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.     

The Influence of the Imperfectness of Contact Conditions on the Critical Velocity of the Moving Load Acting in the Interior of the Cylinder Surrounded with Elastic Medium

The dynamics of the moving-with-constant-velocity internal pressure acting on the inner surface of the hollow circular cylinder surrounded by an infinite elastic medium is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linear theory of elastodynamics. It is assumed that the internal pressure is point-located with respect to the cylinder axis and is axisymmetric in the circumferential direction. Moreover, it is assumed that shear-spring type imperfect contact conditions on the interface between the cylinder and surrounding elastic medium are satisfied. The focus is on the influence of the mentioned imperfectness on the critical velocity of the moving load and this is the main contribution and difference of the present paper the related other ones. The other difference of the present work from the related other ones is the study of the response of the interface stresses to the load moving velocity, distribution of these stresses with respect to the axial coordinates and to the time. At the same time, the present work contains detail analyses of the influence of problem parameters such as the ratio of modulus of elasticity, the ratio of the cylinder thickness to the cylinder radius, and the shear-spring type parameter which characterizes the degree of the contact imperfection on the values of the critical velocity and stress distribution. Corresponding numerical results are presented and discussed. In particular, it is established that the values of the critical velocity of the moving pressure decrease with the external radius of the cylinder under constant thickness of that.     

Inverse Problem of Evaluation of the Coefficient of Friction of Layers According to the Data of Measurements of the Surface Displacements

We consider a one-dimensional model of friction contact of two layers of different nature. The lower surface of the first layer is elastically fixed and the second layer is pressed to the upper surface of the first layer and moves along this surface with variable velocity. As a result of friction, heat is produced on the contact surface according to the Amonton's law. For known boundary and initial conditions, we pose the problem of evaluation of the friction coefficient and the intensity of friction heat flux according to given values of the vertical displacements of the upper surface of the second layer. The posed problem is reduced to the inverse contact problem of thermoelasticity described by the Volterra integral equation of the first kind. The solution of the problem obtained by the method of averaging of functional corrections enables us to study the time behavior of the indicated quantities for the entire period of interaction of the bodies and establish the dependence of the friction coefficient on the basic parameters of the process (sliding velocity, contact pressure, and temperature of the contact surface). The solution of the direct contact problem of thermoelasticity is used to perform the numerical verification of the proposed method for the solution of the inverse problem.     

Transient thermal contact problem for axial crack in a hollow circular cylinder

The transient behavior of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated. It is shown that surface heating may induce compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, this problem can be divided into two parts and solved by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without a crack are obtained by finite element/implicit time integration method. The calculated temperature and thermal stress distributions are in good agreement with the values predicted by the analytical method. Secondly, the opposite senses of the stress distributions along the cracked surfaces, which are obtained previously, are treated as the traction boundary conditions, and the contact length and contact pressure of the real cracked cylinder are obtained by a modified elimination finite element scheme. In this scheme, the concepts of contact-node-pairs' penetration, contact-double-forces and compliance matrix are introduced. The calculated results indicate that the contact length ratio becomes smaller when the crack length ratio increases, and becomes larger as the radius ratio increases. Finally, the normalized stress intensity factor for the crack tip of the cylinder is obtained. It is shown that the larger the crack length ratio the higher the stress intensity factor.     

Coupled transient thermoelastic contact problems for axial cracks in hollow cylinders

A coupled transient thermoelastic behaviour of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated in this study. It is shown that surface heating may induce the compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, we can divide this problem into two parts and solve it by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without crack are obtained by finite element implicit time integration method Secondly, the opposite sense of the stress distributions along the cracked surfaces, which is obtained previously, is treated as the traction boundary conditions; the contact length and contact pressure of the real cracked cylinder are obtained by modified elimination finite element scheme. Finally, we also obtained the normalized stress intensity factor for the crack tip of the cylinder. It is concluded that the effect due to thermoelastic coupling term on stress intensity factor becomes more important for higher coupling coefficient, and this coupling term also results in a small time lag in temperature, thermal stress and stress intensity factor.     

