Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere

Summary By introduction of a special dependent variable and separation of variables technique, the electroelastic dynamic problem of a nonhomogeneous, spherically isotropic hollow sphere is transformed to a Volterra integral equation of the second kind about a function of time. The equation can be solved by means of the interpolation method, and the solutions for displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are presented at the end.The work was supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016).     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

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.     

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.     

The arbitrarily and the periodically laminated elastic hollow sphere: exact solutions and homogenization

Summary The aim of this paper is (1) to develop a rational method for the analysis of an arbitrarily laminated elastic, isotropic or transversely isotropic hollow sphere under internal and/or external pressure, (2) to solve the problem of a periodically layered sphere consisting of many equal groups of n different thin layers. The transfer matrix method is used, and exact closed-form solutions are worked out, supplemented by a numerical example. It turns out that by means of the proposed homogenization an originally (periodically) inhomogeneous isotropic sphere is replaced by a homogeneous anisotropic one belonging to the type of spherical symmetric anisotropy. Received 20 October 1997; accepted for publication 15 December 1997     

Stress Analysis of Rotating Inhomogeneous Transversely Isotropic Hollow Spheres

An approach is proposed to the stress–strain analysis of hollow, arbitrarily inhomogeneous, transversely isotropic, either closed or open spheres that rotate about the axis of symmetry with a constant angular velocity. The stress problem for a rotating single-layer sphere and a segment of a sphere is solved as an example. The distribution of radial stresses and displacements along the radius of the sphere is analyzed depending on thickness variation at arbitrary points of the sphere     

Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres

The effects of material anisotropy and inhomogeneity on void nucleation and growth in incompressible anisotropic nonlinearly elastic solids are examined. A bifurcation problem is considered for a composite sphere composed of two arbitrary homogeneous incompressible nonlinearly elastic materials which are transversely isotropic about the radial direction, and perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Several types of bifurcation are found to occur. Explicit conditions determining the type of bifurcation are established for the general transversely isotropic composite sphere. In particular, if each phase is described by an explicit material model which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials, phenomena which were not observed for the homogeneous anisotropic sphere nor for the composite neo-Hookean sphere may occur. The stress distribution as well as the possible role of cavitation in preventing interface debonding are also examined for the general composite sphere.     

Change in the stress intensity factors for plane cracks in a transversely isotropic medium due to thermal loads

The feasibility of controlling the stress intensity factors for plane cracks of arbitrary form (distributed in a plane perpendicular to isotropy axis in a transversely isotropic material) which are subjected to symmetric mechanical loading by heating the material is demonstrated by using the congruence theorem and making an analogy between isotropic and transversely isotropic materials based on the theory of thermoelasticity. It is shown that a thermal load which fully compensates for the mechanical load can be created within the range in which the material behaves linearly. If it turns out to be technically impossible to create the necessary thermal load, a simpler temperature regime can be chosen that will “cancel” the mechanical force field with a certain factor of safety. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 29–37, September, 1999.     

A class of universal relations in isotropic elasticity theory

A class of universal relations for isotropic elastic materials is described by the tensor equationTB = BT. This simple rule yields at most three component relations which are the generators of many known universal relations for isotropic elasticity theory, including the well-known universal rule for a simple shear. Universal relations for four families of nonhomogeneous deformations known to be controllable in every incompressible, homogeneous and isotropic elastic material are exhibited. These same universal relations may hold for special compressible materials. New universal relations for a homogeneous controllable shear, a nonhomogeneous shear, and a variable extension are derived. The general universal relation for an arbitrary isotropic tensor function of a symmetric tensor also is noted.     

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.     

DYNAMIC CONTACT STIFFNESS OF VIBRATING RIGID SPHERE CONTACTING SEMI-INFINITE TRANSVERSELY ISOTROPIC VISCOELASTIC SOLID

Dynamic contact stiffness at the interface between a vibrating rigid sphere and a semi-infinite transversely isotropic viscoelastic solid is investigated. An oscillating force superimposed onto a static compressive force in the vertical direction excites the vibration of a rigid sphere, which causes variable contact radius and contact pressure distribution in the contact region. The assumption of a sufficiently small oscillating force yields a dynamic contact-pressure distribution of a constant contact radius, which gives dynamic contact stiffness at the interface between the rigid sphere and the semi-infinite solid. Numerical calculations show the influence of vibration frequency of the sphere, and elastic constants of the transversely isotropic solid on dynamic contact stiffness, which benefits quantitative evaluation of elastic constants and orientation of single hexagonal grains by resonance-frequency shifts of the oscillator in resonance ultrasound microscopy.     

