Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory

Here, free vibration analysis of functionally graded piezoelectric (FGP) plates with porosities is carried out based on refined four-unknown plate theory. The present plate theory captures shear deformation impacts needless of shear correction factor. A modified power-law model is adopted to describe the graded material properties of a functionally graded piezoelectric plate. Implementing an analytical approach, which satisfies different boundary conditions, governing equations derived from Hamilton's principle are solved. The obtained results are compared with those provided in the literature. The impacts of applied voltage, porosity distribution, material graduation, plate geometrical parameters, and boundary conditions on vibration of porous FGP plate are discussed.     

Buckling analysis of a porous circular plate with piezoelectric sensor–actuator layers under uniform radial compression

This study presents the buckling analysis of a solid circular plate made of porous material bounded with piezoelectric sensor–actuator patches. The porous material properties vary through the thickness direction of the plate following a given function. The general mechanical nonlinear equilibrium and linear stability equations are derived using the variational formulations to obtain the governing equations of the piezoelectric porous plate. The buckling load is derived for solid circular plates under uniform radial compressive loading for the clamped edge condition. The effects of piezoelectric layers on the buckling load of the plate, piezoelectric layer-to-porous plate thickness ratio, feedback gain, and variation of porosity are investigated. The results are verified with the known results in the literature.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

基于压电传感器获取任意边界板结构体积速度

介绍了一种设计体积速度传感器的新方法。设计策略为将PVDF压电薄膜黏结到结构表面上,两维振动结构的表面振速用Legendre多项式展开,利用两维分布式传感器压电方程,设计PVDF薄膜的形状,使传感器的输出电荷量正比于两维结构的体积速度。这样设计得到的传感器,其输出信号的实时性得到了保证,并且适用于任意边界条件,能测量振动结构的局部体积速度,拓宽了其应用范围。此外,还以简支板及其局部区域为例,通过数值计算,验证了设计方法的可行性。     

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.     

Thin-Film Piezoelectric Actuators of Nonuniform Thickness and Nonhomogeneous Material Properties for Modulating Actuation Stress

The effects of the thickness variation and the material property variation of thin-film piezoelectric actuators on the actuation shear stress when the actuators are attached to an elastic plate are studied. A system of 2D equations for the flexure and shear of an elastic plate with symmetric piezoelectric actuators on the plate surfaces is derived. The equations are reduced to the case of elementary flexure without shear as a special case. The effects of the actuator thickness variation and material property variation on the actuation stress are examined using the equations obtained. It is shown that the distribution of the actuation stress depends on the thickness and material property variations of the actuators, and that actuators with varying thickness or varying material properties can be used to make modal actuators for producing a particular deformation or exciting a particular vibration mode.     

An asymptotic theory for vibrations of inhomogeneous/laminated piezoelectric plates

An asymptotic theory for the vibration analysis of inhomogeneous monoclinic piezoelectric plates is developed by using small parameter expansion. The theory includes the important special case of a laminated plate in which each layer is homogeneous and orthotropic, but distinct from the other layers by having a different material or a different orientation. A hierarchy of two-dimensional equations of the same homogeneous operator for each order is reduced from the three-dimensional framework of linear piezoelectricity. The elasticity version of the leading-order equation is the same as that of the classical Kirchhoff inhomogeneous plate theory and, therefore, is easily solvable. By contrast, it is not straightforward to find solutions of the higher-order equations. The solvability condition is thus established for this purpose, by which higher-order frequency parameters are derived. The present theoretical formulation is examined by comparing the present asymptotic results with an exact three-dimensional solution for a piezoelectric bimorph strip, and excellent agreement is reached. Some new results are presented.     

机电耦合载荷下的压电层合板瞬态响应分析

针对压电层合板在机电耦合激振下的瞬态响应问题, 提出一种高效混合数值计算方法。经过位移场、 电势场在厚度方向的离散, 利用机电耦合理论和哈密顿原理, 推导出结构的运动方程。引入傅里叶变换, 得到波数域内运动控制方程。应用模态分析方法求解波数域内的位移场和电势场, 对结果进行傅里叶逆变换, 得到空间域内的瞬态响应。以PZT-5A/0° PVDF铺层两相材料复合压电层合板为算例, 分析了力、 电耦合线载荷激励下, 位移场和电势场的瞬态响应历程与分布规律, 计算结果给出了该结构的动力学基本特征。该方法结合了有限元法、 傅里叶变换和模态分析法, 计算高频载荷激振下的压电层合板瞬态响应较一般有限元法大幅减少了单元的划分。该方法可推广至分析任意机电载荷下的各类铺层材料压电层合板瞬态响应问题。      

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.     

