

Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 1, pp. 217-233, Warsaw 2015
DOI: 10.15632/jtam-pl.53.1.217

FLEXURAL VIBRATION AND BUCKLING ANALYSIS OF SINGLE-WALLED

CARBON NANOTUBES USING DIFFERENT GRADIENT ELASTICITY

THEORIES BASED ON REDDY AND HUU-TAI FORMULATIONS

Danilo Karličić

Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

Predrag Kozić, Ratko Pavlović

University of Nǐs, Department of Mechanical Engineering, Nǐs, Serbia

e-mail: kozicp@yahoo.com

Theaimof thepresentwork is to analyze free flexural vibrationandbuckling of single-walled
carbonnanotubes (SWCNT)under compressive axial loading based ondifferent constitutive
equations and beam theories. The models contain a material length scale parameter that
can capture the size effect, unlike the classical Euler-Bernoulli or Reddy beam theory. The
equations of motion of the Reddy and the Huu-Tai beam theories are reformulated using
different gradient elasticity theories, including stress, strain and combined strain/inertia.
The equations of motion are derived from Hamilton’s principle in terms of the generalized
displacements. Analytical solutions of free vibration and buckling are presented to bring out
the effect of the nonlocal behavior on natural frequencies and buckling loads. The presented
theoretical analysis is illustrated by a numerical example, and the results are qualitatively
compared by another results.

Keywords: natural frequency, critical buckling load, gradient elasticity theories, nonlocal
behavior

1. Introduction

Vibration and buckling problems of straight carbon nanotubes (CNT) (Spitalsky et al., 2010;
Salvetat et al., 1999)occupy an important place in micro- and nano-scale devices and systems.
Examples include nanosensors (Chopra et al., 2003, nanoactuators (Baughman et al., 1999,
nanooscillators (Nishio et al., 2005),micro-resonators (Bak et al., 2008) andfieldemissiondevices
(De Heer et al., 1995; Saito and Uemura, 2000), etc. In order to make full potential application
of CNT, it is essential to understand theirmechanical behavior well. Inmany papers, analytical
analyses of themechanical behavior of CNT have been proposed besides the experimental work
byRuoff et al. (2003). Carbonnanotubes can bemodeled using atomistic (Zhang et al., 2005) or
continuummechanicsmethods (Li andChou, 2003). The atomicmethods are limited to systems
with a small number ofmolecules or atoms and therefore they are restricted to the studyof small
scale modeling. Unlike atomistic modeling, continuummodels viewCNTas a continuous beam.
For realistic analysis of CNT, onemust incorporate small-scale effects to achieve solutions with
acceptable accuracy (WangandWang, 2007). Since the classical continuummodels are scale free,
for the modeling of CNT structures one can usemodified elasticity theories like Eringen theory
(Eringen, 1983; Eringen and Edelen, 1972) or strain gradient theories (Lam et al., 2003; Kong
et al., 2009; Akgöz and Civalek, 2011). In this way, the internal size scale could be considered
in the constitutive equation simply as amaterial parameter. In the theory of nonlocal elasticity,
the stress at a reference point is considered to be a functional of the strain field at every point
in the body. It can be concluded that continuum mechanics with size-effect could potentially


218 D. Karličić et al.

play a useful role in the analysis related to nanostructures (Adali, 2012;Muc, 2011;Murmu and
Adhikari, 2010a,b, 2011).

