Mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure

The present article studies the mechanical behavior of a FGM micro-beam subjected to a nonlinear electrostatic pressure. The FGM micro-beam is made of metal and ceramic and material properties vary continuously along the beam thickness according to a power-law. The nonlinear equation of dynamic motion of the FGM micro-beam is derived. By solving the equation of the static deflection, equilibrium positions of the micro-beam are determined and shown in the state control space. To study the stability of the fixed points, the trajectories of the beam motion are illustrated in the phase plane for different initial conditions. In order to find the response of the micro-beam to a step DC applied voltage, the nonlinear equation of motion is solved using a Galerkin based reduced order model. Moreover, time histories and phase portraits for different applied voltages are illustrated. The effect of different power law exponent on the stability of the micro-beam is studied.     

On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure

In the present work, mechanical behavior of a functionally graded cantilever micro-beam subjected to a nonlinear electrostatic pressure and temperature changes has been studied. It has been assumed that the top surface is made of pure metal and the bottom surface from a metal–ceramic mixture. The ceramic constituent percent of the bottom surface varies from 0% to 100%. In addition to the Volume Fractional Rule of material, exponential function has been used for representation of continuous gradation of the material properties through micro-beam thickness. Attention being paid to the ceramic constituent percent of the bottom surface, five different types of FGM micro-beams have been investigated. Nonlinear integro-differential thermo-electro mechanical equation based on Euler–Bernoulli beam theory has been derived and solved using Step-by-Step Linearization Method and Finite Difference Method. The effects of temperature changes and the electrostatic pressure on the deflection and stability of FGM micro-beams having various amounts of the ceramic constituent have been studied and normal stress distributions in the cross section along the beam thickness have been given and compared with a classic metal beam.     

A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates

A meshfree model is presented for the static and dynamic analyses of functionally graded material (FGM) plates based on the radial point interpolation method (PIM). In the present method, the mid-plane of an FGM plate is represented by a set of distributed nodes while the material properties in its thickness direction are computed analytically to take into account their continuous variations from one surface to another. Several examples are successfully analyzed for static deflections, natural frequencies and dynamic responses of FGM plates with different volume fraction exponents and boundary conditions. The convergence rate and accuracy are studied and compared with the finite element method (FEM). The effects of the constituent fraction exponent on static deflection as well as natural frequency are also investigated in detail using different FGM models. Based on the current material gradient, it is found that as the volume fraction exponent increases, the mechanical characteristics of the FGM plate approach those of the pure metal plate blended in the FGM.     

Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment

In the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component E_z. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.     

谐变力作用功能梯度旋转圆板强非线性主共振

功能梯度材料作为一种新型材料,具有良好的力学性能,近年来被广泛关注和应用。该文针对金属-陶瓷功能梯度圆板,考虑周边夹支边界约束条件,选取多项式形式的振型函数,利用伽辽金法,推得旋转运动状态和热效应作用下系统的纵横耦合非线性振动方程,求得由旋转及密度差引起的静挠度项。用改进多尺度法求解方程,得到强非线性系统的频幅响应方程和解析解。通过算例,给出功能梯度圆板的幅频曲线、幅值-激励力曲线、幅值-温度曲线,分析了不同物理量对结构共振幅值的影响规律,并且比较了解析解和数值解,两者结果较为吻合。     

受机械力作用金属／陶瓷功能梯度板主共振奇异性

研究机械力作用下金属，陶瓷功能梯度薄板主共振奇异性问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统主共振幅频响应分岔方程并进行奇异性分析，求得幅频响应分岔方程在开折参数平面的转迁集和分岔图。     

随从力作用下功能梯度矩形板的非线性振动

本文研究了切向均布随从力作用下简支FGM矩形板的非线性振动问题。按照材料组份体积分数的简单幂率分布规律,FGM板的材料常数仅沿厚度连续变化。由大挠度的von Karman理论建立了以应力函数和挠度函数表示的运动偏微分方程组,再由Galerkin法转化成非线性常微分方程。对随从力作用下的四边简支陶瓷/金属矩形板,讨论了随从力、梯度指标和边长比对板的动力特性的影响,得到了各种条件下板中心振幅与非线性基频的关系。     

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams 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 to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

受机械力作用金属／陶瓷功能梯度板的超谐共振研究

研究机械力作用下金属／陶瓷功能梯度薄板3次超谐共振问题．按照功能梯度薄板的非线性动力学方程，得到金属／陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统3次超谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统3次超谐共振幅频响应曲线的影响．     

受机械力作用金属／陶瓷功能梯度板的非线性振动模型

研究机械力作用下金属，陶瓷功能梯度薄板的建模问题。应用弹性理论和Galerkin方法建立小挠度金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。     

受机械力作用金属／陶瓷功能梯度板的亚谐共振研究

研究机械力作用下金属，陶瓷功能梯度薄板1／3次亚谐共振问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统1／3次亚谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统1／3次亚谐共振幅频响应曲线的影响。     

非保守功能梯度材料杆的后屈曲分析

对受切向均布随从力作用的右端可移简支功能梯度材料(FGM)杆,基于轴线可伸长杆的大变形理论建立了非线性控制微分方程组,用打靶法对由金属和陶瓷所构成的FGM杆的后屈曲特性进行了数值分析,给出了不同梯度指标下FGM杆的后屈曲特征曲线,并与金属和陶瓷两种单相材料杆的相应特性进行了比较,讨论了FGM杆长高比对其后屈曲特性的影响。     

