Dynamic stress analysis of a functionally graded material plate with a circular hole

This paper is to study the two-dimensional dynamic stress of a functionally graded material (FGM) plate with a circular hole under plane compressional waves at infinity. With using the method of piece-wise homogeneous layers, the dynamic stress distribution of the FGM plate having radial arbitrary material parameters is derived based on the complex variable method. As examples, numerical results are presented for the FGM plate having given radial shear modulus, density and Poisson’s ratio. It is found that the dynamic stress around the circular hole in the FGM plate can be effectively reduced by choosing the proper change ways of the radial material parameters for different frequency incident wave.     

The bifurcation analysis on the circular functionally graded plate with combination resonances

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.     

Several three-dimensional solutions for transversely isotropic functionally graded material plate welded with circular inclusion

The problem of a transversely isotropic functionally graded material (FGM) plate welded with a circular inclusion is considered. The analysis starts with the generalized England-Spencer plate theory for transversely isotropic FGM plates, which expresses a three-dimensional (3D) general solution in terms of four analytic functions. Several analytical solutions are then obtained for an infinite FGM plate welded with a circular inclusion and subjected to the loads at infinity. Three different cases are considered, i.e., a rigid circular inclusion fixed in the space, a rigid circular inclusion rotating about the x-, y-, and z-axes, and an elastic circular inclusion with different material constants from the plate itself. The static responses of the plate and/or the inclusion are investigated through numerical examples.     

热环境中功能梯度圆形薄板的混沌运动

针对陶瓷-金属功能梯度圆板,同时考虑几何非线性、材料物性参数随温度变化且材料组分沿厚度方向按幂律分布的情况,应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下,通过引入位移函数,利用伽辽金方法得到了达芬型非线性动力学方程。利用Melnikov方法,给出了热环境中功能梯度圆板可能发生混沌运动的临界条件。通过数值算例,给出了不同体积分数指数和温度的同宿分岔曲线,平面相图和庞加莱映射图,讨论其对临界条件的影响,证实了系统混沌运动的存在。通过分岔图和与其相对应的最大李雅普诺夫指数图,分析了激励频率和激励幅值对倍周期分岔的影响及变化规律,发现系统可出现周期、倍周期和混沌等复杂动力学响应。     

Stress analysis of a functional graded material plate with a circular hole

This paper is to study the two-dimensional stress distribution of a functional graded material plate (FGMP) with a circular hole under arbitrary constant loads. With using the method of piece-wise homogeneous layers, the stress distribution of the functional graded material plate having radial arbitrary elastic properties is derived based on the theory of the complex variable functions. As examples, numerical results are presented for the FGMPs having given radial Young’s modulus or Poisson’s ratio. It is shown that the stress is greatly reduced as the radial Young’s modulus increased, but it is less influenced by the variation of the Poisson’s ratio. Moreover, it is also found that the stress level varies when the radial Young’s modulus increased in different ways. Thus, it can be concluded that the stress around the circular hole in the FGMP can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

弹性地基上多孔功能梯度材料矩形板的自由振动与临界屈曲载荷分析

多孔功能梯度材料(FGM)构件的特性与孔隙率和孔隙分布形式有密切关系。本文基于经典板理论,考虑不同孔隙分布形式时修正的混合率模型,研究Winkler弹性地基上四边受压多孔FGM矩形板的自由振动与临界屈曲载荷特性。首先利用Hamilton原理和物理中面的定义推导Winkler弹性地基上四边受压多孔FGM矩形板自由振动的控制微分方程并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程和边界条件进行变换,得到计算无量纲固有频率和临界屈曲载荷的代数特征方程。将问题退化为孔隙率为零时的FGM矩形板并与已有文献进行对比以验证其有效性。最后计算并分析了梯度指数、孔隙率、地基刚度系数、长宽比、四边受压载荷及边界条件对多孔FGM矩形板无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

The mixed-mode fracture mechanics analysis of an embedded arbitrary oriented crack in a two-dimensional functionally graded material plate

Mixed-mode fracture mechanics analysis of an embedded arbitrarily oriented crack in a two-dimensional functionally graded material using plane elasticity theory is considered. The material properties are assumed to vary exponentially in two planar directions. Then, employing Fourier integral transforms with singular integral equation technique, the problem is solved. The stress intensity factors (SIFs) at the crack tips are calculated under in-plane mechanical loads. Finally, the effects of crack orientation, material non-homogeneity, and other parameters are discussed on the value of SIF in mode I and mode II fracture.     

