Analytical thermo-elastodynamic solutions for a nonhomogeneous transversely isotropic hollow sphere

Summary  The spherically symmetric dynamic thermoelastic problem for a special nonhomogeneous transversely isotropic elastic hollow sphere is formulated by introduction of a dependent variable and separation of variables technique. The derived solution can be degenerated into that for a homogeneous transversely isotropic hollow sphere, a nonhomogeneous isotropic hollow sphere or a solid sphere. The present method, allow to avoid integral transforms, is suited for a hollow sphere of arbitrary thickness subjected to arbitrary spherical symmetric thermal and mechanical loads, and is convenient in dealing with different boundary conditions of dynamic thermoelasticity . The numerical calculation involved is easy to be performed and its results are also presented. Received 30 October 2001; accepted for publication 21 February 2002 The work was supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016)     

Transient responses of a magneto-electro-elastic hollow sphere for fully coupled spherically symmetric problem

By introducing a dependent variable and a special function satisfying the inhomogeneous mechanical boundary conditions, the governing equation for a new variable with homogeneous mechanical boundary conditions is derived. Then by means of the separation of variables technique and the electric and magnetic boundary conditions, the dynamic problem of a magneto-electro-elastic hollow sphere under spherically symmetric deformation is transformed to two Volterra integral equations of the second kind about two functions of time. Cubic Hermite polynomials are adopted to approximate the two undetermined functions at each time subinterval and the recursive formula is obtained to solve the integral equations successfully. The transient responses of displacements, stresses, electric and magnetic potentials are completely determined at the end. Numerical results are presented.     

Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere

The dynamic solution of a multilayered spherically isotropic piezoelectric hollow sphere subjected to radial dynamic loads is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state-space method, and the dynamic part is obtained by the method of separation of variables coupled with the initial parameter method as well as the orthogonal expansion technique. By using the quasi-static and dynamic parts, the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered spherically isotropic piezoelectric hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric dynamic loads. Finally, numerical results are presented and discussed.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

层合球面各向同性热释电空心球的瞬态响应

运用叠加原理，将层合球面各向同性热释电空心球的球对称动力学问题的解分成准静态和动 态两部分，准静态部分首先运用状态空间法给出了显式表达式，然后运用分离变量法、初参 数法和特征函数展开技术，给出了动态部分的表示式，再结合内外表面上的电学边界条件和 界面上的电学连续条件，导出一个关于时间函数的第二类Volterra积分方程，运用插值法 可成功地给出此积分方程的高精度数值解，最终可求得原问题的位移、应力、电位移以及电 势的响应. 此方法适用任意层数且各层是任意厚度的层合热释电空心球作用随时间以任意形 式变化的球对称温度场. 文中还给出了数值结果.     

Axisymmetric non-Fourier temperature field in a hollow sphere

The non-Fourier axisymmetric (2+1)-dimensional temperature field within a hollow sphere is analytically investigated by the solution of the well-known Cattaneo–Vernotte hyperbolic heat conduction equation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The method of solution is the standard separation of variables method. General linear time-independent boundary conditions are considered. Ultimately, the presented solution is applied to a (1+1)—as well as a (2+1)—dimensional problem, and their respective non-Fourier thermal behavior is studied. The present solution can be reduced to special cases of interest by choosing appropriate boundary conditions parameters. Dedicated to Prof. Gholamali Atefi, with appreciation and admiration on the occasion of his 65th birthday.     

Indentation of a transversely isotropic piezoelectric half-space by a rigid sphere

This paper exactly analyzes the problem of a rigid sphere indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to include the piezoelectric effect. By using the previous results of elasticity, an exact solution is derived. It is found that all the elastoelectric variables can be expressed in terms of elementary functions. The work was partly supported by the National Natural Science Foundation of China (No. 19872060).     

Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material

The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.     

On the free vibrations of a piezoceramic hollow sphere

The aim of the paper is to analyze the free vibrations of a piezoceramic hollow sphere with radial polarization. Using the cnoidal method and a genetic algorithm solves the equations of a radially inhomogeneous spherically isotropic piezoelastic medium. The Reddy and the cosine laws represent the functionally graded property of material. It is seen that for a piezoceramic hollow sphere, the piezoelectric effect consists in increasing the values for the natural frequencies in the specified classes of vibrations.     

