AXISYMMETRIC BENDING OF TWO-DIRECTIONAL FUNCTIONALLY GRADED CIRCULAR AND ANNULAR PLATES

Assuming the material properties varying with an exponential law both in the thick- ness and radial directions,axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper.The de- flections and stresses of the plates are presented.Numerical results show the well accuracy and convergence of the method.Compared with the finite element method,the semi-analytical nu- merical method is with great advantage in the computational efficiency.Moreover,study on ax- isymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials.Two-directional functionally graded material is a potential alter- native to the one-directional functionally graded material.And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.     

A micromechanical model for predicting the fracture toughness of functionally graded foams

In this paper, finite element method based micromechanical analysis is used to understand the fracture behavior of functionally graded foams. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The fracture toughness is evaluated for various crack positions and length within the functionally graded foam. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. Convergence tests for both macro-mechanical and micro-mechanical model analysis were conducted in order to maintain adequate accuracy with reasonable computational time. Fracture toughness of homogenous foams and functionally graded foams for various cases are presented as a function of relative density. This study indicates that the fracture toughness of functionally graded foams mainly depends on the relative density at the crack-tip.     

功能梯度材料构件三维分析的细观元模型

提出一种新颖的功能梯度构件分析的细观元法,给出了方法模型、基本算式及特点与功能。细观元法对构件的常规有限单元内部设置密集细观单元以反映材料特性梯度变化,又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度,再上机计算。这种细观元法既能充分反映材料功能梯度及组分变化特性,而其计算单元与自由度又与常规有限元一样,是一种针对功能梯度构件分析的有效数值方法。算例表明了细观元法对不同情况下功能梯度构件分析的适应性与精度。     

功能梯度材料板件三维分析的半解析梯度有限元法

将半解析有限元与梯度有限元相结合,形成一种半解析梯度有限元来求解功能梯度材料板件问题。该方法兼有有限元法的适应性强、程序统一,半解析有限元法的节省单元与计算工作量,梯度有限元法的适应构件内部材料性能任意梯度分布等特点,并实现用一维数值计算给出构件三维分析结果。算例分析表明了方法的精度、功能与上述特点,充分揭示了功能梯度材料板件力学响应的三维形态。半解析梯度有限元法可推广应用到其他功能梯度材料面结构的各类分析中。     

Coupled thermoelasticity of functionally graded cylindrical shells

The coupled thermoelastic response of a functionally graded circular cylindrical shell is studied. The coupled thermoelastic and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. A second-order shear deformation shell theory that accounts for the transverse shear strains and rotations is considered. Including the thermo-mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain are used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The results are validated with the known data in the literature.     

沿板平面材料梯度变化功能材料板件分析

本文发展一种功能梯度构件分析的细观元法.细观元法在构件的常规有限元内部设置密集观细单元以反映材料特性变化,又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度,再上机计算.这种细观元法既能充分反映材料功能梯度沿各方向任意变化特性;而其计算单元又和常规有限元一样,是一种针对功能梯度结构分析的有效数值方法.现有功能梯度板件分析中无论对不同形状还是不同边界的功能梯度构件,其材料特性均沿板厚度方向梯度变化,本文用细观元法进行计算与分析,给出了目前尚未得到的沿板平面方向功能梯度变化构件的力学量三维分布形态.     

Static analysis of functionally graded doubly-curved shells and panels of revolution

The Generalized Differential Quadrature (GDQ) Method is applied to study four parameter functionally graded and laminated composite shells and panels of revolution. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT), in particular on the Toorani-Lakis Theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element method. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of functionally graded shell structures made of a mixture of ceramics and metal.     

Transient thermal stresses in two-dimensional functionally graded thick hollow cylinder with finite length

In this paper transient thermal stresses in a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on classical theory of thermoelasticity are considered. The volume fraction distribution of materials, geometry and thermal load are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure. Temperature, displacements and stress distributions through the cylinder at different times are investigated. Also the effects of variation of material distribution in two radial and axial directions on the thermal stress distribution and time responses are studied. The achieved results show that using 2D-FGM leads to a more flexible design so that time responses of structure, maximum amplitude of stresses and uniformity of stress distributions can be modified to a required manner by selecting suitable material distribution profiles in two directions.     

The analysis of functionally graded shallow and non-shallow shell panels with piezoelectric layers under dynamic load and electrostatic excitation based on elasticity

Elasticity solution is presented for finitely long, simply-supported, functionally graded shallow and non-shallow shell panel with two piezoelectric layers under pressure and electrostatic excitation. The functionally graded panel is assumed to be made of many sub panels. Each sub panel is considered as an isotropic layer. Material’s properties in each layer are constant and functionally graded properties are resulted by suitable arrangement of layers in multilayer panel. In each interface between two layers, stress and displacement continuities are satisfied. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients for non-shallow panel and constant coefficients for shallow shell panel by means of trigonometric function expansion in circumferential and longitudinal directions. The resulting ordinary differential equations are solved by Galerkin finite element method and Newmark method is used to march in time. Numerical examples are presented for functionally graded shell panel with a piezoelectric layer as an actuator in external surface and a piezoelectric layer as a sensor in internal surface and the results of the shallow and non-shallow panels are discussed.     

黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.     

