Elsevier

Composite Structures

Volume 94, Issue 2, January 2012, Pages 714-723
Composite Structures

Bending response of functionally graded plates by using a new higher order shear deformation theory

https://doi.org/10.1016/j.compstruct.2011.09.007Get rights and content

Abstract

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.

Introduction

Functionally graded materials (FGMs) have been proposed, developed and successfully used in industrial applications since 1980’s [20]. Classical composite 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 particularly under high temperature environments. These problems can be decreased by gradually changing the volume fraction of constituent materials, tailoring the material for desired application. For example, the composition of ceramic and metal can be varied from a ceramic rich surface to a metal rich surface and hence the thermal resistance of the material is increased due to low thermal conductivity of the ceramic and the low toughness problem of ceramics is eliminated by using the metal. 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, encourage researchers in this field.

Many papers, dealing with static and dynamic behaviour of functionally graded materials (FGMs), have been published recently. An interesting literature review of above mentioned work may be found in the paper of Birman and Byrd [3]. Reddy [38] presented Navier’s solutions, and finite element models including geometric non-linearity based on the third-order shear deformation theory for the analysis of FGM plates. Cheng and Batra [10] derived the field equations for a functionally graded plate by utilising the first-order shear deformation theory or the third-order shear deformation theory and simplified them for a simply supported polygonal plate. An exact relationship was established between the deflection of the functionally graded plate and that of an equivalent homogeneous Kirchhoff plate. Vel and Batra [45], [44] developed a three-dimensional analysis of the transient thermal stresses, and the free and forced vibration of simple supported FGM rectangular plates.

Pan [34] proposed an exact solution for functionally anisotropic elastic composite laminates. Pan’s solution extends Pagano’s solution [33], [32] to FGM. A three-dimensional elasticity solution was proposed by Kashtalyan [17] for a functionally graded simply supported plate under transversely distributed load. This solution was extended to a sandwich panel with FG core by Kashtalyan and Menshykova [18]. Qian et al. [35] studied the static and dynamic deformation of thick functionally graded elastic plates by using higher order shear and normal deformable plate theories and meshless local Petrov–Galerkin method. Batra and Jin [2] considered FGM plates, which were obtained by changing the fibre orientation. Free vibration results were provided by using the finite element method. Ferreira et al. [14] used a meshless method for the static analysis of a simply supported functionally graded plate by using a third-order shear deformation theory. The effective material properties were calculated by using the rule of mixtures and the Mori–Tanaka scheme. [13] developed a three-dimensional elasticity solution using the Ritz method for the static response of a clamped rectangular functionally graded plate. Material properties were calculated using a power-law distribution.

The static response of functionally graded plates was presented by Zenkour [48] using his generalised shear deformation theory. Effective material properties were calculated by assuming a power-law. A discrete layer model in conjunction with the Ritz method developed by Ramirez et al. [36] for the approximate solution of a static analysis for the two types of functionally graded plates. In one type, an exponential variation of the material properties through the thickness was assumed, whilst in the second type, the variation was the function of the fibre orientation.

Chi and Chung [11], [12] studied the mechanical behaviour of FGM plates under transverse load. Three evaluations were considered for the FGM properties, which include power-law, sigmoid or exponential function. Sladek et al. [39] presented the static and dynamic analysis of functionally graded plates by the meshless local Petrov–Galerkin method. The Reissner–Mindlin plate bending theory was utilised to describe the plate deformation. Numerical results were presented for simply supported and clamped plates. More information may be found on the review paper given by Birman and Byrd [3] for the manufacturing, design, modelling, testing methods and applications of functionally graded materials between the years 2000 and 2007.

Bo et al. [4] presented the elasticity solutions for the static analysis of functionally graded plates for different boundary conditions. Stress analysis due to thermal and mechanical loads was given by Matsunaga [22] by using a two-dimensional higher-order theory. A power law distribution for the volume fractions of constituents was assumed for the calculation of modulus of elasticity. Navier solution of a simply supported functionally graded plate was provided for stress and deflections. Khabbaz et al. [19] provided a nonlinear solution of FGM plates using the first and third-order shear deformation theories.

