the problem of optimization of a stationary axisymmetric thermal stressed state of a finite solid cylinder by controlling the distribution of volumetric heat sources. The proposed approach is based on the variational method of homogeneous solutions developed earlier for the solution of axisymmetric problems of the theory of elasticity for a cylinder. We study the influence of the height-to-radius ratio of the cylinder on the optimal values of the objective functional and the stressed state.

Hygrothermoelastic response of a finite solid circular cylinder

PurposeIn this paper, a solid circular cylinder of finite length occupying the space 0⩽r⩽1, 0⩽z⩽h is considered. The purpose of this paper is to adopt a linear hygrothermal effect to analyze the unsteady state responses in a finite long solid cylinder subjected to axisymmetric hygrothermal loading T=TR and C=CR at the surface. The analytical solution of temperature, moisture and thermal stresses is obtained by using the integral transform technique. The coupling and uncoupling effects of temperature, moisture and thermal stresses are discussed for a graphite fiber-reinforced epoxy matrix composite material (T300/5208). The numerical results of transient response hygrothermoelastic field are presented graphically.Design/methodology/approachIn the present problem, hygrothermoelastic response of a finite solid circular cylinder has been investigated by integral transform technique consisting of Laplace transform, Hankel transform and Fourier-cosine transform. The problem is investigated subjected to prescribed sources. Numerical algorithm has been developed for numerical computation.FindingsThe analytical solution of temperature, moisture and thermal stresses is obtained by using the integral transform technique. The coupling and uncoupling effects of temperature, moisture and thermal stresses are discussed for a graphite fiber-reinforced epoxy matrix composite material (T300/5208). The numerical results of transient response hygrothermoelastic field are presented graphically.Research limitations/implicationsThe work presented here is mostly hypothetical in nature and totally mathematical.Practical implicationsIt may be useful for composite materials, composite laminated plates in hygrothermal environment. Also it is having the applications in hygrothermal field where porous media exposed to heat and moisture. The problem investigated will be beneficial for the researcher working in the field thermoelastic diffusion and hygrothermoelastic materials.Originality/valueTill date, the other authors did the research work on hygrothermal effect of an infinitely long cylinder without thickness. In this paper, the authors consider finite solid cylinder with finite length and discuss the hygrothermal effect within a small range. Second, the material properties are both homogenous and isotropic and are independent of both temperature and moisture.

Torsion problems of finite cylinders weakened by ring-shaped cracks

Abstract: A finite elastic hollow and solid cylinders are considered. The bottom faces of the cylinders are fixed, the upper faces are free from stress. The tangent axisymmetric loading is applied along their cylindrical surfaces. This leads to the torsion axisymmetric deformation. A system of N ring – shaped cracks is situated inside the cylinders parallel to the cylinder's axis. It is supposed that the branches of the cracks are free from stress. It is necessary to construct the formulas for the stress intensity factor calculation and investigate the stress state of a solid. The initial boundary value problem is reduced with Fourier transformation to a system of integral singular equations with regard to the jumps of the displacements at the cracks' branches. The singularity of the equations kernels is extracted. The system of singular integral equations is solved with the orthogonal polynomial method. The solution of the system is searched as the series by Chebyshev polynomials with the weight function. The realisation of orthogonal polynomial method leads to an infinite system of linear algebraic equations with regard to the unknown coefficients of the series. The formulas for the stresses and displacements of an elastic finite cylinder are presented. The numerical realisation of the proposed method is demonstrated in cases with two and three cracks; the stress state is investigated dependingon the cracks' locations and sizes.

Torsion analysis of finite solid circular cylinders with multiple concentric planar cracks

First, using the separation of variables technique a series solution for torsion of an intact finite cylinder under self‐equilibrating lateral shear tractions is given. Next, an integral form solution for torsion of an intact infinite cylinder under the aforementioned lateral loading is obtained. After that the solution of an axisymmetric rotational Somigliana ring dislocation in the infinite circular cylinder is obtained. The distributed dislocation technique is used to construct integral equations for stress analysis of the infinite cylinder with a set of coaxial axisymmetric cracks. These defects are penny‐shaped, annular and circumferential edge cracks. The infinite cylinder is under the action of the loading similar to the intact cylinder. The integral equations are solved numerically to obtain the dislocation density on the surfaces of the cracks. The stress intensity factors for the cracks are determined by employing the dislocation densities. The problem of a cracked finite cylinder is treated by cutting method. That is, a similar cracked infinite cylinder is sliced by extending two additional annular axisymmetric cracks. To validate the cutting method, the solution of the sliced intact infinite cylinder is compared to that of an intact finite cylinder which both of them are under the same self‐equilibrating lateral shear tractions. The solution is applied to several examples to study the effect of crack type/location on the resulting stress intensity factors at tips of the cracks.

