Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

Summary This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.     

Three-Dimensional Darcy–Brinkman Flow in Sinusoidal Bumpy Tubes

The verified Darcy–Brinkman model and boundary perturbation method are used to study the Brinkman flow in a tube with a bumpy surface, assuming the amplitude of the bumps is small compared to the mean tube radius. This study is important to understand the abnormal flow conditions caused by the boundary irregularities in diseased vessels. The mean rate flow is found, up to second-order correction, as a function of circumferential and longitudinal wave numbers and the permeability parameter of the porous medium. Numerical results displaying the velocity components and bumpiness functions are obtained for various values of the physical parameters of the problem. The results are tabulated and represented graphically for various physical parameters. It is found that, for every permeability parameter and for given bump area, there exists a circumferential wave number, for which the flow resistance is minimized. The limiting cases of Stokes and Darcy’s flows of the bumpiness function are discussed and compared with the available results in the literature.     

Analysis of non-axisymmetric wave propagation in a homogeneous piezoelectric solid circular cylinder of transversely isotropic material

A study concerning the propagation of free non-axisymmetric waves in a homogeneous piezoelectric cylinder of transversely isotropic material with axial polarization is carried out on the basis of the linear theory of elasticity and linear electro-mechanical coupling. The solution of the three dimensional equations of motion and quasi-electrostatic equation is given in terms of seven mechanical and three electric potentials. The characteristic equations are obtained by the application of the mechanical and two types of electric boundary conditions at the surface of the piezoelectric cylinder. A novel method of displaying dispersion curves is described in the paper and the resulting dispersion curves are presented for propagating and evanescent waves for PZT-4 and PZT-7A piezoelectric ceramics for circumferential wave numbers m = 1, 2, and 3. It is observed that the dispersion curves are sensitive to the type of the imposed boundary conditions as well as to the measure of the electro-mechanical coupling of the material.     

Mixed boundary value problems for a cylindrical shell

Two mixed boundary value problems in potential theory for a semi-infinite cylindrical shell are solved. The first is interpreted as a heat conduction problem for an insulated shell containing a circumferential obstruction. The second is the torsion of a cylindrical shell containing a circumferential crack. For both problems the normal derivative of the potential function is taken as zero on the inner and outer shell walls. The boundary conditions at the end of the shell are mixed with respect to the potential function and its normal derivative. The problems are formulated using integral transforms in a manner leading to a singular integral equation which can be solved by numerical means. Intensity factors along the circumference separating the mixed conditions are computed.     

VIBRATIONS OF CIRCULAR CYLINDRICAL SHELLS WITH NONUNIFORM CONSTRAINTS,ELASTIC BED AND ADDED MASS. PART II: SHELLS CONTAINING OR IMMERSED IN AXIAL FLOW

The model introduced in Part I of the present study is extended to take into account a flowing fluid, a mean radial pressure and initial pre-stress in circular cylindrical shells. The axial flow can be external, internal or annular and is described by the potential theory for inviscid and incompressible fluid. The computer program DIVA has been developed. It takes into account all the following complicating effects on the vibrations of circular cylindrical shells: (i) nonuniform boundary conditions around the shell edges including elastic boundary conditions; (ii) fluid–structure interaction including both flowing and quiescent fluids; (iii) internal, external and annular fluids; (iv) effect of a mean radial pressure and initial pre-stress; (v) elastic bed of partial extension in circumferential and longitudinal directions; (vi) intermediate constraints; (vii) added masses. It can be considered the most complete computer program specifically dedicated to dynamics of circular cylindrical shells. The Flügge theory of shells is used to describe the shell deformations. The system has been proved to be conservative for any combination of boundary conditions with restrained displacement at the shell ends. Numerical results show that shells clamped at the upstream end and simply supported at the downstream end have a larger critical velocity than simply supported shells, solving the paradox of Horáček and Zolotarev.     

