Elasticity solution for an FGM cylindrical panel integrated with piezoelectric layers

The three-dimensional elasticity solution for static analysis of a functionally graded material (FGM) cylindrical panel with simply supported edges is developed. The modulus of elasticity varies continuously throughout the thickness direction in the form of an exponential function. The panel is bonded with piezoelectric layers. Using Fourier series expansions in the axial and circumferential directions, the state equations are derived. The stress, displacement and electric potential distributions are obtained by solving these state equations. The influences of the material gradient index, applied voltage, and radius to thickness ratio on the static behavior of FGM shell are also studied.     

Modal parameter prediction for structures with resistive loaded piezoelectric devices

Natural frequencies and damping ratios of a structure, with piezo devices bonded on it and shunted with a resistive load, depend on the electrical load itself. Therefore, several tests (experimental or numerical) ought to be carried out in order to determine the resistor which provides the maximum damping ratio for a mode of interest, and in turn the natural frequency of the whole structure. In this paper we present relationships which allow us to predict the modal parameters mentioned above, by using the natural frequencies of the structure when the external electrical circuit of the piezo device is in short or open conditions. Thus, only two tests would be necessary in order to obtain both the maximum damping ratio, introduced by the piezo device, and the natural frequency of the whole system. Besides, under an acceptable approximation, the resistive load, which should be used to obtain the maximum damping, can be obtained from the natural angular frequencies previously derived.     

压电智能结构荷载识别方法的研究

采用压电智能结构实测荷载的输出响应，基于BP神经网络与有限元逆分析提出一种识别荷 载位置及大小的方法. 首先在结构的不同位置施加单位荷载由有限元方法计算得到网络的学 习样本，经网络作逆分析识别荷载位置，继而通过有限元逆逼近方法确定荷载大小的最小二 乘解. 数值算例表明，该方法计算速度快、精度高，不受结构几何形状和边界条件的限制， 用于识别实际压电智能结构不确定荷载的位置及大小是可行的.     

Approximate dynamic boundary conditions for a thin piezoelectric layer

Approximate dynamic boundary conditions of different orders are derived for the case of a thin piezoelectric coating layer bonded to an elastic material. The approximate boundary conditions are derived using series expansions of the elastic displacements and the electric potential in the thickness coordinate of the layer. All the expansion functions are then eliminated with the aid of the equations of motion and boundary/interface conditions of the layer. This results in boundary conditions on the elastic material that may be truncated to different orders in the thickness of the layer to obtain approximate boundary conditions. The approximate boundary conditions may be used as a replacement for the piezoelectric layer and thus simplify the analysis significantly. Numerical examples show that the approximate boundary conditions give good results for low frequencies and/or thin piezoelectric layers.     

Asymptotic analysis of a multimaterial with a thin piezoelectric interphase

We study the electromechanical behavior of a multimaterial constituted by a linear piezoelectric transversely isotropic plate-like body with high rigidity embedded between two generic three-dimensional piezoelectric bodies by means of the asymptotic expansion method. After defining a small real dimensionless parameter \(\varepsilon \) , which will tend to zero, we characterize the limit model and the associated limit problem. We give also a mathematical justification of the model by means of a functional convergence argument. Moreover, we identify the non classical electromechanical transmission conditions between the two three-dimensional bodies.     

An integrated modelling system for coastal area dynamics

Two-dimensional initial-boundary value problems are considered for the shallow water equations and the equation of advection and dispersion of pollutants. The problems are solved in curvilinear boundary fitted co-ordinates. The transformed equations are integrated on a regular grid by the semi-implicit and implicit finite difference methods. Based on the numerical method, the integrated modelling system Cardinal for coastal area dynamics and pollution processes is developed for application on personal computers. Examples of computations are given.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

A smart single-lap adhesive joint integrated with partially distributed piezoelectric patches

The powerful electro-mechanical coupling attribute of piezoelectric materials enables these materials to act as effective actuators. Using this attribute, a smart single-lap adhesive joint was developed by anti-symmetrically surface bonding piezoelectric patches onto a typical single-lap joint. The forces and bending moments at the edges of the developed smart joint can be adaptively controlled by adjusting the applied electric field in the piezoelectric patches, thus reducing the stress concentration in the joint edges. In order to further verify the effect of surface bonding of the piezoelectric patches, a first-order shear deformation theory based analytical model was developed to evaluate the stress distribution in the adhesive layer. It was established that the piezoelectric patched joint could significantly reduce the stress concentration in the joint edges. The influence of location and size of the piezoelectric patches was also investigated. Furthermore, the state-space method was used to obtain the analytical solution. A series of finite element analyses were also carried out to verify the integrity of the developed solution. Results from the computational analyses were in good agreement with those obtained from the proposed results, thus validating the solutions.     

