压电介质二维边界积分方程中的基本解

由于压电介质的变形－电场耦合效应及压电响应的各向异性，使解析求解压电介质问题的工作变量十分复杂，若采用边界元数值方法求解，必须具备积分方程中的基本解，本文根据电磁场方程及连续介质力学的耦合性质论层出了二维无限域中分别在单位力及单位电荷载作用下的位移场，电势场、应力场和电位移场的解，从而确立了边界积分方程中所必需的八个基本解。     

Exact static analysis of a rotating piezoelectric spherical shell

This paper gives an exact analysis of a rotating piezoelectric spherical shell with arbitrary thickness. At first, three displacement functions are introduced to simplify the basic equations of a spherically isotropic, piezoelectric medium. By expanding the displacement functions as well as the electric potential in terms of spherical harmonics, the basic equations of equilibrium are converted to an uncoupled Euler type, second order ordinary differential equation and a coupled system of three second order ordinary differential equations. A general solution to the homogeneous equations of equilibrium is then derived. The static analysis of a rotating spherical shell is then performed and numerical example is presented. The project supported by the National Natural Science Foundation of China, the Zhejiang Provincial Natural Science Foundation, and the Japanese Committee of Culture, Education and Science.     

PMN-PT层/弹性基底结构中声表面波特性分析

研究了PMN-PT 压电层/弹性(金刚石) 基底结构中表面波的传播特性,压电层表面是机械自由的,电学边界条件分为电学开路和电学短路,压电层与基底之间采用理想连接. 得到了满足控制方程和边界条件的电弹场以及弹性波在结构中传播时的频散方程,通过数值算例分析了压电材料PMN-PT 的极化方向对弹性波频散曲线和机电耦合系数的影响,以及不同极化方向时弹性位移和电势随结构深度方向的变化,结果可为PMN-PT 压电材料在高频声表面波器件中的应用提供有价值的理论参考.      

3D elasticity solution for static analysis of functionally graded piezoelectric annular plates on elastic foundations using SSDQM

Three-dimensional elasticity solution for static analysis of functionally graded piezoelectric (FGP) annular plates with and without elastic foundations through using state-space based differential quadrature method (SSDQM) at different boundary conditions is presented in this paper. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the mechanical behavior of FGP annular plates. The state variables include a combination of electric potential, electric displacement, three mechanical displacement parameters and three stress parameters. Numerical results are given to demonstrate the convergency and accuracy of the present method. Both closed circuit and open circuit effects are studied and the influences of the Winkler and shearing layer elastic coefficients of the foundations, the material property graded index, radius, thickness, mechanical load and boundary conditions on the deflection response of the FGP annular plates are investigated. The new results can be used as a benchmark solutions for future researches.     

功能梯度压电、压磁柱对称问题的振动分析

本文在不考虑体力、体电流和体电荷的情况下, 假定压电、压磁柱壳的材料参数沿圆柱厚度方向呈幂函数分布时, 研究了径向载荷作用下功能梯度压电、压磁空心柱壳的空间柱对称径向振动问题. 首先在柱坐标系下, 由功能梯度材料的参数、本构、梯度和平衡方程推导得出外激励作用下以Bessel函数表示圆柱壳的应力、电势、磁势等物理量的稳态解, 进而对空间柱对称的压电、压磁功能梯度材料动力控制问题进行了理论分析. 可以看出, 当梯度参数时, 即完全退化为横观各项同性压电、压磁柱对称的振动问题, 与文献[16]的基本方程为柱坐标下得出的结果一致. 最后给出数值算例, 结果表明材料不均匀性对沿径向振动各物理量的显著影响, 且用一个特定不均匀性参数值可以优化电磁力耦合的性能, 这在现代工程设计中尤为重要.     

