Exact closed-form solution for the transverse vibration of multi-cracked nanotubes based on the nonlocal Euler–Bernoulli beam

In this study, distributions (generalized functions) are used to investigate transversely vibrating nanotubes in the presence of multiple concentrated cracks. Nanotubes are modeled as Euler–Bernoulli beams based on the theory of nonlocal elasticity; in addition, the concentrated cracks are modeled as a sequence of Dirac’s delta generalized functions which have local effects on the flexural stiffness. The expressions of freevibration of such nanotubes are presented in terms of the four proper fundamental solutions. Thus, unlike previously developed procedures, which usually led to a complicated determinant of order 4(n +1) for solving a beam with n cracks, by this method, after imposing the standard boundary conditions, the explicit frequency equations can be conveniently obtained from a secondorder determinant. To illustrate the precision of the proposed method, the vibration modes of nanotubes with diverse number of cracks, in different positions, and under different boundary conditions have been considered.

Keywords:

Nanotubes , Transverse vibration , Nonlocal Euler– Bernoulli beam , Exact closed-form solution