FUNDAMENTAL SOLUTION BASED GRADED ELEMENT MODEL FOR STEADY-STATE HEAT TRANSFER IN FGM

A novel hybrid graded element model is developed in this paper for investigating thermal behavior of functionally graded materials (FGMs). The model can handle a spatially varying material property field of FGMs. In the proposed approach, a new variational functional is first constructed for generating corresponding finite element model. Then, a graded element is formulated based on two sets of independent temperature fields. One is known as intra-element temperature field defined within the element domain; the other is the so-called frame field defined on the element boundary only. The intra-element temperature field is constructed using the linear combination of fundamental solutions, while the independent frame field is separately used as the boundary interpolation functions of the element to ensure the field continuity over the interelement boundary. Due to the properties of fundamental solutions, the domain integrals appearing in the variational functional can be converted into boundary integrals which can significantly simplify the calculation of generalized element stiffness matrix. The proposed model can simulate the graded material properties naturally due to the use of the graded element in the finite element (FE) model. Moreover, it inherits all the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional FEM and boundary element method (BEM). Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show a good numerical accuracy.     

Dynamic response for functionally graded materials with penny-shaped cracks

This paper provides a method for studying the penny-shaped cracks configuration in functionally graded material(FGM) structures subjected to dynamic or steady loading. It is assumed that the FGMs are transversely isotropic and all the material properties only depend on the axial coordinatez. In the analysis, the elastic region is treated as a number of layers. The material properties are taken to be constants for each layer. By utilizing the Laplace transform and Hankel transform technique, the general solutions for the layers are derived. The dual integral equations are then obtained by introducing the mechanical boundary and layer interface conditions via the flexibility/stiffness matrix approach. The stress intensity factors are computed by solving dual integral equations numerically in Laplace transform domain. The solution in time domain is obtained by utilizing numerical Laplace inverse. The main advantage of the present model is its ability for treating multiple crack configurations in FGMs with arbitrarily distributed and continuously varied material properties by dividing the FGMs into a number of layers with the properties of each layer slightly different from one another. This work was supported by Failure Mechanics Laboratory of State Education Commission and the Post-doctor Research Fund of China.     

功能梯度材料结构分析的半解析数值方法研究

一种典型的半解析数值方法——线法被引入功能梯度材料的结构分析。首先推导了功能梯度材料位移形式的平衡方程和边界条件,然后阐述了线法功能梯度材料结构分析的基本步骤和数值原理。该方法的基本思想是通过有限差分将问题的控制方程半离散为定义在沿梯度方向离散节线上的常微分方程组,然后应用B样条函数Gauss配点法求解该常微分方程组得到问题的解答。为演示线法在功能梯度材料结构分析中的应用,给出了线性梯度和指数梯度功能梯度材料板分别受恒定位移、均匀拉伸载荷和弯曲载荷作用的数值算例。与相应问题解析解和其他数值方法的比较表明,线法的计算结果具有很高的精度,而且不需要任何特殊的考虑就能够有效模拟材料内部物性参数的连续变化,也无需事先选取满足特定条件的待定场函数,是一种非常适合功能梯度材料结构形式和材料特点的半解析数值方法。     

Boundary element analysis of crack problems in functionally graded materials

The present study examines the crack problems in a functionally graded material (FGM) whose upper and bottom surfaces are fully bonded with dissimilar homogeneous materials. A so-called generalized Kelvin solution based boundary element method is used in the numerical examination. The multi-region method and the eight-node traction-singular boundary elements are used for the crack evaluation. The layer discretization technique is utilized to approximate the depth material non-homogeneity of the FGM layer. The proposed method can deal with any depth variations in both the shear modulus and the Poisson ratio of the FGMs. Results of the present analysis are compared very well with the exact analytical solutions available in the literature, which demonstrates that the proposed method can accurately evaluate the stress intensity factors (SIFs) for cracks in FGMs. The paper further evaluates the effect of the functionally graded variations in the Poisson ratio on the stress intensity factors. The paper also assesses the elliptical cracks in the FGM system. The paper presents the influence of both the non-homogeneity and the thickness of the FGM layer on the three SIFs associated with the elliptical cracks.     

Stress analysis of functionally graded rotating discs: analytical and numerical solutions

This study deals with stress analysis of annular rotating discs made of functionally graded materials(FGMs).Elasticity modulus and density of the discs are assumed to vary radially according to a power law function,but the material is of constant Poisson’s ratio.A gradient parameter n is chosen between 0 and 1.0.When n = 0,the disc becomes a homogeneous isotropic material.Tangential and radial stress distributions and displacements on the disc are investigated for various gradient parameters n by means of the diverse elasticity modulus and density by using analytical and numerical solutions.Finally,a homogenous tangential stress distribution and the lowest radial stresses along the radius of a rotating disc are approximately obtained for the gradient parameter n = 1.0 compared with the homogeneous,isotropic case n = 0.This means that a disc made of FGMs has the capability of higher angular rotations compared with the homogeneous isotropic disc.     

