Free axisymmetric vibration of transversely isotropic piezoelectric circular plates

Based on three-dimensional elastic theory of piezoelectric materials, the axisymmetric state space formulation of piezoelectric laminated circular plates is derived. Finite Hankel transforms are used and the boundary variables in free terms are replaced, for two kinds of boundary conditions, to obtain ordinary differential equations with constant coefficients. Regarding the axisymmetric free vibration problem, two exact solutions for two different boundary conditions are found. Discarding piezoelectric effect, the exact solutions for transversely isotropic circular laminates are also obtained through the same procedure. Numerical examples are given and compared with those of Finite Element Method (FEM) .     

压电材料轴对称有限元分析

给出了横观各向同性压电材料轴对称问题有限元法列式,叙述了压电材料有限元法分析中的一些特殊处理步骤,包括无量纲化和特征值问题计算中对电势自由度的凝聚。对压电材料轴对称问题的一些经典静力问题进行有限元计算,同时对压电圆板的轴对称自由振动进行了计算,和解析结果作了比较,二者符合良好。     

压电界面裂纹裂尖强度因子的显式计算公式

对常见横观各向同性压电材料(TIP)中界面裂纹的裂纹面与压电材料的极化方向成任意夹角的一般情况进行了研究,通过推导得到了计算裂尖强度因子的显式外推公式,同时给出了裂纹面与极化方向垂直的典型情况下的外推公式.这些显式计算公式为常见数值方法如有限元法及边界元法在压电材料断裂力学中的应用提供了便利.     

Solutions for transversely isotropic piezoelectric infinite body,semi-infinite body and bimaterial infinite body subjected to uniform ring loading and charge

Based on the fundamental solutions for transversely isotropic piezoelectric materials, the fundamental solutions of axisymmetric problems are derived by integration and explicit expressions for three possible cases of different characteristic roots and multiple roots are all presented. In the case of s₁ ≠ s₂ ≠ s₃ ≠ s₁, based on the Greens functions for semi-infinite piezoelectric body and bimaterial infinite piezoelectric body, the Greens functions for axisymmetric problems of semi-infinite body and bimaterial infinite body are obtained. Taking PZT-4 as an example, numerical computations are conducted by use of the fundamental solutions to axisymmetric problems. Comparison of the calculated results with those of FEM shows good agreement between them.     

轴向载荷作用下压电—弹性层合梁段的二维解析解

本文利用压电弹性介质的二维本构关系，推导出带压电层的弹性梁段受轴力作用的位移和电势分布的解析表达式。为探索压电层的分布感测机理以及验证有限元数值方法提供了参考依据。     

Bi-stable states of initially stressed elastic cylindrical shell structures with two piezoelectric surface layers

A theoretical model is proposed in this paper to predict the bi-stable states of initially stressed cylindrical shell structures attached by surface anisotropic piezoelectric layers.The condition for existence of bi-stability of the shell structural system is presented and analytical expressions for corresponding rolled-up radii of the stable shell are given based on the principle of minimum strain energy.The resulting solution indicates that the shell system may have two stable configurations besides its initial state under a combined action of the actuating electric field and initial stresses characterized by the bending moment.If the piezoelectric layer materials act as only sensor materials without the actuating electric field,initial stresses may produce the bi-stable states,but one corresponding to its initial state.For the shell without initial stresses,the magnitude in the actuating electric field determines the number of the stable states,one or two stable configurations besides the initial state.The theoretical prediction for the bi-stable states is verified by finite element method(FEM) simulation by using the ABAQUS code.     

Finite element analyses of cracks in piezoelectric structures: a survey

Piezoelectric materials have widespread applications in modern technical areas such as mechatronics, smart structures or microsystem technology, where they serve as sensors or actuators. For the assessment of strength and reliability of piezoelectric structures under combined electrical and mechanical loading, the existence of cracklike defects plays an important role. Meanwhile, piezoelectric fracture mechanics has been established quite well, but its application to realistic crack configurations and loading situations in piezoelectric structures requires the use of numerical techniques as finite element methods (FEM) or boundary element methods (BEM). The aim of this paper is to review the state of the art of FEM to compute the coupled electromechanical boundary value problem of cracks in 2D and 3D piezoelectric structures under static and dynamic loading. In order to calculate the relevant fracture parameters very precisely and efficiently, the numerical treatment must account for the singularity of the mechanical and electrical fields at crack tips. The following specialized techniques are presented in detail (1) special singular crack tip elements, (2) determination of intensity factors K _I –K _IV from near tip fields, (3) modified crack closure integral, (4) computation of the electromechanical J-integral, and (5) exploitation of interaction integrals. Special emphasis is devoted to a realistic modeling of the dielectric medium inside the crack, leading to specific electric crack face boundary conditions. The accuracy, efficiency, and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.     

