A sub-layer model for a thick piezoelectric patch bonded on elastic substrate

Summary. In this paper, the two-dimensional electromechanical coupling problems that a piezoelectric patch of finite size bonded to an elastic substrate are considered. A subdivision model that the single physical piezoelectric layer is mathematically divided into a number of thinner layers is proposed to analyze the electromechanical responses of the structures. Within each virtual sub-layer of the piezoelectric patch the electric displacement and normal stress in the axial direction are assumed to be linear functions of the thickness coordinate. Hellinger-Reissner variational principle for elasticity is extended to the systems of piezoelectric multi-materials. The governing equations that comprise one-dimensional differential equations and integro-differential equations are rigorously derived from the stationary conditions of the variational functional along with substitution of the assumed electromechanical fields. The subdivision model satisfies all mechanical and electric continuity conditions across the virtual interfaces and the physical interface of piezoelectrics/substrate. The numerical solutions of the governing equations are conducted, and the convergence of the subdivision model is demonstrated.     

Piezoelectric ceramic rectangular transducers in flexural vibration

Based on the equivalent elastic method and coupled vibration theory, an analytic method is presented to study the flexural vibration of rectangular transducers consisting of piezoelectric ceramic thin plates. By introducing a mechanical coupling coefficient, the flexural vibration of the piezoelectric ceramic rectangular thin plate is reduced to two simple, one-dimensional flexural vibrations of narrow piezoelectric ceramic strips. The resonance frequency equations for the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration are derived under the free and simply supported boundary conditions analytically. The relationship between the resonance frequency and the flexural vibrational order, the geometrical shape, and the dimensions of the piezoelectric ceramic rectangular thin-plate transducer is analyzed. It is demonstrated that the one-dimensional vibrational theory for the flexural vibration of a narrow piezoelectric ceramic strip and the stripe-mode flexural vibrational theory for the piezoelectric ceramic rectangular thin plate can be derived directly from the theory obtained in this paper. Experimental results show that the measured resonance frequencies of the piezoelectric ceramic rectangular thin-plate transducers in flexural vibration under free-boundary conditions are in good agreement with the calculated results. The method presented in this paper can be used in the resonance frequency analysis of vibrating systems in coupled vibration.     

Axisymmetric vibration of infinite piezoelectric cylinders using one-dimensional finite elements

A general finite-element analysis for infinite piezoelectric cylinders has been formulated. The classical three-dimensional elasticity equations of motion are used. The dependence on theta, z, and time are included by assuming appropriate trigonometric functions, and the three-dimensional problem is reduced to a one-dimensional finite element with four degrees of freedom per node. The tabulated results are limited to cylinders with stress-free, shorted electrode (phi=0) boundary conditions at the outside surface of the cylinder. However, solutions for a variety of boundary conditions are possible. Solutions for the piezoelectric cylinder compare favorably with the existing literature. The motion of piezoelectric cylinders with thin coatings is analyzed by modeling the cylinder and thin coating as a layered cylinder.     

Comment on Y.-H. Hsu et al., "Electrical and mechanical fully coupled theory and experimental verification of rosen-type piezoelectric transformers"

This letter discusses the difference between piezoelectric constitutive relations for the case of one-dimensional stress and the case of one-dimensional strain, and its implications in the modeling of Rosen piezoelectric transformers     

Comment on Y.-H. Hsu et al., "electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers" [see reference [1]]

This letter discusses the difference between piezoelectric constitutive relations for the case of one-dimensional stress and the case of one-dimensional strain, and its implications in the modeling of Rosen piezoelectric transformers.     

Flow in the Ekman layer on an oscillating porous plate

Summary.  The Laplace transform technique is utilized to obtain the exact solution for thermally-induced wave propagation in a circular piezoelectric plate. The temperature is assumed to satisfy the one-dimensional heat conduction equation that includes a relaxation time. Displacement and electric potential functions are introduced in order to solve the equations of motion and electrodynamics. Numerical results obtained for both a PZT-5A plate and a non-piezoelectric plate illustrate the effects of relaxation time and piezoelasticity on displacement and stress histories. Received October 3, 2002 Published online: February 10, 2003     

Adaptability of hybrid smart composite plate under low velocity impact

Present study deals with an approach to the detection of the strains and optimization of shape memory alloys (SMA) under low velocity impact. It is assumed that the sensor is mounted on the opposite side of the impact process, and SMA fibers are embedded within the layers of a composite plate. Modeling the SMA fibers constitutive has been derived as the one-dimensional equations. With different skew angles of the piezoelectric sensor, we can illustrate that the measured charge manifests the strain components induced by the impact. By changing the SMA volume fractions and the temperature increments, we can significantly reduce the deflections. It can be shown that the application of piezoelectric sensors is effective in the monitoring of the transient strain. In addition, the optimization of the distribution of volume fraction of SMA fibers provides further benefits in reducing the deflection of the plate.     

