压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Nonlinear electro-mechanical responses of functionally graded piezoelectric beams

This study presents analyses of the nonlinear electro-mechanical responses of functionally graded piezoelectric beams undergoing small deformation gradients. The studied functionally graded beams comprise of electro-active and inactive constituents with gradual compositions varying through the thickness of the beams. Two types nonlinear electro-mechanical responses are considered for the active constituents, which are nonlinear electro-mechanical behaviors for the polarized piezoelectric constituent under electric fields smaller than the coercive limit, and polarization switching responses due to cyclic electric fields with high amplitude. The inactive constituent is modeled with uncoupled linear electro-elastic response. The functionally graded beam is discretized into several graded layers through its thickness. Each layer is comprised of different compositions of the active (piezoelectric) inclusions and conductive matrix. A particle-unit-cell micromechanical model is used to obtain the nonlinear electro-mechanical responses in each layer and is integrated within the laminate theory in order to obtain the overall nonlinear electro-mechanical responses of the functionally graded piezoelectric beams. The numerical predictions are compared with experimental data available in literature. Parametric studies are then performed in order to examine the effects of the thickness of the beam, of the concentration of the constituent, and the frequency of the cyclic electric field on the overall electro-mechanical response of the functionally graded piezoelectric beams.     

Bending analysis of piezoelectric exponentially graded fiber-reinforced composite cylinders in hygrothermal environments

The closed-form solutions of a piezoelectric exponentially graded fiber-reinforced hollow cylinder in hygrothermal environment conditions are investigated. The interaction of electric displacement and electric potentials as well as the elastic deformations is discussed. The present piezoelectric fiber-reinforced circular cylinder is subjected to a mechanical load as well as an electric potential at its lateral surfaces. The resultant constitutive equations of the piezoelectric fiber-reinforced composite cylinder are analytically investigated. The thermal conductivity and the moisture diffusivity coefficient are supposed to change continuously in the radial direction by a simple exponential law. The hygrothermoelastic responses of piezoelectric exponentially graded fiber-reinforced hollow circular cylinders are presented. The significant of influence of the graduation of material, temperature, moisture, pressures and electric parameters is investigated. Concluding remarks and appropriate discussions are presented.     

Dynamic stress from a cylindrical inclusion buried in a functionally graded piezoelectric material layer under electro-elastic waves

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and dynamic stress around a subsurface cylindrical inclusion in a functionally graded piezoelectric material layer bonded to homogeneous piezoelectric materials. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the inclusion. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. Through the numerical solutions of dynamic stress concentration factors around the inclusion, it is found that when the cylindrical inclusion possesses higher rigidity and greater piezoelectric constant than the two phases of functionally graded materials, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion possesses lower rigidity and smaller piezoelectric constant than the two phases of functionally graded materials, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater. The effect of the properties of the inclusion on the dynamic stress around the inclusion is greater than that on the electric field. The effects of wave frequency on the dynamic stress and electric field are also examined.     

功能梯度压电层/压电半空间结构中Love波的传播:一种改进的拉盖尔多项式方法

使用传统的拉盖尔多项式方法求解层状半空间结构时,存在因层间材料差异所造成的应力、电位移不连续的现象。为了克服此方法的不足,提出了一种改进的拉盖尔多项式方法,研究了功能梯度压电层状半空间中Love波的传播特性。与文献中应用WKB法得到的结果进行对比,验证了该方法的正确性。计算和分析了相应的频散曲线、应力和电位移分布曲线。结果表明:该方法能够避免因层间材料差异所造成的应力、电位移不连续现象的出现;高频Love波的应力和电位移主要分布在功能梯度压电层中速度较低的一侧。该研究为基于Love波传感器的设计与优化奠定了一定的理论基础。     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment

In the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component E_z. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.     

Multilayer modal actuator-based piezoelectric transformers

An innovative, multilayer piezoelectric transformer equipped with a full modal filtering input electrode is reported herein. This modal-shaped electrode, based on the orthogonal property of structural vibration modes, is characterized by full modal filtering to ensure that only the desired vibration mode is excited during operation. The newly developed piezoelectric transformer is comprised of three layers: a multilayered input layer, an insulation layer, and a single output layer. The electrode shape of the input layer is derived from its structural vibration modal shape, which takes advantage of the orthogonal property of the vibration modes to achieve a full modal filtering effect. The insulation layer possesses two functions: first, to couple the mechanical vibration energy between the input and output, and second, to provide electrical insulation between the two layers. To meet the two functions, a low temperature, co-fired ceramic (LTCC) was used to provide the high mechanical rigidity and high electrical insulation. It can be shown that this newly developed piezoelectric transformer has the advantage of possessing a more efficient energy transfer and a wider optimal working frequency range when compared to traditional piezoelectric transformers. A multilayer piezoelectric, transformer-based inverter applicable for use in LCD monitors or portable displays is presented as well.     

