Free vibration analysis of a laminated composite sandwich plate with compressible core placed at the bottom of a tank filled with fluid
Introduction
In recent years, sandwich structures have been widely used in various applications of aerospace, marine, mechanical, and civil engineering [1], [2]. The main advantage of these structures is their high strength and stiffness to weight ratios [3], [4], [5]. In many cases, sandwich structures are requested to operate under severe environmental conditions, such as high temperatures and wet environments [6]. These structures are typically made of an inner core and two outer face sheets [7]. Roughly speaking, the core is responsible for shear stiffness and strength, while the face sheets provide bending stiffness and strength. But, the actual behavior of a sandwich plate may be much more complicated due to the high diversity of strength and material properties between the core and face sheets. Cores can be divided into two main categories: hard (incompressible) cores, and soft (compressible) cores. In any case, the mechanical properties of the core play an important role in the overall behavior of a sandwich plate. Therefore, developing a suitable mathematical model is vital to obtain the accurate response of the sandwich plates.
In the literature, two main approaches have been proposed for the analysis of sandwich plates: three-dimensional (3D) elasticity and two-dimensional (2D) structural theories. In turn, two types of models have been used in 2D structural theories, namely the equivalent single layer (ESL) and layer-wise (LW) models.
In the 3D elasticity approach, each layer is represented as a continuous solid body. Therefore, the computational efforts needed to analyze laminated composite plates may be significant. Among the 3D elasticity approaches, it is worth recalling the pioneering work by Pagano [8], who presented the exact solution for the static analysis of composites and sandwich plates. Noor et al. [9] investigated the buckling and free vibration of sandwich plates with composite face sheets by using a 3D solution. Kardomateas [10] described the elastic behavior of sandwich plates by using a 3D solution and considering the effect of normal transverse deformation of the core.
ESL models are used to convert a multi-layered sandwich plate into a single equivalent layer based on constitutive relations between strain and stress. This approach includes the classical lamination theory (CLT) based on Kirchhoff’s assumptions, first-order shear deformation theory (FSDT) based on Reissner–Mindlin assumptions, and higher-order shear deformation theories (HSDTs). Whitney [11] used FSDT to obtain the stress distribution in sandwich plates. Wang [12] applied FSDT for the vibrational analysis of simply supported sandwich plates. Kant and Swaminathan [13], [14] developed an exact solution for the static and free vibration characteristics of simply supported sandwich plates based on a higher-order theory. A similar higher-order theory was used by Swaminathan et al. [15], [16] for the bending and free vibration analysis of antisymmetric angle-ply sandwich plates. Meunier and Shenoi [17], Nayak [18], Babu and Kant [19] used the finite element method (FEM) to investigate the behavior of sandwich plates.
For thick laminated composite sandwich plates, the ESL approach leads to invalid results [20]. Better accuracy can be obtained by using the LW approach, whereas each layer, i.e. core and face sheets, is considered as an individual plate. Rao and Desai [21] used the LW approach along with higher-order theory to study the vibrational behavior of simply supported sandwich plates. Pandey et al. [22] performed both dynamic and static analyses of sandwich plates using a LW finite element formulation. Frostig et al. [23] proposed a higher-order sandwich plate theory (HSAPT), whereas the core is considered to be compressible and its in-plane stresses are neglected. Also, Frostig et al. [24] investigated the non-linear wrinkling of a sandwich panel with functionally graded core by using the extended higher order sandwich panel theory (EHSAPT). Shear buckling response of sandwich panels with compressible and incompressible cores was reported by Frostig [25] by using the HSAPT and EHSAPT methods. Also, regarding the layer-wise method, Khalili et al. [26] presented an analysis of the vibration characteristics of sandwich plate accounting for the influence of temperature. It should be noted that the EHSAPT is the only approach, in which both the in-plane and out-of-plane stresses of the core are considered. Instead, the HSAPT neglects the in-plane stresses of the core. Another important advantage of EHSAPT compared with HSAPT is that EHSAPT includes higher modes through the core depth, which cannot be predicted by HSAPT.
Over the last few decades, the interaction between solid plates and fluids has been increasingly investigated because of its vast applications in various industrial fields, such as space vehicles, shipbuilding, submarines, aeronautics, and ocean engineering [27], [28]. As an example, we recall just a few of such studies without any ambition of being exhaustive. Amabili et al. [29] analyzed the vibrational frequencies of annular plates located in the aperture of an infinite wall. The vibration analysis of the horizontal and vertical rectangular plates submerged in a fluid was studied by Fu and Price [30]. Amabili and Kwak [31] utilized both numerical and analytical methods to obtain the natural frequencies of a circular plate located into the hole of a rigid wall in contact with fluid. Cheung and Zhou [32] used the Ritz approach to investigate the vibrational characteristics of an isotropic plate immersed in a fluid. In [33], the same authors used the Galerkin approach to investigate the free vibrations of a thin circular plate coupled with a fluid. By adopting the Rayleigh-Ritz approach, Amabili [34] investigated the influence of fluid depth on the natural frequencies of circular and annular plates. The size-dependent free vibration analysis of a rectangular microplate in contact with fluid was studied by Omiddezyani et al. [35]. Khorshidi and Karimi [36] presented an analytical solution for a vibrating piezoelectric nano-plate coupled with an inviscid fluid. Also, by using a different modified plate approach, Canales and Mantari [37], [38] obtained the free vibrations of a rectangular laminated composite plate immersed in a fluid. Liao and Ma [39] reported vibration characteristics of a rectangular plate coupled with fluid. Moreover, Watts et al. [40] performed the free vibrational analysis of skew and trapezoidal plates immersed in a fluid based on the element free Galerkin approach.
