Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

WAVE PROPAGATION IN A TRANSVERSELY ISOTROPIC THERMOELASTIC SOLID CYLINDER OF ARBITRARY CROSS-SECTION

The wave propagation in an infinite, homogeneous, transversely isotropic solid cylinder of arbitrary cross-section is studied using Fourier expansion collocation method, within the frame work of linearized, three-dimensional theory of thermoelasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat conduction. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for elliptic and parabolic cross-sectional zinc cylinders. The computed non-dimensional wave numbers are presented in the form of dispersion curves.     

Effect of rotation in generalized thermoelastic solid under the influence of gravity with an overlying infinite thermoelastic fluid

The present problem is concerned with the study of deformation of a rotating generalized thermoelastic solid with an overlying infinite thermoelastic fluid due to different forces acting along the interface under the influence of gravity.The components of displacement,force stress,and temperature distribution are first obtained in Laplace and Fourier domains by applying integral transforms,and then obtained in the physical domain by applying a numerical inversion method.Some particular cases are also discussed in the context of the problem.The results are also presented graphically to show the effect of rotation and gravity in the medium.     

THE CREEPING MOTION IN THE ENTRY REGION OF A SEMI-INFINITE CIRCULAR CYLINDRICAL TUBE

In 1969,Lew and Fung[1]considered the inlet flow into a se-mi-infinite circular cylinder at low Reynolds number.Dagan etal.[2]in1982 obtained a series solution for the creeping motionthrough a pore of finite length directly.The numerical resultsobtained in[1]also describe the entrance flow in a tube of afinite length as the Fourier integrals in the general solutions arereplaced by Fourier series.In the present paper,the Fourier in-tegralss are evaluated numerically and the velocity,pressure dis-tribution and the stream function in the entry region of a semi-infinite circular cylindrical tube is close to the factor1.3 sug-gested by Lew and Fung[1].The collocation technique applied inthe present paper is shown to converge rapidly and it should beuseful in other similar problems.     

坡形加热下的二维广义磁热黏弹性问题研究

运用具有一个热松弛时间的广义热黏弹性理论,研究了处于均布磁场中的二维磁热黏弹 性问题. 运用Laplace变换(对时间变量)和Fourier变换(对于一个空间变量),得到了变 换域内场量的精确表达式,并把结果应用到表面受到坡形加热的半空间问题. 应用 数值逆变换得到了时间-空间域内场量的解,对丙烯酸塑料 给出场量的响应图. 并把运用广义热黏弹性理论所得的结果与传统热黏弹性理 论及热弹性理论下的结果进行了比较.     

State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory [Youssef, H., 2005a. The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, J. Appl. Math. Mech., 26(4), 4827; Youssef, H., 2005b. Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math., 1–8]. The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to thermal shock and traction free. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature parameter.     

A thermoelastic diffusion interaction in an infinitely long annular cylinder

The present paper is aimed at studying a two-dimensional problem for an infinitely long solid conducting circular cylinder with a permeating substance in contact with its bounding surface. The problem is considered in the context of generalized thermoelastic diffusion theory with one relaxation time. The lateral surface of the solid is traction free and subjected to known temperature and chemical potential as functions of time. The solution is obtained by a transform method and a direct approach without the customary use of potential functions. Numerical inversion of the transformed solution is carried out to obtain the temperature, displacement, stress, and concentration of the diffusive material distributions. Numerical results are represented graphically and discussed. The second sound effect and the asymptotic behavior for the solution are discussed.     

Thermoelastic damping in micro-beam resonators

Thermoelastic damping is recognized as a significant loss mechanism at room temperature in micro-scale beam resonators. In this paper, the governing equations of coupled thermoelastic problems are established based on the generalized thermoelastic theory with one relaxation time. The thermoelastic damping of micro-beam resonators is analyzed by using both the finite sine Fourier transformation method combined with Laplace transformation and the normal mode analysis. The vibration responses of deflection and thermal moment are obtained for the micro-beams with simply supported and isothermal boundary conditions. The vibration frequency is analyzed for three boundary condition cases, i.e., the clamped and isothermal, the simply supported and isothermal, and the simply supported and adiabatic. The analytic results show that the amplitude of deflection and thermal moment are attenuated and the vibration frequency is increased with thermoelastic coupling effect being considered. In addition, it can be found from both the analytic results and the numerical calculations that these properties are size-dependent. When the thickness of the micro-beam is larger than its characteristic size, the effect of thermoelastic damping weakens as the beam thickness increases. The size-effect induced by thermoelastic coupling would disappear when the thickness of the micro-beam is over a critical value that depends on the material properties and the boundary conditions.     

