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.     

Fundamental solutions for small deformations superposed on finite biaxial extension of an elastic body

Infinite and semi-infinite isotropic elastic bodies in finite biaxial extension are in addition subjected to infinitesimal concentrated forces parallel and perpendicular to the extension plane respectively. In the semi-infinite case the forces are on the plane boundary. The problems are analogous to those for transversely isotropic bodies subjected to the infinitesimal concentrated forces only. Solutions to both categories of problems are given with the aid of potential functions. To facilitate the evaluation, expressions for the resultant force on a surface are given in terms of line integrals around the bounding curve. The solutions to the semi-infinite body problems are derived from those to the infinite body problems by adding appropriate potential functions so as to meet the boundary conditions on the plane surface. Computations for the semiinfinite cases of the superposition problems are given.     

Indentation of punches into a piezoceramic body: two-dimensional contact problem of electroelasticity

The paper establishes the relationship between the solutions of the static contact problems of elasticity (no friction) for an isotropic half-plane and problems of electroelasticity for a transversely isotropic piezoelectric half-plane with the boundary perpendicular to the polarization axis. This allows finding the contact characteristics in the electroelastic case from the known elastic solution, without the need to solve the electroelastic problem. The contact problems of electroelasticity for different types of wedge-shaped punches (flat punch with rounded one or two edges, half-parabolic punch, and a periodic system of punches) are solved as examples Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 55–70, November 2008.     

Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

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。     

Contact of Transversely Isotropic Bodies in the Hertz Theory

Within the framework of the anisotropic theory of elasticity, a three-dimensional contact problem of interaction of two massive transversely isotropic bodies, whose dimensions substantially exceed the size of the contact region, is investigated. In this case, the isotropy planes of contacting elastic bodies are mutually perpendicular. Exact and numerical solutions of the problem are determined. Calculations for various transversely isotropic materials are carried out.     

横观各向同性材料轴对称问题基本解

从横观各向同性材料的基本解出发,用积分的方法得到了轴对称问题的基本解,对于材料特征Ｓｉ互不相等的两种可能情形都给出了表达式,因此,可直接退化得到各向同性材料轴对称问题基本解。     

Exact Analytical Solutions of Static Electroelastic and Thermoelectroelastic Problems for a Transversely Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads     

A general solution and the application of space axisymmetric problem in piezoelectric material

ＡＧＥＮＥＲＡＬＳＯＬＵＴＩＯＮＡＮＤＴＨＥＡＰＰＬＩＣＡＴＩＯＮＯＦＳＰＡＣＥＡＸＩＳＹＭＭＥＴＲＩＣＰＲＯＢＬＥＭＩＮＰＩＥＺＯＥＬＥＣＴＲＩＣＭＡＴＥＲＩＡＬＷａｎｇＺｉ－ｋｕｎｇ（王子昆）ＣｈｅｎＧｅｎｇ－ｃｈａｏ（陈庚超）（ＸｉａｎＪｉａｏ...     

Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer’s free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. The case of circular domain of contact is considered in detail. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Influence of a thin isotropic coating on the edge effect zone in isotropic and transversely isotropic materials

The effect of a thin isotropic coating on the edge effect zone in a representative element of a coated material is examined. Isotropic and transversely isotropic materials are considered. The transversely isotropic material has the elastic properties of unidirectional glass-fiber-reinforced plastic. The decay of the edge effect in the directions perpendicular to the coating plane and to the plane of isotropy is studied. A boundary-value problem of elasticity for piecewise-homogeneouse orthotropic bodies and a quantitative edge effect decay criterion for normal stresses are used as a design model. The problem is solved using the finite-difference method and base schemes. The results of evaluation of the edge effect zone in homogeneous and inhomogeneous materials are presented. It is shown that the presence of a thin isotropic coating blocks the edge effect, that is, decreases the edge effect zone in both isotropic and transversely isotropic materials __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 61–67, December 2007.     

