Eur. Phys. J. Plus (2020) 135: 318
https://doi.org/10.1140/epjp/s13360-020-00312-z

Regular Article

Torsional Vibration of nanowire with equilateral triangle cross section based on nonlocal strain gradient for various boundary conditions: comparison with hollow elliptical cross section

¹ Department of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran
² Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran
³ Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran

^* e-mail: hosseini@bzte.ac.ir

Received: 13 October 2019
Accepted: 3 March 2020
Published online: 17 March 2020

Abstract

The torsional vibration is an unavoidable part of the mechanical behavior of the nanoscale materials. The nanostructures always are not circular. On the contrary, they have more often noncircular cross sections. In this paper, the size-dependent free torsional vibration of a nanowire with a triangular (equilateral triangle) cross section based on the nonlocal strain gradient theory is investigated for the first time. Three different boundary conditions such as clamped–clamped (C–C), clamped-free (C-F), and clamped–torsional spring (C-T) boundary conditions are utilized. Moreover, the first three natural frequencies of a hollow circular nanowire along with the nanowire with horizontal and vertical hollow elliptical cross sections are evaluated. The equation of motion, as well as the boundary conditions, is extracted by using Hamilton’s principle. An analytical method is employed to reduce the order of the equation of motion and ultimately solve it. The nanowire model contains the material length scale parameter and nonlocal parameter, simultaneously to take into account both stiffness-hardening and stiffness-softening effects on the vibration of the system. The effects of the nondimensional material length scale parameter and nondimensional nonlocal parameter on the first four nondimensional natural frequencies of the nanowire are assessed. The influences of ea/l and nondimensional material length scale parameter, and the effects of l/ea and nondimensional nonlocal parameter on the nondimensional responses are examined. Also, the effects of the nonlocal parameter on the first three natural frequencies of the model having circular and hollow elliptical cross sections are illustrated. The results are proved to be valid compared to another study.