Free vibration of advanced composite plates using a new higher order shear deformation theory

Highlights

• A new shear deformation theory is introduced to conduct free vibration analysis of the functionally graded plates.

• The shear functions presented in this study vary with gradient index and satisfy the stress-free boundary conditions.

• The displacement field is expressed as undetermined integral terms.

• The material properties are assumed to vary through the thickness according to the power-law.

• The amplitude-frequency relationships of the FGM plates are presented and compared with the exsiting results.

Abstract

A new shear deformation theory is introduced to conduct free vibration analysis of the functionally graded plates with simply supported boundary conditions. The shear functions presented in this study vary with gradient index and satisfy the stress-free boundary conditions on the top and bottom surfaces of functionally graded plates without using any shear correction factor. The displacement field is expressed as undetermined integral terms. The governing differential equation and boundary conditions are derived based on Hamilton's principle. The material properties of the functionally graded plates discussed in this paper are assumed to vary through the thickness according to the power-law distribution and the Mori-Tanaka scheme. The amplitude-frequency relationships of the FGM plates are presented and compared with the exsiting results and the obtained numerical results are compared with other 2D and quasi-3D solutions to verify the accuracy and efficiency of the present theory.

Introduction

Functionally graded materials (FGM), are a new category of composite materials, which are isotropic and inhomogeneous and fabricated from two constituent phases with a defined volume fraction. These two material properties vary smoothly and continuously throughout desired directions (usually the thickness direction only). Thus, FGM can not only combine the superior features of ceramic (high temperature and corrosion resistance) and metal (great toughness), but also reduce both thermal and residual stresses which are very common in conventional multilayered composite structures (Koizumi, 1997; Reddy, 2000; Reddy and Cheng, 2002; Thai et al., 2014a; Wang, 1983). In recent years, FGM has gradually become a research hot topic and played an important role in aerospace, nuclear, shipbuilding, automotive and other engineering fields.

In recent years, extensive studies relevant to free vibration analysis of functionally graded plates are carried out due to extensive utilization of FGM plates in the above-mentioned fields. Mantari et al. (2014a, 2012a); Mantari and Guedes Soares, 2013a, Mantari and Guedes Soares, 2013b, 2012a, b, c) investigated the bending and free vibration behavior of functionally graded and multilayered plates using new shear deformation theories. Then, Mantari et al. (2014b, 2014c) presented a new HSDT to investigate the free vibration of FG rectangular plates resting on elastic foundations. Vel and Batra (2004) presented the free vibration analysis of functionally graded rectangular plates by a three-dimensional exact solution. Xiang and Kang (2013) utilized an nth-order shear deformation theory to perform the bending analysis of functionally graded plates. A study on transverse free vibration of Levy-type rectangular plates was provided by Hosseini-Hashemi et al. (2011) by using an exact analytical solution. Thai et al. (2014b) investigated the bending, vibration and buckling behavior of FG sandwich plates using a new first-order shear deformation theory. Duc and Quan (2013a) studied the nonlinear post-buckling behavior of the stiffened FGM double curved shallow shells using the classical shell theory. Kar Jin et al. (2014) investigated the free vibrations of thick FGM plates with general boundary conditions by employing a 3D exact solution. Sahraee and Saidi (2009) developed a fourth-order shear deformation theory to study the axisymmetric bending response of thick functionally graded circular plates. Neves et al. (2013a) assessed the free vibration analysis of FG shells using an HSDT and radial basis functions. Najafizadeh and Heydari (2004) studied thermal buckling responses of FG circular plates based on a higher order shear deformation plate theory. Duc and Quan (2013a, 2013b) analyzed the nonlinear dynamic responses of double curved thin shallow shells are investigated using an HSDT, Bubnov-Galerkin method and Runge-Kutta method. Ebrahimi and Salari (2015) studied the vibration and buckling of functionally graded nano-beams based on a semi-analytical method. Nonlinear static and vibration analysis of FGM double curved shallow shell resting on elastic foundation using Reddy's HSDT and the Galerkin method were studied by Quan and Dinh Duc (2017). Free vibration analysis of FGM plates on Pasternak foundation using a refined shear deformation theory was conducted by Thai and Choi (2012). Hebali et al. (2014) developed a new quasi-3D HSDT for the analysis of static bending and free vibration of FG plates by dividing the transverse displacement into bending, shear, and thickness stretching parts. Chaudhuri and Kabir (1993b), Kabir and Chaudhuri (1994) developed a boundary-continuous displacement-based Navier solution technique to study the free vibration of arbitrarily laminated plates. Free vibration analysis of the FGM plates using a meshless method and a higher order shear deformation theory were investigated by Roque et al. (2007). Vel and Batra (2004) used a three-dimensional analytical solution to research free and forced vibrations of simply supported functionally graded rectangular plates.