Modeling of piezoelectric transformers using finite‐element technique

Abstract

In this work, the modeling of piezoelectric transformers using the finite‐element technique is presented. A 3‐D finite element method solver, which employs 20‐node brick‐element formulation, is developed. Using the solver, the characteristics of piezoelectric transformers under different operating frequencies can be simulated. Also, the solver is capable of accounting for the effects of the electrical loadings attached to the output electrodes of piezoelectric transformers. The modeling results for two different types of piezoelectric transformers, the Rosen‐modal‐type and the unipoled‐disk‐type, are presented. For the Rosen‐modal‐type devices, the simulated voltage gains and the phase differences are validated with our measured results. Also, the simulated results of the unipoled‐disk‐type transformers agree with the measured results found in previously published literature. The effects of electrical loadings on these piezoelectric transformers are also discussed.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Scattering of the SH wave from a crack in a piezoelectric substrate bonded to a half-space of functionally graded materials

The scattering of the SH wave from a crack in a piezoelectric substrate which is bonded to a half-space of functionally graded materials (FGM) is studied. The governing equations along with permeable crack boundary, regularity and continuity conditions across the interface are reduced to a coupled set of Cauchy singular integral equations which are solved approximately by applying Chebyshev polynomials. Numerical results for the normalized dynamic stress intensity factors (NDSIF) and the normalized electric displacement intensity factors (NEDIF) are presented. The effects of the geometric and physical parameters, and the effects of the frequency and the angle of incidence on NDSIF and NEDIF are discussed.     

Normal point force and point electric charge in a piezoelectric transversely isotropic functionally graded half-space

Normal point force and point electric charge acting on surface of a transversely isotropic piezoelectric half-space with a functionally graded transversely isotropic piezoelectric coating are considered. Elastic moduli, piezoelectric constants and dielectric permeabilities of the coating vary with depth according to arbitrary functions. Analytical expressions for the elastic displacements and potential of the electrostatic field are derived for a fixed value of the depth coordinate. Asymptotical analysis of these formulas is derived for small and big values of a radial coordinate. An equilibrium of the half-space under the action of axisymmetric mechanical (normal and tangential) and electric loading is studied and a scheme of reducing the solutions of mixed boundary value problems to integral equations is obtained. As an illustration of the obtained solution, the PZT-5H piezoceramics with typical examples of functionally graded and homogeneous coatings are considered. The results include computations of the profiles of displacements and electric potential for different types of variation of electro-elastic properties in the coating.     

Accurate determination of complex materials coefficients of piezoelectric resonators

This paper presents a method of accurately determining the complex piezoelectric and elastic coefficients of piezoelectric ceramic resonators from the measurement of the normalized electric admittance, Y~, which is electric admittance Y of piezoelectric resonator normalized by the angular frequency /spl omega/. The coefficients are derived from the measurements near three special frequency points that correspond to the maximum and the minimum normalized susceptance (B~) and the maximum normalized conductance (G~). The complex elastic coefficient is determined from the frequencies at these points, and the real and imaginary parts of the piezoelectric coefficient are related to the derivative of the susceptance with respect to the frequency and the asymmetry of the conductance, respectively, near the maximum conductance point. The measurements for some lead zirconate titanate (PZT) based ceramics are used as examples to demonstrate the calculation and experimental procedures and the comparisons with the standard methods.     

压电陶瓷变压器研究和发展现状

概述了压电陶瓷变压器的原理和分类，介绍了各种振动模式的压电陶瓷变压器，并重点叙述了两种电极设计精巧的压电陶瓷变压器，对体型和膜型压电陶瓷变压器的研究进展现状进行了分析。     

Identification of material constants for piezoelectric transformers by three-dimensional,finite-element method and a design-sensitivity method

In this paper, an inversion scheme for piezoelectric constants of piezoelectric transformers is proposed. The impedance of piezoelectric transducers is calculated using a three-dimensional finite element method. The validity of this is confirmed experimentally. The effects of material coefficients on piezoelectric transformers are investigated numerically. Six material coefficient variables for piezoelectric transformers were selected, and a design sensitivity method was adopted as an inversion scheme. The validity of the proposed method was confirmed by step-up ratio calculations. The proposed method is applied to the analysis of a sample piezoelectric transformer, and its resonance characteristics are obtained by numerically combined equivalent circuit method.     

