A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates

Abstract

In this paper, a new higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates is developed. The present theory has only four unknowns, but it accounts for a parabolic variation of transverse shear strains through the thickness of the plate. A shear correction factor is, therefore, not required. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3D and quasi-3D solutions and those predicted by other plate theories. Results show that the present theory can achieve the same accuracy of the existing higher-order shear deformation theories which have more number of unknowns, but its accuracy is not comparable with those of 3D and quasi-3D models which include the thickness stretching effect.

Introduction

Functionally graded materials (FGMs) are a class of composites that have continuous variation of material properties from one surface to another and thus eliminate the stress concentration found in laminated composites. A typical FGM is made from a mixture of ceramic and metal. These materials are often isotropic but nonhomogeneous. The reason for interest in FGMs is that it may be possible to create certain types of FGM structures capable of adapting to operating conditions. The increase in FGM applications requires accurate models to predict their responses. Since the shear deformation has significant effects on the responses of functionally graded (FG) plates, shear deformation theories are used to capture such shear deformation effects. The first-order shear deformation theory (FSDT) accounts for the shear deformation effects by the way of linear variation of in-plane displacements through the thickness. Since the FSDT violates the conditions of zero transverse shear stresses on the top and bottom surfaces of the plate, a shear correction factor which depends on many parameters is required to compensate for the error due to a constant shear strain assumption through the thickness. The higher-order shear deformation theories (HSDTs) account for the shear deformation effects, and satisfy the zero transverse shear stresses on the top and bottom surfaces of the plate, thus, a shear correction factor is not required.

In general, HSDTs are developed based on the assumption of higher-order variations of in-plane displacements [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] or both in-plane and transverse displacements through the thickness [14], [15], [16], [17], [18], [19], [20]. Some of these HSDTs are computational costs because with each additional power of the thickness coordinate, an additional unknown is introduced to the theory (e.g., theories by Pradyumna and Bandyopadhyay [6] and Neves et al. [18], [19], [20] with nine unknowns, Reddy [17] with eleven unknowns, Talha and Singh [16] with thirteen unknowns). Although some well-known HSDTs have the same five unknowns (e.g., third-order shear deformation theory [5], sinusoidal shear deformation theory [8], [9], [10], hyperbolic shear deformation theory [7], [11]), their equations of motion are much more complicated than those of FSDT. Thus, needs exist for the development of HSDTs which are simple to use.

The aim of this paper is to develop a simple HSDT for FG plates. The present theory has only four unknowns and four governing equations, but it satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without requiring any shear correction factors. The displacement field of the proposed theory is chosen based on a constant transverse displacement and cubic variation of in-plane displacements through the thickness. The partition of the transverse displacement into the bending and shear parts leads to a reduction in the number of unknowns and governing equations, hence makes the theory simple to use. Additional feature of the proposed theory is that it has strong similarities with the classical plate theory (CPT) in some aspects such as the governing equations, boundary conditions, and stress resultant expressions. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending and free vibration are obtained. Numerical examples are presented to verify the accuracy of the present theory.