The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane.The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.     

Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.     

Corner stress singularities in an FGM thin plate

Describing the behaviors of stress singularities correctly is essential for obtaining accurate numerical solutions of complicated problems with stress singularities. This analysis derives asymptotic solutions for functionally graded material (FGM) thin plates with geometrically induced stress singularities. The classical thin plate theory is used to establish the equilibrium equations for FGM thin plates. It is assumed that the Young’s modulus varies along the thickness and Poisson’s ratio is constant. The eigenfunction expansion method is employed to the equilibrium equations in terms of displacement components for an asymptotic analysis in the vicinity of a sharp corner. The characteristic equations for determining the stress singularity order at the corner vertex and the corresponding corner functions are explicitly given for different combinations of boundary conditions along the radial edges forming the sharp corner. The non-homogeneous elasticity properties are present only in the characteristic equations corresponding to boundary conditions involving simple support. Finally, the effects of material non-homogeneity following a power law on the stress singularity orders are thoroughly examined by showing the minimum real values of the roots of the characteristic equations varying with the material properties and vertex angle.     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.     

夹层FGM圆柱壳在扭转载荷作用下的弹性稳定性

采用半解析方法研究了两端简支的功能梯度夹层圆柱壳在端部扭转载荷作用下的弹性稳定性.考虑圆柱壳的里外表层为均匀材料,中间层为材料性质沿厚度方向连续变化的功能梯度材料,并且在界面处的材料性质保持连续. 基于Flügge薄壳理论,建立了位移形式的结构静态屈曲控制方程.根据边界条件将位移表示为三角级数形式,获得包含柱壳端部扭转载荷参数的近似线性代数特征值问题,并通过数值方法求得了表征结构失稳特征的临界载荷. 数值结果表明,临界载荷随着半径与厚度比的增加而减小,随着功能梯度中间层的弹性模量的平均值的增加而增加.      

Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements

In this article, post-buckling and non-linear bending analysis of functionally graded annular sector plates based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while Poisson׳s ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio, loading condition and two different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to the stretching–bending coupling effects of the FGMs, the post-buckling behavior of movable simply supported FGM plates is not of the bifurcation-type buckling. Moreover, FGM annular sector plates subjected to uniaxial compression at radial edges show a non-linear bending behavior with unique and stable equilibrium paths following a flattening feature.     

Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations

In this paper, natural frequencies characteristics of a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on three-dimensional equations of elasticity is considered. The axisymmetric conditions are assumed for the 2D-FGM cylinder. The material properties of the cylinder are varied in the radial and axial directions with power law functions. Effects of volume fraction distribution and FGM configuration on the natural frequencies of a simply supported cylinder are analyzed. Also, the effects of length and thickness of the cylinder are considered for different material distribution profiles. Three-dimensional equations of motion are used and the eigen value problem is developed based on direct variational method. Finite element method with graded material characteristics within each element of the structure is used for solution. The study shows that the 2D-FGM cylinder exhibit interesting frequency characteristics when the constituent volume fractions and its configuration are varied.     

Exact solution for in-plane static problems of circular beams made of functionally graded materials

Exact analytical solutions of in-plane static problems of circular beams with uniform cross-section made of functionally graded material (FGM) are obtained. Material properties are assumed to be varying arbitrarily through the thickness. The effects of axial extension and shear deformations are considered. The differential equation system is solved exactly using the initial values method. The circumferential stress distribution on the cross-section is also obtained. The results are compared with those of rather complex approaches in the literature, such as elasticity approach, and the comparison shows an excellent agreement. Effects of power law exponent and radius-to-height ratio of the beam on circumferential stress distribution and displacements are investigated.     

Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory

A generalized refined theory including surface effects is developed for functionally graded ultra-thin films with different surface properties. The classical generalized shear deformable theory is adopted to model the film bulk, while the bulk stresses along the surfaces of the bulk substrate are required to satisfy the surface balance equations of the continuum surface elasticity. As a result, the shape function also shows size-dependence on the film thickness. Since the film is non-homogeneous through the thickness, the state space method and approximate laminate model are employed to derive the variation of shape function through the thickness direction. A simply supported thin film in cylindrical bending is considered as an example to illustrate the application of the present theory. By comparing to the Kirchhoff plate theory including surface effects, the necessity of the present theory for FGM thin films is solidly validated. It is established that the present FGM thin films exhibit significant size-dependence when the thickness approaches to micro-scale. As the gradient index changes, the extent of size-dependence varies accordingly.     

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.     

Rotating sandwich cylindrical shells with an FGM core and two FGPM layers: free vibration analysis

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material(FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded(FG),the material properties vary along the thickness direction as one innovation of this study.Applying the first-order shear deformation theory(FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations(PDEs) using Hamilton's principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material(FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM.     

Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane

The conductive heat transfer between two elastic bodies in the static contact can cause the system to be unstable due to the interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance. This paper investigates the thermoelastic contact instability of a functionally graded material (FGM) layer and a homogeneous half-plane using the perturbation method. The FGM layer and half-plane are exposed to a uniform heat flux and are pressed together by a uniform pressure. The material properties of the FGM layer vary exponentially along the thickness direction. The characteristic equation governing the thermoelastic stability behavior is obtained to determine the stability boundary. The effects of the gradient index, layer thickness and material combination on the critical heat flux are discussed in detail through a parametric study. Results indicate that the thermoelastic stability behavior can be modified by adjusting the gradient index of the FGM layer.     

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．     

Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials

In recent years, functionally graded material (FGM) has been widely explored in coating technology amongst both academic and industry communities. FGM coatings are suitable substitutes for many typical conventional coatings which are susceptible to cracking, debonding and eventual functional failure due to the mismatch of material properties at the coating/substrate interface. In this study, a thick spherical pressure vessel with an inner FGM coating subjected to internal and external hydrostatic pressure is analyzed within the context of three-dimensional elasticity theory. Young’s modulus of the coating is assumed to vary linearly or exponentially through the thickness, while Poisson’s ratio is considered as constant. A comparative numerical study of FGM versus homogeneous coating is conducted for the case of vessel under internal pressure, and the dependence of stress and displacement fields on the type of coating is examined and discussed.     

Three-dimensional elastic deformation of a functionally graded coating/substrate system

The concept of functionally graded material (FGM) is actively explored in coating design for the purpose of eliminating the mismatch of material properties at the coating/substrate interface, typical for conventional coatings, which can lead to cracking, debonding and eventual functional failure of the coating. In this paper, an FGM coating/substrate system of finite thickness subjected to transverse loading is analysed within the context of three-dimensional elasticity theory. The Young’s modulus of the coating is assumed to vary exponentially through the thickness, and the Poisson’s ratio is assumed to be constant. A comparative study of FGM versus homogeneous coating is conducted, and the dependence of stress and displacement fields in the coating substrate/system on the type of coating, geometry and loading is examined and discussed.     

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.     

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

The wave propagation and dynamic response of rectangular functionally graded material plates with completed clamped supports under impulse load

In this paper, the wave propagation and dynamic response of the rectangular FGM plates with completed clamped supports under impulse load are analyzed. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. A complete discussion of dispersion of the FGM plates is given. Using the dispersion relation and integral transforms, exact integral solutions for the FGM plates under impulse load are obtained. The influence of volume fraction distributions on wave propagation and dynamic response of the FGM plates is analyzed.     

Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.     

功能梯度材料平面问题的辛弹性力学解法

将辛弹性力学解法推广用于功能梯度材料平面问题的分析,考虑沿长度方向弹性模量为指数函数变化而泊松比为常数的矩形域平面弹性问题,给出了具体的求解步骤. 提出了移位Hamilton矩阵的新概念,建立起相应的辛共轭正交关系;导出了对应特殊本征值的本征解,发现材料的非均匀特性使特殊本征解的形式发生明显的变化.