Axisymmetric response of circular plates with piezoelectric layers: an exact solution

Electromechanical responses of symmetric circular laminates consisting of piezoelectric layers are studied, and the influence of surface and interlayer electrodes are involved. The laminates are traction-free on the top and bottom surfaces, but may be subjected to external forces at the lateral edge and to voltages applied across certain layers. Under axisymmetric deformation conditions, an approximate model which employs Kirchhoff hypothesis and incorporates the charge equation of electrostatic is established. Then, a closed-form three-dimensional solution of the laminates is generated in a very straightforward manner by the solution of the approximate model. The three-dimensional solution fulfills all field equations and interface or surface conditions as well as the specified electric edge boundary conditions; the only restriction is that the mechanical edge boundary conditions are satisfied in an average manner, rather than point by point. Thus, according to Saint-Venant's principle the proposed solution is exact in the interior region of the laminates.     

Investigating the thermal environment effects on geometrically nonlinear vibration of smart functionally graded plates

An analytical solution for a sandwich circular FGM plate coupled with piezoelectric layers under one-dimensional heat conduction is presented. All materials of the device may be of any functional gradients in the direction of thickness. The solution exactly satisfies all the equilibrium conditions and continuity conditions for the stress, displacement and electric displacement as well as electric potential on the interfaces between adjacency layers. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the FG plate that is subjected to in-plane forces and applied actuator voltage in thermal environment in the case of simply supported boundary conditions. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectric coupled FGM plates are derived. The role of thermal environment as well as control effects on nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators using high input voltages are investigated. Numerical examples are provided and simulation results are discussed. Numerical results for FGM plates with a mixture of metal and ceramic are presented in dimensionless forms. The good agreement between the results of this paper and those of the finite element (FE) analyses validated the presented approach. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage and thermal environment as well as gradient index of FG plate on the dynamics and control characteristics of the structure.     

Boundary conditions for axisymmetric piezoelectric cylinder

For axisymmetric piezoelectric cylinder, the reciprocal theorem and the axisymmetric general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the cylinder of general edge geometry and loadings. A decay analysis technique developed by Gregory and Wan is converted into necessary conditions on the end-data of axisymmetric piezoelectric circular cylinder, and the rapidly decaying solution is established. The prescribed end-data of the circle cylinder must satisfy these conditions in order that they could generate a decaying state within the cylinder. When stress and mixed conditions are imposed on the end of cylinder, these decaying state conditions for the case of axisymmetric deformation of piezoelectric cylinder are derived explicitly. They are then used for the correct formulation of boundary conditions for the theory solution (or the interior solution) of axisymmetric piezoelectric cylinder. The results of the present paper enable us to establish a set of correct boundary conditions, most of which are obtained for the first time.     

确定复合材料对称层板层间应力的边界层近似计算法

提出了一种有效的确定复合材料对称层板自由边附近层间应力分布的边界层近似法。所确定的应力场满足０阶边界层控制方程和自由边上的应力边界条件以及层间界面上的应力连续条件。对几种复合材料层板的分析表明，本文提出的方法可以较好地估算工程应用中几层至几十层的复合材料对称层板听层间应力分布。     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

An analytical solution for free vibration analysis of circular plates in axisymmetric modes based on the two variables refined plate theory

This paper presents, for the first time, an analytical solution for free vibrations of an isotropic circular plate in axisymmetric modes based on the two variables refined plate theory. This theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Governing equations are derived using Hamilton’s principle and an analytical method on the basis of using Bessel functions is introduced to solve them. By this procedure, final form of the governing equations is obtained in matrix form. These equations are solved for classical boundary conditions and comparison studies are performed to verify the validity of the present results. It is found that the results obtained using RPT and TSDT are close to each other. As a benchmark, numerical results are presented in a dimensionless form for various values of thickness to radius ratio.     

Interlaminar stress analysis of general composite laminates

In this study, based on the reduced from of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates under the extension and bending. The constant parameters, which describe the global deformation of a laminate, are properly computed by means of the improved first-order shear deformation theory. Reddy's layerwise theory is subsequently utilized for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. A variety of numerical results are obtained for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates on the boundary-layer stress are examined.     

完全接触条件下轴孔过盈配合的解析解

在过盈配合的设计标准中,将空间问题处理成平面应力问题,必然会产生误差.建立弹性力学轴孔过盈配合空间轴对称模型,采用完全接触条件,用级数法推导了满足边界条件和完全接触条件的空间轴对称问题的Papkovich-Neuber解.得到轴孔过盈配合的解析解,所得的解析解可以作为过盈配合设计的标准.所得的解析解中含有贝塞尔函数,可...     

Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers

In this paper, a free vibration analysis of moderately thick circular functionally graded (FG) plate integrated with two thin piezoelectric (PZT4) layers is presented based on Mindlin plate theory. The material properties of the FG core plate are assumed to be graded in the thickness direction, while the distribution of electric potential field along the thickness of piezoelectric layers is simulated by sinusoidal function. The differential equations of motion are solved analytically for two boundary conditions of the plate: clamped edge and simply supported edge. The analytical solution is validated by comparing the obtained resonant frequencies with those of an isotropic host plate. The emphasis is placed on investigating the effect of varying the gradient index of FG plate on the free vibration characteristics of the structure. Good agreement between the results of this paper and those of the finite element analyses validated the presented approach.     

