Bending, buckling and free vibration of nonlocal FG-carbon nanotube-reinforced composite nanobeams: exact solutions

Abstract

This paper investigates the bending, buckling and free vibration behaviors of functionally graded carbon nanotubereinforced composite (FG-CNTRC) nanobeams by considering small-scale effect. The governing equations of motion of a Timoshenko beam under a general loading are derived utilizing the nonlocal elasticity theory. The equations governing bending and stretching behavior of CNTRC nanobeams are uncoupled to a fifth-order ordinary differential equation with respect to the rotation of cross-section for the static cases of bending and buckling. This uncoupling makes it possible to develop exact solutions for transverse deflection and buckling load of CNTRC nanobeams. Using differential operator method, the decoupled sixth-order differential equations in terms of the kinematic variables are obtained for vibration analysis. By setting the coefficients matrix in the corresponding system of homogenous algebraic equations to zero, an algebraic frequency equation is derived. Finally, based on the presented closed-form solutions, parametric studies are carried out to assess the effects of CNT distribution, nonlocal parameter and type of boundary conditions on the deflection, buckling and natural frequency of CNTRC nanobeams. Findings show that nonlocal effect on the mechanical behavior of nanobeams is strongly dependent on boundary conditions and loadings. It is seen that cantilever nanobeams become harder by taking into account nonlocal effect, contrary to clamped and simply supported nanobeams. In addition, the influence of CNT distribution on the mechanical behavior of cantilever beams is more significant than that of simply supported and clamped beams.

Mechanical behaviour of carbon nanotube composites: A review of various modelling techniques

“…Nonlocal phenomena resulted in a reduction in the critical buckling load, while increasing CNT volume fraction in the beam increased the critical buckling load. Borjalilou et al 107 and Thang et al 108 have conducted studies that indicate opposite trends for deflection and frequencies of beams and plates. In a study by Daghigh et al, 109 the effect of temperature and nonlocal parameter on Young's modulus and natural frequency of CNT composite nanoplates was examined, revealing a reduction of 16—30% and 1—2 %, respectively, with a temperature change of 100 o .…”

A review of size-dependent continuum mechanics models for micro- and nano-structures

“…They showed that the deflection of the plates diminished with an increase of the elastic modulus. Static/dynamic bending and buckling behavior of nonlocal FG-SWCNT-reinforced composite nanobeams with consideration of the Timoshenko beam model was investigated by Borjalilou et al [385]. The governing relations were decoupled to an ordinary differential relation with the fifth-order pertinent to the cross-section rotation associated with the static bending as well as buckling.…”

Investigation of dynamic behaviour of circular plates resting on Winkler and Pasternak foundations

“…Galerkin of weighted residual handles circular plate vibration problem without any manipulation of governing equation with very precise results compared to experimental with few iterations. Other recent publications reported on vibration are [20][21][22][23][24][25].…”