The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

Wave propagation in transversely isotropic cylinders

Various approaches have been used for model1ing problems dealing with interaction of acoustic/elastic waves with transversely isotropic cylinders. The authors developed the first mathematical model for the scattering of acoustic waves from transversely isotropic cylinders [Honarvar, F., Sinclair, A.N., 1996. Acoustic wave scattering from transversely isotropic cylinders. Journal of the Acoustical Society of America 100, 57–63.]. In the current paper, this model is used for derivation of the frequency equations of longitudinal and flexural wave propagation in free transversely isotropic cylinders. Consistency of this model with the physics of the problem is demonstrated and a systematic solution to the corresponding equations is developed. Numerical results obtained for a number of transversely isotopic cylinders are used for verification of the mathematical model.     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

横观各向同性材料三维裂纹问题的数值分析

严格从三维横观各向同性材料弹性空间问题的Green函数出发,采用Hadamard有限部积分概念,导出了三维状态下单位位移间断(位错)集度的基本解.在此基础上,将三维任意形状的片状裂纹问题归结为求解-组以未知位移间断表示的超奇异积分方程;并给出了边界元离散形式.对方程中出现的超奇异积分,采用了Had-alnard定义的有限部积分来处理.论文最后给出了若干典型片状裂纹问题的数值算例,数值结果表明了本文方法是非常有效的.     

Refined Design of Corrugated Noncircular Cylindrical Shells

An approach is proposed to solve static problems for corrugated nonthin cylindrical shells applying spline-approximation in the longitudinal direction and a stable numerical method in the circumferential direction. Solutions are presented in the form of plots and tables for isotropic and transversely isotropic shells of constant and variable thickness__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 18–24, January 2005.     

The state vector methods for space axisymmetric problems in multilayered piezoelectric media

The state vector equations for space axisymmetric problems of transversely isotropic piezoelectric media are established from the basic equations. Using the Hankel transform, the state vector equations are reduced to a system of ordinary differential equations. An analytical solution of the problems in the Hankel transform space is presented in the form of the product of initial state vector and transfer matrix. The transfer matrices are given for the three distinct eigenvalues. Applications of the solutions are discussed. An analytical solution for the transversely isotropic semi-infinite piezoelectric media subjected to concerted point loads on the surface z=0 is presented in the Hankel transform space. Using transfer matrix and the continuity conditions at the layer interfaces, the general solution formulation of N-layered transversely isotropic piezoelectric media is given. A selected set of numerical solutions is presented for a layered semi-infinite piezoelectric solid.     

Anti-plane problems of piezoelectric material with a micro-void or micro-inclusion based on micromorphic electroelastic theory

The linear theory of micromorphic electroelasticity, which incorporate the coupled electromechanical behavior into the framework of micromorphic continuum theory, is used to solve the anti-plane problems of piezoelectric media with a micro-void or micro-inclusion in this paper. The electromechanical field solutions for a transversely isotropic piezoelectric medium are derived in the context of micromorphic electroelasticity and a generalized characteristic length is introduced to describe the size effect. Anti-plane problems of an infinite piezoelectric medium containing a micro-void or micro-inclusion are analyzed. Numerical results reveal that the mechanical and electric fields predicted by the present model highly depend on the relative size of the micro-void or micro-inclusion with respect to the generalized characteristic length, which is obviously different from the classical prediction.     

Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch

This paper presents the fundamental contact solutions of a magneto-electro-elastic half-space indented by a smooth and rigid half-infinite punch. The material is assumed to be transversely isotropic with the symmetric axis perpendicular to the surface of the half-space. Based on the general solutions, the generalized method of potential theory is adopted to solve the boundary value problems. The involved potentials are properly assumed and the corresponding boundary integral equations are solved by using the results in literature. Complete and exact fundamental solutions are derived case by case, in terms of elementary functions for the first time. The obtained solutions are of significance to boundary element analysis, and an important role in determining the physical properties of materials by indentation technique can be expected to play.