The equivalent single layer theories include the classical plate theory (Green, 1944; Whitney, 1971), the first-order shear deformation theory (Chaudhuri and Kabir, 1993a; Kabir et al., 2001; Kabir and Chaudhuri, 1992) and the high-order shear deformation theory (Oktem and Chaudhuri, 2007a, 2007b; Reddy, 2003). According to equivalent single layer theories, all layers are assumed to be one homogeneous single layer to simplify calculation (Thai et al., 2017). So far, the studies on equivalent single layer theory to study the mechanical behavior of functionally graded structures have received much attention in the existing literature. Basset (1890) suggested that the displacements can be expanded in power series of the thickness coordinate for extension and flexure analysis of thin elastic shells. Vu et al. (2019) studied the bending, free vibration and buckling behavior of FG plates using an inverse sin shear deformation plate theory and a meshfree method. Oktem et al. (2012); Oktem and Chaudhuri (2009); Oktem and Guedes Soares (2011,2012) have given bending results for FGM plates, cross-ply plates and doubly-curved shells using higher order shear deformation theories. Duc (2014) employed an HSDT for nonlinear static and dynamic stability analysis of FG plates and shells. Carrera et al. (2011) accounted for the effect of thickness stretching in functionally graded (FGM) plate and shell structures using quasi-3D higher shear deformation theory and Carrera's Unified Formulation. Kharghani and Guedes Soares (2020a, 2020b, 2018) investigated the bending and buckling behavior of rectangular composite laminates by using equivalent single layer and layer-wise theories. Buckling and post-buckling responses of sandwich toroidal shell segments with functionally graded core and homogeneous face sheets based on TSDT and a new Galerkin method were studied by Vuong and Duc (2020). A classic 5-variable shear deformation is presented by Zenkour (2006) to study the static response simply supported FG rectangular plate under a transverse uniform load. Mantari and Guedes Soares (2012b, 2012c), Mantari et al. (2012d) presented new HSDTs for bending analysis of isotropic, laminates and composite sandwich plates. Abualnour (2018) developed a new quasi-3D shear deformation theory for vibration analysis of FGM plates by presenting a new displacement field function that contains undetermined integral terms. A study on bending and free vibration of functionally graded plates resting on Winkler-Pasternak elastic foundation was conducted by Meksi et al. (2015) using a new first-order shear deformation plate theory. Anh and Dinh Duc (2016) used a first-order shear deformation theory and the Galerkin method for the nonlinear stability of FG shallow spherical shells. Li et al. (2020) presented a new shear deformation theory with a changeable shear strain function to study the static response of FG plates. Nguyen et al. (2012) presented a first-order shear deformation theory and a finite element method for the mechanical and thermal loads of FGM plates. Thai and Choi (2013) investigated the bending and free vibration of FGM plates by using the various higher order shear deformation theories with cubic, sinusoidal, hyperbolic, and exponential displacement field functions. Thai and Kim (2013a) studied the bending behavior of FGM plates by using quasi-3D sinusoidal shear deformation theory with fewer unknowns. Mantari and Guedes Soares (2015a, 2014a) applied non-polynomial shear deformation theories and Navier-type solution to determine the bending characteristics of the FGM plates. a higher order shear deformation theory and the boundary-discontinuous double Fourier series technique were applied by Oktem and Chaudhuri (2008a, 2008b) to investigate the effect of various boundary constraints on the response of thick laminates. A vibration solution for isotropic plates resting on multi-segment Winkler-type elastic foundations using classical first-order shear deformation theory has been presented by Xiang (2003). Thermoelastic bending analysis of FGM plates resting on Pasternak foundation using sinusoidal shear deformation theory were investigated by Zenkour (2009). Analysis of vibration behavior of FGM plates with arbitrary boundary conditions using third-order shear deformation theory were conducted by Baferani et al. (2011). The static response of advanced composite plates and shells was presented by Mantari and Guedes Soares (2014b, 2014c, 2015b) using quasi-3D non-polynomial sinusoidal higher-order shear deformation theories. Zenkour (2005a, 2005b); Zenkour and Sobhy (2010) used a sinusoidal shear deformation plate theory for the static, buckling and free vibration analysis of a simply supported functionally graded sandwich plate.

As mentioned above, it can be easily noticed that free vibration of functionally graded plates have been paid much attention due to the importance of vibration in automotive, aerospace, shipbuilding and other engineering fields. However, to the authors' knowledge, there has hardly been an attempt to develop a higher-order shear deformation theory with changeable shear functions and a full-integral displacement field for the free vibration analysis of FGM plates. So, this study presents a new HSDT for free vibration analysis of simply-supported FGM plates. One novelty of this paper is the use of two transverse shear strain functions in which the classical trigonometric and parabolic shear strain shape functions are reformulated with a very small exponential perturbation. The present transverse shear strain functions give a more accurate distribution of the transverse shear stress in the thickness direction without using any transverse shear correction factors. Another novelty of this study is the use of the integral format of the displacement field, which is developed to reduce the differential operations and the probability of errors during the derivation of equations. The equations of motion are derived based on Hamilton's principle and Navier's procedure. The non-dimensional fundamental frequencies of FGM plates are obtained by solving an eigenvalue problem. The accuracy of the proposed HSDT is verified by comparing the obtained numerical results with other results in the existing literature. Moreover, a parametric study will be also carried out to study the effects of side-to-thickness ratios, aspect ratios and power-law indexes on the fundamental frequency of FGM plates.