International Journal of Scientific & Engineering Research, Volume 5, Issue 1, January-2014 2107
ISSN 2229-5518
IJSER © 2014
http://www.ijser.org
Wave Properties of nonlocal Euler beam model
in Carbon nanotubes
B. Yakaiah, A. Srihari Rao
Abstract— In this paper Eringen’s nonlocal theory of elasticity is applied to study the wave properties of single walled carbon nanotubes.
The effects of wave number and nonlocal scaling parameter (e0a) on the frequency, phase velocity and group velocity are discussed
for a particular material of carbon nanotube.
Index Terms— Carbon nanotube, Euler beam theory, Wave propagation, Nonlocal elasticity.
—————————— ——————————
1 INTRODUCTION
The nonlocal theory of elasticity due to Eringen [1] is widely
used in nano mechanical problems of carbon nanotubes such as
crack, wave propagation and vibration analysis. This theory dis-
cuss scale effects and large range atomic interactions. The intro-
ducing of carbon nanotubes by Lijima [2] has motivated a good
research in the field of nanodevices and nanocomposites. Exper-
iments at the nanoscale are very difficult and atomistic modeling
remains is expensive for large sized atomic system. The size de-
pendent continuum models are playing an important role in the
study of carbon nanotubes [3].The main reason these size de-
pendent continuum mechanics are used is because at small length
scales the material properties of microstructures [4, 5, 6] become
importantly significant and the influence of the micro size cannot
be ignored.
In nonlocal elasticity theory the small scale effects are captured
by assuming that the stress at a point as a function not only of the
strain at that point but also of the strains at all other points of the
domain. This is in accordance with predictions from atomic lat-
tice dynamics. It is important to note that the stress tensors de-
fined in the nonlocal elasticity theory are nonlocal ones which is
different from the local stress tensor defined in classical elastici-
ty theory [7].Therefore, it should be kept in mind that in deriving
a nonlocal beam model or any other nonlocal continuum model,
all formulations involving stress components are based on the
nonlocal stress tensor, not on the local ones [8, 9].
The main objective of this paper is to study the wave properties
of single walled carbon nanotube. The related governing equa-
tions for the Euler beam model are derived. The wave and vibra-
tion properties of carbon nanotube based on nonlocal elasticity
are presented. The scale effects on wave speed and group speed
dispersions are studied for a particular material of carbon nano-
tube. The effect of nanoscale parameter on the different disper-
sive properties of wave propagation of the carbon nanotube is
shown numerically.
2 Constituent equations of Nonlocal beam model
The nonlocal beam models in earlier literature are based on a
kind of the nonlocal constitutive relations given by Eringen’s [1]
equations as
. [ ]
kl kl rr kl
t a e µe δ le + = ∇ −
2 2
0
) ( 1
(1)
where
kl
t is the nonlocal stress tensor,
kl
e is the strain tensor, λ
, μ are materials constants, a is an characteristic length and
0
e is
a constant for adjusting the model in matching certain experi-
mental results. For this model, the size in thickness and width are
must be smaller than the length size. Therefore , for the beams
with transverse motion in x-y plane, the nonlocal constitutive
relation (1) can be approximated to one dimensional form as
e E t
x
a e
xx
=
∂
∂
−
2
2
2
0
) ( 1
(2)
γ G t
x
a e
xy
=
∂
∂
−
2
2
2
0
) ( 1 (3)
Where E is the Young’s modulus, G is the shear modulus, є is
the axial strain and ᵞ is the shear strain .These equations consti-
tute nonlocal relation for the nonlocal beam model.
2.1 Equation of motion for Nonlocal Euler beam model
For the Euler beam model, the bending moment M is independent
and the shear force S is related to the bending moment through
IJSER