Numerical study on the free vibration of carbon nanocones resting on elastic foundation using nonlocal shell model
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Numerical study on the free vibration of carbon nanocones resting on elastic foundation using nonlocal shell model Reza Ansari1 • Jalal Torabi1
Received: 28 August 2016 / Accepted: 22 November 2016 Springer-Verlag Berlin Heidelberg 2016
Abstract Employing the variational differential quadrature method, the free vibration of carbon nanocones (CNCs) embedded in an elastic foundation, is studied based on nonlocal elasticity theory. On the basis of the first-order shear deformation theory, the energy functional of the CNC is presented and then discretized by employing the generalized differential quadrature method in the axial direction and periodic differential operators in the circumferential direction. According to Hamilton’s principle and using matrix relations, the reduced forms of mass and stiffness matrices are readily obtained. The results of present study are compared to those obtained by molecular mechanics to verify the proposed approach. In addition, the effects of nonlocal parameter, boundary conditions, semi-apex angle and both Winkler and Pasternak coefficients of elastic foundation are examined on the vibrational behavior of CNCs. The results indicate that the increase in nonlocal parameter and elastic foundation coefficients decreases and increases the fundamental frequency of CNCs, respectively.
1 Introduction By the discovery of the carbon nanotubes, nanorods and nanocones, nanoscale engineering applications have received considerable attention. The high mechanical & Reza Ansari [email protected] & Jalal Torabi [email protected] 1
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
strength, low density and excellent thermal and electrical properties of nanomaterials make them suitable for various usages in micro-electro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS). Ge and Sattler [1] discovered the carbon nanocones (CNCs) in 1994. Subsequently, the existence of only five apex angles of CNCs was verified by Krishnan et al. [2]. Due to the localization of electric field at sharp ends of CNCs, these nanostructures can be used as the high resolution probes in different systems such as atomic force microscopy and field emission devices [3, 4]. There are some experimental studies on the mechanical behavior of nanostructures [5–8]. Due to the difficulties of conducting the controlled experiments at nanoscale, the theoretical modeling is widely used to investigate the mechanical characteristics of nanostructures. There exist three main categories for the theoretical modeling of nanomaterials including atomistic modeling [9–13], hybrid atomistic-continuum mechanics [14, 15] and continuum modeling. For the large-scale nanostructures, the atomistic modelings such as molecular dynamics (MD) have a huge computational cost. However, the continuum mechanics provides a computationally efficient model which makes this category interesting for researchers. Since at nanoscales the mechanical behavior of structures is size dependent [16, 17] an
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