AXISYMMETRIC RESPONSES OF THERMOPIEZOELECTRIC FINITE CYLINDER

This paper is concerned with theoretical treatment of transient responses of finite thermopiezoelectric cylinder under axisymmetric mechanical, electrical and thermal loading. The analytical general solutions for axisymmetric deformations of a thermopiezoelectric medium are obtained in terms of series of Bessel and modified Bessel functions of the first kind. A boundaryvalue problem corresponding to a finite solid thermopiezoelectric cylinder with electrically impermeable boundary condition under applied axisymmetric electromechanical loading and temperature change is solved by expanding the applied load in terms of Fourier-Bessel series. Accuracy of the present numerical solution scheme is confirmed by comparing with existing transient solution of an elastic cylinder. To portray the salient features of the coupling transient responses of finite solid thermopiezoelectric cylinders, selected numerical results are presented for displacements, stresses and temperature distributions inside the cylinders subjected to applied traction together with prescribed temperature at the lateral curve surface of the cylinder.

Equilibrium of an elastic finite cylinder under axisymmetric discontinuous normal loadings

This article presents an analytical technique for solving the axisymmetric elasticity problem for a finite solid cylinder subjected to discontinuous normal loadings on its ends and lateral surface. This technique is based on application of the method of crosswise superposition by representing the solution for stresses in the form of decompositions into Fourier and Bessel–Dini series. For determination of the coefficients in these series, the infinite systems of linear algebraic equations are obtained and solved by means of a modified algorithm of advanced reduction. The technique is numerically validated for typical cases of discontinuous loading. It is shown that the solution procedure is efficient for determination of the stresses in the cylinder including its edges and discontinuity points of normal loadings.

Transient elastodynamic response of finite and infinite solid cylinders

The orthogonal eigensolutions for the vibrations of an isotropic finite solid cylinder with a traction-free lateral boundary and rigid-smooth end boundaries are provided. The transient elastodynamic response of this solid cylinder is then constructed using the method of eigenfunction expansion and further extended explicitly and concisely to that of an isotropic infinite solid cylinder. The numerically evaluated analytical solution is shown to compare favorably with that by finite element method (FEM). The effect of external forces on the excitation of each guided wave mode can be quantitatively investigated on the basis of the present solution.

Finite Cylinders of Si1 − x Ge x Alloy Under the Double-Punch Test and Effect on Three Valence-Bands

An analytical solution for the stress and strain fields within a finite circular solid cylinder of Si1 − xGex alloy under the double-punch test is presented. The method of solution uses the displacement function approach, together with proposing a new expression for the displacement function so that all of the boundary conditions are satisfied exactly. The present solution covers the solution by Wei and Chau [1] for isotropic cylinders under the double-punch test when the isotropic limit case is considered. Numerical results show that strain and stress distributions within the cylinder are not uniform, and strain or stress concentrations are usually developed near the end surfaces, but the strain and stress distributions within the central part are relatively uniform. Moreover, according to quantum mechanics and the energy band theory, the effects of strain on three valence bands, namely the heavy hole (HH) band, the light hole (LH) band and the split-off (SO) band of Si1 − xGex alloy and the corresponding conductivity masses are analyzed. It was found that the quantum behavior of the three valence bands, such as the energy distribution, the shape of constant energy surfaces, the magnitude and the distribution pattern of the conductivity masses can be modified considerably by the external force applied in the double-punch test, depending on the proportion between silicon and Germanium in Si1 − xGex alloy.

