Semi‐analytical analysis for multi‐directional functionally graded plates: 3‐D elasticity solutions

Semi‐analytical 3‐D elasticity solutions are presented for orthotropic multi‐directional functionally graded plates using the differential quadrature method (DQM) based on the state‐space formalism. Material properties are assumed to vary not only through the thickness but also in the in‐plane directions following an exponential law. The graded in‐plane domain is solved numerically via the DQM, while exact solutions are sought for the thickness domain using the state‐space method. Convergence studies are performed, and the present hybrid semi‐analytical method is validated by comparing numerical results with the exact solutions for a conventional unidirectional functionally graded plate. Finally, effects of material gradient indices on the displacement and stress fields of the plates are investigated and discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional analysis of functionally graded plates

Based on the state‐space formalism, a three‐dimensional analysis is presented for orthotropic functionally graded rectangular plates with simply supported edges under static and dynamic loads. The material properties are assumed to be variable through the thickness. The governing equations for the functionally graded material (FGM) are developed on the state‐space approach in the Laplace transform domain. Assuming constant material properties, we derive the analytical solutions that can be used to validate any numerical methods. For FGM plates, the numerical solutions are obtained by the use of radial basis function method. Three examples are presented for the FGMs and laminated composite. The accuracy of the proposed numerical technique has been compared with the exact solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Sensitivity of Dynamic Response of a Simply Supported Functionally Graded Magneto-electro-elastic Plate to its Elastic Parameters

Dynamic response sensitivity of a simply supported functionally graded magneto-electro-elastic plates have been studied by combining analytical method with finite element method. The functionally graded material parameters are assumed to obey exponential law in the thickness direction. A series solution of double trigonometric function agreed with the simply supported boundary condition is adopted in the plane of the plate and finite element method is used across the thickness of the plate. The finite element model is established based on energy variational principle. The coupled electromagnetic dynamic characteristics of a simply supported functionally graded magneto- electro-elastic plate are decided by its dynamics differential equation into which displacement components, electric potential and magnetic potential as nodal degree of freedom are incorporated. Dynamic response sensitivity is defined as a partial differential of dynamic response with respect to material parameter. Sensitivity of dynamic response of a simply supported functionally graded magneto-electro-elastic plate to its elastic parameters has been studied. The influence of the different exponential factor on dynamic response sensitivity has also been investigated.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.     

A higher-order shear and normal deformation functionally graded plate model: some recent results

A higher-order shear and normal deformations plate theory is employed for stress analysis and free vibration of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. This idealization is required in order to obtain the closed-form solutions of some fundamental solid mechanics problems and also simplify the evaluation and development of numerical models of the structures made of FGMs. The material properties of FG plates such as Young’s moduli and material density are considered in this case to vary continuously in the thickness direction according to the volume fraction of constituents and mathematically modeled as exponential and power law functions. Poisson’s ratio is assumed to be constant. The effect of variation of material properties in terms of material grading index on the deformations, stresses, and natural frequency of FG plates is studied. The accuracy of the presented numerical solutions has been established with the solutions available of other models and the exact three-dimensional (3D) elasticity solutions.     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

Transient internal crack problem for a nonhomogeneous orthotropic strip (Mode I)

Supposing the material properties to be one-dimensionally dependent, this paper studied the transient internal crack problem for a functionally graded orthotropic strip. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results show the effects of the material parameter βh, the crack configuration and the isotropic or orthotropic property on the dynamic stress intensity factors.     

Three-dimensional static analysis of thick functionally graded plates by using meshless local Petrov–Galerkin (MLPG) method

In this paper, a version of meshless local Petrov–Galerkin (MLPG) method is developed to obtain three-dimensional (3D) static solutions for thick functionally graded (FG) plates. The Young's modulus is considered to be graded through the thickness of plates by an exponential function while the Poisson's ratio is assumed to be constant. The local symmetric weak formulation is derived using the 3D equilibrium equations of elasticity. Moreover, the field variables are approximated using the 3D moving least squares (MLS) approximation. Brick-shaped domains are considered as the local sub-domains and support domains. In this way, the integrations in the weak form and approximation of the solution variables are done more easily and accurately. The proposed approach to construct the shape and the test functions make it possible to introduce more nodes in the direction of material variation. Consequently, more precise solutions can be obtained easily and efficiently. Several numerical examples containing the stress and deformation analysis of thick FG plates with various boundary conditions under different loading conditions are presented. The obtained results have been compared with the available analytical and numerical solutions in the literature and an excellent consensus is seen.     

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。     

正交各向异性复合柱圆弧型循环对称界面裂纹

讨论反平面载荷作用下空心复合柱的循环对称圆弧型裂纹问题。复合柱由两极正交各向异性功能梯度弹性层粘接而成。采用分离变量将这个混合边值问题转化为Cauchy核奇异积分方程,并用Lobatto-Chebyshev求积法对积分方程进行数值求解,得到了应力强度因子的数值解。分析了梯度非均匀参数,几何与材料参数变动等对应力强度因子的影响。     

Guided waves in functionally graded viscoelastic plates

In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin–Voigt viscoelastic theory. The FGM plate is composed of two orthotropic materials. The material properties are assumed to vary in the thickness direction according to a known variation law. The three obtained wave equations are divided into two groups, which control viscoelastic Lamb-like wave and viscoelastic SH wave, respectively. They are solved respectively by the Legendre orthogonal polynomial series approach. The validity of the method is confirmed through a comparison with the Lamb wave solution of a pure elastic FGM plate and a comparison with the SH wave solution of a viscoelastic homogeneous plate. The dispersion curves and attenuation curves for the graded and homogeneous viscoelastic plates are calculated to highlight their differences. The viscous effect on dispersion curves is shown. The influences of gradient variations are illustrated.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

Analytical solutions for an antiplane problem of two dissimilar functionally graded magnetoelectroelastic half-planes

In this study, the two-dimensional analysis of two bonded dissimilar half-planes for functionally graded magnetoelectroelastic materials subjected to generalized line forces and screw dislocations is presented. The variations of the material properties are assumed to be exponential forms. By using the Fourier-transform technique, Green’s function for the transversely isotropic magnetoelectroelastic nonhomogeneous bimaterial is obtained from the jump and interface continuity conditions. Numerical results for the full-field distributions of the stresses, electric fields, and magnetic fields with different functionally graded parameters are presented. It is noted that the elastic, electric, and magnetic fields are all continuous at the interface in the nonhomogeneous bimaterial if the material constants at the interface are continuous. Furthermore, this functionally graded bimaterial has identical contour slopes for the generalized stresses across the interface.     

Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations

Thermo-mechanical buckling analysis of the orthotropic auxetic plates (with negative Poisson ratios) has not been performed so far, especially, in the hygrothermal environments. The complexity increases when the auxetic plate is fabricated from functionally graded orthotropic materials and surrounded by an elastic foundation. The aforementioned analyses are carried out in the present research, for the first time. The buckling loads may be uniaxial or biaxial ones. Moreover, temperature and moisture dependent material properties are considered. The pre-buckling effects are also considered in the paper. The high-order shear-deformation governing differential equations are solved based on a new differential quadrature method (DQM). The resulting solution may cover many practical simpler applications. A comprehensive parametric study is accomplished for a wide range of geometric and material properties parameters and various boundary conditions. Results reveal that the hygrothermal conditions lead to degradations in the material properties and buckling strengths, especially for higher gradation exponents, the elastic foundation may enhance the buckling behavior through monitoring the buckling pattern, the buckling load decreases as the orthotropy angle increases, and the auxeticity has reduced the buckling strength for the employed material and environmental information.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.