Mathematical Analysis of Piezoelectric Sandwich Torsion Transducers Based on the d36-Effect

In the present article, we analyze a d₃₆-effect piezoelectric torsion transducer following the Saint-Venant torsion theory taking the electrical field into account. A representation of the stress function, the electric potential, and the warping function are derived and solved with finite differences. Then, the one-dimensional governing equations at the structural beam level, including the constitutive relations as well as the balance equations for the dynamics of the transducer, are presented. The axial moment and the total charge are computed as functions of the rate of twist and the applied potential difference. As an example, a cantilevered transducer is studied.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

A one-dimensional model for non-linear behaviour of piezoelectric composite materials

In this paper, a one-dimensional analytical model is proposed to investigate the non-linear behaviour for piezoelectric and piezoelectric fibre reinforced composite (PFRC) materials in the fibre direction. The required linear and non-linear constants for purely piezoelectric materials are obtained using the curve-fitting method and the measured S₃–E₃ non-linear loops of the corresponding piezoelectric materials, whereas those for PFRC materials are determined by employing the quadratic non-linear constitutive equations for a purely piezoelectric material, the iso-field assumptions and linear and non-linear constants of the composite constituents. A numerical study is conducted. The numerical results reveal a significant effect of stress T₃ on S₃–E₃ non-linear behaviour for both soft PZT–5H ceramics and PZN–4.5%PT crystals. It is also found that the piezoelectric fibre volume fraction V_f and strain S₃ can significantly affect the T₃–E₃ non-linear behaviour for both PZT–5H/piezo-polymer polyvinylidene fluoride (PVDF) and PZN–4.5%PT/PVDF PFRC materials. A good correlation is noted between the piezoelectric constants d₃₃ and e₃₃ predicted using the present method and those recommended by manufacturer for PZT–5H ceramics and PZN–4.5%PT crystals.     

State space approach to one-dimensional thermal shock problem for a semi-infinite piezoelectric rod

The theory of generalized thermoelasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a boundary value problem of one-dimensional semi-infinite piezoelectric rod with its left boundary subjected to a sudden heat. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of the modern control theory. Approximate small-time analytical solutions to stress, displacement and temperature are obtained by means of the Laplace transform and inverse transform. It is found that there are two discontinuous points in both stress and temperature solutions. Numerical calculation for stress, displacement and temperature is carried out and displayed graphically.     

Two-dimensional analysis using one-dimensional FEM forstraight-crested waves in arbitrary anisotropic crystal plates andaxisymmetric piezoelectric vibrations in ceramic disks

We have developed a hybrid method applicable to straight-crested waves in arbitrary anisotropic crystal plates and to axisymmetric piezoelectric vibrations in ceramic disks. The solutions to two-dimensional (2-D) equations of motion are described with a linear combination of eigenmodes guided by a pair of parallel edges. The guided eigenmodes and their amplitudes are determined by using one-dimensional (1-D) finite element method (FEM). The method developed here provides rapid convergence with small matrix size compared with 2-D FEM. Computer programs have been developed for three examples, SC- and AT-cut quartz plates and barium titanate (BaTiO₃) disks, for which the frequency spectra and the corresponding mode shapes were calculated. The frequency spectra of AT-cut quartz plates are compared with those obtained from Mindlin's plate equations, with the aim of examining the accuracy of the straight-crested wave solutions for Mindlin's plate equations. A convergence study is also presented for BaTiO₃ disks     

ZnMgO/ZnO异质结构中极化对二维电子气的影响

基于 ZnMgO/ZnO 异质结构模型,从压电极化对应变弛豫度的依赖关系出发,通过自洽求解一维泊松-薛定谔方程,研究了 ZnMgO 势垒层的厚度、Mg 组分和应变弛豫度对 ZnMgO/ZnO 异质界面处二维电子气(2DEG)的分布、面密度等性质的影响,并结合极化和能带偏移对计算结果进行了分析讨论。结果表明通过改变 Mg 组分和应变弛豫度可以调节异质界面两边的极化强度不连续性,进而有效地调控异质结中的二维电子气。     

HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates

The present paper develops a formulation for laminated plates with extensional distributed piezoelectric sensors/actuators. This formulation is based on linear electroelasticity, and an equivalent single layer is used for the mechanical displacement field, applying a Higher-Order Shear Deformation Theory (HSDT), whereas a layerwise discretization is used in the thickness direction for the electric potential. The electric and mechanical local equilibrium equations and local constitutive equations for the problem are identified. The Principle of Virtual Work is used to derive the dynamic equilibrium equations in terms of generalized forces and the consistent boundary conditions. The piezoelectric laminate constitutive equations are built and used to write the equations of motion in terms of generalized displacements. Finally, analytical solutions for simply supported square laminates with piezoelectric layers are developed. The entire laminate, composed of the base structure and piezoelectric layers, can be arbitrary orthotropic. The solution is adequate for an arbitrary number of piezoelectric layers and stacking positions. Moreover, the solution takes into account all material coefficients, whether mechanical, piezoelectric or dielectric. Analytical results are obtained for static bending, both in sensor and actuation modes, and for free vibration of symmetric cross-ply laminates with piezoelectric layers externally bonded to the plate.     