Static deformations of functionally graded polar-orthotropic cylinders with elliptical inner and circular outer surfaces

Functionally graded materials (FGMs) enable one to tailor the spatial variation of material properties so as to fully use the material everywhere. For example, in a hollow circular cylinder one can vary, in the radial direction, the material moduli to make the hoop stress constant. Whereas the problem for a hollow cylinder with the inner and the outer surfaces circular has been studied, that of a cylinder with a circular outer surface and a non-circular inner surface or vice versa has not been investigated. We study here such a plane-strain problem when the cylinder material is polar-orthotropic, material properties vary exponentially in the radial direction, and deformations are independent of the axial coordinate. The problem is challenging since the cylinder thickness varies with the angular position of a point, and the cylinder material is inhomogeneous. Equilibrium equations are solved by expanding the radial and the circumferential displacements in Fourier series in the angular coordinate. The method of Frobenius series is used to solve ordinary differential equations for coefficients of the Fourier series, and boundary conditions are satisfied in the sense of Fourier series. A parametric study has been conducted that delineates effects on stresses of the eccentricity of the ellipse, the material property gradation index and loads applied on boundaries of the cylinder. The analytical solutions presented here will serve as benchmarks for comparing solutions derived by numerical methods.     

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.     

Longitudinal shear of a crack in a multi-material circular cylinder

This is an article dealing with the longitudinal shear of a crack contained in a circular cylinder, which is embedded in and fixed by perfect bonding to a composed hollow circular cylinder, consisting of a number of hollow sub-cylinders of different materials. A rigorous solution to the problem is developed. With the material and geometric constants of the composed hollow circular cylinder as parameters, numerical values of stress intensity factors for the crack are worked out. The delicate behavior in the variation of the stress intensity factors, when the number of the hollow sub-cylinders becomes large, is analyzed and discussed. The solution is developed by utilizing a simplified and improved technique using complex variables.     

Thermoelastic state of a two-component hollow cylinder with edge radial cracks

Stationary two-dimensional axisymmetric problems of thermal conductivity and thermoelasticity for a hollow two-component cylinder with cracks are studied by the method of singular integral equations. The cross section of the cylinder has the form of a circular concentric ring with a layer of another material that also has the form of a concentric ring and contains edge radial cracks. The surfaces of the cylinder are free of stresses. Thermal processes on these surfaces are characterized by temperature conditions of the third kind. Conditions of ideal thermal and mechanical contact are satisfied on the interface of the two media. A numerical solution is obtained for the case where the inner and outer cylindrical surfaces are kept at different constant temperatures. Stress intensity factors near the tip of one or two edge cracks were found for various values of thermal and mechanical characteristics of the cylinder.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 4, pp. 76–80, July – August, 1994.     

A non-axisymmetric cylindrical contact problem

The contact problem under investigation is one whereby a solid circular elastic cylinder of infinite length is rigidly indented by a two piece collar of finite length, each piece being diametrically opposed and extending only partially around one half of the circumference. This case is practically significant in relation to the axisymmetric cylindrical contact problem since in many cases attachment of a component to a cylindrical shaft is achieved by means of a two piece clamp.Shear stresses on the contact interface are taken zero and a radial displacement influence coefficient technique is used to model the integral equation governing this contact problem. Adopting the Papkovich-Neuber solution for the non-axisymmetric cylindrical coordinate case and substituting the appropriate boundary conditions leads to a combined Fourier series, Fourier integral representation for the desired displacements. Convergence of this series—integral is studied and results of interference contact pressure are presented for an illustrative range of the various parameters involved.     

任意轴对称荷载作用下的圆杆分析

锚杆与周围介质相互作用问题涉及圆杆在轴对称荷载作用下的应力应变分析，本文提出一种付里叶方法来分析圆杆在任意轴对称荷载作用下的位移场和应力场，而且考虑了付里叶级数中任意两项的耦合。为了验证这一方法的正确性，本文还导出圆杆在均匀轴对称荷载作用下的理论解。分析成果表明所提方法对于任意轴对称荷载作用下的圆杆问题具有相当好的精度，分析时必须考虑付里叶级数中任意两项的耦合作用。     

Multiple cracks in a half-space under contact loading

This paper develops a semi-analytic solution for multiple cracks in an isotropic half-space under contact loading. The solution takes into account interactions among all the cracks as well as the interactions between the cracks and the loading body. In formulating the governing equations for the subsurface crack problem, each crack of mixed modes I and II is modeled as a continuous distribution of climb and glide dislocations with unknown densities. Such a treatment converts the original contact problem concerning an inhomogeneous half-space into a homogeneous half-space contact problem, for which governing equations with unknown surface contact areas and normal pressure and tangential tractions within the areas can be conveniently formulated. All the unknowns in the governing equations are determined by means of iteration. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the surface contact load and subsurface cracks. The solution is validated by the finite element method. Numerical examples are calculated to demonstrate the generality of the solution.     

