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     

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations. The English text was polished by Keren Wang.     

Propagation of Lamb waves in transversely isotropic thermoelastic diffusive plate

The constitutive relations and field equations for anisotropic generalized thermoelastic diffusion are derived and deduced for a particular type of anisotropy, i.e. transverse isotropy. Green and Lindsay (GL) theory, in which, thermodiffusion and thermodiffusion–mechanical relaxations are governed by four different time constants, is selected for study. The propagation of plane harmonic thermoelastic diffusive waves in a homogeneous, transversely isotropic, elastic plate of finite width is studied, in the context of generalized theory of thermoelastic diffusion. According to the characteristic equation, three quasi-longitudinal waves namely, quasi-elastodiffusive (QED-mode), quasi-massdiffusive (QMD-mode) and quasi-thermodiffusive (QTD-mode) can propagate in addition to quasi-transverse waves (QSV-mode) and the purely quasi-transverse motion (QSH-mode), which is not affected by thermal and diffusion vibrations, gets decoupled from the rest of the motion of wave propagation. The secular equations corresponding to the symmetric and skew symmetric modes of the plate are derived. The amplitudes of displacements, temperature change and concentration for symmetric and skew symmetric modes of vibration of plate are computed numerically. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient and amplitudes of wave propagation are presented graphically in order to illustrate and compare the analytically results. Some special cases of frequency equation are also deduced from the existing results.     

黏弹夹芯板的有限元建模及实验研究

从有限元分析和数值模拟及实验验证的角度研究了黏弹夹芯板的频率依赖振动特性。夹芯板中间层为黏弹性材料,其刚度和阻尼的频率依赖性行为直接影响系统的模态频率和阻尼,并导致振动模式求解的复杂化。采用三阶七参数Biot模型描述黏弹性材料频率相关的黏弹性行为。开发了三层四节点28自由度的夹芯板单元,基于经典板理论和哈密顿原理建立了黏弹夹芯板的有限元动力学方程。通过引入辅助耗散坐标,将Biot模型和黏弹夹芯板的有限元动力学模型结合起来,并将其转化为常规二阶线性系统形式,极大简化了求解非线性振动特性的过程。对一边固定、另三边自由的黏弹夹芯板进行了前三阶固有频率和损耗因子的预测,并与实验结果对比。数值模拟结果和实验结果吻合良好,说明所提有限元方法是正确有效的。     

Dynamic analysis of composite plate with multiple delaminations based on higher-order zigzag theory

In a recent paper, Cho and Kim [Journal of Applied Mechanics] proposed a higher-order cubic zigzag theory of laminated composites with multiple delaminations. The proposed theory is not only accurate but also efficient because it work with a minimal number of degrees of freedom with the application of interface continuity conditions as well as bounding surface conditions of transverse shear stresses including delaminated interfaces. In this work, we investigate the dynamic behavior of laminated composite plates with multiple delaminations. A four-node finite element based on the efficient higher-order zigzag plate theory of laminated composite plates with multiple delaminations is developed to refine the prediction of frequencies, mode shape, and time response. Through the dynamic version of the variational approach, the dynamic equilibrium equations and variationally consistent boundary conditions are obtained. Natural frequency prediction and time response analysis of a composite plate with multiple delaminations demonstrate the accuracy and efficiency of the present finite element method. To prevent penetration violation at the delamination interfaces, unilateral contact constraints by Lagrange multiplier method are applied in the time response analysis. The present finite element is suitable for the prediction of dynamic response of thick composite plates with multiple and arbitrary shaped delaminations.     

功能梯度压电圆板自由振动问题的三维精确分析

本文对周边为广义刚性滑动和广义简支两种边界条件下的功能梯度压电材料圆板自由振动问题进行分析。根据轴对称横观各向同性压电材料基本方程,并利用有限Hankel变换得到了功能梯度压电材料圆板的状态空间方程。假设材料的机械和电学性质均沿板厚方向按统一的指数函数形式梯度分布,从而获得了周边为广义刚性滑动和广义弹性简支两种边界条件下功能梯度压电圆板自由振动问题的三维精确频率方程,该方程是一个关于自由振动频率的超越方程,通过求解该超越方程可得到在不同板厚以及不同的材料性质梯度变化情况下的圆板自由振动频率值,结果表明在相同的材料性质梯度变化情况下频率均随着板厚增加而增大,而在相同的板厚情况下频率则随材料性质梯度变化指数的增大而减小的结论。     

网架结构拟夹层板法的有限元验证

用拟夹层板法和有限元法对网架结构进行分析,对三类屋面网架（正放四角锥网架、两向正交正放网架和正放抽空四角锥网架）进行了均布荷载、局部荷载（半跨均布荷载）作用下的静力分析以及固有振动分析,对三类竖向承重网架墙体进行了稳定性分析。通过与有限元法分析结果的对比,表明了拟夹层板法作为一种简化的计算方法,其精度是比较高的,绝大多数的误差都小于5％,是可以直接用于工程结构设计的一种有效方法。此外,拟夹层板法还可作为一种在宏观上检验有限元建模正确与否的实用方法。     

Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading

The forced monoharmonic bending vibrations and dissipative heating of a piezoelectric circular sandwich plate under monoharmonic mechanical and electrical loading are studied. The core layer is passive and viscoelastic. The face layers (actuators) are piezoelectric and oppositely polarized over the thickness. The plate is subjected to harmonic pressure and electrical potential. The viscoelastic behavior of the materials is described by complex moduli dependent on the temperature of heating. The coupled nonlinear problem is solved numerically. A numerical analysis demonstrates that the natural frequency, amplitude of vibrations, mechanical stresses, and temperature of dissipative heating can be controlled by changing the area and thickness of the actuator. It is shown that the temperature dependence of the complex moduli do not affect the electric potential applied to the actuator to compensate for the mechanical stress __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 79–89, January 2008.     

Modified Mindlin plate theory and shear locking-free finite element formulation

The basic equations of the Mindlin theory are specified as starting point for its modification in which total deflection and rotations are split into pure bending deflection and shear deflection with bending angles of rotation, and in-plane shear angles. The equilibrium equations of the former displacement field are split into one partial differential equation for flexural vibrations. In the latter case two differential equations for in-plane shear vibrations are obtained, which are similar to the well-known membrane equations. Rectangular shear locking-free finite element for flexural vibrations is developed. For in-plane shear vibrations ordinary membrane finite elements can be used. Application of the modified Mindlin theory is illustrated in a case of simply supported square plate. Problems are solved analytically and by FEM and the obtained results are compared with the relevant ones available in the literature.     

Forced axisymmetric vibrations of a thick circular rigidly fixed piezoceramic plate

A new closed-form solution of the axisymmetric nonstationary problem of elasticity is constructed for a circular thick piezoceramic plate whose outer cylindrical surface is rigidly fixed. The use of mixed boundary conditions for a curvilinear plane allows one to obtain sufficiently simple computational relations. The closed-form solution is constructed by the method of expansion in the vector eigenfunctions in the form of a structure algorithm of finite transformations. The obtained solutions are used to determine the natural vibration frequency, the stress-strain state of the considered element, and all characteristics of the induced electric field.     

FINITE ELEMENT MODELING AND ROBUST VIBRATION CONTROL OF TWO-HINGED PLATE USING BONDED PIEZOELECTRIC SENSORS AND ACTUATORS

Active vibration control for a kind of two-hinged plate is developed in this paper. A finite element model for the hinged plate integrated with distributed piezoelectric sensors and actuators is derived, including bending and torsional modes of vibration. In this model, the hinges are simplified as regular plate elements to facilitate operation. The state space representations for bending and torsional vibrations are obtained. Based on two low-order models of the bending and torsional motion, two H ∞ robust controllers are designed for suppressing the vibrations of the bending and torsional modes, respectively. The simulation results indicate the effectiveness and feasibility of the designed H ∞ controllers. The vibration magnitudes of the low-order modes can be reduced without affecting the high frequency modes.     

曲线加筋Kirchhoff-Mindlin板自由振动分析

相比传统加筋板,曲线加筋板能够更充分地发挥材料力学性能.在加筋板力学分析中,厚板通常采用Reissner-Mindlin理论,然而当板厚较薄时易出现剪切自锁,离散的Kirchhoff-Mindlin理论采用假设剪切应变场可避免该问题.针对曲线加筋Kirchhoff-Mindlin板自由振动分析,采用离散的Kirchhoff-Mindlin三角形单元和Timoshenko曲梁单元分别模拟板和加强筋,根据板的位移插值函数及筋板交界面的位移协调条件,建立基于板单元位移自由度的有限元方程.为了验证方法的有效性和准确性,采用直线加筋薄板、曲线加筋薄板和厚板3种模型进行算例研究,通过收敛性和精度分析来选择合理的有限元网格密度.直线加筋薄板前20阶固有频率均与文献结果吻合良好;曲线加筋板算例中,本文方法满足收敛条件的板单元数目为2469,Nastran模型板单元数目为6243;本文所得曲线加筋板固有频率与Nastran计算结果最大误差为3.4%.研究结果表明,本文方法无需筋板单元共节点,可使用较少的有限元网格数量,并能够保证计算精度;在离散Kirchhoff-Mindlin三角形板单元基础上构造Timoshenko梁单元可同时适用于曲线加筋薄板与厚板自由振动分析.     

