Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

Nonlinear bending analysis of FGM rectangular plates with various supported boundaries resting on two-parameter elastic foundations

In this paper, model of the FGM plates resting on two-parameter elastic foundations is put forward by using on physical neutral surface and high-order shear deformation theory. Material properties are assumed to be temperature dependent and vary along the thickness, while Poisson’s ratio depends weakly on temperature change and position and is assumed to be a constant. It is worth noting that physical neutral surface will be changed with temperature. The character of physical neutral surface higher-order shear deformation plate theory is that the displacements have special forms, stretching-bending couplings are eliminated in constitutive equations, and governing equations have the simple and similar forms as homogeneous isotropic plates. The validity of physical neutral surface higher-order shear deformation plate theory can be confirmed by comparing with related researchers’ results. Nonlinear bending approximate solutions of FGM rectangular plates with six cases of boundary conditions are given out using Ritz method, and influences played by different supported boundaries, foundation stiffnesses, thermal environmental conditions, and volume fraction index are discussed in detail.     

功能梯度中厚圆/环板轴对称弯曲问题的解析解

基于一阶剪切变形板理论，导出了热/机载荷作用下，位移形式的功能梯度 中厚圆/环板轴对称弯曲问题的控制方程，获得了问题的位移和内力的一般解析解. 作为特 例，分别研究了边缘径向固定和可动的夹紧和简支的4种实心功能梯度圆板，给出了它们的 解，并分析了热/机载荷作用下解的形态，讨论了横向剪切变形、材料梯度常数和边界条件， 对板的轴对称弯曲行为的影响.     

BENDING RESPONSES OF AN EXPONENTIALLY GRADED SIMPLY-SUPPORTED ELASTIC/VISCOELASTIC/ELASTIC SANDWICH PLATE

The bending response for exponentially graded composite (EGC) sandwich plates is investigated.The three-layer elastic/viscoelastic/elastic sandwich plate is studied by using the sinusoidal shear deformation plate theory as well as other familiar theories.Four types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer.The effective moduli and Illyushin’s approximation methods are used to solve the equations governing the bending of simply-supported EGC fiber-reinforced viscoelastic sandwich plates.Then numerical results for deflections and stresses are presented and the effects due to time parameter,aspect ratio,side-to-thickness ratio and constitutive parameter are investigated.     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

Two new refined shear displacement models for functionally graded sandwich plates

Two refined displacement models, RSDT1 and RSDT2, are developed for a bending analysis of functionally graded sandwich plates. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The developed models are variationally consistent, have strong similarity with classical plate theory in many aspects, do not require shear correction factor, and give rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress-free surface conditions. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher-order models. The functionally graded layers are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson’s ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed models are accurate and simple in solving the bending behavior of functionally graded plates.     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

Differential quadrature method for bending of orthotropic plates with finite deformation and transverse shear effects

Based on the Reddy ‘s theory of plates with the effect of higher-order shear deformations, the governing equations for bending of orthotropic plates with finite deformations were established. The differential quadrature ( DQ ) method of nonlinear analysis to the problem was presented. New DQ approach, presented by Wang and Bert ( DQWB), is extended to handle the multiple boundary conditions of plates. The techniques were also further extended to simplify nonlinear computations. The numerical convergence and comparison of solutions were studied. The results show that the DQ method presented is very reliable and valid. Moreover, the influences of geometric and material parameters as well as the transverse shear deformations on nonlinear bending were investigated. Numerical results show the influence of the shear deformation on the static bending of orthotropic moderately thick plate is significant.     

An efficient and simple refined theory for nonlinear bending analysis of functionally graded sandwich plates

In this paper, an efficient and simple refined theory is presented for nonlinear bending analysis of functionally graded sandwich plates. The theory presented is variationally consistent, does not require the shear correction factor, and gives rise to transverse shear stress variations such that the transverse shear stresses vary parabolically across the plate thickness, satisfying shear-stress-free surface conditions. The energy concept along with the present theory and the first- and third-order shear deformation theories is used to predict the large deflection and the stress distribution across the thickness of functionally graded sandwich plates.     

