Nonlocal modeling of a Carbon Nanotube actuated by an electrostatic force
MATEC Web of Conferences
Nonlocal modeling of a Carbon Nanotube actuated by an electrostatic force
Hassen Ouakad 2
Fehmi Najar 1
Sami El-Borgi 0 1
0 Mechanical Eng. Program, Texas A&M University at Qatar , Eng. Building, P.O. Box 23874, Education City, Doha , Qatar
1 Applied Mechanics and Systems Research Lab., Tunisia Polytechnic School, University of Carthage , La Marsa , Tunisia
2 Mechanical Eng. Dep., King Fahd University of Petroleum and Minerals , P.O. Box 31261, Dhahran , Saudi Arabia
Carbon nanotubes (CNTs) are promising mechanical structures at the nano-scale which have attracted increasing attention due to their amazing mechanical, chemical, thermal, and electrical properties. To take into account size dependence of such small sized structures, the use of nonlocal continuum theory is proposed where intrinsic length scales is taken into account. Based on the Eringen theory, a nonlinear nonlocal model of a clamped-clamped CNT is developed in this study. Static and free vibration responses are simulated and analyzed. The main objective of this work is to study the influence of CNT size and length scale parameter on the static and free vibration response to better understand their e ect on the general behavior of the CNT. It has been found that the nonlocal e ect can largely influence the performance of the CNT and change qualitatively its nonlinear response.
1 Introduction
Carbon nanotubes (CNTs) are viewed as promising
mechanical structures in the nano-scale which have attracted
the engineers and researchers in this field mostly due to
their amazing mechanical, chemical, thermal, and
electrical properties. Many engineering applications have reported
the possible use of CNTs as building block for novel
nanoscale devices [
1–3
]. Therefore, studying properly the
structural behaviors of CNTs under various conditions is a
fundamental problem in any nano?scale investigation. For
example, many conducted experiments on CNT structures
acknowledged their size dependent behavior in the
nanoscale.
On the other hand, the conventional continuum
mechanics, adopted and assumed in many previous
investigations [
4, 5
], fails to predict the size dependence of such
small-sized structures due to lacking of intrinsic length scales.
Therefore, only recently, several higher-order elasticity
theories have been presented to develop size–dependent
continuum models. In what follows is a brief summary of
selected studies in which nonlocal elasticity dierential model
was used to model size dependence in CNTs based nano–
structures. The outcomes of the below investigations are
important in mechanical design considerations of devices
that use CNTs as main building structures.
Numerous studies investigated the linear free and
vibration and wave propagation of CNTs [
6–9
]. Thongyothee
et al. [6] investigated the free vibration problem of CNTs
including the e ect of nonlocal elasticity to study the e ect
of their chirality and various boundary conditions. Ansari
et al. [
7
] investigated the free vibration of double-walled
carbon nanotubes (DWCNTs) using the Eringen’s nonlocal
elastic model along with the Donnell shell model. They
accounted for the van der Waals forces between the inner and
outer nanotubes. Lim and Yang [
8
] discussed the physics
and understanding of nonlocal nanoscale wave
propagation in CNTs based on nonlocal elastic stress field
theory. In this regards, they developed an analytical
nonlocal shear deformable nanobeam model based on the
variational principle for wave propagation in CNTs. Further
works focused on the nonlocal e ect on the buckling
characteristics of CNTs considering linear theory [
9
]. In a more
recent work, Kiani [
10
] developed several novel models,
all based on the nonlocal stress theory, to study the lateral
buckling of groups of vertically aligned single-walled
carbon nanotubes (SWCNTs). In the model, he accounted for
the e ect of vdW forces along with the nonlocal Rayleigh
beam theory.
The previous concisely summarized works are all
assuming linear problems. Those who included
nonlinearity in either the geometry or the actuating force for CNTs
are few [
11–17
]. To mention some, Hosseini-Ara et al.
[
11
] proposed analytical solutions based on nonlocal
Timoshenko kinematics, strain gradient approach and some
variational methods to derive the higher-order boundary
conditions as well as governing for the sake of investigating
the buckling characteristics of CNTs. Karlicic et al. [
12
]
analyzed the free flexural vibration and buckling SWCNT
under compressive axial loading. Ansari et al. [
13
] studied
numerically the torsional vibration behaviors of SWCNTs.
They assumed an Euler?Bernoulli model while including
the material length scale parameters through the strain
gradient elasticity theory as to capture the CNT material size
dependent e ect. Fakhrabadi et al. [
14
] presented an
investigation into the free and forced nonlinear vibration of
CNTs under step e (...truncated)