Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method

The three-dimensional free vibration analysis of a multi-directional functionally graded piezoelectric(FGP) annular plate resting on two parameter(Pasternak)elastic foundations is investigated under different boundary conditions. The material properties are assumed to vary continuously along the radial and thickness directions and have exponent-law distribution. A semi-analytical approach named the state space based differential quadrature method(SSDQM) is used to provide an analytical solution along the thickness using the state space method(SSM) and an approximate solution along the radial direction using the one-dimensional differential quadrature method(DQM).The influence of the Winkler and shear stiffness of the foundation, the material property graded variations, and the circumferential wave number on the non-dimensional natural frequency of multi-directional FGP annular plates is studied.     

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates of transversely isotropic materials is analyzed based on linear three-dimensional theory of elasticity coupled with magnetic and electric fields. The transverse loads are expanded in Fourier-Bessel series and therefore can be arbitrarily distributed along the radial direction. The radial distributions of the displacements are assumed in combination of Fourier-Bessel series and polynomials as well as the electric potential and magnetic potential. If the material properties obey the exponential law along the thickness of the plate, two three-dimensional exact solutions for two unusual boundary conditions can be derived since they satisfy the governing equations and specified boundary conditions point by point. For simply supported or clamped boundary, the obtained solutions satisfy the governing equations exactly and the boundary conditions approximately. A layer wise model is also introduced to treat with the plates whose material property components vary independently and arbitrarily along the thickness of the plates. The numerical results are finally tabulated and plotted to demonstrate the presented method and agree well with those from finite element methods.     

Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations

Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.     

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material properties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England’s method, the problem can be solved by determining the expressions of four analytic functions. Expanding the transverse load in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.     

功能梯度材料矩形板的三维静动态响应分析

基于线弹性理论的基本方程,选用3个位移分量和3个应力分量作为状态变量,利用状态空间法建立了功能梯度矩形板的三维状态方程. 考虑四边简支的边界条件,采用打靶法数值求解了材料常数沿板厚按幂率变化的弯曲问题和自由振动问题,为求解功能梯度材料三维弹性响应提供了一种方法. 并且给出了功能梯度材料三维矩形板的静动态响应受组分材料分布以及板厚长比变化的影响规律.      

3D thermally induced analysis of annular plates of functionally graded materials

Within the framework of three-dimensional elasticity theory, this paper investigates the thermal response of functionally graded annular plates in which the material can be transversely isotropic and vary along the thickness direction in an arbitrary manner. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of annular plates under a through-thickness steady temperature field. The present analytical solutions are validated through comparisons against those available in open literature. A parametric study is conducted to examine the effects of gradient distribution, different temperature fields, different diameter ratio and boundary conditions on the deformation and stress fields of the plate. The results show that these factors can have obvious effects on the thermo-elastic behavior of functionally gradient materials(FGM)annular plates.     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

Semi-analytical solution for three-dimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges

In this paper free vibration of continuous grading fiber reinforced (CGFR) annular plates on an elastic foundation, based on the three-dimensional theory of elasticity, for different boundary conditions at the circular edges is investigated. The foundation is described by the Pasternak or two-parameter model. The CGFR annular plates have an arbitrary variation of fiber volume fraction in the thickness direction. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. Besides, results for CGFR plate with arbitrary variation of fiber volume fraction in the thickness direction of the plate are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on an elastic foundation and comparison with similar discrete laminated composite plate. The interesting and new results show that non-dimensional natural frequency parameters of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. The new results can be taken as the benchmark solutions for those from numerical methods and future researches.     

Three-dimensional vibrations of annular thick plates with linearly varying thickness

