Transient thermal stresses in an orthotropic hollow cylinder for axisymmetric problems

For the thermoelastic dynamic axisymmetric problem of a finite orthotropic hollow cylinder, one comes closer to reality to involve the effect of axial strain than to consider the plane strain case only. However, additional mathematical difficulties should be encountered and a different solution procedure should be developed. By the separation of variables, the thermoelastic axisymmetric dynamic problem of an orthotropic hollow cylinder taking account of the axial strain is transformed to a Volterra integral equation of the second kind for a function of time, which can be solved efficiently and quickly by the interpolation method. The solutions of displacements and stresses are obtained. It is noted that the present method is suitable for an orthotropic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical comparison is made to show the effect of the axial strain on the displacements and stresses. The project supported by the National Natural Science Foundation of China (10172075) and China Postdoctoral Science Foundation (20040350712)     

The transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems

By virtue of the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of the displacements, stresses, electric displacements and electric potentials are finally obtained. The present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are also presented.     

Transient thermal elastic fracture of a piezoelectric cylinder specimen

A finite piezoelectric cylinder with an embedded penny-shaped crack is investigated for a thermal shock load on the outer surface of the cylinder. The theory of linear electro-elasticity is applied to solve the transient temperature field and the associated thermal stresses and electrical displacements without crack. These thermal stresses and electrical displacements are added to the surfaces of the crack to form an electromechanical coupling and mixed mode boundary-value problem. The electrically permeable crack face boundary condition assumption is used, and the thermal stress intensity factor and electrical displacement intensity factor at the crack border are evaluated. The thermal shock resistance of the piezoelectric cylinder is evaluated for the analysis of piezoelectric material failure in practical engineering applications.     

Simulation of cross-flow-induced vibration of cylinder arrays by surface vorticity method

The surface vorticity method (SVM), which is a fast and practical grid-free two-dimensional (2-D) method, and a fluid–structure interaction model incorporating the effects of cylinder motions and displacements is used to simulate the vortex-induced vibration of cylinder arrays at sub-critical Reynolds number Re=2.67×10⁴. The SVM is found to be most suitable for simulating a 2-D cylinder row with large-amplitude vibrations where the vorticity field and the fluid forces of the cylinder row change drastically, and the effect of the stream on the transverse direction vibration is very significant. The fluidelastic instability of a flexible cylinder row at small pitch ratio is also investigated, and the critical reduced velocity of the cylinder row at a reduced damping parameter S_G=1.29 is calculated, which is in good agreement with experimental and analytical results of the unsteady model. Vortex-induced vibration of a staggered cylinder array is simulated using different structural parameters. When the cylinders are relatively more flexible, the flow pattern changes dramatically and the fluid–structure interaction has a dominant impact on the flow field. Compared with grid-based methods, the grid-free SVM is a fast and practical method for the simulation of the FIV of cylinder arrays due to vortex shedding at sub-critical Reynolds numbers.     

Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

In this paper, we propose a method for the solution of the axisymmetric boundary value problem for a finite elastic cylinder with assigned stress and/or displacements acting on the ends and side. The technique utilizes the Love representation, which allows for reduction of the solution of the elastic problem to the search for a biharmonic function on a cylindrical domain. In the solution method suggested here, we write the Love function with a Bessel expansion and analyze in detail the conditions under which it is possible to differentiate the expansion term by term. We show that this is possible only for a restricted class of elastic solutions. In the general case, we introduce two new auxiliary functions of the z-coordinate. In this way, we obtain the general form of the axisymmetric biharmonic function, which is discussed in relation to certain specific boundary conditions applied on the side and ends of the cylinder. We obtain an exact explicit solution of practical interest for a cylinder with free ends and assigned displacements applied to the side.     

The radially nonhomogeneous elastic axisymmentric problem

The plane axisymmetric problem with axisymmetric geometry and loading is analyzed for a radially nonhomogeneous circular cylinder, in linear elasticity. Considering the radial dependence of the stress, the displacements fields and of the stiffness matrix, after a series of admissible functional manipulations, the general differential system solving the problem is developed. The isotopic radially inhomogeneous elastic axisymmetric problem is also analyzed. The exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio and of power law Young’s modulus and constant Poisson’s ratio. For the isotropic elastic axisymmetric problem, a general expression of the stress function is derived. After the satisfaction of the biharmonic equation and making compatible the stress field’s expressions, the stress function and the stress and displacements fields of the axisymmetric problem are also deduced. Applications have been made for a radially nonhomogeneous hollow cylinder where the stress and displacements fields are determined.     

