Free bending vibration analysis of thin bidirectionally exponentially graded orthotropic rectangular plates resting on two-parameter elastic foundations

doi:10.1016/j.compstruct.2017.10.014

Composite Structures

Volume 184, 15 January 2018, Pages 372-377

https://doi.org/10.1016/j.compstruct.2017.10.014 Get rights and content

Abstract

The vibration of biderctionally exponentially graded orthotropic plates (BEGOPs) resting on the two-parameter elastic foundation is studied. Pasternak elastic foundation (PEF) model is used as two-parameter foundation model. The heterogeneity of the orthotropic exponentially changes depending on the axial and thickness coordinates. The motion equation is derived based on the classical plate theory and solved by using Galerkin method. To validate of current results was made a comparison with the previous studies. The effects of material gradient and orthotropy, and the two-parameter elastic foundations on the dimensional frequency parameters (DFPs) are investigated.

Introduction

The wide use of modern composites in various products of modern technology required not only the development of traditional methods for the analysis of thin-walled plates, but also the formulation of new tasks and revealed the need to take into account the new main factors that determine the bearing capacity of structures. Among these factors, anisotropy and heterogeneity of the material occupy an important place. These factors introduce additional complexity into the study of the vibration and stability problems of composite structures. A great contribution to the theory of anisotropic plates was made the work Reddy [1].

Inhomogeneous structures are often used in technical designs that take full advantage of continuous and gradual changes in the physical and mechanical properties of the material. Such structures are widely used in aviation, aerodynamic structure, space vehicles, light-alloy structure of cars and in other engineering structures. Compared to homogeneous orthotropic plates, the adoption of continuous change of material properties can provide important benefits. Indeed, the increase in the number of constructive variables extends the possibilities of advanced composite materials, as well as stability and vibration behaviors may be significantly altered. The reason for the appearance of heterogeneity of the material can be, manufacturing technology, thermal and mechanical treatment, heterogeneity of compositions and a number of other reasons. As a result of the above reasons, the inhomogeneity can simultaneously depend on the spatial coordinates. The basic knowledge on the changes of the material properties is given in the work of Lomakin [2]. Efforts related to the determination of various types of functionally graded anisotropic materials have been the focus of research in recent years [3], [4], [5], [6]. Using above mentioned models, several important problems were solved about the oscillations of the functionally graded orthotropic plates [7], [8], [9], [10], [11], [12].

In many practical applications, composite plates are in contact with soils or other solid particles and can have significant and unavoidable effect on their behaviors. To correctly determine the influence of the elastic foundation, there are various models, among which one of the effective model was proposed by Pasternak, which is called a two-parameter elastic foundation [13]. Besides, a comprehensive review of elastic foundation models is discussed in the Ref. [14]. The vibration of homogeneous orthotropic plates resting on the two-parameter elastic foundations, which has practical applications in civil, mechanical, marine and aerospace engineers have been studied using various analytical and numerical methods [15], [16], [17], [18], [19], [20], [21].

In recent years, the urgency of solving the stability and vibration problems of functionally graded composite plates has increased dramatically. This is explained, first of all, by the continuous expansion of the introduction of inhomogeneous composite plates into load-bearing elements of structures working in contact with different environments. The numerous studies on the vibration of functionally graded orthotropic plates resting on the Pasternak elastic foundation have been published in the literature [22], [23], [24], [25], [26], [27], [28], [29]. In the majority of the above mentioned studies, the change in the elastic properties of FG orthotropic materials was carried out as the function of thickness or axial coordinates, separately. The main contribution to this study is made by the development and implementation of the vibration analysis for thin exponentially graded (EG) orthotropic plates which the material properties vary depending on the axial and thickness coordinates together and resting on the Pasternak elastic foundation.