Electro-elastic analysis of fiber-reinforced multilayered cylindrical composites with integrated piezoelectric actuators

Based on the electro-mechanical coupling theory and the laminate elasticity theory, an electro-elastic solution is obtained for the fiber-reinforced cylindrical composites with integrated piezoelectric actuators when subjected to mechanical and electrical loadings. The hybrid composite is composed of three parts: internal piezoelectric actuator, fiber-reinforced laminated interlayer, and external piezoelectric actuator. The general solution in each piezoelectric smart layer is obtained by introducing three undetermined constants, and the general solutions in the fiber-reinforced laminated interlayer are obtained by means of the state-space method. The mechanical behaviors of the hybrid fiber-reinforced cylindrical composites are investigated. The illustrative examples show that the fiber’s angle, the stacking sequence as well as the applied electric loading strongly affect the physical fields in the fiber-reinforced multilayered cylindrical composites.     

含压电材料智能结构动态特性的研究

本文推导了含压电材料智能结构有限元动力方程,讨论了智能结构系统的动态特性,得到了模态形状和相应的模态电压以及考察了具有分布的压电传感器和执行器的四边简支方板的各阶固有频率随以馈增益的变化规律,这些结果将在智能结构控制的优化设计中起到了很大的作用。     

Modelling of thin piezoelectric layers on plates

The derivation of plate equations for a plate consisting of two layers, one anisotropic elastic and one piezoelectric, is considered. Power series expansions in the thickness coordinate for the displacement components and the electric potential lead to recursion relations among the expansion functions. Using these in the boundary and interface conditions, a set of equations are obtained for some of the lowest-order expansion functions. This set is reduced to six equations corresponding to the symmetric (in-plane) and antisymmetric (bending) motions of the elastic layer. These equations are given to linear (for the symmetric equations) or quadratic (for the antisymmetric equations) order in the thickness. It is noted that it is, in principle, possible to go to any order, and that it is believed that the corresponding equations are asymptotically correct. A few numerical results for guided waves along the plate and a 1D actuator case illustrate the accuracy.     

On the modelling of dynamic structures with discontinuities

A method of simulation is developed for studying the dynamics of the structures with discontinuities using Matlab–Simulink. The concept of dynamic compliance is used for modeling the continuous elements of the structure and the local discontinuity is modeled as nonlinear feedback. As an example, a one-dimensional model of cracked bar under harmonic loading is considered for the simulation purpose.     

Accurate modelling of piezoelectric plates: single-layered plate

Summary  The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates. Received 17 July 2000; accepted for publication 26 September 2000     

具有压电材料薄板弯曲控制的有限元法

考虑一均匀各向同性薄板结构,在其上下表面离散分布状压电执行元件,在横向外载和电压共同作用下,分析板的弯曲变形,并通过变化作用于执行元件上的电压对板的变形进行控制,根据逆压电效应将电压转换成作用于板上的等效作用量,用Ｈａｍｉｌｔｏｎ原理导出压电结构板弯曲变形的有限元公式,并对环形板情况进行数值计算。     

Thermo-electro-elastic transient responses in piezoelectric hollow structures

The paper presents an analytical method to solve thermo-electro-elastic transient response in piezoelectric hollow structures subjected to arbitrary thermal shock, sudden mechanical load and electric excitation. Volterra integral equation of the second kind caused by interaction between elastic deformation and electric field is solved by using an interpolation method. Thus, the exact expressions for the transient responses of displacement, stresses, electric displacement and electric potential in the piezoelectric hollow structures are obtained by means of Hankel transform, Laplace transform, and their inverse transforms. In Section 2, based on spherical coordinates, the governing equation of thermo-electro-elastic transient responses in a piezoelectric hollow sphere is found and the associated numerical results are carried out. In Section 3, based on cylindrical coordinates, the governing equation of thermo-electro-elastic transient responses in a non-homogeneous piezoelectric hollow cylinder is found and the corresponding numerical results are carried out. The results carried out may be used as a reference to solve other transient coupled problems of thermo-electro-elasticity in piezoelectric structures.