The first application of the Eringen nonlocal constitutive relation on the Euler-Bernoulli
beam is the work of Peddinson et al. (2003). They investigated the deflection behavior of the
nonlocal Euler-Bernoulli beam for different boundary conditions and possible application in
microelectromechanical systems (MEMS). A new nonlocal shear deformation beam theory for
bending, buckling and vibration of nanobeams was proposed by Huu-Tai (2012). The author
derived the general equation of motion and took into account a quadratic variation of the shear
strains across the thickness, based on nonlocal constitutive relation of Eringen. In the paper by
Reddy and Pang (2008), the equations of motion of Euler-Bernoulli and Timoshenko beam the-
orieswere reformulatedby theEringennonlocal theory, and thenused to evaluate static bending,
vibrations and buckling response of beamswith various boundary conditions. Recently, nonlocal
Euler-Bernoulli, Timoshenko, Reddy and Levinson beam theories were formulated by Reddy
(2007) in a unifiedmanner using Hamilton’s principle and nonlocal elastic constitutive relation
of Eringen. The analytical solution of natural frequency, critical buckling load and transversal
deflection have been obtained for all presented beam theories. A comparison of stress gradient
(Eringen’s nonlocal theory) and two strain gradient theories applied to free vibration analysis of
Euler-Bernoulli andTimoshenko beamswas carried out byAnsari et al. (2012).Wang andVara-
dan (2006) studied the influence of scale-effect on natural frequencies and comparisonwith local
natural frequencies of both single-walled CNT and double-walled CNT. They concluded that
the classical continuummodels are still valid and convenient for studying vibration responses of
long andwideCNTs, especially for lowermodes. The dynamic behavior of CNTembedded in an
elastic medium (matrix) investigated by using nonlocal Timoshenko beam theory for both the
stress gradient (Eringen nonlocal theory) and strain gradient approachwere considered byWang
andWang (2013). Their results show a significant dependence of frequencies on the surrounding
mediumandnonlocal parameter.Theuse of the nonlocalTimoshenkobeamtheory for analyzing
free vibration and buckling behavior of nano-composite structures reinforced by single-walled
carbon nanotubes (SWCNT) was proposed by Yas and Samadi (2012). They investigated the
influence of geometrical and physical parameters such as nanotube volume fraction, foundation
stiffness parameters, slenderness ratios and boundary conditions on the natural frequencies and
critical buckling load. Liu andReddy (2011) obtained a newmodel for static and free vibrations
problems of a simply supported curved beam based on the nonlocal Timoshenko beam theory.
Static and dynamic analyses of nanobeams based on the nonlocal Euler-Bernoulli, Timoshenko,
Reddy, Levinson and Aydogdu beam theories were presented in Aydogdu (2009). The influence
of the nonlocality and length of a nanobeam on natural frequencies, deflection and critical lo-
ad were investigated in detail for each considered model. Based on the nonlocal elasticity and
Euler-Bernoulli beam theory, the governing equation of transversal vibration of a nonuniform
cantilever nanobeamwas investigated byMurmu andPradhan (2009). They obtained numerical
results for the natural frequency from the governing equation by using the differential quadra-
ture method and analyzing the influence of small-scale effects on the dynamic behavior of the
nanocantilever.In the paper byAskes andAifantis (2009), nonlocal and strain gradient elasticity
theory were employed to obtain equations of motion for Euler-Bernoulli and Timoshenko beam
theory. They investigated the influence of various material parameters of high order continuum
theories on flexural wave dispersion in CNTs, and then compared the results with the results
obtained by molecular dynamic (MD) simulations. Hosseini-Ara et al. (2012) proposed a new
method to investigate the buckling behavior of short clamped CNTs, developed on the basis
of the strain gradient theory andTimoshenko beam kinematics. They determined exact critical
buckling loads using a linear polynomial and also investigated the influence of the scale coeffi-
cients, aspect ratio and transverse shear deformation on buckling of short clampedCNTs.Based
on the strain gradient elasticity theory, the governing equation ofmotion forEuler-Bernoulli and


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 219

higher shear deformation beam theories were derived by Akgöz and Civalek (2012, 2013). Also,
they analyzed the influence of differentmaterial parameters on the dynamic and static behavior
of a micro-size beam for different boundary conditions.
In the present paper, as an extension of the work byAnsari et al. (2012), we apply different

gradient elasticity theories on the Reddy and Huu-Tai beam theories and obtain the governing
equation for freeflexural vibrations andbucklingof SWCNTunderaxial loading.Thediscussions
are limited only to the case of a simply supported straight nanotube. The natural frequencies
and critical buckling load are obtained in the analytical form, based on Hamilton’s principle by
making use the stress gradient (nonlocal Eringen theory) and both strain gradient and combi-
ned strain/inertia gradient theories. The resulting equation for natural frequencies and critical
buckling load contains a scale parameter and can capture the size effect. Thedifferences between
the natural frequencies and critical buckling load for stress gradient, strain gradient, combined
strain/inertia theory and classical elasticity theory are shown and compared with the results by
Ansari et al. (2012), and excellent agreement is shown.

