Abstract
This paper studies the influence of the variable nonlocal parameter and porosity on the free vibration behavior of the functionally graded nanoplates with porosity. Four patterns of distribution of the porosity through the thickness direction are considered. The classical nonlocal elasticity theory is modified to take into account the variation of the nonlocal parameter through the thickness of the nanoplates. The governing equations of motion are established using simple first-order shear deformation theory and Hamilton’s principle. The closed-form solution based on Navier’s technique is employed to solve the governing equations of motion of fully simply supported nanoplates. The accuracy of the present algorithm is proved via some comparison studies in some special cases. Then, the effects of the porosity, the variation of the nonlocal parameter, the power-law index, aspect ratio, and the side-to-thickness ratio on the free vibration of nanoscale porous plates are investigated carefully. The numerical results show that the porosity and nonlocal parameter have strong effects on the free vibration behavior of the nanoplates.
Highlights
Shock and Vibration nonlocal elasticity theory and Levy type solution
A rectangular Functionally graded materials (FGMs) nanoplate with porosity is considered. e dimensions of the plate are a × b, and the thickness is h. e Cartesian coordinate xyz is placed at the middle surface of the FGM nanoplates as shown in Figure 1. e distribution of the ceramic and metal components through the thickness of the plate is described via the volume fractions with the power-law function. e porosity is distributed through the thickness of the plate with four different patterns, namely, patterns A, B, C, and D, and they are described by the following formula [58, 71]:
To present the validity of the current algorithm, the free vibration of fully supported square porous FGM plates is considered. e plates are made of Al/Al2O3, the dimensions of the plate are a b 1, and the thickness is h. e material properties of the perfect plates are calculated via power-law function, and the porosity is normally distributed through the thickness direction
Summary
Shock and Vibration nonlocal elasticity theory and Levy type solution. Shen et al [22] investigated the free vibration of a single-layered graphene sheet-based nanomechanical sensor using nonlocal Kirchhoff plate theory. To take into account the thickness stretching effects, Sobhy and Radwan [51] developed a new quasi-3D nonlocal plate theory to investigate the vibration and buckling behavior of functionally graded nanoplates. MoradiDastjerdi et al [59,60,61] analyzed thermo-electro-mechanical, free vibration, and buckling behaviors of advanced smart sandwich plates including the effects of porosity. In these studies, Young’s modulus, mass density, and Poisson’s ratio are assumed to depend on the individual material properties and porosity. Is is the main aim of the present study, in which the classical Eringen’s elasticity theory is modified to consider the variable nonlocal parameter for free vibration analysis of FGM porous nanoplates. A rectangular FGM nanoplate with porosity is considered. e dimensions of the plate are a × b, and the thickness is h. e Cartesian coordinate xyz is placed at the middle surface of the FGM nanoplates as shown in Figure 1. e distribution of the ceramic and metal components through the thickness of the plate is described via the volume fractions with the power-law function. e porosity is distributed through the thickness of the plate with four different patterns, namely, patterns A, B, C, and D, and they are described by the following formula [58, 71]:
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