Vibration band gap properties of periodic Mindlin plate structure using the spectral element method

In this paper, vibration band gap properties of periodic Mindlin plate structure with two simply supported opposite edges are studied using the spectral element method (SEM). Based on the Mindlin plate theory (MPT), the spectral Mindlin plate stiffness matrix is deduced. The spectral equation of the whole periodic structure is further established. Frequency responses are calculated by the spectral equations to illustrate band gap properties. Compared with the results based on the classic plate theory (CPT), the effects of the shear and rotary inertia on the band gap properties are analyzed. The results are also compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. Furthermore, the influences of the material properties and structure dimension on the band gap behaviors are investigated.     

Non-linear principal resonance of an orthotropic and magnetoelastic rectangular plate

Based on the von Karman plate theory of large deflection, we have derived a non-linear partial differential equation for the vibration of a thin orthotropic plate under the combined action of a transverse magnetic field and a transverse harmonic mechanical load. The influence of the magnetic field is due to the magnetic Lorentz force induced by the eddy current. By employing the Bubnov-Galerkin method, the non-linear partial differential equation is transformed into a third-order non-linear ordinary differential equation. The amplitude-frequency equations are further derived by means of the multiple-scale method. As numerical examples for an orthotropic plate made of silver, the influence of the magnetic field, orthotropic material property, plate thickness, and the mechanical load on the principal resonance behavior is investigated. The higher-order effect and stability of the solution are also discussed.     

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electro- magnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period- doubling motion and quasi-period motion are discussed.     

Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates

In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have been carried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finite element procedure is adopted across the thickness of the plate such a way that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with displacement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are calculated by post-computation through constitutive equation. The functionally graded material is assumed to be exponential in the thickness direction. The numerical results obtained by the present model are in good agreement with available functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., in press. Green’s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Solids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic properties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition further study has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available in the literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates].     

Unified 2D continuous and reduced order modeling of thermomechanically coupled laminated plate for nonlinear vibrations

A unified formulation of the thermomechanical problem of laminated plates with von Karman nonlinearities, undergoing finite amplitude vibrations, is presented. It integrates mechanical and thermal aspects, by addressing them in parallel via the introduction of generalized 2D variables and equations also for the latter. The formulation virtually embeds a multitude of possible models, resulting from different assumptions about the plate mechanical and thermal configurations. The obtained continuous model is then subjected to a minimum reduction via Galerkin procedure. Some analyses of free and forced nonlinear vibrations under variable mechanical and/or thermal excitations are also carried out, to get some hints on the importance of different thermal aspects associated with membrane and bending dynamics, and on the possibility to catch them via variably simplified models.     

不同理论下广义压电热弹性问题的有限元求解

基于G-L和L-S广义压电热弹性理论研究了无限大厚压电板在上下表面受到条带状热冲击时的广义压电热弹性问题。在时间非常短的情况下，为避免积分变换求解带来的精度丢失，采用有限元方法对问题在时间域进行直接求解，获得压电板在热冲击作用下的温度、位移、应力及电势等，并将结果与经典压电热弹性理论进行比较。结果表明，直接求解方法可以准确描述热在介质中以有限的速度传播。     

Vibration analysis of periodic cellular solids based on an effective couple-stress continuum model

Cellular solids are usually treated as homogeneous continuums with effective properties. Nevertheless, these mechanical properties depend strongly on the ratio of the specimen size to the cell size. These size effects may be accounted for according to preliminary static analysis of effective continuums based on couple-stress theory. In this paper an effective dynamic continuum model, based on couple-stress theory, is proposed to analyze the behavior of free vibrations of periodic cellular solids. In this continuum model, the effective mechanical constants of the effective continuum are deduced by an equivalent energy method. The cellular solid structure is then replaced with the equivalent couple-stress continuum with same overall dimension and shape. Moreover, the finite element formulation of the couple-stress continuum for the generalized eigenvalue analysis is developed to implement the free vibration analysis. The eigenfrequencies of the effective continuum are then obtained via the shear beam theory or the finite element method. A conventional finite element analysis by discretizing each cell of the cellular solids is also carried out to serve as an exact solution. Several structural cases are calculated to demonstrate the accuracy and effectiveness of the proposed continuum model. Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.     

Two-dimensional thermoelastic problem of an infinite magneto-microstretch homogeneous isotropic plate

In this work, the magneto-thermoelastic problem of an infinite microstretch homogeneous isotropic plate placed in a transverse magnetic field is studied in the context of different theories of generalized thermoelasticity. The upper surface of the infinite plate is subjected to a zonal time-dependent heat shock. The problem is investigated by applying finite element method. The solution is obtained by solving finite element governing equations of the problem in time domain directly. The results, including temperature, stresses, displacements, microrotation, microstretch, induced magnetic field, and induced electric field, are presented graphically. Comparison is made in the results predicted by different theories of generalized thermoelasticity, to show that the micropolar effect has a slight influence on the results while the microstretch effect has a great influence on the results. Finally, a parameter study provides an idea about the influence of the respective terms of the theories.