Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.     

Electro-elastic interaction between a piezoelectric screw dislocation and collinear rigid lines

Electro-elastic stress investigation on the interaction between a piezoelectric screw dislocation and collinear rigid lines under anti-plane mechanical and in-plane electrical loading is carried out. The lines are considered, respectively, as dielectrics or conductors. The screw dislocation is subjected to a line charge and a line force at the core. Closed-form analytical solutions are derived by means of complex variable method. Explicit expressions for the field variables, the singularity of the field variables at the line tip and the force on the dislocation are obtained for a single rigid line.     

On refined nonlinear theories of laminated composite structures with piezoelectric laminae

In this paper geometrically nonlinear theories of laminated composite plates with piezoelectric laminae are developed. The formulations are based on thermopiezoelectricity, and include the coupling between mechanical deformations, temperature changes, and electric displacements. Two different theories are presented: one based on an equivalent-single-layer third-order theory and the other based on the layerwise theory, both of which were developed by the senior author for composite laminates without piezoelectric laminae. In the present study, they are extended to include piezoelectric laminae. In both theories, the electric field is expanded layerwise through the laminate thickness. The dynamic version of the principle of virtual displacements (or Hamilton’s principle) is used to derive the equations of motion and associated boundary conditions of the two theories. These theories may be used to accurately determine the response of laminated plate structures with piezoelectric laminae and subjected to thermomechanical loadings.     

Electro-elastic stress analysis for a wedge-shaped crack interacting with a screw dislocation in piezoelectric solid

The electro-elastic stress investigation on the interaction problem of a screw dislocation near the tip of a semi-infinite wedge-shaped crack in piezoelectric material has been carried out. Explicit closed-form analytical solutions are obtained for the stress intensity factor (SIF) and the electric displacement intensity factor (EDIF) of the crack, as well as the force on dislocation. The derivation is based on the conformal mapping method and the perturbation technique. The dislocation has Burger's vector normal to the isotropic basal plane, with a line force and a line charge being applied at the core of the dislocation. The influence of the location and the wedge angle of the crack on the image force of the dislocation has been discussed in detail. At the same time the effect of the dislocation on the crack behavior has been also examined under different configurations. Two types of PZT materials are used to numerically illustrate the influences of the wedge angle and the location of the dislocation on the image force and the crack intensities. Results obtained in the current study can be fully reduced to various special cases available in the literatures.     

Electro-elastic stress analysis for a screw dislocation interacting with a coated inclusion in piezoelectric solid

Summary. The electro-elastic stress investigation on the interaction problem of a piezoelectric screw dislocation near a coated inclusion in a piezoelectric material has been carried out. In our study, three dissimilar material phases are involved: the matrix, the inclusion and the coating layer. All the three materials are piezoelectric and with different material constants. Explicit closed-form analytical solutions for the stress and electric displacement fields are obtained by using the complex variable method. The image force acting on the screw dislocation is calculated by using the generalized Peach-Koehler formula. Numerical examples for different material constant combinations are performed. The influences of material properties of the inclusion and the coating layer on the image forces are examined and discussed.     

The primary resonance of laminated piezoelectric rectangular plates

Based on Hamilton's principle and the Rayleigh- Ritz method, a model of a nonlinear dynamic laminated piezoelectric rectangular plate was established, and the governing equations were derived and solved for both the thin-plate and thick-plate models. In the present investigation, the nonlinear constitutive relations of piezoelectric materials were considered and the effects of the nonlinearity on the response of the plate were discovered. The primary resonance of rectangular plate is investigated with the use of the method of multiple scales. The results obtained in the present paper agree very well with the experiment results.     

The resonance frequency of laminated piezoelectric rectangular plates

Based on the theory of elasticity and piezoelectricity, a dynamic model of laminated elastic-piezoelectric rectangular plates is considered. The bending equations are established and solved, taking the nonlinear behavior of the piezoelectric material into account. The resonance frequency of a laminated piezoelectric rectangular plate with four kinds of different boundary conditions is then investigated. The present results agree very well with the experimental findings and can be extended to practical applications, such as considering the effect of an epoxy package on the resonance of piezoceramic plate.     

