Strain Gradient Finite Element Analysis For Vibration Of Carbon Nanotubes

Researches on carbon nanotubes (CNTs) have kept grown due to their remarkable electronic,chemical and mechanical properties since their discovery. At present, nonlocal elasticity whichconsiders the small scale effect has been widely applied to study the vibration and wave propagationof CNTs. In this thesis, we build the Euler-beam model for CNTs based on the strain gradient nonlocaltheory, and then a2-nodes6-degree of freedom (DOF) beam element is developed through theprinciple of virtual work to study the vibration behaviors of CNTs. Main works of this thesis include:(1) The strain gradient theory is applied to build the dynamic equation of single-walled carbonnanotubes (SWCNTs). A2-nodes6-DOF beam finite element is developed by the theory ofvirtual work. Numerical simulations are carried out to study the small scale effect on both naturalfrequencies and vibration modes of SWCNT with different natural boundary conditions andlengths. The results of finite element method (FEM) are in good agreement with theoreticalresults in analyzing simply-supported SWCNT, which indicates that the small scale effect givenby concerning strain gradient increase with increase of the mode order and decrease of the length.The comparison of vibration modes shows that vibration modes have been obviously affectedwhen the two adjacent nodes distance of the mode is closed to interatomic distance.(2) A double-Euler-beam which concerns van der Waals interaction between tubes is built based onthe strain gradient theory. Then a2-node6-DOF beam element is applied to deal with the higherorder gradient of strain due to the consideration of small scale effect. FEM results are in goodagreement with theoretical results in analyzing simply-supported double-walled carbon nanotubes(DWCNTs). It indicates the small scale effect increase with the increase of the mode order. It alsoshows nanostructure characterized by strain gradient theory has obvious effect on the vibrationalmode of clamped-clamped DWCNTs.(3) The embedded CNTs, including SWCNTs and DWCNTs, are studied through the strain gradientfinite element analysis. Numerical results are in good agreement with theoretical results whichindicates such element can been used to simulate the vibration of embedded strain gradient beamseffectively. Numerical results of SWCNTs show that the effect from the substrate starts whenWinkler constant increases to1010Pa, and the lower natural frequencies first affected. The studyof DWCNTs shows the similar results, and the effect of Winkler constant has critical values forthe lower branch of frequencies.