Third Day Poster Session, 17 June 2010 - NanoTR-VI

Poster Session, Thursday, June 17
Theme F686 - N1123
Free vibration analysis of carbon nanotubes based on nonlocal continuum and gradient
elasticity theories
Ömer Civalek 1 , Bekir Akgöz, Hakan Ersoy
1 Akdeniz University, Civil Engineering Department, Division of Mechanics
Antalya-TURKIYE, Tel: + 90- 242-310 6319, Fax: + 90-242-310 6306
Abstract- Free vibration analysis of single walled carbon nanotubes (CNT) is presented based on the Euler-Bernoulli beam
theory. The size effect is taken into consideration using the Eringen’s non-local elasticity theory. Gradient elasticity theory is
also adopted for modeling. The governing differential equations for CNT vibration is being solved using the differential
quadrature (DQ) method. Numerical results are presented to show the effect of nonlocal behavior on frequencies of CNT.
The concept of carbon nanotubes (CNTs) was first
introduced in 1991 by Iijima [1] in Japan. Reviews on the
development and application of such nano structures have
been presented [2]. So, the studies of mechanical behaviors
of carbon nanotubes have being attracted more and more
attentions of scientists in the world and also have become a
new research area of applied mechanics [3,4]. In the
present work, the consistent governing equations for the
beam model for CNTs are derived for free vibration
analysis. Nonlocal beam and couple stress beam theories
are adopted for modeling. It is known that, the stress state
of any body at a point x is related to strain state at the same
point x in the classical elasticity. But this theory is not
conflict the atomic theory of lattice dynamics and
experimental observation of phonon dispersion. As stated
by Eringen [5] the linear theory of nonlocal elasticity leads
to a set of integropartial differential equations for the
displacements field for homogeneous, isotropic bodies.
According to the nonlocal elasticity theory of Eringen’s,
the stress at any reference point in the body depends not
only on the strains at this point but also on strains at all
points of the body. This definition of the Eringen’s
nonlocal elasticity is based on the atomic theory of lattice
dynamics and some experimental observations on phonon
dispersion. In the present manuscript two different
approaches are used for modeling of carbon nanotubes.
Euler-Bernoulli beam-nonlocal model [5]
4
2
W
2
2 W
EI A W ( e0a)
A 0 (1)
4
2
x
x
Euler-Bernoulli beam-gradient elasticity theory [6]
W
x
W
x
4
4
2
2
EI g EI A 0 (2)
4
4
7
Table 1. First three frequencies (10
) of S-S carbon
8
nanotubes via gradient theory ( L 510
m ,
3
12 2
2300kg
/ m , m , t 510 10
E 10 N /
m )
Mode
g/L (DQ results)
0.005 0.015 0.125
1 0.10388 0.10669 0.11374
2 0.41065 0.41103 0.42301
3 0.91863 0.92007 0.93485
7
Table 2. First three frequencies (10
) of S-S carbon
8
nanotubes via nonlocal theory( L 510
m ,
3
12 2
2300kg
/ m , m , t 510 10
E 10 N /
m )
Mode (e 0 a) 2 (DQ results)
0 2 4
1 0.10273 0.10158 0.09962
2 0.40967 0.40863 0.40553
3 0.9172 0.90864 0.90637
[1] S. Iijima, Nature, 354, 56 (2001).
[2] D. Qian, G.J. Wagner, W.K. Liu, Appl. Mech. Rev.,
55, 495(2002).
[3] C.M. Wang, V.B.C. Tan, T.Y. Zhang, J. Sound Vib.
294, 1060 (2006).
[4] J.N. Reddy, S.D. Pang, J. Appl. Phys. 103, 023511
(2008).
[5] A.C. Eringen, J. Appl. Phys., 54, 4703 (1983).
[6] S.P. Beskou, D. Polyzos, D.E. Beskos, Struct. Eng.
Mech. 15, 705(2003).
[7] Ö. Civalek, Engineering Structures, 26, 171(2004).
The results obtained by differential quadrature (DQ)
method [7] using two higher order elasticity theories are
listed in Tables 1-2. In table 1, first three frequencies of
simple supported (S-S) carbon nanotubes are listed for
different gradient parameter. It is shown that, the
frequencies are increased gradually with the increasing
value of g for all modes. Nonlocal parameter also affected
on frequencies (Table 2). When the nonlocal parameters
are increased, the values of frequencies are decreased,
significantly. It is possible to say that, the classical beam
theories can not to capture to size effect on mechanical
behavior of nano sized structures. So, it is suitable to use
some higher order continuum theory such as nonlocal
elasticity theory or gradient strain theory to investigate the
size effect on mechanical behaviour of nano/micro
structures.
6th Nanoscience and Nanotechnology Conference, zmir, 2010 705