Nonlinear thermal bending response of FGM plates due to heat conduction

Nonlinear thermal bending analysis is presented for a simply supported, shear deformable functionally graded plate without or with piezoelectric actuators subjected to the combined action of thermal and electrical loads. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations of an FGM plate are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. A two step perturbation technique is employed to determine the thermal load–deflection and thermal load–bending moment curves. The numerical illustrations concern nonlinear bending response of FGM plates without or with surface bonded piezoelectric actuators due to heat conduction and under different sets of electric loading conditions. The results reveal that for the case of heat conduction the nonlinear thermal bending responses are quite different to those of FGM plates subjected to transverse mechanical loads, and the temperature-dependency of FGMs could not be neglected in the thermal bending analysis.     

Bending analysis of functionally graded piezoelectric plates via quasi-3D trigonometric theory

Abstract

This work is devoted to the bending analysis of functionally graded piezoelectric (FGP) material plate by using a simple quasi-3D sinusoidal shear deformation theory under simply supported edge conditions. The governing equations and boundary conditions are derived by using the principle of virtual work. The impact of piezoelectric, electric loading, and gradient index on the displacement, electric displacement, electric potential, and stresses are explored numerically presented and discussed in detail. To check the accuracy and validity of bending results obtained from this analysis of FGP plates, results are compared with the analytical solution obtained by 3D, quasi-3D, and higher order shear deformation theories. Parametric studies are then performed to examine the effects of the thickness of the plate and the electric field on the overall electro-mechanical response of the FGP plates. The presented results are useful in design processes of smart structures and analysis from piezoelectric materials.     

A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators

Compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise are presented for a simply supported, shear deformable functionally graded plate with piezoelectric fiber reinforced composite (PFRC) actuators. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and PFRC layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. A two step perturbation technique is employed to determine buckling loads (temperature) and postbuckling equilibrium paths. The numerical illustrations concern the compressive and thermal postbuckling behaviors of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded PFRC actuators under different sets of thermal and electric loading conditions. The results for monolithic piezoelectric actuator, which is a special case in the present study, are compared with those of PFRC actuators. The results reveal that, in the compressive buckling case, the applied voltage usually has a small effect on the postbuckling load–deflection curves of the plate with PFRC actuators, whereas in the thermal buckling case, the effect of applied voltage is more pronounced for the plate with PFRC actuators, compared to the results of the same plate with monolithic piezoelectric actuators.     

通过PVDF传感器测量振动板结构的声辐射模态伴随系数

基于声辐射模态理论进行ASAC控制系统中的一个重要内容就是获得声辐射模态伴随系数。以简支板为例,设计了一种新的PVDF(POLYVINYLIDENE FLUORIDE聚偏氟乙烯)压电传感器,用来测量板的前3阶声辐射模态伴随系数。由仿真计算结果表明,在中低频率,测量第一阶声辐射模态伴随系数时,只需要在板布置两条PVDF传感器即可;测量第二、三阶声辐射模态伴随系数时,可在板上布置多条传感器能得到比较精确的结果。同时PVDF传感器测量的结果与理论相符,说明这种新型PVDF压电传感器的设计是可行的。     

Magnetic field-induced strain and magnetoelectric effects in sandwich composite of ferromagnetic shape memory Ni-Mn-Ga crystal and piezoelectric PVDF polymer

A sandwich composite consisting of one layer of ferromagnetic shape memory Ni-Mn-Ga crystal plate bonded between two layers of piezoelectric PVDF polymer film was fabricated, and its magnetic field-induced strain (MFIS) and magnetoelectric (ME) effects were investigated, together with a monolithic Ni-Mn-Ga crystal, as functions of magnetic fields and mechanical load. The load-free dc- and ac-MFISs were 0.35 and 0.05% in the composite, and 5.6 and 0.3% in the monolithic crystal, respectively. The relatively smaller load-free MFISs in the composite than the monolithic crystal resulted from the clamping of martensitic twin-boundary motion in the Ni-Mn-Ga plate by the PVDF films. The largest ME coefficient (α(E)) was 0.58 V/cm·Oe at a magnetic bias field (H(Bias)) of 8.35 kOe under load-free condition. The mechanism of the ME effect originated from the mechanically mediated MFIS effect in the Ni-Mn-Ga plate and piezoelectric effect in the PVDF films. The measured α(E)-H(Bias) responses under different loads showed good agreement with the model prediction.     

A closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature-dependent properties

A thermal buckling analysis is presented for functionally graded rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The temperature-dependent material properties of the functionally graded plate are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the third-order shear deformation plate theory. Results for the critical buckling temperatures are obtained in closed-form solution, which are convenient to be used in engineering design applications. The effects of the applied actuator voltage, plate geometry, and volume fraction exponent of the functionally graded material on the buckling temperature are investigated.     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

功能梯度压电双材料板中厚度-扭曲波的传播

分析了厚度-扭曲波在无限大功能梯度压电双材料板中的传播性能,板的上表面和下表面是机械自由和电学开路的,材料常数在厚度方向按指数规律变化。首先推导了满足控制方程和边界条件的电弹场,然后利用界面条件得到了厚度-扭曲波传播所应满足的关系。通过算例表明了材料梯度变化对厚度-扭曲波传播性能的影响,结果对功能梯度压电材料在声波器件中应用有参考价值。     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.