A novel higher-order shear deformation theory with stretching effect for functionally graded plates

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Abstract

This paper presents an analytical solution to the static analysis of functionally graded plates (FGPs) by using a new trigonometric higher-order theory in which the stretching effect is included. The governing equations and boundary conditions of FGPs are derived by employing the principle of virtual work. Navier-type solution is obtained for FGPs subjected to transverse bi-sinusoidal load for simply supported boundary conditions. Benchmark results for the displacements and stresses of geometrically different plates are obtained. The results are compared with 3D exact solution and with other higher-order shear deformation theories, and the superiority of the present theory can be noticed.

Introduction

Functionally graded materials (FGMs) are advanced composite materials, in which the material properties are varied in a predetermined manner. In the nature, FGM characteristics are found in sea shells, bones, etc., and the understanding of the highly complexity of such kind of materials is contributing to the synthesizing of new kind of materials. In the industry, FGMs have been proposed, developed and successfully used since 1984 [1]. Nowadays FGMs are alternative materials widely used in aerospace, nuclear reactor, energy sources, biomechanical, optical, civil, automotive, electronic, chemical, mechanical, and shipbuilding industries. Functionally graded materials (FGMs) are both macroscopically and microscopically heterogeneous advanced composites which are normally made from a mixture of ceramics and metals with continuous composition gradation from pure ceramic on one surface to full metal on the other. Such gradation leads to smooth change in the material profile as well as the effective physical properties.

Classical composites structures suffer from discontinuity of material properties at the interface of the layers and constituents of the composite. Therefore the stress fields in these regions create interface problems and thermal stress concentrations under high temperature environments. Furthermore, large plastic deformation of the interface may trigger the initiation and propagation of cracks in the material [2]. These problems can be decreased by gradually changing the volume fraction of constituent materials and tailoring the material for the desired application. In fact, FGMs are materials with spatial variation of the material properties. However, in most of the applications available in the literature, as in the present work, the variation is through the thickness only. Therefore, the early state development of improved production techniques, new applications, introduction to effective micromechanical models and the development of theoretical methodologies for accurate structural predictions, have encourage researchers and opened several research topics in this field.

Plates are often subjected to combinations of lateral pressure and thermal loading. However, plates and shells with FGM properties are frequently under a thermal load to utilize perfect thermal resistance of FGMs. Therefore, it is interesting to analyze plates and shells under a general thermal load. Tauchert [3] gave a nice overview of thermally induced flexure, buckling and vibration applied to classical composite plates described by the Kirchhoff theory. With knowledge gained in classical composites, studies on advanced composites are largely devoted to thermal stress analysis [4], [5], [6], and also to the fracture analysis of FG plates and shells [7], [8]. Finot and Suresh [9] presented a closed form solution based on the classical Kirchhoff’s theory of thin plates for the analysis of multilayered and functionally graded material plates, subjected to thermal loading. The dynamic thermoelastic response of the functionally graded cylinders and plates are obtained by Reddy and Chin [10]. Praveen and Reddy [11] obtained the nonlinear transient thermoelastic response of the functionally graded ceramic metal plates using a plate finite element method, employing transverse shear strain, rotary inertia and von-Karman nonlinearity. Reddy [12] presented Navier’s solutions, and finite element models including geometric non-linearity based on the third-order shear deformation theory for the analysis of FGPs. Cheng and Batra [13] derived the field equations for a functionally graded plate by utilizing the first-order shear deformation theory and the third-order shear deformation theory and simplified them for a simply supported polygonal plate. Literature survey shows that many papers dealing with static and dynamic behaviour of functionally graded materials (FGMs), have been published recently. An interesting literature review of abovementioned work may be found in the paper of Birman and Byrd [14]. Therefore, for completeness, in the present article, the relevant work from 2007 up to now is described. For relevant papers perhaps not included in the above mentioned review paper, readers may consult Carrera et al. [15] and Mantari et al. [16].

Zenkour [17] investigated the static problem of transverse load acting on exponentially graded (EG) rectangular plates using both 2D trigonometric plate theory (TPT) and 3D elasticity solution. Sladek et al. [18] presented the static and dynamic analysis of functionally graded plates by the meshless local Petrov–Galerkin method. The Reissner–Mindlin plate bending theory was utilized to describe the plate deformation. Numerical results were presented for simply supported and clamped plates. Later the author applied the same method to solve problems plates and shells under thermal loading in Sladek et al. [19], [20]. Abrate [21] deduced, by using the classical plate theory, that no special tools are required to analyze functionally graded plates, because FGPs behave like homogeneous plates.

Bo et al. [22] presented the elasticity solutions for the static analysis of functionally graded plates for different boundary conditions. Stress, free vibration and buckling analysis due to mechanical and thermal loads were given by Matsunaga [23], [24], [25] by using a kind of generalized two-dimensional higher-order theory. This interesting theory was obtained by using the method of power series expansion of continuous displacement components. Khabbaz et al. [26] provided a nonlinear solution of FGPs using the first and third-order shear deformation theories.