功能梯度压杆在均布载荷作用下的后屈曲形态

对受均布载荷作用功能梯度材料(FGM)压杆的屈曲及后屈曲行为进行了分析。基于杆的大变形理论, 考虑杆的轴线伸长, 建立了受均布载荷作用下细长FGM压杆的几何非线性平衡方程, 其中假设FGM杆的性质沿厚度方向按照幂函数连续变化。采用打靶法和解析延拓法数值求解非线性两点边值问题, 获得了一端自由一端固定FGM杆的后屈曲数值解。给出了不同梯度指标下FGM杆的后屈曲特征曲线, 并与金属和陶瓷两种单相材料杆的相应特性进行了比较, 分析和讨论了材料的梯度性质参数对杆变形的影响。结果表明: FGM杆后屈曲行为与各向同性均质杆有很大区别, 梯度指数对杆的屈曲载荷以及后屈曲形态有明显的影响。      

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.     

Static response and free vibration of two-dimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions

The free vibration and static response of a two-dimensional functionally graded (2-D FGM) metal/ceramic open cylindrical shell are analyzed using 2-D generalized differential quadrature method. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. This paper presents a novel 2-D power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multifunctional requirements. Various material profiles in two radial and axial directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori–Tanaka scheme. The 2-D generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the mechanical stresses and natural frequency than conventional 1-D FGM. The achieved results confirm that natural frequency and mechanical stress distribution can be modified to a required manner by selecting an appropriate volume fraction profile in two directions.     

Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM

The effects of three-parameter elastic foundations and thermo-mechanical loading on axisymmetric large deflection response of a simply supported annular FGM plate are investigated. An annular FGM plate, resting on a three-parameter elastic foundation under a transverse uniform loading and a transverse non-uniform temperature, is considered. The mechanical and thermal properties of the FGM plate 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 mathematical modeling of the plate and the resulting nonlinear governing equations of equilibrium are derived based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions. A polynomial-based differential quadrature method is used as a simple but powerful numerical technique to discretize the nonlinear governing equations and to implement the boundary conditions. Finally, the effects of certain parameters, such as nonlinear foundations stiffness, volume fraction index, and temperature, on the axisymmetric large deflection response of the FGM plate are obtained and discussed in detail.     

On the size-dependent behavior of functionally graded micro-beams

In this paper, the size-dependent static and vibration behavior of micro-beams made of functionally graded materials (FGMs) are analytically investigated on the basis of the modified couple stress theory in the elastic range. Functionally graded beams can be considered as inhomogeneous composite structures, with continuously compositional variation from usually a ceramic at the bottom to a metal at the top. The governing equations of motion and boundary conditions are derived on the basis of Hamilton principle. Closed-form solutions for the normalized static deflection and natural frequencies are obtained as a function of the ratio of the beam characteristic size to the internal material length scale parameter and FGM distribution functions of properties. The results show that the static deflection and natural frequencies developed by the modified couple stress theory have a significant difference with those obtained by the classical beam theory when the ratio of the beam characteristic size to the internal material length scale parameter is small.     

Stochastic Finite element analysis of the free vibration of functionally graded material plates

The superior properties of functionally graded materials (FGM) are usually accompanied by randomness in their properties due to difficulties in tailoring the gradients during manufacturing processes. Using the stochastic finite element method (SFEM) proved to be a powerful tool in studying the sensitivity of the static response of FGM plates to uncertainties in their material properties. This tool is yet to be used in studying free vibration of FGM plates. The aim of this work is to use both a First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM), combined with a nine-noded isoparametric Lagrangian element based on the third order shear deformation theory to investigate sensitivity of the fundamental frequency of FGM plates to material uncertainties. These include the effect of uncertainties on both the metal and ceramic constituents. The basic random variables include ceramic and metal Young’s modulus and Poisson’s ratio, their densities and ceramic volume fraction. The developed code utilizes MATLAB capabilities to derive the derivatives of the stiffness and mass matrices symbolically with a considerable reduction in calculation time. Calculating the eigenvectors at the mean values of the variables proves to be a reasonable simplification which significantly increases solution speed. The stochastic finite element code is validated using available data in the literature, in addition to comparisons with results of the well-established Monte Carlo simulation technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration of FGM plates, when compared to the slower Monte Carlo simulation techniques.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches

A generic static and dynamic finite element formulation is derived for the modelling and control of piezoelectric shell laminates under coupled displacement, temperature and electric potential fields. The base shell is of functionally graded material (FGM) type, which consists of combined ceramic–metal materials with different mixing ratios of the ceramic and metal constituents. A multi‐input–multi‐output (MIMO) system is applied to provide active feedback control of the laminated shell using self‐monitoring sensors and self‐controlling actuators through a close loop. Numerical studies clearly show the influence of the positional configurations of sensor/actuator pairs on the effectiveness of static and dynamic control for the shell laminates. The effects of the constituent volume fractions on the static and dynamic responses of the shell laminate are also elucidated. Copyright © 2002 John Wiley & Sons, Ltd.