A half-space problem in the theory of generalized thermoelastic diffusion

In this work we consider the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques.The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

Generalized thermoelastic functionally graded solid with a periodically varying heat source

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity without energy dissipation (TEWOED). The governing equations of generalized thermoelasticity without energy dissipation (GN model type II) for a functionally graded materials (FGM) (i.e. material with spatially varying material properties)are established. The governing equations are expressed in Laplace–Fourier double transform domain and solved in that domain. Now, the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand is obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking nonhomogeneity parameter suitably. Finally the results obtained are presented graphically to show the effect of nonhomogeneity on displacement, temperature, stress and strain.     

Creeping flow of viscoplastic materials through a planar expansion followed by a contraction

In this work, the creeping flow of a viscoplastic fluid through a planar channel with an expansion followed by a contraction is analyzed numerically. The solution of the conservation equations of mass and momentum is obtained via the finite volume method. In order to model the non-Newtonian behavior of the fluid, it was used the generalized Newtonian fluid constitutive equation. The viscosity function was the one proposed by Souza Mendes and Dutra [Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids. Appl. Rheol. 14, 296–302]. The yielded and unyielded regions are obtained for several combinations of rheological parameters. The influence of these parameters on pressure drop through the cavity is also obtained and analyzed.     

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.     

Fractional-order generalized thermoelastic diffusion theory

The present work aims to establish a fractional-order generalized themoelastic diffusion theory for anisotropic and linearly thermoelastic diffusive media. To numerically handle the multi-physics problems expressed by a sequence of incomplete differential equations, particularly by a fractional equation, a generalized variational principle is obtained for the unified theory using a semi-inverse method. In numerical implementation, the dynamic response of a semi-infinite medium with one end subjected to a thermal shock and a chemical potential shock is investigated using the Laplace transform. Numerical results, i.e., non-dimensional temperature, chemical potential, and displacement, are presented graphically. The influence of the fractional order parameter on them is evaluated and discussed.     

Thermoelastic stability of a double-layered spherical shell

In the paper the axi-symmetrical thermoelastic stability of free double-layered spherical shells at uniform temperature is considered. The basic equations are derived from Reissner's theory of thin shells and assumptions that materials of the layers are linear thermoelastic. The conditions for the buckling of shells are determined. The temperature-deflection diagrams are calculated by using the collocation method.     

To the theory of analysis of a two-layer cylindrical bearing

We consider the plane contact problem of elasticity concerning the interaction between an absolutely rigid cylinder and the internal cylindrical surface of the cylindrical base, which consists of two circular cylindrical layers with different elastic constants. The base external surface is fixed, the layers are rigidly connected with each other, and the friction forces are absent in the contact region. Such problems sufficiently well model the operation of a composite cylindrical slider bearing, especially in the case of loads for which the angular dimension of the contact site is commensurable with the bearing width and the moduli of the insert liner and of the support are different and significantly less than the modulus of elasticity of the other details of the bearing.For the above-stated problem of elasticity, we first construct integral equations, which are solved by the direct collocation method [1, 2] and by the asymptotic method [3, 4].In contrast to the similar problems considered earlier (e.g., see [3, 4]) for a single-layer cylinder, the collocation method used here permits studying the problem practically for any parameter values. The asymptotic approach gives an efficient solution in the case of relatively thin layers in simple analytic form. We also compare the two solutions numerically and determine the scope of the asymptotic method.     

