Abstract
This article presents a mathematical continuum model to analyze the free vibration response of cross-ply carbon-nanotube-reinforced composite laminated nanoplates and nanoshells, including microstructure and length scale effects. Different shell geometries, such as plate (infinite radii), spherical, cylindrical, hyperbolic-paraboloid and elliptical-paraboloid are considered in the analysis. By employing Hamilton’s variational principle, the equations of motion are derived based on hyperbolic sine function shear deformation theory. Then, the derived equations are solved analytically using the Galerkin approach. Two types of material distribution are proposed. Higher-order nonlocal strain gradient theory is employed to capture influences of shear deformation, length scale parameter (nonlocal) and material/microstructurescale parameter (gradient). Temperature-dependent material properties are considered. The validation of the proposed mathematical model is presented. Detailed parametric analyses are carried out to highlight the effects of the carbon nanotubes (CNT) distribution pattern, the thickness stretching, the geometry of the plate/shell, the boundary conditions, the total number of layers, the length scale and the material scale parameters, on the vibrational frequencies of CNTRC laminated nanoplates and nanoshells.
Highlights
Carbon nanotubes (CNTs) discovered in 1991 by Iijima, are graphite sheets rolled to cylindrical geometry with 1 nm diameter and lengths up to micrometres [1]
The effective temperature-dependent material properties of the CNTs are given as the following expression [62]
shells considering different boundary conditions including: simple-simple (SSSS), R x /a = Ry /b = 5, N = 10 Vcnt In Figure 8, we show the radii of curvature (R ⁄a) on the dimensionless frequencies
Summary
Carbon nanotubes (CNTs) discovered in 1991 by Iijima, are graphite sheets rolled to cylindrical geometry with 1 nm diameter and lengths up to micrometres [1]. Dindarloo and Li [55] studied the 3D vibrational response of FG-CNTRC doubly curved, nonlocal nanoshells, based on a new higher-order shear deformation theory. Tran et al [58] extended four-unknown, higher-order shear deformation nonlocal theory to study the bending, buckling and free vibration of FG porous nanoshell resting on an elastic foundation. Twinkle and Pitchaimani [59] developed a semi-analytical, nonlocal model to investigate the static stability and vibration behavior of FG-CNTRC nano cylinders under non-uniform edge loads This manuscript aims, for the first time, to investigate the impact of the length scale, as well as the microstructure, on the natural frequencies of sandwich FG-CNTRC nonlocal strain gradient of nanoshell, using kinematic higher-order hyperbolic shear function.
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