Electro-elastic stress analysis for a Zener-Stroh crack interacting with a coated inclusion in a piezoelectric solid

Summary. The problem of a Zener-Stroh crack initiated near a coated circular inclusion in a piezoelectric medium is investigated in this paper. By using the solution of a single piezoelectric screw dislocation near a coating inclusion as the Greens function, a Mode III displacement loaded crack is investigated. The proposed problem is formulated as a set of singular integral equations which are solved by numerical techniques. The influence of various parameters, such as the material constants of the inclusion, the coating, the matrix, the coating layer thickness, etc., on the crack behavior is studied. The stress and electric displacement intensity factors of the crack are derived. Several numerical examples are given and the results obtained are discussed in detail.     

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.     

压电复合材料层合板的混沌动力学研究

压电复合材料具有独特的机电耦合性、比强度高、比刚度大和抗疲劳性能好等优点,越来越广泛地被应用于航空航天等工程领域中。根据Reddy的三阶剪切变形层合板理论,研究了受面内横向外激励力、面内纵向参数激励和面外参数激励以及压电参数激励联合作用下四边简支矩形复合材料层合板在1:2:4内共振情况下的六维平均方程。考虑平均方程存在一对双零特征值和两对纯虚特征值的情况,利用规范形理论进行化简,得到与原方程拓扑等价的最简规范形,然后在此基础上利用能量相位法从理论上分析得到系统会产生同宿的多脉冲跳跃现象;同时,基于平均方程,通过数值仿真,发现系统会产生混沌运动,具有跳跃现象。     

An exact analysis of a rectangular plate piezoelectric generator

We study thickness-twist vibration of a finite, piezoelectric plate of polarized ceramics or 6-mm crystals driven by surface mechanical loads. An exact solution from the three-dimensional equations of piezoelectricity is obtained. The plate is properly electroded and connected to a circuit such that an electric output is generated. The structure analyzed represents a piezoelectric generator for converting mechanical energy to electrical energy. Analytical expressions for the output voltage, current, power, efficiency, and power density are given. The basic behaviors of the generator are shown by numerical results.     

Transient piezothermoelastic problem for a laminated composite strip composed of angle-ply and piezoelectric laminae

This paper is concerned with the theoretical treatment of transient piezothermoelastic problem which is developed for a laminated composite strip composed of angle-ply laminae and a piezoelectric material of crystal class mm2, subject to nonuniform heat supply in the width direction. We obtain the exact solution for the two-dimensional temperature change in a transient state and the transient piezothermoelastic response of a simple supported composite strip under the state of generalized plane deformation. As an example, numerical calculations are carried out for a angle-ply laminated composite strip made of an alumina fiber reinforced aluminum composite, associated with a piezoelectric layer of a cadmium selenide solid. Some numerical results for temperature change, displacement, stress and electric potential distributions in a transient state are shown in figures. Furthermore, the influence of the thickness of the piezoelectric layer on the thermal stress or electric field is investigated.     

复合材料中压电螺型位错与含刚性核夹杂的电弹干涉

为了研究压电复合材料中位于基体的压电螺型位错与含共焦椭圆导电刚性核椭圆夹杂的电弹相互作用, 基于复变函数方法, 获得了基体和夹杂区域的精确级数形式解析解。运用广义Peach-Koehler公式, 导出了作用在位错上像力的解析表达式。在此基础上讨论了椭圆刚性核和材料电弹特性对位错像力以及位错平衡位置的影响规律, 同时讨论了压电夹杂和弹性基体的复合情况。结果表明: 椭圆刚性核对位错有着明显的排斥作用, 可以增强硬夹杂对位错的排斥, 减弱软夹杂对位错的吸引; 对于软夹杂, 在界面附近位错存在一个不稳定的平衡位置; 在基体和夹杂的界面上, 像力迅速增大; 当夹杂的剪切模量远小于基体时, 界面附近不会出现位错的平衡位置。     