Finite element simulation of piezoelectric transformers

Piezoelectric transformers are nothing but ultrasonic resonators with two pairs of electrodes provided on the surface of a piezoelectric substrate in which electrical energy is carried in the mechanical form. The input and output electrodes are arranged to provide the impedance transformation, which results in the voltage transformation. As they are operated at a resonance, the electrical equivalent circuit approach has traditionally been developed in a rather empirical way and has been used for analysis and design. The present paper deals with the analysis of the piezoelectric transformers based on the three-dimensional finite element modelling. The PIEZO3D code that we have developed is modified to include the external loading conditions. The finite element approach is now available for a wide variety of the electrical boundary conditions. The equivalent circuit of lumped parameters can also be derived from the finite element method (FEM) solution if required. The simulation of the present transformers is made for the low intensity operation and compared with the experimental results. Demonstration is made for basic Rosen-type transformers in which the longitudinal mode of a plate plays an important role; in which the equivalent circuit of lumped constants has been used. However, there are many modes of vibration associated with the plate, the effect of which cannot always be ignored. In the experiment, the double resonances are sometimes observed in the vicinity of the operating frequency. The simulation demonstrates that this is due to the coupling of the longitudinal mode with the flexural mode. Thus, the simulation provides an invaluable guideline to the transformer design     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

压电功能梯度层合梁的力-电-热耦合梁单元及最优振动控制

给出了一个压电功能梯度层合梁振动分析的两节点力-电-热耦合梁单元,并将其用于功能梯度层合梁的振动最优控制。在这个多场耦合梁单元中,功能梯度材料的等效力学性能用Voigt或Mori-Tanaka模型表征;梁的位移场用Shi改进的三阶剪切变形板理论描述;压电层的电势场用Layer-wise理论分层表征,且呈高阶非线性电势场的压电层可离散成数个子层。用Hamilton原理推导了压电功能梯度梁的力-电-热耦合单元列式,用拟协调元法给出了多场耦合梁单元的高计算效率的显式单元刚度矩阵,以及采用线性二次型(LQR)最优控制算法进行压电功能梯度层合梁的最优振动控制。使用所得力-电-热耦合梁单元进行了压电功能梯度层合梁的静力和动力分析。数值算例表明,所得力-电-热耦合梁单元可靠、准确和高效,LQR最优控制算法得到最优控制电压可有效抑制功能梯度梁的振动且实现控制系统能量的优化。     

The analysis of time-dependent creep in FGPM thick walled sphere under electro-magneto-thermo-mechanical loadings

Time-dependent creep response of a smart sphere made of functionally graded piezoelectric material (FGPM) is investigated. The vessel is subjected to an internal pressure, a uniform temperature field, an electric potential and a uniform magnetic field. Under such a loading condition initial elastic stresses are locked in the vessel at zero time. Due to high temperature, creep evolution causes stress redistribution in the sphere which is followed by electric potential redistribution across the thickness of the sphere. History of radial stresses is always reflected by history of electric potential which can be used for condition monitoring of the smart sphere. From the initial elastic stresses it has been found that imposing an electric potential decreases effective stresses. It has also been concluded from history of electric potential that electric potential redistribution is decreasing due to creep evolution and therefore this is followed by increasing effective stresses.     

基于等几何方法的压电功能梯度板动力学及主动振动控制分析

针对表面粘贴有压电层的功能梯度板的动力学及主动振动控制问题,建立了一种基于三阶剪切变形理论的等几何分析求解方法。其中,功能梯度板的材料属性为板厚方向的幂函数分布,并假设电势沿着压电层的厚度方向呈线性变化。利用线性压电本构方程以及哈密顿变分原理,推导了压电功能梯度板的相关等几何分析有限元方程。通过分析压电智能结构的静态弯曲行为验证了该方法的有效性与精确性。运用模态叠加技术与Newmark-β直接积分法分析了两种不同结构的压电功能梯度板的动力学响应与主动振动控制问题。在主动振动控制分析中,引入了物理中面的概念避免当传感器与驱动器分别粘贴于功能梯度的上、下表面时,由拉伸-耦合效应引起的控制不稳定的问题,并着重分析了振动控制过程中两种结构传感器层和驱动器层的电压响应。