Nonlinear analysis of a thin circular functionally graded plate and large deflection effects on the forces and moments

The dynamic von Karman equations are used for nonlinear analysis of a thin circular plate made of a functionally graded material. The thickness of the plate is constant and the properties of the functionally graded material depend on temperature and vary throughout the thickness. It is assumed that the plate oscillates with large amplitudes. The forces and moments in the plate are determined in solving the equations for harmonic vibrations. Relevant results are obtained in the case of stead-state free vibrations. These results indicate that the volume fraction has a strong effect on the forces, moments, and material properties Published in Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 134–144, June 2008. An erratum to this article can be found at     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

多孔功能梯度材料Timoshenko梁的自由振动分析

基于Timoshenko梁理论研究多孔功能梯度材料梁(FGMs)的自由振动问题.首先,考虑多孔功能梯度材料梁的孔隙率模型,建立了两种类型的孔隙分布.其次,基于Timoshenko梁变形理论,给出位移场方程、几何方程和本构方程,利用Hamilton原理推导多孔功能梯度材料梁的自由振动控制微分方程,并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,得到含有固有频率的等价代数特征方程.最后,计算了固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)三种不同边界下多孔功能梯度材料梁自由振动的无量纲固有频率.将其退化为均匀材料与已有文献数据结果对照,验证了正确性.讨论了孔隙率、细长比和梯度指数对多孔功能梯度材料梁无量纲固有频率的影响.     

Non-linear analysis of functionally graded circular plates under asymmetric transverse loading

Based on the first-order shear deformation plate theory with von Karman non-linearity, the non-linear axisymmetric and asymmetric behavior of functionally graded circular plates under transverse mechanical loading are investigated. Introducing a stress function and a potential function, the governing equations are uncoupled to form equations describing the interior and edge-zone problems of FG plates. This uncoupling is then used to conveniently present an analytical solution for the non-linear asymmetric deformation of an FG circular plate. A perturbation technique, in conjunction with Fourier series method to model the problem asymmetries, is used to obtain the solution for various clamped and simply supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified by comparison with the existing results in the literature. The effects of non-linearity, material properties, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. It is found that linear analysis is inadequate for analysis of simply supported FG plates which are immovable in radial direction even in the small deflection range. Furthermore, the responses of FG materials under a positive load and a negative load of identical magnitude are not the same. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effect in clamped FG plates being stronger than that in simply supported plates.     

Natural dynamic characteristics of a circular cylindrical Timoshenko tube made of three-directional functionally graded material

The natural dynamic characteristics of a circular cylindrical tube made of three-directional(3 D) functional graded material(FGM) based on the Timoshenko beam theory are investigated. Hamilton’s principle is utilized to derive the novel motion equations of the tube, considering the interactions among the longitudinal, transverse,and rotation deformations. By dint of the differential quadrature method(DQM), the governing equations are discretized to conduct the analysis of natural dynamic charact...     

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。     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

Abstract

The article addresses development of the Transfer Matrix Method (TMM) for free vibration of cracked continuous Timoshenko beam made of Functionally Graded Material (FGM). The governing equations of free vibration are established for the beam based on the power law of material grading, actual position of neutral plane and double spring model of crack. There is conducted frequency equation of the beam with intermediate rigid supports using the TMM after the transverse displacements at rigid supports have been disregarded. Therefore, the frequency equation is simplified and becomes more useful to compute natural frequencies of continuous FGM Timoshenko beam with a number of cracks. The obtained numerical results show the essential effect of cracks, material properties and also number of spans on natural frequencies of the beam.     

Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations

In the present paper, the differential transformation method is employed to develop a semi-analytical solution for free vibration and modal stress analyses of two-dimensional functionally graded circular plates resting on two-parameter elastic foundations. Simultaneous variations of the material properties in the radial and transverse directions are described by a general function. Some comprehensive sensitivity analyses are performed, and the natural frequencies and the modal stresses are extracted for free, simply supported, and clamped boundary conditions and different combinations of the geometric, material, and foundation parameters. Therefore, very complex combinations of the material properties, boundary conditions, and parameters of the elastic foundation are considered in the present semi-analytical solution approach. Thus, many novelties are included in the present research. Comparisons made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. Moreover, the paper treats some interesting problems, for the first time.