计及压电效应时矩形振子的三维振动研究

在计及压电效应的情况下，本文利用解析方法对有限尺寸压电陶瓷矩形振子的三维振动进行了研究，推出了振子耦合振动的频率方程，并对振子的振动模式进行了分析。理论研究表明，利用本文中的解析法研究振子的耦合振动，计算简单、物理意义明显。与传统的一维理论分析方法及数值方法相比，由于本研究考虑了振子的压电效应以及不同振动模式不同的相互耦合，因此，振子的理论计算频率与测量值更加符合。     

On free vibration of a functionally graded piezoelectric rectangular plate

Summary On the basis of three-dimensional theory equations of transversely isotropic piezoelasticity, two independent state equations with variable coefficients are derived. To this end, separation formulae for displacements and shear stresses are employed. A laminated approximation is used to transform the state equations to the ones with constant coefficients in each layer. The free vibration problem of a piezoelectric rectangular plate with a functionally graded property is then investigated. Discussion on the boundary conditions is presented.     

单向轴压下压电矩形薄板的动态后屈曲问题

利用基于5自由度一阶剪切变形和von Karman几何非线性应变的压电薄板理论,通过双重F0u rier级数展开、Galerkin方法和多尺度方法,获得单向轴压和简支边界条件下压电矩形薄板动态后屈曲问题的解析解.给出了压电薄板动态后屈曲的共振特性曲线,对动态后屈曲的跳跃现象进行了研究.数值分析表明,几何参数和材料参数都对压电板的跳跃特性有显著影响.     

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

Summary The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.     

Resonance and antiresonance of symmetric and asymmetric cantilevered piezoelectric flexors

The resonances of dynamically excited symmetric piezoelectric bimorphs have been determined from the equations of state. Under the effect of sinusoidal stimuli: a moment exerted at the tip M, a force exerted perpendicular to the plane of the bimorph also applied at the tip F, a uniformly applied pressure p, and an electrode voltage V, they respond with a sinusoidal tip rotation alpha, tip deflection delta, volume displacement nu, and electrode charge Q. All of the former are related to all of the latter through a dynamic admittance matrix B. The antiresonance frequency of the capacitance C have been found while also antiresonance in off-diagonal elements have been determined. The latter indicate that at these frequencies the bimorph does not work as an actuator or sensor in the particular domain of the off-diagonal. The mode shape at these antiresonance frequencies has been determined. The antiresonance of b(14) determines that for this frequency the tip has deflection but no rotation, while the antiresonance of b(24 ) indicates that the tip has rotation but no deflection. No antiresonance in the volume displacement is found, indicating that the bimorph is a pressure converter (microphone) at all frequencies. Micromachined piezoelectric heterogeneous bimorphs have been fabricated using the techniques of I.C. fabrication. Their deflections have been measured as a function of frequency and applied voltage, while these have been compared with the theoretical predictions. An anomalously large quadratic deflection has been found, superimposed on the linear piezoelectric behavior. The agreement between the linear part of the experimental deflection and the theory was quite good.     

Design of large-displacement asymmetric piezoelectric microgripper based on flexible mechanisms

The output displacement of the traditional symmetrical microgripper is large, but its micro-components or parts are easily damaged due to the uneven force exerted on the left and right jaws of the gripper. The output force of the traditional asymmetric microgripper is stable. However, its output displacement is small, typically half the output displacement of the symmetric microgripper. To solve these problems, in this study, we designed a large-displacement asymmetric microgripper. First, we calculated the relationship between the theoretical input and output variables based on their geometric relationship. Then, we analyzed the performance of the microgripper using finite element software. Lastly, we used a piezoelectric actuator as the input driver of the microgripper. The errors associated with the theoretical and simulated output displacements were 7.05% and9.24%, respectively. At 150 V of driving voltage, the maximum output displacement was 224 μm, and the actual magnification was 11.2 times. Microparts can be gripped in parallel and stably, which confirms the validity of the design.