2. Structural model and theoretical formulation

TheReddy and Huu-Tai beam theories are adopted in this study. These theories, which do not
require shear correction factor, account for both small the scale effect and quadratic variation
of shear strains and, consequently, shear stresses through thickness of the beam. In order to
derive the equation of motions, we define the rectangular Cartesian coordinate system Oxyz.
The x-coordinate is taken along the length of the beam, the z-coordinate along the thickness of
the beam, and the y-coordinate along the width of the beam. We consider free vibration and
buckling in the xz-plane.

2.1. Constitutive relations

According to thenonlocal theory, stressatapointdependsnotonlyonthe strainat thatpoint
but also on strains at all other points of a body. The differential form of nonlocal constitutive
relations for a one-dimensional structure was proposed by Eringen (1983) as

σxx−µ
d2σxx
dx2

=Eεxx σxz−µ
d2σxz
dx2

=Gγxz (2.1)

whereE andG are the elasticmodulus and shearmodulus of the beam, respectively, µ=(e0a)
2

is the nonlocal parameter (length scales), e0 is a constant to adjust the model to match the
reliable results by experiments or microscopic models, a is the internal characteristic length
(e.g. lattice parameter, granular distance, wavelength) which can be identified from atomistic
simulations or by using a dispersive curve of the Born-Karmanmodel of lattice dynamics.
The combined strain/inertia constitutive relations for the one dimensional case, according

to the papers by Ansari et al. (2012), Askes and Aifantis (2009) and Hosseini-Ara et al. (2012)
are

σxx =E
(
εxx+µ

d2εxx
dx2

)
+ρµmε̈xx σxz =G

(
γxz +µ

d2γxz
dx2

)
+ρµmγ̈xz (2.2)

where ρ is the mass density and µm = l
2
m
and µ = l2 are related to inertia gradients and

strain gradient length scales, respectively. It should be noted that for µm = 0 the combined
strain/inertia theories are reduced to strain gradient theories, and for µ=0 the strain gradient
theory is reduced to the classical elasticity theory. The inertia gradient length scale factors µm,
for the representative volume element (RVE) size, are related to the dynamic case, which tends
to be larger than length scale factors µ for the static case. Accurate dynamical analysis of CNT


220 D. Karličić et al.

is obtained using the inertia gradients length scales µm. More details can be found in the paper
by Askes and Aifantis (2009).

2.2. The Reddy beam theory

Based on the Reddy beam theory, the axial displacements u(x,z,t) and transverse displace-
ments w(x,z,t) of any point of the beamare given by Reddy (2007) as

u(x,z,t) =u0(x,t)+zφx(x,t)− c1z
3
(
φx(x,t)+

∂w0(x,t)

∂x

)

v(x,z,t) = 0 w(x,z,t) =w0(x,t)
(2.3)

where c1 =4/(3h
2) andh is the height of the beam,w0(x,t) andφx(x,t) are the transversal and

rotation components of the displacement. The nonzero strains of the proposed beam theory are

εxx =
∂u0
∂x
+z
∂φx
∂x
− c1z

3
(∂φx
∂x
+
∂2w0
∂x2

)

γxz =(1− c2z
2)
(
φx+

∂w0
∂x

) (2.4)

where c2 = 3, c1 = 4/h
2. In this case, the component u0 of the axial displacement u(x,z,t) is

neglected.

Based on Hamilton’s principle which states that the motion of an elastic structure during
the time interval 0 < t < T is such that the time integral of the total potential is extremum
(Reddy, 2007) one writes

T∫

0

(δU + δV − δK) dt=0 (2.5)

where δU is the variation of strain energy, δV is the virtual work of external forces and δK is
the variation of kinetic energy of the nanobeam.