Thermal stresses in infinite circular cylinder subjected to rotation

The present investigation is concerned with the effect of rotation on an infinite circular cylinder subjected to certain boundary conditions.An analytical procedure for evaluation of thermal stresses,displacements,and temperature in rotating cylinder subjected to thermal load along the radius is presented.The dynamic thermal stresses in an infinite elastic cylinder of radius a due to a constant temperature applied to a variable portion of the curved surface while the rest of surface is maintained at zero temperature are discussed.Such situation can arise due to melting of insulating material deposited on the surface cylinder.A solution and numerical results are obtained for the stress components,displacement components,and temperature.The results obtained from the present semi-analytical method are in good agreement with those obtained by using the previously developed methods.     

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。     

Effect of elastomer slight compressibility

This paper is concerned with the constitutive equation for slightly compressible elastic material under finite deformations. We show that material slight compressibility can be effectively taken into account in the case of high hydrostatic pressure or highly confined material. In all other situations the application of the incompressible and nearly incompressible material theories gives practically the same results. Therefore it is of interest to consider the problem in which allowing for material slight compressibility leads to results essentially different from those obtained with help of the incompressible material model. In the present paper this difference has been demonstrated for the problem of high hydrostatic pressure causing an increase of the ‘bulk’ and ‘shear’ material moduli. The behavior of a long hollow cylinder of real material under finite deformations is analyzed. The cylinder is subjected to internal pressure and axial and circular displacements at the outer surface. This problem has been solved analytically using the small parameter method. The solution obtained predicts a decrease of the axial and circular displacements of the outer surface under fixed shear (axial and circular) forces when the internal pressure is applied.     

轴对称一维六方准晶圆柱的精化理论

将轴对称圆柱的精化分析推广到一维六方准晶中轴对称圆柱的研究当中。利用准调和函数的Bessel算子函数表示以及一维六方准晶中的通解,在不做任何预先假设的情况下,给出了一维六方准晶中轴对称圆柱的精化理论。首先,根据准调和函数的Bessel算子函数表示,利用三个一维待定函数,表示出声子场和相位子场的位移场和应力分量。再根据非齐次边界条件,推导出柱面受径向外载时的精化方程。通过舍弃高阶项,推导出了在径向方向受到柱面载荷的近似解。     

Evaluation of finite-element calculations in a part-circular crack by coherent optics techniques

Two techniques, speckle photography and holographic interferometry, were used to test three-dimensional finite-element calculations in an internally pressurized cylinder with an external part-circular crack. Opening displacements along the crack line were measured by speckle photography. Radial displacements were obtained from holographic fringe patterns. Good agreement between experimental and numerical data is obtained. Stress-intensity factor variations along the crack front are calculated from numerical results.     

Thermoelastic Contact of a Shroud Ring and a Cylinder under Unsteady Friction-Induced Heat Release

Mathematical formulation is performed and a solution is found for a quasi-static thermoelastic problem of contact interaction of an elastic shroud ring and a hollow circular cylinder inserted into this ring, which are compressed by a load varied along the axis of the system, under the condition of an unloaded contact over the ring surface or over the circumference contour. The radial displacements of the contact surface of the shroud ring are approximated by displacements of the surface of a long circular hollow cylinder. Unsteady friction-induced heat release caused by the action of friction forces owing to shroud ring rotation over the cylinder with a time-dependent low angular velocity is taken into account. The problem is reduced to a system of integral equations whose structure is determined by the form of thermophysical contact conditions. A numerical algorithm of the solution is proposed, and the influence of the problem parameters on the contact pressure and temperature distributions is considered. Based on an analysis of results, a conclusion is made that the character of axial variation of the compressing load has a significant effect on the distribution of contact pressure in describing the kinematic condition of interaction of bodies in accordance with Hertz’s theory.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 161–178, July– August, 2005.     

Free vibrations of an elastic cylinder in a cross flow and their effects on the near wake

