Application of higher-order structural theory to bending and free vibration analysis of sandwich plates with CNT reinforced composite facesheets
Introduction
Engineered materials, mostly inspired from nature have been replacing the conventional materials due to their high strength-to- and stiffness-to-weight ratios. Among the various structural constructions, the sandwich type of structures are more attractive due to its outstanding bending rigidity, low specific weight, excellent vibration characteristics and good fatigue properties. These sandwich constructions can be considered for the requirement of light weight and high bending stiffness in design by appropriate choice of materials. The response of such structures depends on the bonding characteristics. A typical sandwich structure may consist of a homogeneous core with face sheets. To improve the characteristics, the face sheets can be laminated composites [1], functionally graded materials [2] or polymer matrix with reinforcements [3]. Recent introduction of carbon nanotubes (CNTs) as reinforcement has attracted researchers to investigate the responses of such structures [4]. Experimental investigations show that the CNTs have extraordinary mechanical properties than those of carbon fibers [5]. The CNTs are seen as promising reinforcement for sandwich face sheets. Thostenson and Chou [6] showed that the addition of nanotubes increases the tensile modulus, yield strength and ultimate strength of the polymer films. Their study also showed that the polymer films with aligned nanotubes as reinforcements yield superior strength to randomly oriented nanotubes. The properties of the polymer films can also be optimized by varying the distribution of CNTs through the thickness of the film. Moreover, Formica et al. [7] highlighted that the CNT reinforced plates can be tailored to respond to an external excitation. This has generated interest among researchers. For predicting the realistic behavior of sandwich structures, more accurate analytical/numerical models based on the three-dimensional models may be computationally involved and expensive. Hence, among the researchers, there is a growing appreciation of the importance of applying two-dimensional theories with new kinematics for the evolution of the accurate structural analysis. Various structural theories proposed for the laminated structures have been examined and some of the important contributions pertaining to the sandwich laminated plates are discussed here.
Based on the first-order shear deformation theory, Zhu et al. [8] studied the static and free vibration of CNT reinforced plates. They considered polymer matrix with CNT reinforcement, neglecting the temperature effects. It was predicted that the CNT volume fraction has greater influence on the fundamental frequency and the maximum center deflection. Wang and Shen [9] studied the large amplitude vibration of nanocomposite plates resting on elastic foundation using a perturbation technique. The governing equations were based on higher-order shear deformation theory and the composite plates were reinforced with carbon nanotubes. Their study brought out that while the linear frequencies decrease with the addition of CNTs, the non-linear to linear frequency ratio increased, especially when increasing the temperature or by decreasing the foundation stiffness. Arani et al. [10], Liew et al. [11] and Lei et al. [12] studied the buckling and post-buckling characteristics of CNT reinforced plates using the finite element method and meshless methods, respectively. It was shown that the reinforcement with CNT increases the load carrying capacity of the plate. Aragh et al. [13] used generalized differential quadrature and obtained a semi-analytical solution for 3D vibration of cylindrical panels. It was shown that graded CNTs with symmetric distribution through the thickness have high capabilities to alter the natural frequencies when compared to the uniformly distributed or asymmetrically distributed.
It is observed from these investigations that first-order shear deformation theory has been widely employed by many researchers whereas higher-order model considering variation in in-plane displacements has been used by few authors for the analysis of CNT reinforced plates. However, the available literature pertaining to sandwich structures with CNT reinforced face sheets is rather limited compared to plates. Also, to the authors knowledge, the theories accounting the variation of in-plane displacement through the thickness, and the possible discontinuity in slope at the interface, and the thickness stretch affecting transverse deflection is not exploited while investigating the structural behavior of CNT reinforced sandwich structures. A layerwise theory is the possible candidate for this purpose, but it may be computationally expensive as the number of unknowns to be solved increases with increase in the number of mathematical or physical layers. Ali et al. [14] and Ganapathi and Makhecha [15] have used an alternative higher-order plate theory based on global approach, for multi-layered laminated composites by incorporating the realistic through the thickness approximations of the in-plane and transverse displacements by adding a zig-zag function and higher-order terms, respectively. This formulation has proved to give very accurate results for the composite laminates. Such a model for the current problems is considered while comparing with the other approaches available in the literature.
Approach. In this paper, a 8-noded quadrilateral plate element with 13 degrees of freedom per node [15], [16], [17] based on higher order theory [14] is employed to study the static deflection and free vibration analysis of thick/thin sandwich carbon nanotube reinforced functionally graded material plates. The efficacy of the present formulation, for the static analyses subjected to mechanical and thermal loads, and the free vibration study is illustrated through the numerical studies by employing various structural models deduced from the present higher-order accurate theory.
Outline. The paper is organized as follows. The computation of the effective properties of carbon nanotube reinforced composites are discussed in the next section. Section 3 presents the higher order accurate theory to describe the plate kinematics and Section 4 describes the 8-noded quadrilateral plate element employed in this study. The numerical results for the static deflection and the free vibration of thick/thin sandwich carbon nanotube reinforced functionally graded plates are given in Section 5, followed by concluding remarks in the last section.
Section snippets
Theoretical Formulation
Consider a rectangular sandwich plate with co-ordinates x and y along the in-plane directons and z along the thickness direction as shown in Fig. 1. The core-to-facesheet thickness ratio is , where, is the core thickness and is the facesheet thickness. The length, the width and the total thickness of the plate are and h (see Fig. 1). A Cartesian coordinate system is assumed and the origin is located at the corner of the plate on the middle plane. We assume that the CNT reinforced
Higher order accurate theory
The sandwich plate is assumed to be made up of three discrete layers with a homogeneous core. The in-plane displacements and , and the transverse displacement for the layer, are assumed as [14], [16]:The terms with even powers of z in the in-plane displacements and odd powers of z occurring in the expansion
Element description
In this paper, continuous, eight-noded serendipity quadrilateral shear flexible plate element is used. The finite element represented as per the kinematics based on Eq. (5) is referred to as HSDT13 with cubic variation. The 13 dofs are: . Five more alternate discrete models are proposed to study the influence of higher-order terms in the displacement functions, whose displacement fields are deduced from the original element by deleting the appropriate
Numerical results and discussion
In this section, we present the static response and the natural frequencies of sandwich plates with homogeneous core and CNT reinforced facesheets using the eight-noded quadrilateral element. The effect of plate side-to-thickness ratio, thermal environment, CNT volume fraction on the global response is numerically studied. In this study, only simply supported boundary conditions are considered and are as follows:where a and b refer
Conclusions
The static and dynamic responses of sandwich plates with CNT reinforced facesheets are studied considering various parameters such as the sandwich type, temperature effects, the thickness ratio and the volume fraction of the CNT. Different plate models are employed in predicting the global structural behavior and their through thickness variations in the sandwich plate. From a detailed parametric investigation on the effectiveness of higher-order models (HSDT13, HSDT11A and HSDT11B) over the
Acknowledgement
S. Natarajan would like to acknowledge the financial support of the School of Civil and Environmental Engineering, The University of New South Wales for his research fellowship for the period September 2012 onwards.
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