The variation of strain energy of the beam is

δU =

A∫

0

L∫

0

(σxxδεxx+σxzδγxz) dxdA

=

L∫

0

[
(Mxx− c1Pxx)δ

∂φx
∂x
− c1Pxxδ

∂2w0
∂x2
+(Qx− c2Rx)δ

(
φx+

∂w0
∂x

)]
dx

(2.6)

whereMxx, Pxx,Qx andRx are the stress resultants defined as

(Mxx,Pxx)=

A∫

0

(z,z3)σxx dA (Qx,Rx)=

A∫

0

(1,z2)σxz dA (2.7)

The variation of potential energy of external forces can be expressed as

δV =−

L∫

0

(
q(x)δw0+ N̂0

∂w0
∂x
δ
∂w0
∂x

)
dx (2.8)

where q(x) is the continual transversal load and N̂0 is the axial load.


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 221

The variation of kinetic energy is obtained as

δK =

A∫

0

L∫

0

ρ(u̇δu̇+ ẇδẇ) dxdA=

L∫

0

[
Î2φ̇xδφ̇x− c1Î4

(
φ̇x+

∂ẇ0
∂x

)
δφ̇x

− c1Î4φ̇xδ
(
φ̇x+

∂ẇ0
∂x

)
+ c21Î6

(
φ̇x+

∂ẇ0
∂x

)
δ
(
φ̇x+

∂ẇ0
∂x

)
+ Î0ẇ0δẇ0

]
dx

(2.9)

where ρ is the mass density and ∂w0/∂t = ẇ0 is the time dderivative of the displacement w0
and (Î0, Î2, Î4, Î6) are the mass inertias defined as

(Î0, Î2, Î4, Î6)=

A∫

0

(1,z2,z4,z6)ρ dA (2.10)

To derive the equations of motion associated with the present model, we substitute the expres-
sions for δU, δV and δK from Eqs. (2.6), (2.8) and (2.9) into Eq. (2.5), and after integrating
by parts and then collecting the coefficients of δw(0) and δφx, the following equations of motion
are obtained

δw0 :
∂Q
x

∂x
+ c1
∂2Pxx
∂x2

+ N̂0
∂2w0
∂x2

= Î0ẅ0+ Î4
∂φ̈x
∂x
− c21Î6

(∂φ̈x
∂x
+
∂2ẅ0
∂x2

)

δφx :
∂Mxx
∂x
−Q

x
= φ̈xÎ2− c1Î4

(
2φ̈x+

∂ẅ0
∂x

)
+ c21Î6

(
φ̈x+

∂ẅ0
∂x

) (2.11)

where

Q
x
=Qx− c2Rx Mxx =Mxx− c1Pxx (2.12)

The boundary conditions of the model are

w0 or Vx = c1
∂Pxx
∂x
+Q

x
− N̂0

∂w0
∂x
−c1Î4φ̈x+ c

2
1Î6
(
φ̈x+

∂ẅ0
∂x

)

∂w0
∂x

or Pxx

φx or Mxx =Mxx− c1Pxx

(2.13)

where Vx denotes the equivalent shear force.

By substitutingEq. (2.4) into Eq. (2.1) and the subsequent results into Eq. (2.7) and (2.12),
the stress resultants for the nonlocal Eringen theory (stress gradient) are obtained as

Pxx−µ
d2Pxx
dx2

=E(I4−c1I6)
∂φx
∂x
− c1I6E

∂2w0
∂x2

Q
x
−µ
d2Qx
dx2

=GÂ
(
φx+

∂w0
∂x

)

Mxx−µ
d2Mxx
dx2

=−Ec1I4
(
2
∂φx
∂x
+
∂2w0
∂x2

)
+EI2

∂φx
∂x
+Ec21I6

(∂φx
∂x
+
∂2w0
∂x2

)

(2.14)

where

(A,I2,I4,I6)=

A∫

0

(1,z2,z4,z6) dA Â=(A−2c2I2+ c
2
2I4) (2.15)


222 D. Karličić et al.

The equations of motion can be expressed in terms of the displacement (w0,φx) for nonlocal
constitutive relations (stress gradient). By substituting Eq. (2.14) into Eq. (2.11), we get the
following equations of motion

N0
∂2w0
∂x2
+ Î0ẅ0−K1

∂φ̈x
∂x
− c21Î6

∂2ẅ0
∂x2
−µ

(
N0
∂4w0
∂x4
+ Î0
∂2ẅ0
∂x2
−K1

∂3φ̈x
∂x3
− c21Î6

∂4ẅ0
∂x4

)