直角坐标下横观各向同性饱和半空间体动力响应问题的解析解

基于双重Fourier变换技术，成功求解了直角坐标系下，横观各向同性弹性饱和多孔介质的三维Biot动力方程，得到了以固体骨架位移分量和孔隙流体压力为基本未知量的积分形式一般解．进而，用一般解给出了饱和介质的总应力分量表达式。在此基础上，研究了在任意分布的表面竖向和水平谐振力作用下，横观各向同性饱和半空间体的动力响应问题。数值结果表明，采用各向同性饱和介质的动力学模型，不能准确描述具有明显各向异性特性的饱和土地基的动力性能。     

横观各向同性热弹性梁的精化理论

本文从横观各向同性梁的二维问题出发,研究了横观各向同性热弹性梁的精化理论。首先,在不作任何预先假设的条件下,利用横观各向同性热弹性理论和Lur’e算子函数,获得了由梁中线上的物理量表示的位移场和应力场。对热弹性梁上下表面承受非齐次边界条件的情况,推导出梁的近似控制微分方程。再舍去温度项,则横观各向同性热弹性梁的精化理论退化为横观各向同性梁的精化理论。     

3-D DYNAMIC RESPONSE OF TRANSVERSELY ISOTROPIC SATURATED SOILS

Introduction Thedynamicanalysisofsaturatedporoelasticmediaisofbroadmeaninginmanyaspectssuchasseismology,earthquakeengineering,soilmechanicsandgeophysics,etc.Thetransverselyisotropicsaturatedsoilsissolid fluidcoupledtwophasemediainwhichthe soilskeletondisp…     

Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder

A theoretical method for analyzing the axisymmetric plane strain elastodynamic problem of a non-homogeneous orthotropic hollow cylinder is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions and the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to homogeneous ones. By virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally obtained, which can be degenerated in a rather straightforward way into the solution for a homogeneous orthotropic hollow cylinder and isotropic solid cylinder as well as that for a non-homogeneous isotropic hollow cylinder. Using the present method, integral transform can be avoided and it can be used for hollow cylinders with arbitrary thickness and subjected to arbitrary dynamic loads. Numerical results are presented for a non-homogeneous orthotropic hollow cylinder subjected to dynamic internal pressure. The project supported by the National Natural Science Foundation of China (10172075 and 10002016)     

Acoustic wave interaction with a laminated transversely isotropic spherical shell with imperfect bonding

An exact analysis is carried out to study interaction of a time-harmonic plane-progressive sound field with a multi-layered elastic hollow sphere made of spherically isotropic materials with interlaminar bonding imperfections. A modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics, ultimately leading to calculation of a global transfer matrix. A linear spring model is adopted to describe the interlaminar adhesive bonding whose effects are incorporated into the global transfer matrix by introduction of proper interfacial transfer matrices. The solution is first used to correlate the perturbation in the material elastic constants of an evacuated and water submerged steel (isotropic) spherical shell to the sensitivity of resonances appearing in the backscattered amplitude spectrum. The backscattering form function, in addition to the acoustic radiation force acting on selected transversely isotropic spherical shells with distinct degrees of material anisotropy, is subsequently calculated and discussed. An illustrative numerical example is given for a multi-layered hollow sphere with two distinct interlaminar interface conditions (i.e., perfectly and imperfectly bonded layers). Limiting cases are considered and fair agreements with solutions available in the literature are established.     

横观各向同性超弹性球壳的有限振动

应用有限变形动力学理论研究了一种横观各向同性不可压超弹性材料球壳在表面突加均布拉伸载荷作用下的有限振动问题．给出了球壳振动的振幅和外加载荷之间的关系,得到了,球壳振动的相同和近似的周期,讨论了球壳振动的振幅、相图及振动的周期和材料各向异性程度的关系．     

横观各向同性饱和层状地基的三维稳态动力响应

首先引入状态向量,将直角坐标系下横观各向同性饱和土的Biot波动方程转化为一组状态方程,然后基于双重Fourier变换,求解了状态方程,得到传递矩阵.进而利用传递矩阵,并结合饱和地基的边界条件、排水条件及层间接触和连续条件,求解了横观各向同性饱和层状地基的稳态动力响应问题.     

Stress Solution for Transversely Isotropic Corrugated Hollow Cylinders

The stress problem for corrugated hollow transversely isotropic cylinders is solved in three-dimensional formulation for certain end conditions. Discrete Fourier series are used to make the problem one-dimensional, which is then solved by the stable discrete-orthogonalization method. Examples of analysis are given__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 62–69, March 2005.     

Non-axisymmetrical vibration of elastic circular plate on layered transversely isotropic saturated ground

The non-axisymmetrical vibration of elastic circular plate resting on a layered transversely isotropic saturated ground was studied.First,the 3-d dynamic equations in cylindrical coordinate for transversely isotropic saturated soils were transformed into a group of governing differential equations with 1-order by the technique of Fourier ex- panding with respect to azimuth,and the state equation is established by Hankel integral transform method,furthermore the transfer matrixes within layered media are derived based on the solutions of the state equation.Secondly,by the transfer matrixes,the general solutions of dynamic response for layered transversely isotropic saturated ground excited by an arbitrary harmonic force were established under the boundary conditions, drainage conditions on the surface of.ground as well as the contact conditions.Thirdly, the problem was led to a pair of dual integral equations describing the mixed boundary- value problem which can be reduced to the Fredholm integral equations of the second kind solved by numerical procedure easily.At the end of this paper,a numerical result concerning vertical and radical displacements both the surface of saturated ground and plate is evaluated.