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Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

Applied Mathematical Modelling (Impact Factor: 2.25). 05/2010; 34:1276-1291. DOI: 10.1016/j.apm.2009.08.008

ABSTRACT

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.

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Available from: Shahrokh Hosseini-Hashemi
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    • "Zenkour [17] presented a generalized shear deformation theory in which the in-plane displacements are expanded sinusoidally across the thickness. Hosseini-Hashemi et al. [18] presented free vibration of functionally graded rectangular plates resting on elastic foundations, using first-order shear deformation plate theory. More recently, Thai et al. [19] formulated a simple first order shear deformation theory for the bending and free vibration analysis of FGMs with only four unknowns, by defining the shear strains in terms of transverse displacement. "
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    ABSTRACT: The vibration, sound radiation and transmission characteristics of plates with various functionally graded materials (FGM) are explored and a detailed investigation is presented on the influence of specific material properties on structural-acoustic behavior. An improved model based on a simplified first order shear deformation theory along with a near-field elemental radiator approach is used to predict the radiated acoustic field associated with a given vibration and acoustic excitation. Various ceramic materials suitable for engineering applications are considered with aluminum as the base metal. A power law is used for the volume fraction distribution of the two constitutive materials and the effective modulus is obtained using the Mori-Tanaka homogenization scheme. The structural-acoustic response of these FGM plates is presented in terms of the plate velocity, radiated sound power, sound radiation efficiency for point and uniformly distributed load cases. Increase in both vibration and acoustic response with increase in power law index is observed for the lower order modes. The vibro-acoustic metrics such as root-mean-squared plate velocity, overall sound power, frequency averaged radiation efficiency and transmission loss, are used to rank these materials for vibro-acoustically efficient combination. Detailed analysis has been made on the factors influencing the structural-acoustic behavior of various FGM plates and relative ranking of particular ceramic/metal combinations.
    Full-text · Article · Oct 2015 · International Journal of Applied Mechanics
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    • "Hosseini-Hashemi et al. [14] presented the free vibration of FG rectangular plates resting on elastic foundation for various boundary conditions using the first order shear deformation theory. They neglected the in-plane displacement components for finding the governing equations and obtained the natural frequencies of the plate with this simplification. "
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    ABSTRACT: Vibration analysis of a functionally graded rectangular plate resting on two parameter elastic foundation is presented here. The displacement filed based on the third order shear deformation plate theory is used. By considering the in-plane displacement components of an arbitrary material point on the mid-plane of the plate and using Hamilton’s principle, the governing equations of motion are obtained which are five highly coupled partial differential equations. An analytical approach is employed to decouple these partial differential equations. The decoupled equations of functionally graded rectangular plate resting on elastic foundation are solved analytically for levy type of boundary conditions. The numerical results are presented and discussed for a wide range of plate and foundation parameters. The results show that the Pasternak (shear) elastic foundation drastically changes the natural frequency. It is also observed that in some boundary conditions, the in-plane displacements have significant effects on natural frequency of thick functionally graded plates and they cannot be ignored.
    Full-text · Article · Jun 2011 · Composite Structures
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    • "(1) and (2) to describe the material properties and volume fraction of plate (e.g. see [14] [17] [21] [23] [27]). Therefore, FGM plates with power-law function will be considered in this paper. "
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    ABSTRACT: In this article, a new exact closed-form procedure is presented to solve free vibration analysis of functionally graded rectangular thick plates based on the Reddy’s third-order shear deformation plate theory while the plate has two opposite edges simply supported (i.e., Lévy-type rectangular plates). The elasticity modulus and mass density of the plate are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents whereas Poisson’s ratio is constant. Based on the present solution, five governing complicated partial differential equations of motion were exactly solved by introducing the auxiliary and potential functions and using the method of separation of variables. The validity and high accuracy of the present solutions are investigated by comparing some of the present results with their counterparts reported in literature and the 3-D finite element analysis. It is obvious that the present exact solution can accurately predict not only the out of plane, but also the in-plane modes of FG plate. Furthermore, a new eigenfrequency parameter is defined having its special own characteristics. Finally, the effects of boundary conditions, thickness to length ratio, aspect ratio and the power law index on the frequency parameter of the plate are presented.
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