纵横载荷作用下功能梯度梁的弯曲和过屈曲

基于一阶非线性梁理论和物理中面概念,导出了纵横向载荷作用下功能梯度材料(FGM)梁非线性弯曲和过屈曲问题的控制方程,并获得了该问题的精确解;据此解研究了梯度材料性质、外载荷、横向剪切变形以及边界条件等因素对功能梯度材料梁非线性力学行为的影响,分析中假设功能梯度材料性质只沿梁厚度方向,并按成分含量的幂指数函数形式变化。结果表明:纵横载荷共同作用下,功能梯度梁的弯曲构形将有无限多个;随着梯度指数的增大,梁的变形减小,临界载荷升高;随着长高比的增大,横向剪切变形的影响减小。     

功能梯度与均匀圆板弯曲解的线性转换关系

基于一阶剪切理论, 研究了功能梯度材料圆板与均匀圆板轴对称弯曲解之间的线性转换关系. 通过理论分析和比较 功能梯度材料圆板和均匀圆板在一阶剪切理论下的位移形式的轴对称弯曲控制方程, 发现了功能梯度材料圆板的转角与均匀圆板的转角之间的相似转换关系. 在假设材料性质沿板厚连续变化的情况下, 给出了相似转换系数的解析表达式. 在此基础上, 进一步导出了一阶剪切理论下功能梯度圆板的挠度与经典理论下, 均匀圆板的挠度之间的线性关系. 从而, 可将功能梯度材料圆板在一阶剪切理论下的弯曲问题求解, 转化为相应均匀薄圆板在经典理论下的弯曲问题求解, 以及转换系数的计算问题. 这一方法为功能梯度非均匀中厚度圆板的求解提供了简捷途径, 而且更便于工程应用. 作为例子, 采用上述方法分别求得了周边简支和夹紧条件下, 梯度圆板在均布横向载荷作用下的弯曲解析解, 该解答与Reddy得到的结果完全吻合.      

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material (FGM) plate by Reissner’s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson’s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner’s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner’s effect when the in-homogeneity parameter approaches zero.     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton’s principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-order theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.     

An efficient shear deformation theory for vibration of functionally graded plates

This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton??s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

On the simple and mixed first-order theories for functionally graded plates resting on elastic foundations

In this article, the bending response of a functionally graded plate resting on elastic foundations and subjected to a transverse mechanical load is investigated. An accurate solution for the functionally graded plate with simply supported edges resting on elastic foundations is presented. The interaction between the plate and the elastic foundations is considered and included in the equilibrium equations. Pasternak’s model is used to describe the two-parameter elastic foundations, and get a special case of Winkler’s model by considering one-parameter of elastic foundation. A relationship between the simple and mixed first-order transverse shear deformation theories is presented. Numerical results for deflections and stresses of functionally graded plates are investigated. Comparisons between the results of the simple and mixed first-order theories are made, and appropriate conclusion is formulated. Additional boundary conditions at the edges of the present plates are investigated.     

High-order shear theory for static analysis of functionally graded plates with porosities

The bending responses of porous functionally graded (FG) thick rectangular plates are investigated according to a high-order shear deformation theory. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates are derived. The solution to the problem is derived by using Navier's technique. Numerical results have been reported and compared with those available in the open literature for non-porous plates. The effects of the exponent graded and porosity factors are investigated.     

剪切变形对FGM圆板轴对称屈曲的影响

基于一阶剪切变形板理论，推导了功能梯度材料圆形板在边界面内均布压力作用下的轴对称屈曲方程。在推导过程中，忽略了前屈曲耦合变形。利用一阶板理论与经典板理论屈曲方程之间在数学形式上的相似性，得到了一阶板理论下功能梯度材料圆板与经典板理论下均匀圆板临界屈曲载荷之间的解析关系。利用这个解析关系，可以直接从已有的较为简单的经典理论的结果，获得一阶板理论下功能梯度材料板的临界屈曲载荷。