The Rayleigh-Ritz method is commonly adopted to obtain the free vibration response of structures. In this approach, often trigonometric functions are used to represent the displacement parameters. For instance, Aiello and Ombres [41] used trigonometric functions to analyze the buckling and vibration of unsymmetric laminates resting on an elastic foundation. Other commonly used base functions are polynomials. For instance, 2-D orthogonal polynomials were used by Chow et al. [42] to investigate the vibrational characteristics of symmetrically laminated plates. Orthogonal two-variable polynomials were produced by using the Gram-Schmidt process by Bhat [43] to analyze the flexural vibration of polygonal plates. The free vibrations of a rectangular plate were studied by Liew [44] by using a set of orthogonal plate functions constructed using the Gram-Schmidt process. Kumar [45], Pablo et al. [46], and Chakraverty et al. [47] presented a comprehensive review of the Rayleigh-Ritz method and its application. As mentioned by Nallim et al.[48], convergence occurs much faster when orthogonal polynomials functions are used.
This paper investigates the free vibration problem of a rectangular laminated composite sandwich plate with compressible core, placed at the bottom of a tank filled in with fluid.
To the best of the authors’ knowledge, although the vibrational behavior of some types of plates, such as isotropic and laminated composite plates in the presence of fluid has been studied by other researchers, the vibrational characteristics of sandwich plates with flexible core which is placed at the bottom of a tank filled with fluid has not yet been obtained in the literature, thus the present paper is an effort to fill this gap. The main objective of the current paper is to achieve the natural frequencies of the fluid-plate coupled system. To this aim, the EHSAPT [24] is used to consider both the in-plane and out-of-plane stresses of the sandwich core. In this respect, the distributions of the core in-plane and transverse displacements are supposed to be cubic and quadratic, respectively. Besides, a single series is used with two-variable orthogonal polynomials (orthogonal plate functions) [44] to apply the Rayleigh-Ritz method.
The rest of this paper is organized as follows. In Section 2, the mathematical formulation for the fluid-plate system is presented. Section 3 deals with the proposed solution method. In Section 4, the convergence studies are presented, as well as comparisons of the obtained numerical results with alternative methods. In Section 5, numerical results and discussion are reported. Finally, Section 6 includes the conclusion of the present paper.
Section snippets
Sandwich plate model
A schematic of a tank filled in with fluid is illustrated in Fig. 1. Here, c, d, and e are the length, width, and height of the tank, respectively. A coordinate system is fixed with the origin located at a corner of the tank bottom. The tank bottom and walls are assumed to be rigid, except for a part of the bottom with length a and width b, which is modeled as a flexible rectangular sandwich plate with compressible core. The fluid is assumed to be inviscid, irrotational, and
Governing differential equations and corresponding boundary conditions
The Hamilton’s principle for the considered case can be written as follows:where and denote the strain and kinetic energies of the sandwich plate, respectively. It is noted that and are the summation of and , respectively. Finally, by substituting these energies into Hamilton’s principle, the governing differential equations of motion and the corresponding boundary conditions are derived (see Appendix A).
Convergence and comparison studies
This section aims to provide convergence and comparison studies. At first, a convergence study is presented. This was performed by comparing the first six natural frequencies of sandwich plates. Then, comparison studies were conducted to compare the results of this study with those obtained by other methods proposed in the literature. The material properties used for the convergence and comparison studies are listed in Table 1.
Numerical results and discussion
Here, numerical results are presented based on the proposed analytical solution for the free vibrations of sandwich plates with three different boundary conditions. In terms of the environmental conditions, the numerical results are presented for the sandwich plates in dry environment and in contact with a fluid. The rest of this section is divided into six parts.
Conclusions
In this paper, the fluid–structure interaction is examined of a rectangular sandwich plate with compressible core placed at the bottom of a rigid tank filled in with fluid. The extended higher order sandwich plate theory (EHSAPT) has been used to model the plate. Besides, the fluid has been supposed to be inviscid, incompressible, and irrotational. To obtain the vibrational characteristics of the plate, the Rayleigh-Ritz method with two-variable orthogonal polynomials has been used. In the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The first and second authors acknowledge the funding support of Babol Noshirvani University of Technology through Grant program No. BNUT/964113035/97. Likewise, the third author acknowledges the funding support of the University of Pisa through the PRA 2018-2019 project“ Multiscale Modelling in Structural Engineering”
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