Nonlocal effect on the free vibration of short nanotubes embedded in an elastic medium

Abstract

In this paper, the small size effect on the free vibration behavior of finite length nanotubes embedded in an elastic medium is investigated. The problem is formulated based on the three-dimensional (3D) nonlocal elasticity theory. Since the 3D nonlocal constitutive relations in a cylindrical coordinate system are used, in addition to displacement components, the stress tensor components are chosen as degrees of freedom. The surrounding elastic medium is modeled as the Winkler's elastic foundation. The differential quadrature method as an efficient and accurate numerical tool in conjunction with the series solution is used to discretize the governing equations. Very fast rate of convergence of the method is demonstrated. The effects of the nonlocal parameter together with the other geometrical parameters and also the stiffness parameter of the elastic medium on the natural frequencies are studied.

Dynamic stability of cylindrical nanoshells under combined static and periodic axial loads

“…Since it is difficult to solve analytically the coupled equations of motion under different boundary conditions, the differential quadrature method (DQM) as an accurate and low computational efforts numerical tool [9,[34][35][36][37][38][39][40][41][42][43][44] is applied to discretize the equations of motion and the related boundary conditions in the spatial domain. The main advantage of this method is that the equations of motion and the boundary conditions are discretized in their strong form and only limited number of grid points is required to achieve accurate converged results.…”

“…To reduce the computational efforts, one can easily eliminate the boundary degrees of freedom from the discretized equations of motion by using the discretized boundary conditions [35]. After doing this, the resulting system of linear (25) x…”

“…Hence, in this paper, the dynamic stability of nanoshells is presented based on the two-dimensional nonlocal elasticity of Eringen in conjunction with the first-order shear deformation theory (FSDT) of shells. The differential quadrature method (DQM) as an efficient and accurate numerical method [9,[34][35][36][37][38][39][40][41][42][43][44] is utilized to spatially discretize the equations of motion subjected to different boundary conditions. The discretized governing partial differential equations are converted to a system of Mathieu-Hill-type equations.…”

Resonant frequency tuning of nanobeams by piezoelectric nanowires under thermo‐electro‐magnetic field: a theoretical study

“…Generally, modified continuum-based theories such as the couple stress theory [6,7] and the strain gradient elasticity [8,9] have been applied to microscale structural elements while the dynamics and statics of nanostructures have been formulated with the help of the non-local elasticity theory [10][11][12][13][14]. For instance, Malekzadeh et al [15] studied the effect of length scale on the frequency response of short nanotubes resting on an elastic foundation. Aydogdu [16] developed a general non-classical beam model for the static deformation, stability and vibration of nanoscale beams based on the non-local version of continuum mechanics.…”

DQ thermal buckling analysis of embedded curved carbon nanotubes based on nonlocal elasticity theory

“…In general, 0 e a is believed to be less than 2 for the case of CNTs (Setoodeh et al, 2011;Malekzadeh et al, 2012). A homogeneous isotropic nanotube is assumed here while an axial load is perpendicularly exerted on the nanotube cross section.…”