More recently, Aghdam et al. [1] presented a static analysis of fully clamped functionally graded plates and doubly curved panels by using the extended Kantorovich method. Wu and Li [46] used a RMVT-based third-order shear deformation theory of multilayered FGM plates 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 [41] investigated the free vibration and static analysis of functionally graded plates using the finite element method by employing a higher order shear deformation theory. Vaghefi et al. [43] presented a three-dimensional static solution for thick functionally graded plates by utilising a meshless Petrov–Galerkin method. An exponential function was assumed for the variation of Young’s modulus through the thickness of the plate, whilst 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 FGM plates were presented by Wu et al. [47] and solutions obtained in this paper are in excellent agreement with 3D solutions.

Carrera et al. [9] studied the effects of thickness stretching in FGM plates and shells. The importance of the transverse normal strain effects in mechanical prediction of stresses for FGM plates was pointed out. In fact, this work is an extension of several FGM paper published by using Carrera’s Unified Formulation (CUF), as described in Carrera et al. [8], Brischetto [6] and Brischetto and Carrera [5], [7].

In the present paper, an analytical solution to the static analysis of functionally graded plates is developed using a higher order shear deformation theory (HSDT) recently developed by Mantari et al. [21]. Other HSDT’s available in the literature [37], [42], [40], [16], are presented and studied for comparisons. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, so that a shear correction factor is not required.

The mechanical properties of the panels are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of the functionally graded plate and the boundary conditions are derived by employing the principle of virtual work. These equilibrium equations are then solved using Navier solution method. Static results are presented for plates for simply supported boundary conditions. FGM plates are subjected to transverse bi-sinusoidal and distributed loads. Results are provided for thick to thin plates and for different volume fraction distributions. The accuracy of the present model is verified by comparing it with available above mentioned results in the literature.

Section snippets

Theoretical formulation

The rectangular plate of uniform thickness h made of a functionally graded material is shown in Fig. 1. The rectangular Cartesian coordinate system x, y, z, has the plane z = 0, coinciding with the mid-surface of the plate. The material is inhomogeneous and the material properties vary through the thickness with a simple power-law distribution, which is given below:P(z)=(Pt-Pb)V+Pb,V=zh+12n,where P denotes the effective material property, Pt and Pb denote the property of the top and bottom faces

Solution procedure

Exact solutions of the partial differential Eqs. (10a-e) on arbitrary domain and for general boundary conditions are difficult. Although the Navier type solutions can be used to validate the present theory, and more general boundary conditions will require solution strategies involving, e.g., boundary discontinuous double Fourier series approach (see Oktem and Chaudhuri [23], [24], [25], [26], [27], [28], [29], [30], Oktem and Guedes Soares [31]).

Solution functions that completely satisfy the

Bending analysis of FGM Al/ZrO2plates

The results are presented for the simply supported plates under bi-sinusoidal and uniformly distributed transverse loads of intensity q. The static analysis was conducted using Aluminum (bottom, Al) and Zirconia (top, ZrO2). The following material properties are used for computing the numerical results.Et=151GPa,υt=0.3;Eb=70GPa,υb=0.3.The following non-dimensional quantities are used:w¯=D(2,2)qa4wa2,b2,0,σ¯xx=σxxhqa2a2,b2,h2,σ¯xy=σxyhqa2(0,0,0),σ¯xz=σxzhqa20,b2,0.where q denotes the transverse

Conclusions

The static response of functionally graded plates is analyzed using a new higher order shear deformation theory (HSDT). A simple power-law distribution in terms of the volume fractions of the constituents for the calculation of mechanical properties of the panels is assumed. 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. Static results are presented for two different

Acknowledgment

The first and the second author have been financed by the Portuguese Foundation of Science and Technology under the contract numbers SFRH/BD/66847/2009 and SFRH/BPD/47687/2008, respectively.

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