Free vibration analysis of carbon nanotube RC nanobeams with variational approaches

There is not enough mixed finite element method (MFEM) model developed for dynamic analysis of carbon nanotube reinforced (CNTRC) composite beams in the literature. In the present study, free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) nanobeams is carried out in the framework of variational formulations. The rule of mixture is employed to estimate the effective material properties of single-walled CNT reinforced nanobeams. Four kinds of CNT distribution of un-axially aligned reinforcement material are investigated in the through thickness direction of the nanobeams. There are the uniform distribution (UD) and functionally graded distributions FG-O, FGX and FG-Lambda of CNTs in the thickness direction of the nanobeams (z axis direction) are assumed here for the analysis. The Hamilton's principle is used to derive governing differential equations based on trigonometric shear deformation beam theory. The effective functional has been constituted for FG-CNTRC nanobeams through a scientific procedure based on the Gateaux differential. A simple mixed finite element formulation is utilized for the formulation of free vibration problems of FG-CNTRC nanobeams with different boundary conditions. The results of the present method are compared with others from the literature where a good agreement has been found. An effective energy functional and the mixed finite element formulation for FGCNTRC nanobeams are the original contributions of this study.