压电薄板屈曲有限元分析及DKQ单元

在机电耦合本构方程基础上,利用Hamilton原理推导了压电薄板屈曲分析的有限元特征方程和机电耦合的内力计算公式,在有限元实现中选择了基于Kirchhoff薄板假定的四边形薄板单元（DKQ单元）,并给出该单元的几何刚度阵及其数值积分方法。在大型通用有限元分析和优化设计软件系统JIFEX中实现了该方法。给出的数值验证了DKQ单元在屈曲分析和压电薄板静力分析中具有较高精度和收敛性,通过机械荷载和电荷载联合作用下的临界荷载计算,表明压电耦合效应能够影响结构的稳定性,可以通过改变外加电压对结构稳定性进行控制。     

Analysis of electric boundary condition effects on crack propagation in piezoelectric ceramics

There are three types of cracks: impermeable crack, permeable crack and conducting crack, with different electric boundary conditions on faces of cracks in piezoelectric ceramics, which poses difficulties in the analysis of piezoelectric fracture problems. In this paper, in contrast to our previous FEM formulation, the numerical analysis is based on the used of exact electric boundary conditions at the crack faces, thus the common assumption of electric impermeability in the FEM analysis is avoided. The crack behavior and elasto-electric fields near a crack tip in a PZT-5 piezoelectric ceramic under mechanical, electrical and coupled mechanical-electrical loads with different electric boundary conditions on crack faces are investigated. It is found that the dielectric medium between the crack faces will reduce the singularity of stress and electric displacement. Furthermore, when the permittivity of the dielectric medium in the crack gap is of the same order as that of the piezoelectric ceramic, the crack becomes a conducting crack, the applied electric field has no effect on the crack propagation. The project supported by the National Natural Science Foundation of China (19672026, 19891180)     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary

In this paper, the problem of a crack embedded in a half-plane piezoelectric solid with traction-induction free boundary is analyzed. A system of singular integral equations is formulated for the materials with general anisotropic piezoelectric properties and for the crack with arbitrary orientation. The kernel functions developed are in complex form for general anisotropic piezoelectric materials and are then specialized to the case of transversely isotropic piezoelectric materials which are in real form. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors for transversely isotropic piezoelectric materials.     

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。     

Theoretical transient analysis and wave propagation of piezoelectric bi-materials

The transient response of piezoelectric bi-materials subjected to a dynamic anti-plane concentrated force or electric charge with perfectly bonded interface is examined in the present study. The problem is solved by using the Laplace transform method and the inverse Laplace transform is evaluated by means of Cagniard’s method. Exact transient full-field solutions of the contribution for each wave are expressed in explicit closed forms. The transient behavior of field quantities is examined in detail by numerical calculations. The existence condition of a propagating surface wave along the interface is discussed in detail. A surface wave can be guided by the interface of two semi-infinite materials in contact if one, at least, of these two materials is piezoelectric. The propagation velocity of the surface wave is explicitly expressed and is found to be less than the lower shear wave velocity of the two materials. The existence of the surface wave for piezoelectric–piezoelectric bi-materials is restricted to the situation that the shear waves of the two piezoelectric materials are very close. The possibility for the existence of the surface wave for piezoelectric–elastic bi-materials is much greater than that of the piezoelectric–piezoelectric bi-materials.     

梁板结构埋入压电片的深度和厚度优化研究

根据梁的弯曲变形理论，以及压电材料的压电效应，对压电复合材料梁结构的应变和应力状态进行了分析。推导了压电材料埋入梁结构时其埋入深度和压电材料本身厚度的优化目标函数。根据优化目标函数，绘制出了在给定的基体材料和压电材料弹性模量比之下，压电材料的驱动力矩随埋入深度和压电材料厚度变化的三维曲面和等高线图，直观地表示出了埋入型压电材料智能结构获得最佳驱动力矩的配置方案。同时也分析了结构和压电材料两者的弹性模量之比对其结果的影响。结果表明，压电材料对于结构的作动力矩与其埋入基体材料的深度、本身厚度以及基体材料和压电材料的弹性模量之比都有着密切的联系。同时将结果也推广到了压电板结构，此时压电材料和基体材料的泊松比时其结果也有一定的影响。     

