Nonlocal effects of longitudinal vibration in nanorod with internal long-range interactions

The physically-based nonlocal model is used to investigate influences of the nonlocal long-range interactions on the longitudinal vibration of nanorod. The exact solution of the vibration is determined under the condition of a uniform nonlocal kernel. Nonlocal effects in the vibration of the nanorod are examined in detail. The results show that the nanorod becomes stiffer due to the internal long-range interactions. Meanwhile, an upper bound of the material parameter characterizing the long-range interactions is found. The low-frequency insulating effect induced by the long-range interactions is predicted. This effect shows that there exists a forbidden band of basic frequency within which external excitation is not transmitted in the nanorod. The Lagrangian formulations of the physically-based nonlocal theory are established based on a new definition of nonlocal variable. By these formulations, the physically-based nonlocal model can be conveniently expanded into beam, plate and shell.