A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates
Introduction
Functionally graded materials (FGMs) are a class of composites that have continuous variation of material properties from one surface to another, thus eliminating the stress concentration found in laminated composites. A typical FGM is made from a mixture of two material phases, for example, a ceramic and a metal. The reason for the increasing use of FGMs in a variety of aerospace, automotive, civil, and mechanical engineering structures is that their material properties can be tailored to different applications and working environments. The increase of FGM applications requires the development of accurate theories to predict their responses. It should be noted that the classical plate theory (CPT), which is based on the Kirchhoff hypothesis, is suitable for thin plates, but inadequate for thick plates or plates made of advanced composites like FGMs. The first-order shear deformation theory (FSDT) [1], [2], [3] accounts for the shear deformation effect by the way of linear variation of in-plane displacements through the thickness. Thus, a shear correction factor is required to compensate for the difference between the actual and assumed constant stress states [4], [5]. The higher-order shear deformation theory (HSDT) [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23] accounts for the shear deformation effect by the way of a higher-order variation of in-plane displacements through the thickness, and hence, the shear correction factor is not required. It should be noted that the above-mentioned 2-D plate theories (i.e., CPT, FSDT, and HSDT) discard the thickness stretching effect (i.e., εz = 0) due to assuming a constant transverse displacement through the thickness. This assumption is appropriate for thin or moderately thick functionally graded (FG) plates, but is inadequate for thick FG plates [24]. The importance of the thickness stretching effect in FG plates has been pointed out in the work of Carrera et al. [25]. This effect plays a significant role in moderately thick and thick FG plates and should be taken into consideration.
Quasi-3D theories are HSDTs with higher-order variations through the thickness for the transverse displacement. In general, quasi-3D theories can be implemented using the unified formulation initially proposed by Carrera [26], [27], [28] and recently extended by Demasi [29], [30], [31], [32], [33], [34]. More detailed information and applications of the unified formulation can be found in the recent books by Carrera et al. [35], [36]. Since the quasi-3D theory accounts for a higher-order variation of both in-plane and transverse displacements through the thickness, both the shear deformation effect and the thickness stretching effect are considered. Many quasi-3D theories have been proposed in the literature [37], [38], [39], [40], [41], [42], [43], [44]. These theories are cumbersome and computationally expensive due to having a host of unknowns (e.g., theories by Talha and Singh [39] with thirteen unknowns; Chen et al. [38] and Reddy [41] with eleven unknowns; and Ferreira et al. [40] and Neves et al. [42], [43], [44] with nine unknowns). Although some well-known quasi-3D theories developed by Zenkour [45] and recently by Mantari and Guedes Soares [46], [47] have six unknowns, they are still more complicated than the FSDT. Thus, developing a simple quasi-3D theory is necessary.
This paper aims to develop a simple quasi-3D theory with only five unknowns. The displacement field is chosen based on a sinusoidal variation of in-plane and transverse displacements through the thickness. The partition of the transverse displacement into the bending and shear parts leads to a reduction of the number of unknowns, and subsequently, makes the new theory simple to use. Governing equations and boundary conditions are derived using the principle of virtual displacements. Analytical solutions for deflections and stresses are obtained for a simply supported rectangular plate. Numerical examples are presented to verify the validity of the present theory.
Section snippets
Kinematics
The aim of this paper is to develop a simple quasi-3D theory in which in-plane and transverse displacements are expanded as a sinusoidal variation through the thickness. The advantages of the sinusoidal functions over the polynomial functions are that they are simple and accurate, and the stress-free boundary conditions on the top and bottom surfaces of the plate can be guaranteed [48]. In fact, the use of sinusoidal functions was first proposed by Levy [49] and assessed by Stein [50], and
Analytical solutions
Consider a simply supported rectangular plate with length a and width b under transverse load q as shown in Fig. 1. Based on the Navier approach, the solutions are assumed to bewhere α = mπ/a, β = nπ/b, (Umn, Vmn, Wbmn, Wsmn, ϕzmn) are coefficients. The transverse load q is also expanded in the double-Fourier sine series as follows:
Numerical results
In this section, the bending results of simply supported FG plates under sinusoidal loads are presented. For verification purpose, the obtained results are compared with the exact solution of 3D elasticity theory and those predicted by FSDT, HSDT, and quasi-3D theories. The description of various displacement models and their corresponding number of unknowns are listed in Table 1. Poisson’s ratio ν is assumed to be constant and equal to 0.3.
Conclusions
A simple quasi-3D sinusoidal shear deformation theory is developed for FG plates. The governing equations and boundary conditions are derived from the principle of virtual displacements. Analytical solutions are obtained for simply supported rectangular plates. By making further simplifying assumptions to the quasi-3D theory of Zenkour [45], the number of unknowns of the new quasi-3D is reduced by one, and hence, makes the new theory simple and efficient to use. Numerical results show that
Acknowledgements
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0030847), and a Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20104010100520).
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