Elsevier

Composites Part B: Engineering

Volume 75, 15 June 2015, Pages 148-155
Composites Part B: Engineering

Dynamic analysis of functionally graded plates using a novel FSDT

https://doi.org/10.1016/j.compositesb.2015.01.028Get rights and content

Abstract

The purpose of this paper is to study the vibrational behavior of advanced composite plates by using a novel first shear deformation theory (FSDT). This theory contains only four unknowns, with is even less than the classical FSDT. The governing equations are derived by employing the Hamilton's principles and solved via Navier's solution. The present results were validiated by comparing it with the 3D, classical FSDT and other solutions available in the literature. Shear correction factor apper to be unfovarable in some cases (case dependent). Finally, authors recommend further study of this new manner to model the displacement field.

Introduction

Functionally graded materials (FGMs) can be defined as advanced materials having graded transition in mechanical properties, either continuous or in fine, discrete steps, across the interface. This material is produced by mixing two or more materials in a certain volume ratio (commonly ceramic and metal). FGMs have been proposed [1], developed and successfully used in industrial applications since 1980's [2]. These materials were initially designed as a thermal barrier for aerospace structures and fusion reactors. They are now being developed for general use as structural components subjected to high temperatures. Classical composites structures such as fiber reinforced plastic (FRP) suffer from discontinuity of material properties at the interface of the layers and constituents. Therefore the stress fields in these regions create interface problems and thermal stress concentrations under high temperature environments. Furthermore, large plastic deformation of the interface may trigger the initiation and propagation of cracks in the material [3]. These problems can be decreased by gradually changing the volume fraction of constituent materials and tailoring the material for the desired application. The areas where FGM offer potential improvements and advantages in engineering applications include a reduction of in-plane and transverse through-the-thickness stresses, prevention or reduction of the delamination tendencies in laminated or sandwich structures, improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors [4].

Many authors have investigated the dynamic behavior of functionally graded plates (FGPs), mostly, by means of both the classical first-order shear deformation theory (FSDT) and the higher-order shear deformation theories (HSDT). In this paper, relevant works on vibrational analysis of FGM based on the classical and modified FSDTs were reviewed and presented in what follows.

Malekzadeh and Alibeygi [5] presented the free vibration analysis of functionally graded (FG) arbitrary straight-sided quadrilateral plates under thermal environment based on the FSDT employing differential quadrature method (DQM). Hosseini et al. [6] analyzed the free vibration of rectangular plates by using the FSDT and exact close-form solution procedure. Zhu et al. [7] presented the free vibration analysis of FGPs using the local Kriging meshless method. The governing equations of free vibration problem are obtained based on the FSDT and the local Petrov-Garlekin formulation.

Natarajan et al. [8] studied the bending and free vibration behavior of functionally graded sandwich plates employed a C0 8-noded quadrilateral plate element based on HSDT, results based on FSDT were also presented. Valizadeh et al. [9] studied the static and dynamic behavior of FGPs using a non-uniform rational B-spline based iso-geometric finite element method, where the plate kinematics is based on FSDT. Thai and Choi [10] presented a simple FSDT with four unknowns for bending and free vibration analysis of FGPs. The authors divided the transverse displacement “w” into bending and shear parts, “wb” and “ws”.

In the present paper, the free vibration analysis of functionally graded (FG) sandwich and single plates are studied. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power law distribution or Mori-Tanaka homogenization method in terms of the volume fractions of the constituents. The governing equations of the plates are derived by employing the Hamilton's principle. These equations are then solved via Navier solution. The fundamental frequencies are found by solving eigenvalue problem. The accuracy of the present code is verified by comparing it with other HSDTs and 3D and quasi-3D solutions available in literature. Although similar results as the classical FSDT are found, the reduced number of unknowns of this theory plays a key importance in the performance. Consequently, the numerical solution may be of paramount interesting for futures works.

Section snippets

Analytical modelling

The mathematical model was built to solve both FG sandwich and single plates. Plates of uniform thickness “h”, length “a”, and width “b” are shown in Fig. 1a, b. The rectangular Cartesian coordinate system x, y, z, has the plane z = 0, coinciding with the mid-surface of the plates.

The material properties for the single plate (Fig. 1a) vary through the thickness with a power law distribution, which is given below (Fig. 2a):P(z)=(PtPb)V(z)+Pb,V(z)=(zh+12)p,h2zh2.where P denotes the effective

Solution procedure

For the analytical solution of the partial differential equations (Eqs. (10a), (10b), (10c), (10d)), the Navier method, based on double Fourier series, is used under the specified boundary conditions. Using Navier's procedure, the solution of the displacement variables satisfying the simple supported boundary conditions can be expressed in the following Fourier series:u(x,y)=m=1n=1Umncos(αx)sin(βy)eiωt,0xa;0ybv(x,y)=m=1n=1Vmnsin(αx)cos(βy)eiωt,0xa;0ybw(x,y)=m=1n=1Wmnsin(αx)

Numerical results and discussions

In this section the accuracy of the present FSDT which has a displacement field with four unknowns, is evaluated. Numerical examples for free vibration analysis of FG sandwich and single plates with various indexes that specify the material variation profile through the thickness and several values of the side-to-thickness ratio “a/h” and aspect ratio “a/b” are also presented. Typical mechanical properties for metal and ceramics used in the numerical examples are listed in Table 1. In the

Conclusions

This paper presents a free vibrational analysis for FG sandwich and single plates using an original FSDT with 4 unknowns. The governing equations are obtained through the Hamilton's principle. These equations are solved via Navier's method. The fundamental frequencies are found by solving the eigenvalue problem. The results were compared with the solutions of several theories. It is concluded that the results of the present theory with shear correction factor K = 5/6 has an excellent agreement

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