Non-Fourier heat conduction in an exponentially graded slab

The present article investigates one-dimensional non-Fourier heat conduction in a functionally graded material by using the differential transformation method. The studied geometry is a finite functionally graded slab, which is initially at a uniform temperature and suddenly experiences a temperature rise at one side, while the other side is kept insulated. A general non-Fourier heat transfer equation related to the functionally graded slab is derived. The problem is solved in the Laplace domain analytically, and the final results in the time domain are obtained by using numerical inversion of the Laplace transform. The obtained results are compared with the exact solution to verify the accuracy of the proposed method, which shows excellent agreement.     

A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials

A robust unstructured control-volume finite element method is presented for the solution of two-dimensional transient heat conduction in functionally graded materials (FGMs) with isotropic properties. The material properties at a point in the domain vary exponentially to spatial coordinates. A triangular mesh is chosen for spatial discretization and a fully implicit scheme is adapted for time discretization. Several problems are investigated and the results are successfully validated by using analytical and other numerical solutions available in the literature.     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials

The application of the method of fundamental solutions to the Cauchy problem for steady-state heat conduction in two-dimensional functionally graded materials (FGMs) is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, regularization is required in order to solve this system of equations in a stable manner. This is achieved by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.     

Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic–plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.     

动态载荷下功能梯度复合材料的圆币形裂纹问题

研究了动态载荷下功能梯度材料中的圆币形裂纹问题．假设材料为横观各向同性,并且含有多个垂直于厚度方向的裂纹,材料参数沿轴向（与裂纹面垂直的方向）为变化的,沿该方向将材料划分为许多单层,各单层材料参数为常数,利用Hankel变换祛,在Laplace域内推导出了控制问题的对偶积分方程组．利用Laplace数值反演,得出了裂纹尖端的动态应力强度因子和能量释放率．研究了含两个裂纹的功能梯度接头结构,分析了材料非均匀性参数对应力强度因子和能量释放率的影响．     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

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.     

Parameters controlling fracture resistance in functionally graded materials under mode I loading

In this investigation the fracture behavior of functionally graded materials (FGMs) was studied by means of experiments carried out on model polymer-based FGMs. Model graded materials were manufactured by selective ultraviolet irradiation of ECO [poly(ethylene carbon monoxide)], a photo-sensitive ductile copolymer that becomes more brittle and stiffer under exposure to ultraviolet light. The mechanical response of the graded material was characterized using uniaxial tensile tests. Single edge notched tension graded ECO specimens possessing different spatial variations of Young’s modulus, failure stress and failure strain were tested under remote opening loading. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material. The measured displacement field was then used to extract fracture parameters such as stress intensity factor and T-stress, and thus construct resistance curves for crack growth in the FGMs. For this loading configuration it was found that the nonsingular T-stress term in the asymptotic expansion for stresses needs to be accounted for in order to accurately measure the fracture resistance in FGMs. In addition, the influence of local failure properties (i.e., failure stress and failure strain) on crack growth resistance was investigated in detail. It was found that depending on the combined effects of the spatial variation of these two failure parameters, regardless of the spatial variation of the Young’s modulus, the FGM fracture resistance can either increase, decrease or remain constant with continued crack growth.     

求解动荷载作用下多层粘弹性半空间轴对称问题的精确刚度矩阵法

利用Laplace—Hankel联合积分变换，推导出了单层粘弹性半空间轴对称问题在动荷载作用下，层间完全接触情况的刚度矩阵，然后按传统的有限元方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程就可解出动荷载作用下多层粘弹性半空间轴对称问题的矩阵。由于刚度矩阵的元素中只含有负指数项，计算时不会出现溢出的现象。本文还成功地应用了Durbin的Laplace逆变换的数值方法，求解出了多层粘弹性体的时域解。最后，文中还给出了路面动弯沉的计算结果与实测结果的对比来证明推导结果的准确性。     

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.     

Coupled thermal/mechanical analysis for the fracture of functionally graded materials under transient thermal loading

A comprehensive treatment of fracture of functionally graded materials (FGMs) is provided. It is assumed that the material properties depend only on the coordinate perpendicular to the crack surfaces and vary continuously along the crack faces. By using a laminated composite plate model to simulate the material non-homogeneity, an algorithm for solving the system based on Laplace transform and Fourier transform techniques is presented. Unlike earlier studies that considered certain assumed property distributions and a single crack problem, the current investigation studies multiple crack problem in the FGMs with arbitrarily varying material properties. Transient thermal stresses are presented. Project supported by the National Natural Science Foundation of China (Nos 10102004 and 19902003).     

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials(FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement,strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.     

The elastic–viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties

This paper revisits the elastic–viscoelastic correspondence principle for non-homogeneous materials. Several recent publications discussed this principle for functionally graded materials (FGMs) with time translation invariant viscoelastic properties. It was demonstrated that the correspondence principle is valid only for the FGMs with separable relaxation moduli (moduli in separable form in space and time). This paper reconsiders this issue. It shows that the correspondence principle is valid even for non-homogeneous materials with separable relaxation moduli even if the time-dependences of the relaxation moduli in shear and dilatation are not necessarily time translation invariant. The property of similarity of Volterra operators is used to obtain the corresponding elastic solution. The correspondence is established between the elastic solution and the operator-transformed viscoelastic solution. The transformation operators are combinations of the Laplace transform operator and additional integral operators.