An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

Abstract

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. 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 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.

Introduction

With the ever-increasing demand for high strength-to-weight ratio materials, engineering materials are continuously replacing the conventional materials in the automotive, nuclear and aerospace industries. In general, most engineered materials are inspired from nature. A new class of materials was introduced by Japan scientists [1] to decrease the thermal stresses in propulsion and airframe structural systems of astronautical flight vehicles. This class of engineered materials was designated as functional graded material (FGM). FGMs are a class of composites that exhibit continuous variation of material properties from one surface to another and thus eliminate the stress concentration generally encountered in laminated composites. A typical FGM is synthesized from a mixture of ceramic and metal. These materials are often isotropic but non-homogeneous. The reason for sustained 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 mathematical models to predict their responses.

In the past three decades, researches on functionally graded material (FGM) plates have received substantial attention, and an extensive spectrum of plate theories has been introduced based on the classical plate theory and shear deformation plate theory. The classical plate theory (CPT) neglects shear deformations and can lead to inaccurate results for moderately thick plates. This theory has been implemented for buckling analysis of FGM plates by Feldman and Aboudi [2], Abrate [3], Mahdavian [4], and Mohammadi et al. [5]. However CPT under-predicts deflections and over-predicts frequencies as well as buckling loads of moderately thick plate [6]. First-order shear deformation theory (FSDT) [7], [8] considers the transverse shear deformation effects, but needs a shear correction factor in order to satisfy the zero transverse shear stress boundary conditions at the top and bottom of the plate. Many studies of the mechanical behavior of plates have been carried out using FSDT [9], [10], [11], [12], [13], [14], [15]. To avoid the use of shear correction factors, several higher-order shear deformation plate theories (HSDTs) have been proposed such as the theory propounded by Nelson and Lorch [16] with nine unknowns, Lo et al. [17] with eleven unknowns, Bhimaraddi and Stevens [18] with five unknowns, Reddy [19] with five unknowns, Kant and Pandya [20] with seven unknowns, Kant and Khare [21] with nine unknowns and Talha and Singh [22] with eleven unknowns. Some higher order theories based on Carrera’s Unified Formulation (CUF) such as proposed in Refs. [23], [24], [25], [26], [27], [28], [29], [30], [31] have been used also to study FGM structures.

The majority of HSDTs used to investigate FGM plate mechanics have the same five unknowns (e.g., third-order shear deformation theory [32], sinusoidal shear deformation theory [33], hyperbolic shear deformation theory [34], [35]). The resulting equations of motion are much more complicated than those yielded with FSDT. In addition, it should be noted that the above-mentioned two-dimensional plate theories (i.e., CPT, FSDT, and HSDT) discard the thickness stretching effect (i.e. they assume ${\epsilon }_z=0$) as they consider a constant transverse displacement through the thickness. This assumption is appropriate for thin or moderately thick FGM plates, but is inadequate for thick FGM plates [36]. The importance of the thickness stretching effect in FGM plates has been identified succinctly in the work of Carrera et al. [37]. This effect plays a significant role in thick FGM plates and should be taken into consideration.

In general, higher order shear and normal deformation theories which consider thickness stretching effect can be implemented using the unified formulation initially proposed by Carrera [38], [39], [40] and recently extended by Demasi [41], [42], [43], [44], [45]. More detailed information and applications of the unified formulation can be found in the recent books by Carrera et al. [46], [47]. Many higher order shear and normal deformation theories have been proposed in the literature [48], [49], [50]. These theories are cumbersome and computationally expensive since they invariably generate a host of unknowns (e.g., theories by Talha and Singh [49] with thirteen unknowns; Reddy [50] with eleven unknowns; and Neves et al. [24], [25], [26] with nine unknowns). Although some well-known quasi-three-dimensional theories developed by Zenkour [51] and recently by Mantari and Guedes Soares [52], [53] have six unknowns, they are still more complicated than the FSDT. Thus, there is a scope to develop an accurate higher order shear and normal deformation theory, which is relatively simple to use and simultaneously retains important physical characteristics. Indeed, Huu and Seung [54] presented recently a quasi-3D sinusoidal shear deformation theory with only five unknowns for bending behavior of FGM plates.

In the present article, a new higher order shear and normal deformation theory with only five unknowns is developed for FGM plates. Contrary to the four-variable refined theories elaborated in [55], [56], [57], [58], [59], [60], [61], [62], [63], where the stretching effect is neglected, in the current investigation this so-called “stretching effect” is taken into consideration. The displacement field is chosen based on a hyperbolic variation of in-plane and transverse displacements through the thickness. Partitioning the transverse displacement into the bending, shear and thickness stretching components leads to a reduction in the number of unknowns, and consequently, makes the new theory much more amenable to mathematical implementation. The 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. The current study is relevant to aero-structures.