= c1
[
E(I4−c1I6)

∂3φx
∂x3
− c1I6E

∂4w0
∂x4

]
+G

(∂φx
∂x
+
∂2w0
∂x2

)
Â

G
(
φx+

∂w0
∂x

)
Â+K2φ̈x+K1

∂ẅ0
∂x
−µ

(
K2
∂2φ̈x
∂x2
+K1

∂3ẅ0
∂x3

)

=EI2
∂2φx
∂x2
−EI4c1

(
2
∂2φx
∂x2
+
∂3w0
∂x3

)
+ c21EI6

(∂2φx
∂x2
+
∂3w0
∂x3

)

(2.16)

where N̂0 =−N0 is the applied axial compressive force, and

K1 = c
2
1Î6−c1Î4 K2 = Î2−2c1Î4+ c

2
1Î6 (2.17)

In a very similar wayas in the previouscase, we get the stress resultants and equation of motion
for the case of the combined strain/inertia gradient theory of elasticity. By substitutingEq. (2.4)
into Eq. (2.2) and the subsequent results into Eqs. (2.7) and (2.12), the stress resultants for the
combined strain/inertia gradient theory are obtained in the following form

Pxx =EI4
∂φx
∂x
− c1EI6

(∂φx
∂x
+
∂2w0
∂x2

)
+µEI4

∂3φx
∂x3
−µc1EI6

(∂3φx
∂x3
+
∂4w0
∂x4

)

+ρµmI4
∂φ̈x
∂x
−ρµmc1I6

(∂φ̈x
∂x
+
∂2ẅ0
∂x2

)

Qx =GÂ
(
φx+

∂w0
∂x

)
+µGÂ

(∂2φx
∂x2
+
∂3w0
∂x3

)
+ρµmÂ

(
φ̈x+

∂ẅ0
∂x

)

Mxx =EI2
∂φx
∂x
− c1EI4

(
2
∂φx
∂x
+
∂2w0
∂x2

)
+µEI2

∂3φx
∂x3
−µc1EI4

(∂3φx
∂x3
+
∂4w0
∂x4

)

+ c21EI6
(∂φx
∂x
+
∂2w0
∂x2

)
−µc1EI4

∂3φx
∂x3
+µc21EI6

(∂3φx
∂x3
+
∂4w0
∂x4

)
+ρµmI2

∂φ̈x
∂x

−ρµmc1I4
(
2
∂φ̈x
∂x
+
∂2ẅ0
∂x2

)
+ρµmc

2
1I6
(∂φ̈x
∂x
+
∂2ẅ0
∂x2

)

(2.18)

By substituting Eq. (2.18) into Eq. (2.11), we obtain the following equation of motion for the
combined strain/inertia gradients constitutive relation in terms of generalized displacements as

GÂ
(∂φx
∂x
+
∂2w0
∂x2

)
+µGÂ

(∂3φx
∂x3
+
∂4w0
∂x4

)
+ρµmÂ

(∂φ̈x
∂x
+
∂2ẅ0
∂x2

)
+ c1EI4

∂3φx
∂x3

− c21EI6
(∂3φx
∂x3
+
∂4w0
∂x4

)
+µc1EI4

∂5φx
∂x5
−µc21EI6

(∂5φx
∂x5
+
∂6w0
∂x6

)
+ρµmc1I4

∂3φ̈x
∂x3

−ρµmc
2
1I6
(∂3φ̈x
∂x3
+
∂4ẅ0
∂x4

)
=N0

∂2w0
∂x2
+ Î0ẅ0−K1

∂φ̈x
∂x
− c21Î6

∂2ẅ0
∂x2

EI2
∂2φx
∂x2
− c1EI4

(
2
∂2φx
∂x2
+
∂3w0
∂x3

)
+µEI2

∂4φx
∂x4
−µc1EI4

(∂4φx
∂x4
+
∂5w0
∂x5

)