EXPERIMENTAL INVESTIGATIONS ON DYNAMIC CHARACTERISTICS OF A MULTILAYER PIEZOELECTRIC STACK ACTUATOR

Multilayer piezoelectric stack actuators are widely used in many industrial applications and the investigation on the dynamic behavior of this element is needed. In this study, two optical interferometric techniques called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV) are used to experimentally investigate the vibration characteristics of a single-layer piezoelectric disc and a multilayer piezoelectric stack actuator. These two techniques are full-field measurement for AF-ESPI and point-wise displacement measurement for LDV. Because the clear fringe patterns obtained by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding vibration mode shapes of the piezoelectric disc and the multilayer piezoelectric stack actuator are obtained simultaneously by the AF-ESPI method. Interferometric fringe patterns for both the in-plane and out-of-plane vibration mode shapes are demonstrated. In addition to the proposed two optical techniques, numerical computations based on a commercially available finite element package are presented for comparison with the experimental results. Good agreement between the measured data by experimental methods and the numerical results predicted by FEM is found in resonant frequencies and mode shapes for the single-layer piezoelectric disc. However, some discrepancies are observed for the results obtained by AF-ESPI and impedance analysis for the multilayer piezoelectric stack actuator. A detailed discussion is made to address important issues of this problem.     

带剪切型压电激励器的板的高阶剪切变形理论

利用压电材料固有的正，逆压电效应可以对结构变形和振动进行控制。与外加电场与极化方向平行于板厚度的压电材料的拉伸作动机制相比，外加电场与极化方向垂直的压电材料的剪切作动机制可以在作动器内产生较小的应力，从而降低作动器边界产生分层破坏的危险。本文对于压电材料的剪切作动机制进行研究，应用三阶剪切变形理论建立带剪切型压电激励器的智能层合板模型。采用哈密顿原理导出带剪切型压电激励器的层合板的控制方程。采用空间法得到了各种边界条件组合条件下板的解析解。数值算例对一三层板采用高阶和一阶剪切变形理论进行计算，结果表明两种理论所得的变形曲线很相似。但对于厚度剪切型激励器而言，由于激励器是引起板的剪切变形，而高阶剪切变形理论比一阶剪切变形理论能更好地反映结构的剪切应变能，因此高阶剪切变形理论可以提供板变形的更为精确的解。因此，对于厚度剪切型激励器，剪切变形理论的选取对于板变形结果的好坏有重要的作用。     

Numerical analysis for a crack in piezoelectric material under impact

In this paper, a numerical analysis of impact interfacial fracture for a piezoelectric bimaterial is provided. Starting from the basic equilibrium equation, a dynamic electro-mechanical FEM formulation is briefly presented. Then, the path-independent separated dynamic J integral is extended to piezoelectric bimaterials. Based on the relationship of the path-independent dynamic J integral and the stress and electric displacement intensity factors, the component separation method is used to calculate the stress and electric displacement intensity factors for piezoelectric bimaterials in this finite-element analysis. The response curves of the dynamic J integral, the stress and electric displacement intensity factors are obtained for both homogeneous material (PZT-4 and CdSe) and CdSe/PZT-4 bimaterial. The influences of the piezoelectricity and the electro-mechanical coupling factor on these responses are discussed. The effects of an applied electric field are also discussed.     

不同压电介质界面上的反平面运动裂纹

利用积分变换技术,得到不同压电介质界面上的平面运动裂纹问题的分析解。结果表明应力及电位移强度因子均与界面裂纹扩展速度及材料参数相关,这不同于均匀压电介质中运动裂纹的结论,当两种压电介质完全相同时,本文结果将退化为均匀压电介质中反平面运动裂纹问题的解。     

加热压电纤维复合材料圆板的横向自由振动

基于经典薄板理论和极正交各向异性材料的本构理论,建立了加热压电纤维复合材料圆板的线性振动控制微分方程。采用打靶法分别获得了加热压电纤维复合材料圆板在周边固支和简支情况下,无量纲固有频率随温度和电场强度变化的关系曲线,并分析了压电纤维体积分数、刚度参数、电场强度和温度变化对压电纤维复合材料圆板无量纲固有频率的影响。结果表明,一定体积分数或者电场强度下,压电纤维复合材料圆板的无量纲固有频率都随温度的升高而单调下降;同一温度下,刚度参数越小,无量纲固有频率越低;电场强度越大,无量纲固有频率越高。     

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

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