INTERACTION OF ELECTRIC CHARGES IN A PIEZOELECTRIC WITH RIGID EXTERNAL CRACKS

IntroductionRecently much attention has been paid to Green’s functions[1 -7]for anisotropic mediabecause of their important applications in boundary element methods used widely inengineering analysis. Generally speaking, it is relatively easy to obtain G…     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

Green's functions for anisotropic piezoelectric bimaterials with a slipping interface

An explicit full-field expression of the Green's functions for anisotropic piezoelectric bimaterials with a slipping interface is derived. When the electro-elastic singularity reduces to a pure dislocation in displacement and electric potential, interaction energy between the dislocation and the bimaterials is obtained explicitly while the generalized force on the dislocation is given in a real form which is also valid for degenerate materials. The investigation demonstrates that the boundary conditions at the slipping interface between two piezoelectric materials will exert a prominent influence on the mobility of the dislocation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

饱和土埋置力源的三维动力Lamb问题解答

基于一组弹性土波动方程，应用Fourier级数展开和Hankel积分变换，得到了三维问题饱和土骨架与孔隙水的应力及位移分量在变换域内的积分形式通解．考虑地基表面透水情形，由边界条件导出了半空间饱和土体在埋置力源作用下的三维动力Lamb问题的解答．给出了埋置水平力作用下地基表面竖向位移、径向位移及周向位移的数值解．该研究为运用边界元法求解饱和地基的动力响应课题奠定了理论基础．     

Green''s functions for transversely isotropic piezoelectric solids

Closed-form expressions are obtained for the infinite-body Green's functions for a transversely isotropic piezoelectric medium. The four Green's functions represent the coupled elastic and electric response to an applied point force or point charge. The Green's functions are obtained using a formulation where the three displacements and the electric potential are derivable from two potential functions. When piezoelectric coupling is absent, the results reduce to those for uncoupled elasticity and electrostatics.     

Thermoelectroelastic Green's function and its application for bimaterial of piezoelectric materials

Summary For a two-dimensional piezoelectric plate, the thermoelectroelastic Green's functions for bimaterials subjected to a temperature discontinuity are presented by way of Stroh formalism. The study shows that the thermoelectroelastic Green's functions for bimaterials are composed of a particular solution and a corrective solution. All the solutions have their singularities, located at the point applied by the dislocation, as well as some image singularities, located at both the lower and the upper half-plane. Using the proposed thermoelectroelastic Green's functions, the problem of a crack of arbitrary orientation near a bimaterial interface between dissimilar thermopiezoelectric material is analysed, and a system of singular integral equations for the unknown temperature discontinuity, defined on the crack faces, is obtained. The stress and electric displacement (SED) intensity factors and strain energy density factor can be, then, evaluated by a numerical solution at the singular integral equations. As a consequence, the direction of crack growth can be estimated by way of strain energy density theory. Numerical results for the fracture angle are obtained to illustrate the application of the proposed formulation. Received 10 November 1997; accepted for publication 3 February 1998     

GREEN''''S FUNCTIONS FOR 2D PROBLEMS OF ANISOTROPIC PIEZOELECTRIC MATERIAL WITH A STRIP REGION

By using Stroh's formalism and the conformal mapping technique, we derive the simple explicit elastic fields of a generalized line dislocation and a generalized line force in a general anisotropic piezoelectric strip with fixed surfaces, which are two fixed conductor electrodes. The solutions obtained are usually considered as Green's functions which play important roles in the boundary element methods. The Coulomb forces of the distributed charges along the region boundaries on the line chargeq atz ⁰ are analysed in detail. The results are valid not only for plane and antiplane problems but also for the coupled problems between inplane and outplane deformations.     

A general solution for static piezothermoelastic problems of crystal class 6mm solids

In this paper, a general solution for three-dimensional static piezothermoelastic problems of crystal class 6mm solids is presented. The general solution involves four piezoelastic potential functions and a piezothermoelastic potential function, of which four piezoelastic potential functions are governed by weighted harmonic differential equations. Compared with the general solution given by Ashida et al., in which seven potential functions are introduced, the general solution proposed in the present paper is more rigorously derived. Moreover, it has a simple form and is convenient for application. As an illustrative example, the problem of a pyroelectric half-space subjected to axisymmetric heating is studied. Numerical results of displacements stresses, electric potential and electric displacements are obtained for a cadmium selenide half-space. Thermally induced displacements and stress distribution are compared with those obtained for the same material without piezoelectric and pyroelectric effects. Project supported by the National Natural Science Foundation of China (19872060).     

A penny-shaped crack in a transversely isotropic piezoelectric solid: Modes II and III problems

An exact analysis of the modes II and III problems of a pennyshaped crack in a transversely isotropic piezoelectric medium is performed in this paper. The potential theory method is employed based on the general solution of three-dimensional piezoelasticity and the four harmonics involved are represented by one complex potential. Previous results in potential theory are then utilized to obtain the exact solution that is expressed in terms of elementary functions. Comparison is made between the current results with those published and good agreement is obtained. The project supported by the National Natural Science Foundation of China (No. 19872060)     

