Shock and Vibration

Volume 2016, Article ID 1403856, 15 pages

http://dx.doi.org/10.1155/2016/1403856

## Three-Dimensional Elasticity Solutions for Sound Radiation of Functionally Graded Materials Plates considering State Space Method

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

Received 8 June 2015; Revised 17 August 2015; Accepted 19 August 2015

Academic Editor: Sergio De Rosa

Copyright © 2016 Tieliang Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an analytical study for sound radiation of functionally graded materials (FGM) plate based on the three-dimensional theory of elasticity. The FGM plate is a mixture of metal and ceramic, and its material properties are assumed to have smooth and continuous variation in the thickness direction according to a power-law distribution in terms of volume fractions of the constituents. Based on the three-dimensional theory of elasticity and state space method, the governing equations with variable coefficients of the FGM plate are derived. The sound radiation of the vibration plate is calculated with Rayleigh integral. Comparisons of the present results with those of solutions in the available literature are made and good agreements are achieved. Finally, some parametric studies are carried out to investigate the sound radiation properties of FGM plates.

#### 1. Introduction

Functionally graded materials (FGM) are the heterogeneous composite materials with material properties varying smoothly and continuously in one or more directions, and this characteristic properties are usually achieved through continuous change of the volume fraction of the constituent phases [1–3]. The concept of FGM was originally proposed in 1984 by a group of material scientists in Japan as thermal barrier materials, and the superiority to conventional laminated composites of such as eliminating the stress concentration leads to a wider applications of FGM in areas such as aeronautics, astronautics, nuclear, biology, navigation [4]. In practical applications, FGM structures subjected to dynamic load internal or external will generate noise and radiate sound into the surrounding medium, which may result in less comfort. On the other hand, the radiated sound carries useful information of the FGM structures that can be used for nondestructive evaluation or estimation of material properties. Therefore, the investigation of sound radiation of FGM structures is of great importance from the academic or engineering applications point of view.

Plates are one of the most widely used structural components in industrial applications. Sound radiation from panel structures is a practical engineering problem that has been studied extensively. Numerical methods such as finite element method (FEM) and boundary element method (BEM) [5–7] are always utilized to estimate the sound radiation of structures. However, these methods are computationally expensive, especially at high frequency domain. An infinite extent of plate is always obtained in the research of sound radiation of plate. However, the plate structures are of finite size in practice. Several theories were proposed to take into account the finite size of a plate structure in sound radiation calculation, for example, the spatial windowing technique presented by Villot et al. [8]. As a more general method, the vibration plates are assumed to be placed in an infinite rigid baffle, which, of course, differs from the actual situation in practice, and some authors concentrated on sound radiation from unbaffled plates [9, 10]. One of the authors of this paper studied the sound radiation characteristic from unbaffled rectangular plates [11]. More recently, Putra and Thompson [12, 13] also investigated this problem. However, the assumption of an infinite rigid baffle makes the sound radiation problem easier to solve for the fact that the velocity field equals zero everywhere except for the plate surface. In this paper, the assumption of an infinite rigid baffle is also obtained for the convenience in studying the main trends of sound radiation from FGM plate.

Two-dimensional (2D) plate theories such as classic plate theory (CPT), first-order shear deformation theory (FSDT), and higher order shear deformation theories are always utilized in the research of sound radiation of plate-like structures; however, acoustic model based on three-dimensional (3D) elasticity theory may be the sole ultimate choice for more precise calculations. Due to the simplicity, the well-known classic plate theory [14–16] is utilized conventionally to model the plate in the calculation of sound radiation from plate structures. The CPT is based on the Kirchhoff–Love hypothesis that straight lines perpendicular to the plane of the undeformed plate remain straight and inextensible and rotate such that they always remain perpendicular to the midplane of the plate after deformation. Nevertheless, neglecting the transverse shear effects and rotary inertia leads to overestimating the natural frequencies of the plate, especially for thick plates or structural response in high frequency range. In order to consider the transverse shear effects on isotropic plates, Reissner [17] and Mindlin [18] developed the FSDT. Hashemi et al. [19] investigated acoustic radiation of rectangular Mindlin plates in different combinations of classical boundary conditions. Cao et al. [20] investigated the sound radiation from infinite stiffened laminated plates theoretically based on the FSDT, and the comparison with Yin’s method [21], which is based on CPT, shows that the model based on FSDT and CPT shows a good agreement about the estimation of sound pressure level in the low and medium frequency range, but discrepancies can be found near the coincidence frequency and in the high frequency range. Chandra et al. [22] analytically studied the vibroacoustic response and sound transmission loss characteristics of FGM plates based on a simple FSDT, which was presented by Thai and Choi [23]. As is well-known to all, a shear force correction factor is required in FSDT to take in account the nonuniformity of the shear strain distribution through the thickness. The shear correction factor is typically 5/6 [17] or [18] for isotropic homogeneous plates; however, the constant shear correction factor is not appropriate for FGM plates due to the material properties and geometric dimension of FGM plates [24, 25]. In order to overcome this shortcoming of FSDT, third-order shear deformation theory (TSDT) or higher order shear deformation theory such as Reddy [26, 27] was developed, and no shear correction factors are required and better precision can be achieved. Daneshjou et al. [28] studied sound transmission through relatively thick FGM cylindrical shells based on third-order shear deformation shell theory, and their work denotes that there are some discrepancies between FSDT and TSDT at high frequencies for relatively thick FGM cylindrical shells. However, studies of sound radiation from plate structures based on 3D elasticity may be the sole ultimate choice, which is, however, relatively scarce. Hwang et al. [29] presented an elasticity theory solution for acoustic radiation by a point- or line-excited fluid-loaded laminated plate. Shen et al. [30] studied acoustic radiation from multilayered anisotropic plates based on 3D elasticity model. Hasheminejad and Keshavarzpour [31] studied the active sound radiation control of thick piezolaminated smart plate, and the orthotropic laminated plate is modeled based on 3D piezoelasticity theory with the use of state space formulation. Huang and Nutt [32] developed a unique analytical formulation for sound transmission of unbounded FGM panels by employing three-dimensional theory of elasticity and state space method.

In the present paper, the sound radiation of FGM plates with arbitrary thickness is investigated. An analytical model of sound radiation of FGM plates based on the three-dimensional theory of elasticity is proposed. By means of state space formulation and modal expansion, the three-dimensional governing equations of elastodynamics are converted into a set of ordinary differential equations with variable coefficients for FGM plate. The solution of the ordinary differential equations results in the transfer matrix relating the top and bottom surface of the plate. Some similar procedures were also used in prior plates studies such as vibration [33–35], wave propagation [36–38], and sound transmission [32]. The external loads including point load, line load, and distributed load are expanded into a double Fourier series form and applied on the surface. Applying the continuity of displacement and stresses at the interfaces of plate leads to global governing equations of the vibroacoustic system, and then sound radiation of the plate is calculated using Rayleigh integral with a primitive numerical scheme.

#### 2. Theoretical Formulation

Consider a rectangular FGM plate with length , width , and an arbitrarily constant thickness , as shown in Figure 1. A Cartesian coordinate system is located at the corner of the panel on the bottom surface of the plate. The plate is assumed to be placed in an infinite rigid baffler. The FGM plate is under the excitation of external load on the surface and the fluid structure interaction between the plate and the surrounding air is not considered.