GN理论下受到移动内热源的半空间问题的研究

运用无能量耗散的热弹性GN理论研究了受到移动内热源的半空间问题.通过势函数法使问题 转化成一组偏微分方程,采用Laplace变换和Fourier变换法得到问题在变换域内表面位移 精确解. 运用级数展开法得到在小时间范围内表面位移的近似解.给出近似解的适用范围,同时给出热 源固定不动和非耦合理论下问题的解.并对铜介质进行了数值计算.     

Propagation of discontinuities in electromagneto generalized thermoelasticity in cylindrical regions

In this work we study wave propagation for a problem of an infinitely long solid conducting circular cylinder whose lateral surface is traction free and subjected to known surrounding temperatures in the presence of a uniform magnetic field in the direction of the axis. The problem is in the context of generalized magneto-thermo-elasticity theory with one relaxation time. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Wave propagation in the elastic medium and in the free space, bounding it, is investigated.     

Ground vibration due to a high-speed moving harmonic rectangular load on a poroviscoelastic half-space

The transmission of vibrations in the ground, due to a high-speed moving vertical harmonic rectangular load, is investigated theoretically. The problem is three-dimensional and the interior of the ground is modelled as a totally or partially saturated porous viscoelastic half-space, using the complete Biot theory. The solutions in the transformed domain are obtained using a double Fourier transform on the surface spatial variables. A modified hysteretic damping model defined in the wavenumber domain is used, first presented by Lefeuve-Mesgouez et al. [Lefeuve-Mesgouez, G., Le Houédec, D., Peplow, A.T., 2000. Vibration in the vicinity of a high-speed moving harmonic strip load. Journal of Sound and Vibration 231(5) 1289–1309]. Numerical results for the displacements of the solid and fluid phases, over the surface of the ground and in depth, are presented for loads moving with speeds up to and beyond the Rayleigh wave speed of the medium.     

Study on generalized magneto-thermoelastic problems by FEM in time domain

This paper presents an investigation of temperature, displacement, stress, and induced magnetic field in a half space perfectly-conductive plate. Finite element equations regarding generalized magneto-thermoelasticity problems with two relaxation times (i.e., the G-L theory) are derived using the principle of virtual work. For avoiding numerical complication involved in inverse Laplace and Fourier transformation and low precision thereof, the equations are solved directly in time-domain. As a numerical example, the derived equation is used to investigate the generalized magneto-thermoelastic behavior of a semi-infinite plate under magnetic field and subjecting to a thermal shock loading. The results demonstrate that FEM can faithfully predict the deformation of the plate and the induced magnetic field, and most importantly can reveal the sophisticated second sound effect of heat conduction in two-dimensional generalized thermoelastic solids, which is usually difficult to model by routine transformation methods. A peak can be observed in the distribution of stress and induced magnetic field at the heat wave front and the magnitude of the peak decreases with time, which can not be obtained by transformation methods. The new method can also be used to study generalized piezo-thermoelastic problems.The project supported by the National Natural Science Foundation of China (10132010 and 10472089)The English text was polished by Yunming Chen     

Simulation of non‐linear free surface motions in a cylindrical domain using a Chebyshev–Fourier spectral collocation method

When a liquid is perturbed, its free surface may experience highly non‐linear motions in response. This paper presents a numerical model of the three‐dimensional hydrodynamics of an inviscid liquid with a free surface. The mathematical model is based on potential theory in cylindrical co‐ordinates with a σ‐transformation applied between the bed and free surface in the vertical direction. Chebyshev spectral elements discretize space in the vertical and radial directions; Fourier spectral elements are used in the angular direction. Higher derivatives are approximated using a collocation (or pseudo‐spectral) matrix method. The numerical scheme is validated for non‐linear transient sloshing waves in a cylindrical tank containing a circular surface‐piercing cylinder at its centre. Excellent agreement is obtained with Ma and Wu's [Second order transient waves around a vertical cylinder in a tank. Journal of Hydrodynamics 1995; Ser. B4 : 72–81] second‐order potential theory. Further evidence for the capability of the scheme to predict complicated three‐dimensional, and highly non‐linear, free surface motions is given by the evolution of an impulse wave in a cylindrical tank and in an open domain. Copyright © 2001 John Wiley & Sons, Ltd.