The free vibration of annular thick plates with linearly varying thickness along the radial direction is studied, based on the linear, small strain, three-dimensional (3-D) elasticity theory. Various boundary conditions, symmetrically and asymmetrically linear variations of upper and lower surfaces are considered in the analysis. The well-known Ritz method is used to derive the eigen-value equation. The trigonometric functions in the circumferential direction, the Chebyshev polynomials in the thickness direction, and the Chebyshev polynomials multiplied by the boundary functions in the radial direction are chosen as the trial functions. The present analysis includes full vibration modes, e.g., flexural, thickness-shear, extensive, and torsional. The first eight frequency parameters accurate to at least four significant figures for five vibration categories are obtained. Comparisons of present results for plates having symmetrically linearly varying thickness are made with others based on 2-D classical thin plate theory, 2-D moderate thickness plate theory, and 3-D elasticity theory. The first 35 natural frequencies for plates with asymmetrically linearly varying thickness are compared to the finite element solutions; excellent agreement has been achieved. The asymmetry effect of upper and lower surface variations on the frequency parameters of annular plates is discussed in detail. The first four modes of axisymmetric vibration for completely free circular plates with symmetrically and asymmetrically linearly varying thickness are plotted. The present results for 3-D vibration of annular plates with linearly varying thickness can be taken as benchmark data for validating results from various plate theories and numerical methods.     

功能梯度板条的热弹性力学解

本文利用推广后的Mian 和Spencer 功能梯度板理论,研究了功能梯度板条在非均布温度场作用下的热弹性问题．采用该理论中的位移展开公式,在板厚度方向上考虑热传导引起的稳态温度场,材料常数沿板厚方向可以任意连续变化,从而得到了基于弹性理论的功能梯度板条在温度场作用下的解析解．通过数值算例分析,验证了本文理论的正确性并讨论了边界条件和梯度变化程度对功能梯度板条热弹性响应的影响．     

NONLINEAR DYNAMICS OF A FGM PLATE WITH TWO CLAMPED OPPOSITE EDGES AND TWO FREE EDGES

This paper deals with the nonlinear forced vibration of FGM rectangular plate with a boundary of two edges clamped opposite and the other two free. The plate is subjected to transversal and in-plane excitations. The present research treats the material properties of the FGM plates as temperature-dependent and graded continuously throughout the thickness direction, following the volume fraction of the constituent materials according to the power law. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. In the framework of geometrical nonlinearity the plate is modeled and the equations of motion are obtained on Hamilton＇s principle. With the help of Galerkin discretization, the nonlinear ordinary differential equations describing transverse vibration of the plate are proposed. By the numerical method, the nonlinear dynamical responses of the FGM plate with two clamped opposite and two free edges are analyzed.     

基于物理中面FGM中厚板自由振动的有限元分析

基于物理中面和一阶剪切变形板理论,研究了不同边界条件下功能梯度材料(FGM)中厚板的自由振动问题．假设功能梯度板的材料性质沿厚度方向按幂函数规律连续变化．根据哈密顿原理建立了FGM板有限元形式的自由振动方程,利用MATLAB软件编写程序进行了计算．通过数值算例,讨论了不同边界条件下FGM中厚板的无量纲频率随材料梯度指数和厚宽比的变化情况,并与经典板理论下的频率进行了比较．     

Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.     

功能梯度材料明德林矩形微板的热弹性阻尼

功能梯度材料微板谐振器热弹性阻尼的建模和预测是此类新型谐振器热?弹耦合振动响应的新课题. 本文采用数学分析方法研究了四边简支功能梯度材料中厚度矩形微板的热弹性阻尼. 基于明德林中厚板理论和单向耦合热传导理论建立了材料性质沿着厚度连续变化的功能梯度微板热弹性自由振动控制微分方程. 在上下表面绝热边界条件下采用分层均匀化方法求解变系数热传导方程, 获得了用变形几何量表示的变温场的解析解. 从而将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的偏微分方程. 然后,利用特征值问题在数学上的相似性,求得了四边简支条件下功能梯度材料明德林矩形微板的复频率解析解, 进而利用复频率法获得了反映谐振器热弹性阻尼水平的逆品质因子. 最后, 给出了材料性质沿板厚按幂函数变化的陶瓷?金属组分功能梯度矩形微板的热弹性阻尼数值结果. 定量地分析了横向剪切变形、材料梯度变化以及几何参数对热弹性阻尼的影响规律. 结果表明, 采用明德林板理论预测的热弹性阻尼值小于基尔霍夫板理论的预测结果, 而且两者的差别随着相对厚度的增大而变得显著.      