+ c21EI6
(∂2φx
∂x2
+
∂3w0
∂x3

)
−µc1EI4

∂4φx
∂x4
+µc21EI6

(∂4φx
∂x4
+
∂5w0
∂x5

)
+ρµmI2

∂2φ̈x
∂x2

−ρµmc1I4
(
2
∂2φ̈x
∂x2
+
∂3ẅ0
∂x3

)
+ρµmc

2
1I6
(∂2φ̈x
∂x2
+
∂3ẅ0
∂x3

)
=GÂ

(
φx+

∂w0
∂x

)

+µGÂ
(∂2φx
∂x2
+
∂3w0
∂x3

)
+ρµmÂ

(
φ̈x+

∂ẅ0
∂x

)
+K2φ̈x+K1

∂ẅ0
∂x

(2.19)


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 223

The equation of motion for the strain gradients constitutive relation, in terms of generalized
displacements, is obtained from Eqs. (2.19) by setting the parameter µm equal to zero, as

GÂ
(∂φx
∂x
+
∂2w0
∂x2

)
+µGÂ

(∂3φx
∂x3
+
∂4w0
∂x4

)
+ c1EI4

∂3φx
∂x3
− c21EI6

(∂3φx
∂x3
+
∂4w0
∂x4

)

+µc1EI4
∂5φx
∂x5
−µc21EI6

(∂5φx
∂x5
+
∂6w0
∂x6

)
=N0

∂2w0
∂x2
+ Î0ẅ0−K1

∂φ̈x
∂x
− c21Î6

∂2ẅ0
∂x2

EI2
∂2φx
∂x2
− c1EI4

(
2
∂2φx
∂x2
+
∂3w0
∂x3

)
+µEI2

∂4φx
∂x4
−µc1EI4

(∂4φx
∂x4
+
∂5w0
∂x5

)

+ c21EI6
(∂2φx
∂x2
+
∂3w0
∂x3

)
−µc1EI4

∂4φx
∂x4
+µc21EI6

(∂4φx
∂x4
+
∂5w0
∂x5

)

=GÂ
(
φx+

∂w0
∂x

)
+µGÂ

(∂2φx
∂x2
+
∂3w0
∂x3

)
+K2φ̈x+K1

∂ẅ0
∂x

(2.20)

The equation ofmotion of the local beam theory can be obtained fromEq. (2.20) by setting the
nonlocal parameter µ to be equal to zero.

2.3. The Huu-Tai beam theory

The displacement field of the Huu-Tai beam theory proposed byHuu-Tai (2012) is based on
the following assumptions:

(1) the axial and transverse displacements consist of bending and shear components in which
the bending components do not contribute toward shear forces and, likewise, the shear
components do not contribute toward bendingmoments;

(2) the bending component of the axial displacement is similar to that given by the Euler-
Bernoulli beam theory;

(3) the shear component of the axial displacement gives rise to the parabolic variation of the
shear strain and hence to the shear stress through thickness of the beam in such a way
that the shear stress vanishes on the top and the bottom surface.

According to these assumptions, we get the next displacement field

u(x,z,t) =u0(x,t)−z
∂wb(x,t)

∂x
+
[1
4
z−
5

3
z
(z
h

)2]∂ws(x,t)
∂x

v(x,z,t) = 0 w(x,z,t) =wb(x,t)+ws(x,t)
(2.21)

whereh h is the height of the beam,wb(x,t) andws(x,t) are the bending and shear components
of the transverse displacement and u0(x,t) is the axial displacement along the midplane of the
beam. According to the proposed beam theory, the strain-displacement relations are given by

εxx =
∂u0
∂x
−z
∂2wb
∂x2
−f
∂2ws
∂x2

γxz = g
∂ws
∂x

(2.22)

where f = −z/4+5z(z/h)2/3 and g = 5/4− 5(z/h)2. In this case, the component u0 of the
axial displacement u(x,z,t) is neglected.

