Longitudinally varying magnetic field influenced transverse vibration of embedded double-walled carbon nanotubes Keivan Kiani n Department of Civil Engineering, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran article info Article history: Received 7 August 2013 Received in revised form 2 April 2014 Accepted 21 April 2014 Available online 29 April 2014 Keywords: Double-walled carbon nanotube (DWCNT) Free transverse vibration Longitudinally varying magnetic field Nonlocal beam theories Reproducing kernel particle method (RKPM) abstract Using nonlocal Rayleigh, Timoshenko, and higher-order beam theories, the free dynamic deflection of elastically supported double-walled carbon nanotubes (DWCNTs) subjected to a longitudinally varying magnetic field (LVMF) is examined. By employing reproducing kernel particle method (RKPM), the equations of motion of the magnetically affected DWCNT (MADWCNT) for each model are reduced to a set of algebraic equations. For four common boundary conditions, namely fully simple, fully clamped, simple-clamped, cantilevered supports, the dominant frequencies of the nanostructure are calculated. In particular cases, the predicted results by the RKPM are compared with those of the exact solution. Additionally, the convergence checks of the proposed numerical models are performed. The effects of the innermost radius, slenderness ratio, small-scale parameter, maximum strength of the LVMF, transverse and rotational stiffness of the surrounding medium on the fundamental frequency of the MADWCNT are addressed. The capabilities of the proposed models in predicting the characteristics of free vibration are also discussed. Further, the limitations of the local analysis as well as the classical beam theory in capturing the lateral vibrations of the MADWCNTs are revealed. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Assemblage of nanoscale materials over macroscopic dimen- sions as well as progress of novel nano-electro-mechanical systems (NEMS) hold great promise for a diverse range of applications in nanotechnology. However, there has been a great challenge to find an efficient way to control both assembling of such materials and their dynamic response at the nanoscale level. Carbon nanotubes (CNTs) are considered as one of the most promising materials in nanotechnology. Such an enormous confidence in their usage is chiefly indebted to their excellent electrical, mechanical, chemical, and physical properties [1–5]. One practical way to reach the above- mentioned goals of interest would be applications of appropriate electrical and magnetic fields. The experimentally observed data have revealed that when the CNTs-matrix is acted upon by a magnetic field, CNTs could be aligned along the direction of the applied magnetic field [6–8]. Additionally, further studies show that the electro-thermo- mechanical properties of CNTs and their composites are generally enhanced by magnetic field processing [9–13]. Thereby, realization of vibration behavior of magnetically affected CNTs-composites would be of great significance from two standpoints: optimal design and vibration control. A class of CNTs family which is of concern in the present work is double-walled carbon nanotubes (DWCNTs). Such nanostructures are composed of two coaxial single-walled carbon nanotubes (SWCNTs), one nested in another at the vicinity of each other. The morphology and properties of DWCNTs are mostly identical to those of SWCNTs, however, their resistance to chemicals is highly enhanced. This is mainly related to this fact that in SWCNTs, covalent functionalization will break some C-C bonds causing some damages within the nanostructure. Thereby, both electrical and mechanical properties would be modified. In the case of DWCNTs, only the properties of the outer wall are modified. Such an important advantage of DWCNTs provides them as a more efficient functionalized nanostructure in comparison to SWCNTs, particularly when adding new proper- ties to the nanostructure is of concern. In this work, only transverse vibration of a system of DWCNTs-matrix is of concern. Therefore, the magnetically affected DWCNTs (MADWCNTs) can be reasonably simulated via well-known beam theories and their interactions with its surrounding matrix are taken into account by continuous transverse and rotational springs. Further, we study the transverse vibration of an individual DWCNT embedded in an elastic matrix subjected to an arbitrarily longitudinal magnetic field. For a group of DWCNTs embedded in matrix (i.e., an ensemble of DWCNTs), more precise interactions between them Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijmecsci International Journal of Mechanical Sciences http://dx.doi.org/10.1016/j.ijmecsci.2014.04.018 0020-7403/& 2014 Elsevier Ltd. All rights reserved. n Tel.: þ98 21 88779473; fax: þ98 21 88779476 E-mail addresses: k_kiani@kntu.ac.ir, keivankiani@yahoo.com International Journal of Mechanical Sciences 87 (2014) 179–199