Research Papers

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

[+] Author and Article Information
S. Narendar

 Defence Research and Development Laboratory, Kanchanbagh, Hyderabad-500 058, Andhra Pradesh, Indiasnarendar@aero.iisc.ernet.in/nanduslns07@gmail.com

S. Gopalakrishnan

 Structures Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore-560 012, Karnataka, Indiakrishnan@aero.iisc.ernet.in

J. Appl. Mech 78(6), 061018 (Aug 26, 2011) (9 pages) doi:10.1115/1.4003909 History: Received May 23, 2010; Revised February 14, 2011; Published August 26, 2011; Online August 26, 2011

In this article, the Eringen’s nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

(a) A nanorod, showing length L, Young’s modulus E, density ρ, cross-sectional area A and longitudinal displacement u; (b) nodal loads and degree of freedom for the longitudinal spectral element

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Figure 2

Wavenumber dispersion (spectrum relation) for a nanorod obtained from both local and nonlocal elasticity models

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Figure 3

Escape frequency variation with nonlocal scaling parameter of the axial wave in a nanorod

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Figure 4

Frequency dependent exact shape function G∧1(x) of the nanorod for (a) local elasticity e0a=0, (b) nonlocal elasticity e0a=0.5nm and (c) e0a=1.0nm

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Figure 5

Frequency dependent exact shape function G∧2(x) of the nanorod for (a) local elasticity e0a=0, (b) nonlocal elasticity e0 a = 0.5 nm and (c) e0 a = 1.0 nm.

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Figure 6

Dynamic stiffness comparison between spectral finite element method (SFEM) and conventional finite element method (FEM)

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Figure 7

Dynamic stiffness (k∧11) behavior at (a) lower frequencies and (b) higher frequencies

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Figure 8

Dynamic stiffness (k∧12) behavior at (a) lower frequencies and (b) higher frequencies




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