Analytical solution of a two variable refined plate theory for bending analysis of orthotropic Levy-type plates

This paper presents analytical solutions of deflection and stress for orthotropic plates using a two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factor. Additional features of the theory are that it has strong similarity with classical plate theory in many aspects, and the number of involved variables is only two as against three in case of other shear deformation theories. The Levy-type solution procedure in conjunction with the state space concept is used to determine the closed-form solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are performed to verify the validity of the present results. Finally, the effects of thickness ratio, modulus ratio and aspect ratio on the deflection and stress of orthotropic plates are investigated and discussed.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

轴向预压缩双晶片动力学模型与特性分析

基于欧拉梁的假设,将双晶片简化为3层压电层合梁,由Hamilton原理建立系统的动力学模型,得到描述轴向预压缩压电双晶片的偏微分方程及特定边界条件,通过求解上述微分方程的边值问题得到双晶片在不同轴向力下的输出特性的解析表达式。利用有限元软件对双晶片进行仿真,验证了理论分析的正确性,增加理论分析结果的可信度。最后通过实验的方法研究双晶片的静、动力学特性,证明了施加轴向力可以降低双晶片的弯曲刚度和固有频率,显著增加其机电耦合效率,提高力和位移输出能力。轴向力虽然会影响双晶片启动时的位移峰值,但是对响应时间和带宽影响较小,大轴向力下双晶片仍具有带宽高与响应快的优势。     

基于压电悬臂梁的驱动器与传感器性能分析的精确解析模型

三层压电梁结构在电场作用下发生变形后会产生诱发电势,进而改变材料整体电势分布,本文考虑此变形和电势耦合效应,基于欧拉-伯努利梁变形理论,推导出能够准确预测压电智能悬臂梁传感器与驱动器性能的解析表达式。考虑压电梁结构弯曲变形后产生的电场影响,建立了三层压电梁结构的控制方程;建立了压电梁作为驱动器时端部输出位移、驱动力矩与输入电压之间联系的解析表达式,以及作为传感器时输出电压与端部作用力之间联系的解析表达式。通过与ANSYS有限元模拟结果以及传统的驱动器和传感器性能表达式的对比,验证了所推导的解析表达式的准确性。     

Free vibration of laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

Nonlinear viscoelastic analysis of thick-walled cylindrical composite pipes

This paper analyzes the effect of the polymer matrix non-viscoelastic behaviour in the mechanical behaviour of thick multilayered cylinders. The original contribution of this work is to provide novel approximate analytical solutions to compute the time-dependent internal stress state through the pipe thickness within the framework of nonlinear viscoelasticity theory. The structures considered are thick, multilayered anisotropic infinitive long cylinders subjected to axisymmetric mechanical loading. Under such conditions there is an exact elastic solution which naturally satisfies equilibrium, strain-displacement, compatibility and boundary conditions for the stated constitutive equations and loading. Due to the continuous stress variations through the cylinder thickness, the proposed nonlinear viscoelastic solution assumes the averaged stress state to calculate the nonlinear elastic and viscoelastic factors in each layer. Furthermore, the solution is obtained assuming that the creep strains, within each layer, are constant through the thickness. The proposed algorithm converges to the exact solution when the number of layers is artificially increased. For the linear viscoelastic case, the proposed solution proved to match the exact known solution for isotropic viscoelastic materials. Finally several invented cases are run to illustrate the importance of the viscoelasticity phenomenon on the internal stress field in thick-laminated cylinders.     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

Dynamic analysis of smart composite beams by using the frequency-domain spectral element method

To excite or measure the dynamic responses of a laminated composite structure for the active controls of vibrations or noises, wafer-type piezoelectric transducers are often bonded on the surface of the composite structure to form a multi-layer smart composite structure. Thus, for such smart composite structures, it is very important to develop and use a very reliable mathematical and/or computational model for predicting accurate dynamic characteristics. In this paper, the axial-bending coupled equations of motion and boundary conditions are derived for two-layer smart composite beams by using the Hamilton??s principle with Lagrange multipliers. The spectral element model is then formulated in the frequency domain by using the variation approach. Through some numerical examples, the extremely high accuracy of the present spectral element model is verified by comparing with the solutions by the conventional finite element model provided in this paper. The effects of the lay-up of composite laminates and surface-bonded wafer-type piezoelectric (PZT) layer on the dynamics and wave characteristics of smart composite beams are investigated. The effective constraint forces at the interface between the base beam and PZT layer are also investigated via Lagrange multipliers.     

Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.     

Modelling of cross-ply piezoelectric composite laminates in cylindrical bending with interfacial shear slip

An approximate theory for cross-ply piezoelectric composite laminates in cylindrical bending with interfacial shear slip is developed. This theory uses only 4 displacement and potential variables, the number of which is independent of the number of layers involved. The displacement and electric potential fields are depicted by the displacement and electric potential distribution functions through thickness, respectively. The two functions are formulated according to particular solutions to the three-dimensional elasticity equilibrium equations and electrostatics charge equation. In this shear slip modelling interfacial opening is neglected. The interfacial bonding conditions are characterised by a linear slip law and an electrically permeable assumption. A corresponding finite element is also developed to deal with piezoelectric laminates with local shear slip. The accuracy and effectiveness of the present theory are demonstrated in numerical examples.