Free non-axisymmetric oscillations of a thick-walled,nonhomogeneous, transversally isotropic,hollow sphere

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 5, pp. 12–17, May, 1988     

The general solution of spherically isotropic magnetoelectroelastic media and its applications

A general solution of the three-dimensional equilibrium problem of spherically isotropic magnetoelectroelastic media is presented. Base on the obtained general solution, exact and compact form solutions are obtained for (1) a spherically isotropic magnetoelectroelastic cone subjected to concentrated force, concentrated couple, a point charge and a point electric current at its apex; (2) a spherically isotropic magnetoelectroelastic space with a concentrated force at the origin; (3) a spherical shell under spherically symmetric deformation; and (4) stress concentration around a spherical cavity subjected to remote uniform tensile force, electric charge and electric current.     

Benchmark analytic solution of the dynamic response of a spherical shell composed of a transverse isotropic elastic material

Developing benchmark analytic solutions for problems in solid and fluid mechanics is very important for the purpose of testing and verifying computational physics codes. In order to test the numerical results of physics codes, we consider the geometrically linear dynamic sphere problem. We present an exact solution for the dynamic response of a spherical shell composed of a linearly elastic material exhibiting transverse isotropic symmetry. The solution takes the form of an infinite series of eigenfunctions. We demonstrate, both qualitatively and quantitatively, the convergence of the computed benchmark solution under spatial, temporal, and eigenmode refinement.     

Completeness of general solutions for three-dimensional transversely isotropic piezoelectricity

In this paper, a general solution for three-dimensional transversely isotropic piezoelectricity in terms of four quasi-quadri-harmonic functions is established first. Owing to complexity of the higher-order equation, it is difficult to obtain rigorous analytic solutions and in most cases this general solution is not applicable. By virtue of the generalized Almansi’s Theorem, the simplified generalized LHN solution and E–L solution expressed by lower order functions are achieved, respectively, by taking a decomposition and superposition technique. In the absence of piezoelectric coupling, these two simplified general solutions can be degenerated into those for transversely isotropic elasticity, i.e. LHN and E–L solutions. More importantly, the completeness of these two generalized solutions is proved if the domain is z-convex, no matter whether the characteristic roots are distinct or possibly equal to each other.     

Generalized magneto-thermoelasticity in a nonhomogeneous isotropic hollow cylinder using the finite element method

In this paper, we constructed the equations of generalized magneto-thermoelasticity in a perfectly conducting medium. The formulation is applied to generalizations, the Lord–Shulman theory with one relaxation time, and the Green–Lindsay theory with two relaxation times, as well as to the coupled theory. The material of the cylinder is supposed to be nonhomogeneous isotropic both mechanically and thermally. The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution, displacement, radial stress, and hoop stress are represented graphically. The results indicate that the effects of nonhomogeneity, magnetic field, and thermal relaxation times are very pronounced. In the absence of the magnetic field or relaxation times, our results reduce to those of generalized thermoelasticity and/or classical dynamical thermoelasticity, respectively. Results carried out in this paper can be used to design various nonhomogeneous magneto-thermoelastic elements under magnetothermal load to meet special engineering requirements. An erratum to this article can be found at     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

The elastodynamic solution for a solid sphere and dynamic stress-focusing phenomenon

This paper presents an analytical method of solving the elastodynamic problem of a solid sphere.The basic solution of the elastodynamic problem is decomposed into a quasi-static solution satisfying the inhomogeneous compound boundary conditions and a dynamic solution satisfying the homogeneous compound boundary conditions.By utilizing the variable transform,the dynamic equation may be transformed into Bassel equation.By defining a finite Hankel transform,we can easily obtain the dynamic solution for the inhomogeneous dynamic equation.Thereby,the exact elastodynamic solution for a solid sphere can be obtained.From results carried out,we have observed that there exists the dynamic stress-focusing phenomenon at the center of a solid sphere under shock load and it results in very high dynamic stress-peak.     

Analytical solution of a simply supported piezoelectric beam subjected to a uniformly distributed loading

IntroductionIntelligentstructureisakindofnewstructuremodelsandhasbeenreceivedmuchattentioninrecentyears.Usedaspiezoelectricsensorsandactuators,thiskindofintelligentstructureshasmanyadvantagessuchasthepromptnessofresponseandtheconvenienceforthesignalcontrol.Soitisfollowedwithinterestbothinthetheoreticalstudyandintheengineeringapplications[1].Forexample ,theresearchoffundamentalsolution[2 ,3]andellipsoidalinclusion[4 ]forthree_dimensionalpiezoelectricmaterialbyuseofFouriertransformation ;thestu…     

Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders

Exact solutions are obtained for transient torsio- nal responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free-free, free-fixed and fixed-fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.