The Generalized Lame´ Problem—Part I: Coupled Poromechanical Solutions

Cylindrical geometries are known to present special problem simulation capabilities in engineering design. (For example, solid and hollow cylinder tests are routinely studied in soil and rock mechanics to gain insights into the geomechanical properties and to assess the stability of boreholes and cylindrical openings in the project design geomedia.) This paper identifies a unified and universal solution to all the three recognized right-cylindrical problem objects in poromechanics. A closed-form solution to the problem of the finite, homogeneous, isotropic, fully saturated, thick-walled hollow cylinder subjected to various loading modes is readily presented and described. The assumed loading modes encompass arbitrary temporal functions of uniformly distributed inner/outer pore pressure, inner/outer confining pressure, inner/outer deviatoric stress, and end axial compaction or extension. The time-dependent response derivations are outlined within the frameworks of the Biot’s theory of linear poroelasticity and facilitated by the governing generalized plane-strain (GPS) principle. The (as presented) solution is shown to converge asymptotically to those of the two essential problem setups in geomechanics: the finite solid cylinder and the borehole core in an infinite medium. As such, a complete/explicit solution to a generalized statement of the Lame´ problem is presented. The solution utilizes a fairly simple loading decomposition scheme which leads to two basic problem forms: a generalized poroelastic axisymmetric problem and a generalized, plane-strain, poroelastic deviatoric problem.

The Generalized Lame´ Problem—Part II: Applications in Poromechanics

The cylinder is the geometry most widely used in laboratory testing procedures for rocks and other geomaterials. This paper applies a unified and universal Lame´ solution to all the three recognized right-cylindrical problems in poromechanics. As such, the solution of the hollow-cylinder features itself converging asymptotically to the exact values predicted by the solutions of the two other essential problem setups in geomechanics; namely, the finite solid cylinder case and the borehole core in an infinite medium. The time-dependent response derivations were “scripted” within the frameworks of the Biot’s theory of linear poroelasticity and facilitated by the governing generalized plane-strain (GPS) principle.

A general integration method for radiative transfer in 3-D non-homogeneous cylindrical media with anisotropic scattering

A general set of integral equations is presented to solve 3-D radiative heat transfer problems in emitting, absorbing and linear anisotropic scattering finite hollow or solid cylinders with non-homogeneous media. By tracing a ray to compute the intensity,it is much easier to handle the spatial change properties including extinction coefficient. Both the continuous change property and step-change property are dealt with without difficulties. The solid angle integration in getting the incident radiation and heat fluxes is represented by the bounding surface integration. In order to avoid the singularity problem near the bounding surface, the surface integrations are transformed to new modified integral equations by mathematical methods. By doing so, we get more flexible general integral equations applicable to all cases (3-D solid cylinders, 3-D hollow cylinders, finite cylinders or infinite cylinders). This scheme has been verified by comparing the results with published data in the literature. It is believed that this method will be useful in combined radiation and convection heat transfer problems.

An efficient method to calculate surface currents on a PEC cylinder with flat end caps

In this paper, an efficient method is developed to model the scattering of electromagnetic waves by large three‐dimensional perfectly conducting (PEC) bodies of revolution with sharp edges. The magnetic field integral equation (MFIE) is implemented and applied to a finite solid cylinder, including the upper and lower end caps. Point matching is used to construct the interaction matrix. The numerical instabilities that generally occur to the current behavior near sharp edges are completely eliminated by the proper choice of special base functions near the edges. Not taking the singular behavior of the currents into account leads to a bad convergence or to incorrect results.

Finite solid circular cylinders subjected to arbitrary surface load. Part I — Analytic solution

This paper presents a general framework for obtaining analytic solutions for finite elastic isotropic solid cylinders subjected to arbitrary surface load. The method of solution uses the displacement function approach to uncouple the equations of equilibrium. The most general solution forms for the two displacement functions for solid cylinders are proposed in terms of infinite series, with z- and θ-dependencies in terms of trigonometric and hyperbolic functions, and with r-dependency in terms of Bessel and modified Bessel functions of the first kind of fractional order. All possible combinations of odd and even dependencies of θ and z are included; and the curved boundary loads are expanded into double Fourier Series expansion, while the end boundary loads are expanded into Fourier–Bessel expansion. It is showed analytically that only one set of the end boundary conditions needs to be satisfied. A system of simultaneous equations for the unknown constants is given independent of the type of the boundary loads. This new approach provides the most general theory for the stress analysis of elastic isotropic solid circular cylinders of finite length. Application of the present solution to the stress analysis for the double-punch test is presented in Part II of this study.

