Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

Abstract

In this article, the free vibrations of graded porous circular nanoplates subjected to various boundary conditions have been investigated. It is assumed that the performance of graded porous materials varies continuously in the whole thickness, and two cosine forms of non‐uniform porosity distribution along its thickness are considered. In order to capture the size effect, the Eringen's nonlocal elastic theory is applied. Then, Mindlin plate theory, combined with Hamilton's principle, is utilized to derive the governing equations of motion. Various types of boundary conditions are assumed for the plate with a combination of clamped, simply supported and free edges. Finally, the governing equations of motion are solved numerically using the shooting technique. The effects of various factors such as porosity coefficient, porosity distribution pattern, thickness‐diameter ratio, and the nonlocal scale effect and boundary conditions on the natural frequencies of grade porous circular nanoplates are discussed in detail.

The influence of surface effects on axisymmetric vibration of graded porous Mindlin circular nanoplate in a temperature field

“…According to Mindlin plate theory and applying the physical surface concept, the axisymmetric displacement field could take the following forms [43] 𝑈 𝑟 = (𝑧 − 𝑧 0 ) 𝜓 (𝑟, 𝑡) , 𝑈 𝜃 = 0𝑈 𝑧 = 𝑤 (𝑟, 𝑡) (14) in which 𝑈 𝑟 , 𝑈 𝜃 , 𝑈 𝑧 represent the total displacements of the radial, circumferential, and transverse of the FG porous circular nanoplate, respectively. 𝑤 is the transverse deflection, and 𝜓 is the rotation of a transverse normal line, 𝑡 is the time variable.…”

“…According to Mindlin plate theory and applying the physical surface concept, the axisymmetric displacement field could take the following forms [43] $$$$\begin{equation}{U_r} = \left( {z - {z_0}} \right)\psi \left( {r,t} \right),\,{U_\theta } = {\mathrm{0}}{U_z} = w\left( {r,t} \right)\end{equation}$$$$in which $${U_r}$$, $${U_\theta }$$, $${U_z}$$ represent the total displacements of the radial, circumferential, and transverse of the FG porous circular nanoplate, respectively. w is the transverse deflection, and ψ is the rotation of a transverse normal line, t is the time variable.…”

Free vibration analysis of FG‐GPL and FG‐CNT hybrid laminated nano composite truncated conical shells using systematic differential quadrature method

“…They considered FGMs and effects of some environmental factors on the analysis like temperature and humidity. Qinglu Li et al [34] investigated of free vibrations of graded porous circular nanoplates subjected to various boundary conditions. They discussed about of effects of porosity coefficient, porosity distribution pattern, thickness-diameter ratio, and the nonlocal scale effect and boundary conditions on the natural frequencies of grade porous circular nanoplates.…”

Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

“…[25][26][27][28][29][30][31][32] modified couple stress theory have been considered to show the responses of scale-dependent nano/micro elements. Nonlocal elasticity theory has been used by researchers [33][34][35][36][37][38][39][40][41][42] to investigate the different effects on the nano/micro-scaled beams, rods, etc. Strain gradient elasticity has been adopted the buckling of nano/microbeams in refs.…”