Dynamics of nonlocal strain gradient nanobeams with longitudinal magnetic field

Abstract

Flexural dynamics of carbon nanotubes under the longitudinal magnetic field have been investigated in the paper. Carbon nanotube has been assumed as a hollow beam with circular cross section. Nonlocal strain gradient model has been employed in the modeling of nanobeam. Transverse component of Lorentz force, which occurs with longitudinal magnetic field, has been considered as an external force on nanobeam. Stress and strain energy functionals have been defined for the nonlocal strain gradient model using Hamilton principle. Higher‐order governing differential equation of motion for the present problem has been solved with Differential Quadrature Method. Lorentz force and nonlocal strain gradient parameters have been considered in the standard and nonstandard boundary conditions in the present model formulation. Length scale and magnetic field parameters effect on the flexural vibration response of nanobeam have been investigated. Mode shapes of nonlocal strain gradient nanobeam have been depicted on various cases. Softening nonlocal strain gradient model gives physically consistent results and should be used in the analysis. Magnetic field effect shifts the mode shapes of clamped‐free nanobeam. Present study could give useful results for designing of magnetically actuated nanomotors.

Application of the modified Fourier series method and the genetic algorithm for calibration of small‐scale parameters in the nonlocal strain gradient nanobeams

“…It is worth noting that the high-order or nonclassical boundary conditions are appeared when the differential form of the NSGT is applied. 47,64 Hence, it is of great interest to determine that which one of the high-order boundary conditions is more consistent with the MD results. This point has not been addressed yet, and this paper attempts to fill this gap.…”

“…Numerous studies have been conducted by researchers using the NSGT. [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] Li et al 54 studied wave propagation in viscoelastic CNTs in the framework of the NSGT. Zhu and Li 55 employed the NSGT for analyzing the nanoscaled rods in tension.…”

“…In the NSGT, the size‐dependent effects are considered applying two parameters, namely, the nonlocal and the strain gradient parameters. Numerous studies have been conducted by researchers using the NSGT 37–53 . Li et al 54 studied wave propagation in viscoelastic CNTs in the framework of the NSGT.…”

“…As reviewed above, a few research works have been carried out for the calibration of small‐scale parameters in the NSGT. It is worth noting that the high‐order or nonclassical boundary conditions are appeared when the differential form of the NSGT is applied 47,64 . Hence, it is of great interest to determine that which one of the high‐order boundary conditions is more consistent with the MD results.…”

A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

“…It can be remarked that modified strain gradient theory 66 is also employed to investigate the mechanical behavior of small-scaled structures as bars, 67,68 beams, [69][70][71][72] and plates. 73,74 Moreover, the longitudinal vibration analysis of axial rods, 75 free vibration of sinusoidal shear deformable beams, 76 wave propagation analysis of viscoelastic carbon nanotubes exposed to fluid conveying, 77 functionally graded Timoshenko nanobeams, 78,79 and vibration and buckling of microtubule bundles 80 are presented by utilizing nonlocal strain gradient approach 81 which examines the effect of two different small-scale parameters.…”