Section snippets

Formulation of the problem

The configuration of rectangular biderctionally exponentially graded orthotropic plate (BEGOP) with the length $a$ , the breadth $b$ and the thickness $h$ and resting on the Pasternak elastic foundation (PEF) is illustrated in Fig. 1. The plate referred to a system of rectangular coordinate system Oxyz. The mid-plane being $z = 0$ and the origin is at one corners of the orthotropic plate. The $x$ and $y$ axes are taken along the principle directions of orthotropy and $z$ axis is normal to the them. The reaction

Basic equation

Based on the classical plate theory (CPT), the relationships between the stresses and strains at an arbitrary point of the BEGOPs are written in the following form [2], [3], [4], [5], [6], [7]: $σ_{11} = \frac{E_{1}^{0} α_{1}^{X} α_{2}^{Z + 05}}{1 - ν_{12} ν_{21}} (ε_{11} + ν_{12} ε_{22}), σ_{22} = \frac{E_{2}^{0} α_{1}^{X} α_{2}^{Z + 05}}{1 - ν_{12} ν_{21}} (ε_{22} + ν_{21} ε_{11}) σ_{12} = G_{12}^{0} α_{1}^{X} α_{2}^{Z + 05} ε_{12}$

Let us assume that the Kirchhoff-Love hypotheses are valid for the BEGOPs, and have [1] $ε_{11} = e_{11} - z \frac{\partial^{2} w}{\partial x^{2}}, ε_{22} = e_{22} - z \frac{\partial^{2} w}{\partial y^{2}}, ε_{12} = e_{12} - 2 z \frac{\partial^{2} w}{\partial x \partial y}$ where $e_{11}, e_{22}, e_{12}$ are the strains in the mid-plane.

The force and moment

The solution of equation of motion

We assume that the boundary conditions for the bending of continuous the BEGOP coincide with the usual ones in the homogeneous isotropic plate.

We take the harmonic solution of Eq. (13) in the form [15], [16] $w (x, y, t) = A \sin \frac{m π x}{a} \sin \frac{n π y}{b} e^{i ω t}$ which satisfies the movable simply-supported boundary conditions edges of the BEGOPs, here $i = \sqrt{- 1}$ , $m$ and $n$ positive integers and A is the unknown amplitude.

Substituting (14) into Eq. (13) and applying Galerkin method, after integrating we obtain expression for the

Comparative studies

In first example, the values of the DFP, $ω_{1} = ω (a^{2} / h) \sqrt{ρ^{0} / E_{2}^{0}}$ , for square HOPs without elastic foundations for $a / h = 100$ are compared with the Levy type solution of Thai and Kim [22] and presented in Table 1. The following material properties were used in the comparison: $E_{1}^{0} / E_{2}^{0} = 10, 25, 40; G_{12}^{0} = 0.5 E_{2}^{0}; ν_{12} = 0.25; ρ_{0} = 1 kg / m^{3} .$ The values obtained in this study are in a good agreement with those obtained in the study of the Thai and Kim [22].In second example, the values of dimensionless frequency parameter, $ω_{1} =$

Conclusions

Based on the CPT, the free vibration of BEGOPs resting on the two-parameter elastic foundations is studied. The PEF model is used as two-parameter elastic foundation model. The biderctionally graded profiles of the orthotropic materials vary depending on the axial and thickness coordinates. The motion equation is derived based on the classical plate theory and solved by using Galerkin method. To validate of current results was made a comparison with the previous studies. The effects of material

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Free bending vibration analysis of thin bidirectionally exponentially graded orthotropic rectangular plates resting on two-parameter elastic foundations

Abstract

Introduction

Section snippets

Formulation of the problem

Basic equation

The solution of equation of motion

Comparative studies

Conclusions

J Sound Vib

Compos Struct

Eng Anal Boundary Ele

Thin-Walled Struct

J Sound Vib

Compos Struct

Int J Eng Sci

Int J Mech Sci

Computer Struct

Compos Struct

Compos B Eng

Compos Struct

Compos B Eng

Int J Mech Sci