Study On The Buckling And Vibration Behaviors Of Nanobeams

In this thesis,we develop a continuum model for the harmonic forced vibration of the nanobeams on elastic substrates,based on which the buckling,free and forced vibration problems are studied.The nanobeam is modeled as a bulk material plus an elastic surface layer.The Winkler,Pasternak and Generalized foundation models are used to describe the elastic substrates of various kinds.By introducing a transition parameter,two assumptions on the axial force directions in the classical Timoshenko beam model are unified into the present model.Changing the values of the transition parameter,we could selectively modulate the axial force direction between the basic two directions.By means of the Hamilton's principle,the present model is developed.The present model for the forced vibration,which takes the effects of the axial force and elastic substrates into consideration,could be degenerated to study the buckling and free vibration problems of nanobeams.Substituting the general solutions of the buckling and free vibration problems into the boundary conditions and then solving the eigenvalue problems,we determine the critical buckling forces for the nanobeams with three typical boundaries(simply supported,clamped-clamped and clamped-free)and the natural frequencies for the three typical nanobeams on elastic substrates.By means of the Laplace transform and the inverse transform,we obtain the general solution of the forced vibration problem in the form of the Green's function.By substituting the general solution into the boundary conditions,the dynamical responses are determined for the three typical nanobeams on elastic substrates.We validate the present solutions by numerically comparing the present solutions for the buckling and vibration problems with those reported in the literature.The comparison between the critical buckling forces from the present solutions and the molecular dynamic simulation shows that the present model is better at the prediction of the critical buckling loads of nanobeams than the continuum models in the literature.The transition parameter is observed to have a great effect on the critical buckling forces of the nanobeams,of which the shear deformation accounts for a large proportion among the total deformation.The mode shapes of nanobeams are found to be changed owing to the presences of elastic substrates and surface energy.Moreover,the negative(positive)surface elastic modulus and surface residual stress could increase(decrease)the effect of elastic substrates on the natural frequencies and dynamical responses of nanobeams,and the elastic substrates could decrease the surface effect on the natural frequencies and dynamical responses of nanobeams,which are related to the length-to-height ratio and mode order of nanobeams and the types of elastic substrates and so on.