Dynamic analysis of functionally graded plates using the hybrid Fourier-Laplace transform under thermomechanical loading

In this study, the analytical solution is presented for dynamic response of a simply supported functionally graded rectangular plate subjected to a lateral thermomechanical loading. The first-order and third-order shear deformation theories and the hybrid Fourier-Laplace transform method are used. The material properties of the plate, except Poisson’s ratio, are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The plate is subjected to a heat flux on the bottom surface and convection on the upper surface. A third-order polynomial temperature profile is considered across the plate thickness with unknown constants. The constants are obtained by substituting the profile into the energy equation and applying the Galerkin method. The obtained temperature profile is considered along with the equations of motion. The governing partial differential equations are solved using the finite Fourier transformation method. Using the Laplace transform, the unknown variables are obtained in the Laplace domain. Applying the analytical Laplace inverse method, the solution in the time domain is derived. The computed results for static, free vibration, and dynamic problems are presented for different power law indices for a plate with simply supported boundary conditions. The results are validated with the known data reported in the literature. Furthermore, the results calculated by the analytical Laplace inversion method are compared with those obtained by the numerical Newmark method.     

非均匀复合材料的动态热弹性断裂力学分析

对非均匀复合材料的动态热弹性断裂力学问题进行了研究,假设材料参数沿厚度方向为变化的,沿该方向将复合材料划分为许多单层,取每一单层材料参数为常数,应用Fourier变换法,在Laplace域内推导出了控制问题的奇异积分方程组,给出了热应力强度因子的表达式,然后利用Laplace数值反演,得出了裂纹尖端的动态应力强度因子．本文的方法具有以下特点：（1）多个垂直于厚度方向的裂纹,（2）材料可以为正交各向异性：（3）考虑了惯性效应．作为算例,研究了带有两个裂纹的功能梯度结构,分析了材料参数的变化对应力强度因子的影响．     

3D elasticity solution for static analysis of functionally graded piezoelectric annular plates on elastic foundations using SSDQM

Three-dimensional elasticity solution for static analysis of functionally graded piezoelectric (FGP) annular plates with and without elastic foundations through using state-space based differential quadrature method (SSDQM) at different boundary conditions is presented in this paper. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the mechanical behavior of FGP annular plates. The state variables include a combination of electric potential, electric displacement, three mechanical displacement parameters and three stress parameters. Numerical results are given to demonstrate the convergency and accuracy of the present method. Both closed circuit and open circuit effects are studied and the influences of the Winkler and shearing layer elastic coefficients of the foundations, the material property graded index, radius, thickness, mechanical load and boundary conditions on the deflection response of the FGP annular plates are investigated. The new results can be used as a benchmark solutions for future researches.     

Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach

This paper presents an analytical investigation on the buckling analysis of symmetric sandwich plates with functionally graded material (FGM) face sheets resting on an elastic foundation based on the first-order shear deformation plate theory (FSDT) and subjected to mechanical, thermal and thermo-mechanical loads. The material properties of FGM face sheets 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. The core layer is still homogeneous and made of an isotropic material. An analytical approach is used to reduce the governing equations of stability and then solved using an analytical solution which is named as power series Frobenius method for symmetric sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of the plate aspect ratio, side-to-thickness ratio, loading type, sandwich plate type, volume fraction index, elastic foundation coefficients and boundary conditions on the buckling response of FGM sandwich plates. This has not been done before and serves to fill the gap of knowledge in this area.     

Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case Of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.     

轴对称自由振动功能梯度材料微圆板中的热弹性阻尼

基于经典弹性薄板理论和单向耦合热传导理论,研究了材料性质沿厚度连续变化的功能梯度微圆板的热弹性阻尼特性．首先,考虑热力耦合效应,建立了功能梯度微圆板轴对称横向自由振动微分方程．然后,忽略温度梯度在面内的变化,建立了单向耦合变系数一维热传导方程．采用分层均匀化近似方法,将变系数热传导方程转化为一系列常系数的微分方程,利用上下表面的热边界条件和层间连续性条件获得了微圆板温度场解析解．将所得温度场代入微圆板的自由振动微分方程,得到了包含热弹性阻尼的复频率,从而获得了反映热弹性阻尼水平的逆品质因子．最后,针对材料性质沿板厚按幂函数变化的陶瓷-金属功能梯度微圆板,定量地分析材料梯度指数、几何尺寸、边界条件、温度环境等对微圆板热弹性阻尼的影响．