曲梁压电俘能器强迫振动的格林函数解

本文运用格林函数法求解了曲梁压电俘能器在强迫振动下的解析解.运用微分法分析了压电层合曲梁结构面内各内力,根据曲梁压电 俘能器的动力学方程组,基于压电本构关系,建立了包含径向阻尼但不考虑俘能器曲梁结构部分的轴向力以及轴向惯性项的Prescott力 电耦合模型. 采用Laplace变换法求得了耦合振动方程的格林函数解.根据叠加原理和格林函数的物理意义,对耦合的系统方程解耦进而 求得强迫振动下曲梁压电俘能器的输出电压. 数值计算中,通过与现有文献的解析解进行对比,验证了本文解析解的有效性,并研究了阻 尼、电阻等重要物理参数对压电函数和谐振频率的影响.通过与有关传统直梁压电俘能器研究成果的对比,体现了曲梁压 电俘能器Prescott模型的高效集能特性. 数值分析研究表明:(1)使得曲梁俘能器达到最大输出电压时连接的最优负载电 阻为1 M$\Omega$;(2)通过更换适当的基底材料,降低材料的弹性模量,可以改变曲梁俘能器的高基频现象,以使结构适应 更复杂的工作环境,但这会导致俘能器的工作效率降低.      

The effective properties of piezocomposites, part I: Single inclusion problem

The problem of a piezoelectric ellipsoidal inclusion in an infinite nonpiezoelectric matrix is very important in engineering. In this paper, it is solved via Green's function technique. The closed-form solutions of the electroelastic Eshelby's tensors for this kind of problem are obtained. The electroelastic Eshelby's tensors can be expressed by the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem. Since the closed-form solutions of the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem can be given by theory of elasticity and electrodynamics, respectively, the electroelastic Eshelby's tensors can be obtained conveniently. Using these results, the closed-form solutions of the constraint elastic fields and the constraint electric fields inside the piezoelectric ellipsoidal inclusion are also obtained. These expressions can be readily utilized in solutions of numerous problems in the micromechanics of piezoelectric solids, such as the deformation and energy analysis, damage evolution and fracture of the piezoelectric materials. The project supported by the National Natural Science Foundation of China     

ANALYSIS OF A SCREW DISLOCATION INSIDE A CIRCULAR INCLUSION WITH INTERFACIAL CRACKS IN PIEZOELECTRIC COMPOSITES

IntroductionPiezoelectric materials have potentials for use in many modern devices and compositestructures. The presence of various defects, such as inclusions, holes, dislocations andcracks, can greatly influence their characteristics and coupled behavio…     

Electroelastic field of a piezoelectric annular finite cylinder

This paper presents a theoretical study of a piezoelectric annular cylinder under axisymmteric electromechanical loading. The piezoelectric material is assumed to be transversely isotropic and the general solutions of the governing equations are obtained in terms of a Fourier–Bessel series containing Bessel functions of the first and second kind. The boundary-value problems for vertical pressure and an electric charge loading applied to the ends of an annular cylinder are solved by expanding the applied loading in terms of a Fourier–Bessel series. Selected numerical results for the electroelastic field of an annular cylinder are presented for different aspect ratios of a cylinder and material properties.     

Analysis of laminated piezoelectric cylindrical shells

A new method is developed for three-dimensional stress analysis of laminated piezoelectric cylindrical shell with simple support. The shell can be subjected to various applied loadings, including distributed body force, inner and outer surface traction and potential. Each layer of the shell can be piezoelectric or elastic/dielectric, with perfect bonding assumed between each interface. The governing equations are solved by the state-space technique. Numerical results are presented to show the sensing and actuating effects of three-layered piezoelectric cylindrical shell. The project supported by the National Natural Science Foundation of China (19572027)     

Why traction-free? Piezoelectric crack and Coulombic traction

The non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces. An analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap. As the crack in such materials can be thought of as a low-capacitance medium carrying a potential drop, the Coulombic traction always pulls the two opposite surfaces of the crack together. It is proved that under relatively larger mechanical loading and relatively smaller electrical field, the Coulombic traction may be negligible and the previous investigations under the traction-free crack condition may be accepted in a tolerant way, otherwise the Coulombic traction may lead to some erroneous results with over 10% relative errors. It is also shown that, unlike the traction-free crack condition, the applied electric field does change the Mode I stress intensity factor (SIF) for a central crack in an infinite plane piezoelectric material, and in this way may significantly influence piezoelectric fracture. It is also concluded that the variable tendencies of the normalized SIF and the ERR against the applied electric field depend on the mechanical loading levels. This load-dependence feature may lead to a transformation of the normalized SIF and the ERR from an even functional dependence to an odd functional dependence on the applied electric field.     

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…     

Application of local co-ordinate systems to the solution of integral equations referred to plane fluid flow problems

Two-dimensional fluid flow problems expressed in terms of velocity potentials or stream functions are often summarized as boundary-value problems for the Laplace or Poisson equations, or the homogeneous or non-homogeneous biharmonic equations. Simple local co-ordinate systems have been applied to the solution of integral equations associated with these boundary-value problems. This procedure has been shown to be an efficient technique in the numerical solution of fluid flow problems.     

Functionally graded piezoelectric cantilever beam under load

Summary In the present paper, the problem of a functionally graded piezoelectric cantilever beam subjected to different loadings is studied. The piezoelectric beam is characterized by continuously graded properties for one elastic parameter and the material density. A pair of stress and induction functions in the form of polynomials is proposed and determined. Based on these functions, a set of analytical solutions for the beam subjected to different loadings is obtained. As particular cases, series of solutions for some canonical problems can be directly obtained from the solutions of the present paper, such as for the problems of a piezoelectric cantilever beam with constant body force or without body forces, etc.This research work is supported by the National Natural Science Foundation of China (50272003). Support was also given by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.