功能梯度压电、压磁柱对称问题的振动分析

本文在不考虑体力、体电流和体电荷的情况下, 假定压电、压磁柱壳的材料参数沿圆柱厚度方向呈幂函数分布时, 研究了径向载荷作用下功能梯度压电、压磁空心柱壳的空间柱对称径向振动问题. 首先在柱坐标系下, 由功能梯度材料的参数、本构、梯度和平衡方程推导得出外激励作用下以Bessel函数表示圆柱壳的应力、电势、磁势等物理量的稳态解, 进而对空间柱对称的压电、压磁功能梯度材料动力控制问题进行了理论分析. 可以看出, 当梯度参数时, 即完全退化为横观各项同性压电、压磁柱对称的振动问题, 与文献[16]的基本方程为柱坐标下得出的结果一致. 最后给出数值算例, 结果表明材料不均匀性对沿径向振动各物理量的显著影响, 且用一个特定不均匀性参数值可以优化电磁力耦合的性能, 这在现代工程设计中尤为重要.     

Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials

Using the hypersingular integral equation method based on body force method, a planar crack in a three-dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed. This crack problem is reduced to solve a set of hypersingular integral equations. Compare with the crack problems in elastic isotropic materials, it is shown that for the impermeable crack, the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials. Based on the exact analytical solution of the singular stresses and electrical displacements near the crack front, the numerical method of the hypersingular integral equation is proposed by the finite-part integral method and boundary element method, which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied. Finally, the numerical solutions of the stress and electric field intensity factors of some examples are given.     

Spherical-symmetric steady-state response of piezoelectric spherical shell under external excitation

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Polynomial approach for modeling a piezoelectric disc resonator partially covered with electrodes

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach.     

含有多个涂层压电纤维复合材料的平面问题研究

压电纤维在未来的复合材料结构健康监测中具有重要作用．本文基于横观各向同性压电材料位移和应力连续条件以及经典的复势函数理论,讨论了同时受到平面内机械载荷和出平面电载荷作用时含有多个带涂层压电纤维的无限大线弹性基体的平面力学问题．首先将线弹性基体、涂层和压电纤维的应力场、位移场表示成复势函数,然后通过横观各向同性压电材料和线弹性材料的位移和应力连续条件确定复势函数表达式．将得到的复势函数表达式代入线弹性基体、涂层和压电纤维的的应力场、位移场公式可确定其应力场和位移场．最后,通过定量的案例讨论了涂层的材料属性对线弹性基体应力场的影响．案例分析表明涂层的材料属性对压电复合材料的应力场有重要的影响．     

Coupled refined layerwise theory for dynamic free and forced response of piezoelectric laminated composite and sandwich beams

This work extends a previously presented coupled refined layerwise theory to dynamic analysis of piezoelectric laminated composite and sandwich beams. Contrary to most of the available theories, all the kinematic and stress boundary conditions are satisfied at the interfaces of the piezoelectric layers with the non-zero longitudinal electric field. Moreover, both electrical transverse normal strains and transverse flexibility are taken into account for the first time in the present theory. In the presented formulation a high-order polynomial, an exponential expression and a layerwise term containing the electric field are included in the describing expression of the in-plane displacement of the beam. For the transverse displacement, the coupled refined model uses a combination of continuous piecewise fourth-order polynomials with a layerwise representation of electrical unknowns. The electric field is also approximated as linear across the thickness direction of piezoelectric layers. One of advantages of the present theory is that the mechanical number of the unknown parameters is very small and is independent of the number of the layers. For validation of the proposed model, various free and forced vibration tests for thin and thick laminated/sandwich piezoelectric beams are carried out. For various electrical and mechanical boundary conditions, excellent correlation has been found between the results obtained from the proposed formulation with those resulted from the three-dimensional theory of piezoelasticity.     

Generalized Bleustein-Gulyaev type waves in layered porous piezoceramic structure

The propagation of a Bleustein-Gulyaev (B-G) type wave in a structure consisting of multiple layers and a half-space of porous piezoelectric materials is theoretically studied. The solutions of the problem in terms of the mechanical displacements and electric potential functions are obtained for each layer and the half-space. The dispersion equation is obtained for electrically open and shorted boundary conditions by use of the transfer matrix method. A peculiar kind of B-G waves is investigated, which can propagate only in the layer over the half-space. The relationship between the piezoelectric constants and the dielectric constants is found for the existence of a peculiar kind of propagation modes. The numerical results in terms of the phase velocity and the electromechanical coupling factor with different thicknesses of the layer stack are presented.     

Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL-Ⅰ: GENERALIZED VARIATIONAL THEORETICAL MODELING

Based on the generalized variational principle of magneto-thermo-elasticity of the ferromagnetic elastic medium, a nonlinear coupling theoretical modeling for a ferromagnetic thin shell is developed. All governing equations and boundary conditions for the ferromagnetic shell are obtained from the variational manipulations on the magnetic scalar potential, temperature and the elastic displacement related to the total energy functional. The multi-field couplings and geometrical nonlinearity of the ferromagnetic thin shell are taken into account in the modeling. The general modeling can be further deduced to existing models of the magneto-elasticity and the thermo-elasticity of a ferromagnetic shell and magneto-thermo-elasticity of a ferromagnetic plate, which axe coincident with the ones in literature.     

Penny shaped crack in a piezoelectric cylinder surrounded by an elastic medium subjected to combined in-plane mechanical and electrical loads

The fracture problem of a penny shaped crack in a piezoelectric ceramic cylinder surrounded by an infinite elastic medium under in-plane normal mechanical and electrical loads is considered with the electric continuous boundary conditions on the crack surface. By using the potential theory and Hankel transform, a system of dual integral equations is obtained, and expressed to a Fredholm integral equation of the second kind. The mechanical and electrical field equations and all sorts of field intensity factors of mode I are obtained, and the numerical values of various field intensity factors for PZT-6B piezoelectric ceramic surrounded by several different elastic media are graphically shown for a uniform load and a ring-shaped load, respectively. And the effects of the size of the piezoelectric cylinder and the elastic material properties on various field intensity factors are obtained.     

Love waves in functionally graded piezoelectric materials

To investigate the features of Love waves in a layered functionally graded piezoelectric structure, the mathematical model is established on the basis of the elastic wave theory, and the WKB method is applied to solve the coupled electromechanical field differential equation. The solutions of the mechanical displacement and electrical potential function are obtained for the piezoelectric layer and elastic substrate. The dispersion relations of Love waves are deduced for electric open and short cases on the free surface respectively. The actual piezoelectric layer–elastic substrate systems are taken into account, and some corresponding numerical examples are proposed comparatively. Thus, the effects of the gradient variation about material constants on the phase velocity, the group velocity, the coupled electromechanical factor and the cutoff frequency are discussed in detail. So the propagation behaviors of Love waves in inhomogeneous medium is revealed, and the dispersion and the anti-dispersion are analyzed. The conclusions are significant both theoretically and practically for the surface acoustic wave devices.     

Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

Summary This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.     

压电弹性层合板静力机耦合特性的解析解

应用经典权壳理论和压电理论对层合板结构进行简化。对四边简支压电层合板在不同电学边界条件下,包括四边电学开路和电学短路。上下表面受外电压及无外加电压作用,进行分析,求得了电势和挠度的解析表达式,给出了压电层和基体的挠度、电势分布图。     

Steady state thermal and mechanical stresses of a poro-piezo-FGM hollow sphere

In this study, an analytical method is developed to obtain mechanical and thermal stress and electrical potential functions, electrical and mechanical displacement in two dimensional steady (r,θ) stat a functionally graded piezo electric porous material hollow sphere (FGPPM). It is assumed that properties of poro, piezoelectric and FGM material is changed through thickness according to power law functions, Heat conduction equation is obtained for obtaining temperature distribution and Navier equations analytically using Legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different power indices law and is drawn on a graph. These results are confirmed with the obtained in formations in the paper.     

Penny-shaped crack in a piezoelectric cylinder under electro-mechanical loads

Summary The fracture behavior of a penny-shaped crack in a axisymmetrical piezoelectric ceramic cylinder of finite radius under mechanical and electrical loads is analyzed under electric continuous boundary conditions on the crack surface. The potential theory and Hankel transform are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. Singular mechanical and electrical fields and field intensity factors of mode I are obtained. The numerical values of various field intensity factors for PZT-6B piezoelectric ceramics are graphically shown for a uniform load and a ring-shaped load, respectively. The effects of the radius of the cylinder on the field intensity factors are investigated. This work was supported by the Korea Research Foundation Grant (KRF-2001-041-E00057).