More recently, the thermo-bending problems of FGM sandwich plate, consisting of a homogeneous core-layer bounded with two FGM face-sheet layers, were studied by Zenkour and Alghamdi [27], using a variety of equivalent single-layer theories (ESLTs), in which the material properties of the face-sheet layers were assumed to obey a power-law distribution of the volume fractions of the constituents through the thickness coordinate.

Aghdam et al. [28] presented a static analysis of fully clamped functionally graded plates and doubly curved panels by using the extended Kantorovich method. Wu and Li [29] used a RMVT-based third-order shear deformation theory of multilayered FGPs under mechanical loads. The exponent-law distributions through the thickness and the power-law distributions of the volume fractions of the constituents were used to obtain the effective properties. Talha and Singh [30] investigated the free vibration and static analysis of functionally graded plates using the finite element method by employing a quasi-3D higher-order shear deformation theory. Vaghefi et al. [31] presented a three-dimensional static solution for thick functionally graded plates by utilizing a meshless Petrov–Galerkin method. An exponential function was assumed for the variation of Young’s modulus through the thickness of the plate, while the Poison’s ratio was assumed to be constant. RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGPs and circular hollow cylinders were presented by Wu et al. [32] and Wu and Yang [33]. Benachour [34] developed a novel four variable refined plate theory for free vibration analysis of plates made of functionally graded materials with an arbitrary gradient. In that HSDT, the number of DOFs involved is only four. Thai and Choi [35] studied the free vibration analysis of FG plates on elastic foundation by using similar refined plate theory as the one presented in [34]. Reddy and Kim [36] proposed a general nonlinear third-order plate theory that accounts for geometric nonlinearity, microstructure-dependent size effects, and two-constituent material variation through the plate thickness using the principle of virtual displacements and Hamilton’s principle.

Carrera et al. [15] studied the effects of thickness stretching in FG plates and shells. The importance of the transverse normal strain effects in mechanical prediction of stresses for FGPs was pointed out. In fact, this work is an extension of several FGM papers published by using Carrera’s Unified Formulation (CUF), as described in [37], [38], [39], [40]. Mantari et al. [16] presented bending results of FGPs by using a new non-polynomial HSDT, different to the one presented in here. Recently, Neves et al. [41] presented a quasi-3D hybrid polynomial and trigonometric shear deformation theory for the static and free vibration analysis of functionally graded plates by using collocation with radial basis functions.

In the present paper, an analytical solution to the static analysis of functionally graded plates is developed using a new trigonometric HSDT. In fact, an improved version of this HSDT [42], including the so-called “stretching effect”, is presented. 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. The plate material is exponentially graded in the thickness direction. The plate’s governing equations and its boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for plates subjected to transverse bi-sinusoidal load for simply supported boundary conditions. Benchmark results for the displacement and stresses of an exponential graded rectangular plate are obtained. The results are compared with 3D exact solution and with other higher-order shear deformation theories, and the superiority of the present theory can be noticed.

Section snippets

Theoretical formulation

A rectangular plate of uniform thickness h made of a functionally graded material is shown in Fig. 1, in which the rectangular Cartesian coordinate system x, y, z, with the plane z = 0, coincident with the mid-surface of the plate, is shown. The material is inhomogeneous and the material properties vary exponentially through the thickness, as indicated in:P(z)=g(z)Pb,g(z)=enzh+12,Pt=enPb,where P denotes the effective material property, Pt and Pb, denote the property of the top and bottom faces of

Solution procedure

Exact solutions of the partial differential Eqs. (10a-f) on arbitrary domain and for general boundary conditions are difficult. Although the Navier-type solutions can be used to validate the present theory, more general boundary conditions will require solution strategies involving, e.g., boundary discontinuous double Fourier series approach (see for example Oktem and Guedes Soares [44]).

Solution functions that completely satisfy the boundary conditions in Eqs. (14a), (14b), (14c), (14d), (14e)

Numerical results and discussions

In the present paper, results of simply supported EGPs by using a new trigonometric HSDT are presented. These results are compared with: (a) 3D exact solutions [17]; (b) the well-known trigonometric plate theory (TPT), which includes sinus function originally developed by Levy [45], corroborated and improved by Stein [46], extensively used by Touratier [47] and recently adapted to FGM and exponentially graded material (EGM) by Zenkour [17], [48]; (c) a HSDT results also provided by Zenkour [17]

Conclusions

The static response of exponentially graded plates is presented by using a new trigonometric higher-order shear deformation theory that includes stretching effect. The plate material is exponentially graded in the thickness direction. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved via a Navier-type, closed form solution, for FG plates subjected to transverse bi-sinusoidal load for simply supported boundary

Acknowledgment

The first author has been financed by the Portuguese Foundation of Science and Technology under the Contract Number SFRH/BD/66847/2009.

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