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)