Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory

Abstract

An analytical solution of the static governing equations of exponentially graded plates obtained by using a recently developed higher order shear deformation theory (HSDT) is presented. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of exponentially graded plates and boundary conditions are derived by employing the principle of virtual work. A Navier-type analytical solution is obtained for such plates subjected to transverse bi-sinusoidal loads for simply supported boundary conditions. Results are provided for thick to thin plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and with other well-known trigonometric shear deformation theory. From the obtained results, it can be concluded that the present HSDT theory predict with good accuracy inplane displacements, normal and shear stresses for thick exponentially graded plates.

Introduction

Nowadays functionally graded materials (FGMs) are an alternative materials widely used in aerospace, nuclear, civil, automotive, optical, biomechanical, electronic, chemical, mechanical and shipbuilding industries. In fact, FGMs have been proposed, developed and successfully used in industrial applications since 1980’s [1]. 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, encourage researchers in this field.

Many papers, dealing with static and dynamic behavior of 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 [4] 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 [5] derived the field equations for a functionally graded plate by utilizing 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 [2], [6] developed a three-dimensional analysis of the transient thermal stresses, and the free and forced vibration of simply supported FGM rectangular plates. Pan [7] proposed an exact solution for functionally anisotropic elastic composite laminates. Pan’s solution extends Pagano’s solution [8], [9] to FGM. A three-dimensional elasticity solution was proposed by Kashtalyan [10] 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 [11]. Qian et al. [12] studied the static and dynamic deformation of thick functionally grade elastic plates by using higher order shear and normal deformable plate theories and meshless local Petrov–Galerkin method. Batra and Jin [13] considered FGM plates which were obtained by changing the fiber 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. Elishakoff et al. [15] 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 [16] using the generalized shear deformation theory developed by the author.

A discrete layer model in conjunction with the Ritz method was developed by Ramirez et al. [17] 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, while in the second type, the variation was the function of the fiber orientation. Chi and Chung [18], [19] studied the mechanical behavior of FGM plates under transverse load. Three evaluations were considered for the FGM properties, which include power-law, sigmoid or exponential function. Zenkour [20] investigated the static problem of transverse load acting on EGM rectangular plates using both 2D trigonometric plate theory (TPT) and 3D elasticity solution. Sladek et al. [21] 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.

More information may be found on the review paper given by Birman and Byrd [3] for the manufacturing, design, modeling, testing methods and applications of functionally graded materials between the years 2000 and 2007. Bo et al. [22] 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 [23] 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. Khabbaz et al. [24] provided a nonlinear solution of FGM plates using the first and third-order shear deformation theories.

More recently, Aghdam et al. [25] presented a static analysis of fully clamped functionally graded plates and doubly curved panels by using the extended Kantorovich method. Wu and Li [26] 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 [27] 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. [28] 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 FGM plates were presented by Wu and Li [29] and solutions obtained in this paper are in excellent agreement with 3D solutions. Carrera et al. [30] 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, the work is an extension of several FGM paper published by using the Carrera’s Unified Formulation (CUF), see Carrera et al. [31], Brischetto [32] and Brischetto and Carrera [33], [34], for more details.

In the present paper, an analytical solution of the static governing equations of exponentially graded plates is obtained by using a new trigonometric higher order shear deformation theory (HSDT) recently developed by Mantari et al. [35]. In order to compare results, the well-known trigonometric higher order shear deformation theory, which includes sinus function originally developed by Levy [36], corroborated and improved by Stein [37], extensively used by Touratier [38] and recently adapted to FGM and exponentially graded material (EGM) by Zenkour [16], [20] was successfully reproduced and extended to cover EGM (without considering stretching effect). The reason to reproduce such results is because in Zenkour [16] only FGM was covered (without considering stretching effect) and in Zenkour [20] only EGM was studied, but this time considering stretching effect. These theories account for adequate distribution of the transverse shear stresses through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary exponentially in the thickness direction. The governing equations of the exponentially graded plate and the boundary conditions are derived by employing the principle of virtual work. These equilibrium equations are then solved via Navier solution method. Static results are presented for plates for simply supported boundary conditions. EGM plates are subjected to transverse bi-sinusoidal loads. Results are provided for thick plates and for different values of the parameter n, which dictates the material variation profile through the plate thickness. The accuracy of the present code is verified by comparing it with 3D elasticity solution and the other well-known trigonometric shear deformation theory.