Dimensional reduction study of piezoelectric ceramics constitutive equations from 3-D to 2-D and 1-D

Accurate performance evaluation is crucial to the design and development of macro/micro-sized piezoelectric devices, and key to this is the proper use of the stiffness/ compliance and piezoelectric coefficients of the piezoelectric ceramics involved. Although the literature points out effective piezoelectric coefficients e_31,f and d_33,f for thin film materials and reduced dimensionality of equations for bulk material, the elastic and piezoelectric coefficients remain unchanged from the 3-D equations in most reported 1-D and 2-D analyses of the macro/micro-sized devices involving the e form of the constitutive equations. The use of unchanged coefficients leads to variations between numerically predicted and experimental results in most devices. To understand effects of the dimensional reduction from 3-D to 2-D and 1-D on stiffness/compliance and piezoelectric coefficients, this paper derives the 2-D and 1-D constitutive equations from the 3-D equations, focusing on the discussion of often-required device configurations for sensor and actuator design and analysis. Two modified coefficients are proposed, termed reduced and enhanced, which enable better understanding of effects of the dimensional reduction and also effects on the design and analysis of sensors and actuators.     

Electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers

A piezoelectric transformer is a power transfer device that converts its input and output voltage as well as current by effectively using electrical and mechanical coupling effects of piezoelectric materials. Equivalent-circuit models, which are traditionally used to analyze piezoelectric transformers, merge each mechanical resonance effect into a series of ordinary differential equations. Because of using ordinary differential equations, equivalent circuit models are insufficient to reflect the mechanical behavior of piezoelectric plates. Electromechanically, fully coupled governing equations of Rosen-type piezoelectric transformers, which are partial differential equations in nature, can be derived to address the deficiencies of the equivalent circuit models. It can be shown that the modal actuator concept can be adopted to optimize the electromechanical coupling effect of the driving section once the added spatial domain design parameters are taken into account, which are three-dimensional spatial dependencies of electromechanical properties. The maximum power transfer condition for a Rosen-type piezoelectric transformer is detailed. Experimental results, which lead us to a series of new design rules, also are presented to prove the validity and effectiveness of the theoretical predictions.     

Flexural vibration of a lithium niobate piezoelectric plate with a ferroelectric inversion layer

Abstract

Two-dimensional equations for flexural motions of a lithium niobate piezoelectric plate are derived from three-dimensional equations. The plate has a ferroelectric inversion layer where the piezoelectric constants reverse signs. The equations show an unconventional behavior that a uniform electric field along the plate thickness can produce bending. This offers many possibilities for new designs of devices. Waves in unbounded plates are examined for the accuracy of the equations. A piezoelectric energy harvester based on such a plate is analyzed as an example of the application of the equations derived in vibrations of finite plates.     

Partially debonded circular inclusion in one-dimensional quasicrystal material with piezoelectric effect

International Journal of Mechanics and Materials in Design - In this first effort, a partially-debonded circular inclusion in a one-dimensional quasicrystal material with piezoelectric effect is...     

间隙波在功能梯度压电板和压电半空间结构中的传播

研究了间隙波在功能梯度压电板和压电半空间结构中的传播性质.功能梯度压电板的材料性能沿x2方向呈指数变化,首先推导了间隙波传播时的解析解,利用界面条件得到了间隙波的频散方程,基于推导的频散方程,结合数值算例分析了功能梯度压电材料的梯度、压电层厚度以及材料性能对间隙波相速度的影响,研究结果对功能梯度压电材料的覆层结构在声波器件中的应用具有重要的参考价值.     

一维六方压电准晶中带三条不对称裂纹的圆形孔口的反平面问题

本文利用复变函数方法和保角映射,研究一维六方压电准晶材料中带不对称三裂纹的圆形孔口的的断裂问题。根据准晶压电材料基本方程的基础上,利用点群的对称性和一维六方准晶的线性压电效应,导出了一维六方准晶压电材料反平面问题的控制方程,并结合Cauchy积分公式,得到电非渗透与电渗透边界条件下的裂纹尖端场强度因子的解析表达式。当改变裂纹长度和孔口半径时,所得结果可以模拟出一些新裂纹模型。在不考虑电载荷作用时,所得结果和原有结果是一致的。通过数值算例讨论了材料的几何参数对场强度因子的影响,得出水平裂纹长度和圆半径可以促进裂纹增长。本研究为工程中材料的制备与应用将提供可靠的理论价值。