Finite solid circular cylinders subjected to arbitrary surface load. Part II — Application to double-punch test

This paper derives the stress distributions within a finite isotropic solid circular cylinder of diameter 2 b and length 2 h under the double-punch test, which was introduced by Chen (1970) for the determination of the indirect tensile strength of concrete. The stresses induced by the two rigid circular punches of diameter 2 a at the top and bottom of the solid cylinder are modeled by considering contact problem. The general stress analysis discussed in a companion paper (Part I) is used to obtain the stress field within the solid. In general, tensile stress concentrations are developed beneath the punches compared to the roughly uniform tensile stress at the central portion of the axis of the cylinder. The maximum tensile stress in the tensile zone decreases with the increase of Poisson’s ratio and a/b, but is roughly independent of h/b. For small Poisson’s ratio (say about 0.1) and a/b (say smaller than 0.1), the assertion made by Chen (1970) and Marti (1989) that a uniform tensile stress field, similar to that of the Brazilian test, is developed along the axis of symmetry is incorrect. The tensile strength interpreted from the present analysis is found comparable to the formula proposed by Bortolotti (1988) for a/b>0.2 and agrees well with the experimental data, and thus provides an improvement over Chen’s (1970) original formula.

Backscattering enhancements from Rayleigh waves on the flat face of a tilted solid cylinder in water

When the face of a finite solid elastic cylinder is ensonified by an acoustic wave, a variety of backscattering contributions associated with acoustic wave coupling into elastic waves are observed. A significant backscattering enhancement is observed for tilts such that the acoustic wave is incident on the face of the cylinder in the vicinity of the coupling angle for launching Rayleigh waves across the face. The observed backscattering indicates that the Rayleigh waves are reflected at the edge of the face and subsequently radiate acoustic waves in the backscattering direction. The measured backscattering is compared to an approximate theoretical prediction. Approximating the focusing of the Rayleigh wave after reflection at the (circular) edge by a Gaussian beam pressure distribution on the cylinder's face yields simple expressions for the amplitude which are consistent with the measurements. In the vicinity of end-on incidence, other backscattering contributions due to the reflection of waves traveling down the length of the cylinder are observed. There is also evidence of a face-traversing longitudinal wave for slightly tilted cylinders.

A Stress Analysis of a Taper Hub Flange With a Bolted Flat Cover

A bolted connection consisting of a cover on a pressure vessel flange with a metallic flat gasket on raised faces is analyzed as a four-body contact problem using axisymmetrical theory of elasticity. The contact stress distribution, the load factor (the relationship between an increment of bolt axial force and an internal pressure), and the gasket properties (the gasket seating width and the moment arm) are examined. In the analysis, the cover is replaced with a finite solid cylinder. The metallic flat gasket, the flange, and the hub are replaced with finite solid cylinders. The effects of the stiffness and the thickness of various size gaskets on the contact stress distribution are obtained by numerical calculations. Experiments were carried out to obtain the load factor, the maximum stress produced in bolts, and the stress produced on the hub. The analytical results obtained are shown to be consistent with the experimental results.

Rotation of a Cylinder in a Casing at Zero Reynolds Number

A finite solid cylinder rotates inside a larger, fluid-filled cylindrical casing. The Stokes equation is solved by an efficient method using domain decomposition, eigenfunction expansion, and collection. The resistive torque is found as a function of the geometric parameters. The torque due to the rotation of a finite cylinder in an infinite fluid is extrapolated.

Static Axisymmetric End Problems in Semi-infinite and Finite Solid Cylinders

Our earlier technique for a semi-infinite strip (Kim and Steele, 1990) is extended to study general end problems and corner singularities for semi-infinite and finite solid cylinders with free walls. For handling general end conditions, we expand the displacement and stress in term of the Dini series which are the solutions of the cylinders with mixed wall conditions. The relation between the harmonic coefficients of the end displacement and stress is then formed, which we call the end stiffness matrix. One advantage of the end stiffness matrix approach is that the procedure for finite cylinders can be easily built up from that of semi-infinite cylinders. For some end conditions which may yield singular stresses, the nature of the singularity is investigated by the asymptotic analysis of the Dini series coefficients of the stresses. The problems studied by Benthem and Minderhoud (1972) and Robert and Keer (1987) are solved with the present approach.