Fluid-structure interactions resulting from the free vibrations of a two-dimensional elastic cylinder in a cross flow are not well understood. Experimental data pertaining to the interaction behavior is rather scarce, especially that related to the phenomenon of synchronization where the vortex shedding frequency is approximately equal to the natural frequency of the fluid-structure system. The present investigation attempts to examine this problem experimentally using a laser vibrometer to assess the bending displacements and a laser Doppler anemometer to measure the velocities in the wake. Experiments were carried out over a range of reduced velocities. The reduced velocity was first varied by using cylinders of different materials and then by changing the free stream velocity, while maintaining the cylinder diameter constant. A proper choice of materials and reduced velocity allowed the synchronization phenomenon to be investigated. For the range of reduced velocity investigated, the vibration amplitude of the cylinder is finite, even at synchronization, and increases with reduced velocity. The results further show that more than one mode of vibration is excited away from synchronization; however, only the first mode is evident at synchronization. In addition, the near-wake flow behind the elastic cylinder, at three different Reynolds numbers in the sub-critical range, was measured in detail and the data was used to analyse the vibration effects on the mean and turbulence field compared to those measured behind a relatively rigid cylinder at the same Reynolds numbers. It is found that cylinder vibrations have little or no effect on the mean drag and the normalized mean field. However, cylinder vibrations enhance turbulent mixing, thus resulting in a substantial increase in the turbulent intensities. This implies that the large-scale vortical motion is also affected. Nevertheless, turbulence structure in the inertial sub-range is not affected by cylinder vibrations. The slopes of the velocity spectra in this range is still measured to be −5/3 for the freely vibrating cylinders investigated. Received: 20 December 1998/Accepted: 20 September 1999     

Elastic field of a composite cylinder with a spatially varying dynamic eigenstrain

The analytical solutions of displacements and stresses for an eigenstrain problem in a composite bi-layered coaxial cylinder are presented in this article. The inner cylinder is assumed to undergo a dynamic, spatially varying eigenstrain. The spatial distribution of the eigenstrian is taken to be a quadratic polynomial with arbitrary coefficients along the radial direction. Furthermore, the eigenstrain is assumed to be harmonically time-dependent. Elasticity equations are constructed to directly solve the problem. The effect of spatial distribution, as well as the angular frequency of the eigenstrain on the elastic response of the composite cylinder has been illustrated graphically.     

Collision of an elastic finite-length cylinder with a rigid barrier

The paper proposes an approximate solution describing a collision of an elastic finite-length cylinder with a rigid barrier when the lateral boundary conditions of the first fundamental problem of elasticity are satisfied. A finite-difference approach with respect to time and the integral transform method are used to reduce the original initial-boundary-value problem to a one-dimensional one. It is solved using the matrix Green’s function. The final expressions for displacements are obtained by solving a singular integral equation by the orthogonal-polynomial method. The values of displacements and strains are analyzed for short periods of time __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 74–82, September 2007.     

Some problems of linear elasticity for cylinders in micropolar orthotropic material

Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.     

Transient thermoelastic waves in an anisotropic hollow cylinder due to localized heating

This paper presents a theoretical study of transient ultrasonic guided waves generated by concentrated heating of the outer surface of an infinite anisotropic hollow circular cylinder. Generalized thermoelastic theory proposed by Lord and Shulman is adopted to model the dynamic thermoelastic behavior of the cylinder. The concentrated heat source model used is to represent heating due to a pulsed laser beam, which is focused on the outer surface of the cylinder. A semi-analytical finite element (SAFE) method is employed to evaluate guided wave modes in the cylinder. Using integral transform techniques, the modal wave forms are obtained in frequency and wave number domains. Time histories of the propagating modes are then calculated by applying inverse Fourier transformation in the time domain. Numerical results showing the dispersion curves for the group velocities of the propagating modes and transient radial displacements are presented. For this purpose it is assumed that the cylinder is made of transversely isotropic silicon nitride (Si₃N₄). Attention is focused on the propagation characteristics of longitudinal and flexural modes separately.     

Plastic zone correction in a stretched cylinder with an external circumferential crack

The Dugdale hypothesis is adapted to the problem of an external circumferential crack in a stretched cylinder. The lateral surface of the cylinder is stress free and restrained from radial displacements. An external circumferential edge crack in the cylinder which is considered elastic-perfectly plastic is envisaged with the assumption that the plastic zone forms a very thin in-plane layer surrounding the crack. The solution of the problem is reduced to the solution of dual Dini series which, in turn, is reduced to a Fredholm integral equation of the second kind. Solving this integral equation numerically and using the boundedness of the axial stress, the size of the plastic zone correction is obtained.     

Transient thermal stresses in two-dimensional functionally graded thick hollow cylinder with finite length

In this paper transient thermal stresses in a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on classical theory of thermoelasticity are considered. The volume fraction distribution of materials, geometry and thermal load are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure. Temperature, displacements and stress distributions through the cylinder at different times are investigated. Also the effects of variation of material distribution in two radial and axial directions on the thermal stress distribution and time responses are studied. The achieved results show that using 2D-FGM leads to a more flexible design so that time responses of structure, maximum amplitude of stresses and uniformity of stress distributions can be modified to a required manner by selecting suitable material distribution profiles in two directions.