Nonlinear free vibration of spinning cylindrical shells with arbitrary boundary conditions

The aim of the present study is to investigate the nonlinear free vibration of spinning cylindrical shells under spinning and arbitrary boundary conditions. Artificial springs are used to simulate arbitrary boundary conditions. Sanders' shell theory is employed, and von Kármán nonlinear terms are considered in the theoretical modeling. By using Chebyshev polynomials as admissible functions, motion equations are derived with the Ritz method. Then, a direct iteration method is used to obtain the nonlinear vibration frequencies. The effects of the circumferential wave number, the boundary spring stiffness, and the spinning speed on the nonlinear vibration characteristics of the shells are highlighted. It is found that there exist sensitive intervals for the boundary spring stiffness, which makes the variation of the nonlinear frequency ratio more evident. The decline of the frequency ratio caused by the spinning speed is more significant for the higher vibration amplitude and the smaller boundary spring stiffness.     

压电材料中切口/接头端部平面电弹性场奇异性有限元分析

为了对平面载荷作用下压电材料中切口或接头端部附近电弹性场奇异性问题进行分析，首先以应力平衡方程、Maxwell方程和和边界条件为基础，得到一种求解压电材料特征问题的弱式方程；其次，假定楔形切口或接头端部附近单元内位移和电势沿径向分布为指数形式，而周向方向分布则采用泡函数插值，将其代入弱式方程，建立一种只需对楔形切口或接头端部附近周边进行离散的一维简单有限元方法．压电材料的极化轴可以是任意方向．利用该有限元模型讨论了楔形切口角度、极化轴方向和边界条件对奇性场的影响．通过和其它特定情况下的现有解相比，证实了该文有限元数值方法的有效性，而且精度很高．     

Elastic interface acoustic waves in twinned crystals

A new type of interface acoustic waves (IAW) is presented, for single-crystal orthotropic twins bonded symmetrically along a plane containing only one common crystallographic axis. The effective boundary conditions show that the waves are linearly polarized at the interface, either transversally or longitudinally. Then the secular equation is obtained in full analytical form using new relationships for the displacement–traction quadrivector at the interface. For gallium arsenide and for silicon, it is found that the IAWs with transverse (resp. longitudinal) polarization at the interface are of the Stoneley (resp. leaky) type.     

A closed-form solution procedure for circular cylindrical shell vibrations

A systematic procedure for obtaining the closed-form eigensolution for thin circular cylindrical shell vibrations is presented, which utilizes the computational power of existing commercial software packages. For cylindrical shells, the longitudinal, radial, and circumferential displacements are all coupled with each other due to Poissons ratio and the curvature of the shell. For beam and plate vibrations, the eigensolution can often be found without knowledge of absolute dimensions or material properties. For cylindrical shell vibrations, however, one must know the relative ratios between shell radius, length, and thickness, as well as Poissons ratio of the material. The mode shapes and natural frequencies can be determined analytically to within numerically determined coefficients for a wide variety of boundary conditions, including elastic and rigid ring stiffeners at the boundaries. Excellent agreement is obtained when the computed natural frequencies are compared with known experimental results.     

Effects of exponential volume fraction law on the natural frequencies of FGM cylindrical shells under various boundary conditions

In the present work, vibration characteristics of thin functionally graded cylindrical shells are studied under the influence of various boundary conditions. Fabrication of FGM cylindrical shell is carried out by using exponential volume fraction law. Strain- and curvature-displacements relationships are taken from Love’s thin shell theory. The frequency equation in the form of eigenvalue problem is obtained by adapting the Rayleigh-Ritz method. Characteristic beam functions are assumed to approximate the axial modal dependence. Effects of exponential volume fraction law on the natural frequencies of the FGM cylindrical shells for various boundary conditions are studied against circumferential wave number, length to radius ratio and thickness to radius ratio for different values of power law exponents. Results evaluated show good agreement with those available in the literature.