Non-axial-symmetric thermal stresses in circular discs or cylinders

A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.     

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.     

Dynamics of the Moving Ring Load Acting in the System “Hollow Cylinder + Surrounding Medium” with Inhomogeneous Initial Stresses

This paper studies the influence of the inhomogeneous initial stress state in the system consisting of a hollow cylinder and surrounding elastic medium on the dynamics of the moving ring load acting in the interior of the cylinder. It is assumed that in the initial state the system is compressed by uniformly distributed normal forces acting at infinity in the radial inward direction and as a result of this compression the inhomogeneous initial stresses appear in the system. After appearance of the initial stresses, the interior of the hollow cylinder is loaded by the moving ring load and so it is required to study the influence of the indicated inhomogeneous initial stresses on the dynamics of this moving load. This influence is studied with utilizing the so-called three-dimensional linearized theory of elastic waves in elastic bodies with initial stresses. For solution of the corresponding mathematical problems, the discrete-analytical solution method is employed and the approximate analytical solution of these equations is achieved. Numerical results obtained within this method and related to the influence of the inhomogeneous initial stresses on the critical velocity of the moving load and on the response of the interface stresses to this load are presented and discussed. In particular, it is established that the initial inhomogeneous initial stresses appearing as a result of the action of the aforementioned compressional forces cause to increase the values of the critical velocity of the moving load.     

The temperature field in a hollow cylinder due to a source moving along a helix

We solve the problem of the temperature distribution in a hollow cylinder due to a source moving along a helix on the outer surface with boundary conditions of the second kind. The solution can be used to calculate temperature fields on a computer for various forms of processing materials.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 1, pp. 154–156, January, 1971.     

Multiple 3D inhomogeneous inclusions in a half space under contact loading

A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading.     

Oblique impact of similar bodies with circular contact

Summary The general solution for tangential loading histories of similar bodies with circular areas of contact is outlined and applied to the tangential impact. Classical numerical methods use an interpolation function for the stress distribution on a set of points and obtain the stress distribution for given displacements by inversion of a linear equation system. Our solution avoids large systems of simultaneous equations and gives exact solutions for loading histories in form of finite, successive displacement increments. The general equations of motion for the impact are derived and applied to the case, where the normal and the tangential equations of impact are uncoupled. The Nassi-Shneiderman diagrams of the contact algorithm and the impact algorithm are presented. The solution is compared with other results from earlier publications. A tangential coefficient of restitution, which determines the behaviour of the bodies after impact, is plotted as a function of two parameters.     

Stable crack growth in a surface compression-strengthened hollow cylinder

In this paper the static fatigue problem for a circumferentially cracked hollow cylinder is examined. For this particular configuration, stable crack growth, in the absense of any external forces, is determined for cylinders with axial components of residual stress which are compressive on the inner and outer radial surfaces and tensile in the cylinder wall. An initial surface crack which is deep enough to penetrate the compression strengthened surface region and enters the tensile zone may propagate in a stable manner until either sudden spontaneous failure occurs or the crack arrests. Since a portion of the crack near the cylinder surface will be closed because of the compressive residual stress field, an additional unknown in the problem is the extent of the crack surface contact. This crack surface contact length is determined by iteration on the integral equation which arises in the mathematical derivation for an embedded circumferential crack in a hollow cylinder. As an illustration of stable crack growth for this geometry with a realistic residual stress distribution, numerical results are presented for a hollow, soda-lime glass cylinder, based on crack growth rates in soda-lime glass exposed to water at 25‡ C. Using the fracture toughness and slow crack growth characteristics for soda-lime glass, the conditions for no crack propagation, crack propagation leading to crack arrest, and catastrophic failure are established.