Bending and vibration analysis of functionally graded trapezoidal nanocomposite plates reinforced with graphene nanoplatelets (GPLs)
Introduction
Functionally graded materials (FGMs) characterized by continuous variations in material composition and properties along one or more dimensions [1] have great potential in various engineering applications such as the heat-resistant components for aerospace structures [2], [3], wear-resistant materials [4], [5], sensors [6], [7] and implant materials [8]. Comparing to traditional composites, FGMs offer remarkably improved mechanical properties, in particular, excellent resistance to crack and delamination due to the significantly reduced stress mismatch in the interface between two dissimilar materials [9], [10]. Moreover, FGMs can also be designed and optimized to meet the desired structural performances by making the best use of the advantages of each material constituent [11], [12].
During the past decades, carbon-based fillers, including carbon black (CB) [13], [14], carbon fibers (CFs) [15], [16], carbon nanotube (CNTs) [17], [18], have been widely used to develop FGMs. Ahankari et al. [14] experimentally found the storage modulus of functionally graded nanocomposites reinforced with carbon black exhibited an improvement of 273% compared to that of the homogeneous composites. Yang and his co-workers [18], [19] found that functionally graded CNT reinforced nanocomposites have better vibration resistance than those with random and uniform distribution of CNT fillers. Recently, graphene and its derivatives, such as graphene oxide (GO) and graphene platelets (GPLs), demonstrate great potential as reinforcing fillers to develop high-performance nanocomposites for light weight structures since its discovery in 2004 [20]. This can be attributed to the excellent intrinsic mechanical and physical properties and improved reinforcing effects of graphene and its derivatives, for example, the Young’s modulus and tensile strength of graphene are found to be as high as 1 TPa and 130 GPa [21], respectively. A very small amount of graphene and its derivatives added into polymers can significantly improve the mechanical properties of the composite materials. Rafiee et al. [22] experimentally found that an addition of 0.1 wt.% GPL into epoxy matrix can increase the Young’s modulus of the composite by 31% while only an increase of 3% can be achieved by CNT fillers. More experimental and theoretical studies on the reinforcing effects of graphene and its derivative can be found in [23], [24], [25]. The mechanisms underlying the improved reinforcing effects of graphene and its derivatives can be explained by their unique 2D structure and extremely high surface area (almost twice the surface area of CNTs) which provide excellent load transfer from the matrix to the reinforcements [26], [27], [28], [29], [30], [31].
To make the most effective use of the excellent reinforcing effect of graphene and its derivatives, Yang et al. [32] proposed a novel multilayer functionally graded nanocomposite in which GPLs are non-uniformly distributed along the thickness direction. Their study showed that a small loading of GPLs can lead to dramatically improved bulking and post bulking performance of the GPL-reinforced nanocomposite beam and the performance is also highly sensitive to GPL distribution. Wu et al. [33] and Song et al. [34] demonstrated that a significantly enhanced dynamic behavior was achieved by incorporating 1.0 wt.% GPLs into polymer matrix in a non-uniform manner. Other most recent studies on the mechanical performances of functionally graded nanocomposites reinforced with graphene and its derivatives could be found in [35], [36], [37].
Trapezoidal plates have been extensively applied in various fields, especially serving as aircraft wings and tails in aerospace engineering. Given the excellent and unique mechanical attributes, functionally graded graphene reinforced nanocomposite becomes one of the most promising material candidates for engineering structures. Since 1992, Liew et al. [38], [39], [40], [41] conducted a series of mathematical studies on the static and dynamic behavior of isotropic and anisotropic trapezoidal plates by employing Rayleigh-Ritz method. Karami et al. [42] employed differential quadrature method to analyze the static, free vibration and stability of trapezoidal laminated trapezoidal. Torabi et al. [43] investigated the free vibration of cantilevered thick trapezoidal plates theoretically. Although works have been done on trapezoidal plates, to the best of the authors’ knowledge there are no studies on the trapezoidal plates reinforced with functionally graded distribution of graphene and its derivatives.
This paper presents the first attempt to investigate the bending, free and forced vibration of functionally graded GPL reinforced trapezoidal plates by employing the finite element method (FEM). The material properties of the nanocomposite are determined based on modified Halpin-Tsai model and the rule of mixture. The effects of the distribution, concentration and dimension of GPLs and the plate geometry on the static and dynamic behaviors of such trapezoidal plates are comprehensively discussed through a parametric study.
Section snippets
Effective material properties
A functionally graded GPL reinforced nanocomposite trapezoidal plate is shown in Fig. 1 where a and b are the lengths of the parallel sides, h and t are the height and thickness of the plate, α and β denote the two angles at the base, respectively. P is the center point of the top surface and L is the line connecting the points A and B at the bottom surface. The bending and vibration deflections of point P and von Mises stress along line L will be presented for parametric study.
In this study,
Finite element implementation
In this present study, ABAQUS is selected as the FEM package, in which 3D element C3D8R with eight nodes is used to create and mesh the trapezoidal plate model. Fig. 3a shows the multi-layered structure with perfect bonding between neighboring layers and Fig. 3b is the model after meshing.
The displacement vector for the eight-node element C3D8R can be written aswhere {δi} = {ui, vi, wi}T with ui, vi and wi being the displacements of node i, and [Ni] is the shape function
Numerical results and discussion
In this section, bending and vibration behaviors of cantilevered functionally graded GPL reinforced trapezoidal plates are studied. The dimensions of the trapezoidal plate as shown in Fig. 1 are selected as a = 0.2 m, b = 1 m, h = 1 m and t = 0.2 m, respectively. GPLs and epoxy are chosen as the reinforcing fillers and polymer matrix, respectively, whose material properties are set as the same as those reported in [22], [34], i.e. Em = 3.0 GPa, νm = 0.34, ρm = 1200 kg/m3, EGPL = 1010 GPa, νGPL = 0.186, ρGPL = 1060 kg/m3.
Conclusions
Static and vibration characteristics of functionally graded GPL reinforced polymer trapezoidal plates are investigated by using finite element analysis. The influences of GPL distribution, concentration and dimension as well as the plate geometry on the bending and vibration behaviors of the trapezoidal plates are discussed. Numerical results show that apart from GPL concentration, the distribution pattern plays an essential role in the structural performance. GPLs with fewer graphene layers
Acknowledgements
The work described in the present paper is fully funded by research grants from the Australian Research Council under Discovery Project scheme (DP160101978) and Linkage Project scheme (LP150100103, LP140100747). The authors are grateful for the financial support.
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