Size dependent torsional vibration of a restrained single walled carbon nanotube (SWCNT) via nonlocal strain gradient approach

A torsional vibration model of a restrained nano-sized rod in which deformable boundary effect is considered in the context of nonlocal strain gradient theory is proposed in this work. The strain gradient nonlocal theory introduced from Eringen's nonlocal model contains additional small scale gradient parameter that represent the high-order strain gradient effects besides the Eringen's size effect parameter depicting interactions of particles. Within the framework of Fourier infinite series and Stokes' transform, an eigenvalue problem is employed to obtain the torsional vibration frequencies. The validity of the presented work is checked by comparing the calculated numerical results by giving proper values to elastic spring parameters and good agreement is achieved. This proposed model can be effectively applied to practical and non-idealized torsionally vibrating problems involving deformable boundary conditions. The two size parameters influencing on torsional dynamics of restrained nanorods are study in detail, meanwhile, validation and comparisons of present work with other non-classical elastic models, and the relations between rigid and deformable boundary conditions are shown in this paper.