Two-dimensional modeling of heterogeneous structures using graded finite element and boundary element methods

In the present work, graded finite element and boundary element methods capable of modeling behaviors of structures made of nonhomogeneous functionally graded materials (FGMs) composed of two constituent phases are presented. A numerical implementation of Somigliana’s identity in two-dimensional displacement fields of the isotropic nonhomogeneous problems is presented using the graded elements. Based on the constitutive and governing equations and the weighted residual technique, effective boundary element formulations are implemented for elastic nonhomogeneous isotropic solid models. Results of the finite element method are derived based on a Rayleigh–Ritz energy formulation. The heterogeneous structures are made of combined ceramic–metal materials, in which the material properties vary continuously along the in-plane or thickness directions according to a power law. To verify the present work, three numerical examples are provided in the paper.     

THREE-DIMENSIONAL STATIC ANALYSES FOR FGM PLATES WITH MEDIUM COMPONENTS AND DIFFERENT NET STRUCTURES

A new microelement method for the analyses of functionally graded structures was proposed. The key of this method is the maneuverable combination of two kinds of elements. Firstly, the macro elements are divided from the functionally graded material structures by the normal finite elements. In order to reflect the functionally graded distributions of materials and the microconstitutions in each macro-element, the microelement method sets up the dense microelements in every macro-element, and translates nodes to the same as the normal finite elements by the degrees of freedom of all microelemental the compatibility conditions. This microelement method can fully reflect the micro constitutions and different components of materials, and its computational elements are the same as the normal finite elements, so it is an effective numerical method for the analyses of the functionally graded material structures. The three-dimensional analyses of functionally graded plates with medium components and different micro net structures are given by using the microelement method in this paper. The differences of the stress contour in the plane of functionally graded plates with different net microstructures are especially given in this paper.     

Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method

The work deals with the development of an effective numerical tool in the form of pseudospectral method for wave propagation analysis in anisotropic and inhomogeneous structures. Chebyshev polynomials are used as basis functions and Chebyshev–Gauss–Lobatto points are used as grid points. The formulation is implemented in the same way as conventional finite element method. The element is tested successfully on a variety of problems involving isotropic, orthotropic and functionally graded material (FGM) structures. The formulation is validated by performing static, free vibration and wave propagation analysis. The accuracy of the element in predicting stresses is compared with conventional finite elements. Free vibration analysis is carried out on composite and FGM beams and the computational resources saved in each case are presented. Wave propagation analysis is carried out using the element on anisotropic and inhomogeneous beams and layer structures. Wave propagation in thin double bounded media over long propagating distances is studied. Finally, a study on scattering of waves due to embedded horizontal and vertical cracks is carried out, where the effectiveness of modulated pulse in detecting small cracks in composites and FGMs has been demonstrated.     

具有不同功能梯度分布函数的板件的三维分析

发展了一种新颖的功能梯度结构分析的细观元法。细观元法对结构的常规有限单元内部设置密集细观单元以反映材料特性梯度变化,又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度,再上机计算。这种细观元法既能充分反映材料功能梯度变化特性;而其计算单元又和常规有限元一样,是一种针对功能梯度结构分析的有效数值方法。文中通过细观元技术进行计算、分析,给出了具有不同功能梯度分布函数板件的力学量三维分布形态。     

Sliding frictional contact between a rigid punch and a laterally graded elastic medium

Analytical and computational methods are developed for contact mechanics analysis of functionally graded materials (FGMs) that possess elastic gradation in the lateral direction. In the analytical formulation, the problem of a laterally graded half-plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered. The governing partial differential equations and the boundary conditions of the problem are satisfied through the use of Fourier transformation. The problem is then reduced to a singular integral equation of the second kind which is solved numerically by using an expansion–collocation technique. Computational studies of the sliding contact problems of laterally graded materials are conducted by means of the finite element method. In the finite element analyses, the laterally graded half-plane is discretized by quadratic finite elements for which the material parameters are specified at the centroids. Flat and triangular punch profiles are considered in the parametric analyses. The comparisons of the results generated by the analytical technique to those computed by the finite element method demonstrate the high level of accuracy attained by both methods. The presented numerical results illustrate the influences of the lateral nonhomogeneity and the coefficient of friction on the contact stresses.     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

功能梯度板动力分析的光滑有限元法研究

为提高功能梯度板动响应问题求解精度,基于一阶剪切变形板理论,提出了求解功能梯度板自由振动问题的Cell-Based 光滑有限元格式．功能梯度板材料属性沿厚度方向呈梯度连续变化,计算系统刚度矩阵时在光滑域内进行光滑梯度操作,可提高求解精度．采用Cell-Based 光滑有限元法,讨论了长厚比、形状因子和边界条件对两种典型功能梯度方板无量纲频率参数的影响,并与FEM法的解和文献中的解做了对比．数值算例的结果表明,光滑梯度操作可改善有限元系统的刚度,Cell-Based 光滑有限元法的计算结果更加逼近真实解,从而为功能梯度材料的进一步应用打下基础．     

热环境下功能梯度夹层双曲壳三维自由振动分析

功能梯度夹层双曲壳结构广泛应用在航空航天、海洋工程等领域中,对于该类结构的动力学特性研究非常重要。本文以热环境下功能梯度夹层双曲壳为研究对象,在三阶剪切变形理论的基础上,考虑横向拉伸作用的影响提出了一种新的位移场,假设材料的物性参数与温度有关,且沿厚度方向表示为幂律函数。利用Hamilton原理得到简支边界条件下功能梯度夹层双曲壳三维振动系统动力学方程,利用Navier法求得两种不同夹层类型的系统固有频率。研究了几何物理参数和温度场对功能梯度夹层双曲壳自由振动固有频率的影响。