A nonlocal Huu-Tai beammodel is developed using a different gradient elasticity theory and
Hamilton’s principle Eq. (2.5). The variation of strain energy of the Huu-Tai beam theory is

δU =

A∫

0

L∫

0

(σxxδεxx+σxzδγxz) dxdA=

L∫

0

(
−Mbδ

∂2wb
∂x2
−Msδ

∂2ws
∂x2
+Qδ

∂ws
∂x

)
dx (2.23)


224 D. Karličić et al.

whereMb,Ms andQ are the stress resultants defined as

(Mb,Ms)=

A∫

0

(z,f)σxx dA Q=

A∫

0

gσxz dA (2.24)

The variation of potential energy of external forces can be expressed as

δV =−

L∫

0

[
q(x)δ(wb+ws)+N0

∂(wb+ws)

∂x
δ
∂(wb+ws)

∂x

]
dx (2.25)

where q(x) is the continual transversal load andN0 is the axial load.
The variation of kinetic energy is obtained as

δK =

A∫

0

L∫

0

ρ(u̇δu̇+ ẇδẇ) dxdA

=

L∫

0

[
Î0(ẇb+ ẇs)δ(ẇb+ ẇs)+ Î2

∂ẇb
∂x
δ
∂ẇb
∂x
+
Î2
84

∂ẇs
∂x
δ
∂ẇs
∂x

]
dx

(2.26)

where Î0 and Î2 are defined in the preceding section, Eq. (2.10), ẇb and ẇs are time derivatives
of the bending and shear components of the transverse displacement, respectively.
Substituting the expressions for δU, δV and δK from Eqs. (2.23), (2.25) and (2.26) into

Hamilton’s principle Eq. (2.5) and integrating by parts, and then collecting the coefficients of
δwb and δws, we obtain

δwb :
∂2Mb
∂x2

−N0
∂2(wb+ws)

∂x2
= Î0(ẅb+ ẅs)− Î2

∂2ẅb
∂x2

δws :
∂2Ms
∂x2

+
∂Q

∂x
−N0

∂2(wb+ws)

∂x2
= Î0(ẅb+ ẅs)−

Î2
84

∂2ẅs
∂x2

(2.27)

The boundary conditions involve specification of one element of each of the following four pairs
at x=0 and x=L

wb or Vb =
∂Mb
∂x
−N0

∂(wb+ws)

∂x
+ Î2
∂ẅb
∂x

ws or Vs =
∂Ms
∂x
+Q−N0

∂(wb+ws)

∂x
+
Î2
84

∂ẅs
∂x

∂wb
∂x

or Mb

∂ws
∂x

or Ms

(2.28)

where Vb and Vs are the equivalent transversal forces on the ends of the beam.
By substituting Eq. (2.22) into Eq. (2.1) and the subsequent results into Eq. (2.24), the

stress resultants for the nonlocal Eringen theory (stress gradient) are obtained as

Mb−µ
d2Mb
dx2

=−EI2
d2wb
dx2

Ms−µ
d2Ms
dx2

=−
EI2
84

d2ws
dx2

Q−µ
d2Q

dx2
=
5GA

6

∂ws
∂x

(2.29)

whereA and I2 are defined in the previous section Eq. (2.15).


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 225

By substitutingEq. (2.29) intoEq. (2.27), the nonlocal equations ofmotion can be expressed
in terms of the displacements wb andws as

−EI2
∂4wb
∂x4
−N0

[∂2(wb+ws)
∂x2

−µ
∂4(wb+ws)

∂x4

]

= Î0
[
(ẅb+ ẅs)−µ

∂2(ẅb+ ẅs)

∂x2

]
− Î2

(∂2ẅb
∂x2
−µ
∂4ẅb
∂x4

)

−
EI2
84

∂4ws
∂x4
+
5GA

6

∂2ws
∂x2
−N0

[∂2(wb+ws)
∂x2

−µ
∂4(wb+ws)

∂x4

]

= Î0
[
(ẅb+ ẅs)−µ

∂2(ẅb+ ẅs)

∂x2

]
−
Î2
84

(∂2ẅs
∂x2
−µ
∂4ẅs
∂x4

)

(2.30)

When the shear deformation effect of the nonlocal parameter is neglected (ws = 0, µ = 0),
the equations of motions Eq. (2.30) and boundary conditions in Eq. (2.28) are reduced to the
Euler-Bernoulli beam theory.
The equation of motion for the case of the combined strain/inertia gradient theory of ela-

sticity, according to the Huu-Tai beam theory, is obtained by substituting Eqs. (2.22) into Eqs.
(2.2), then subsequent results into Eq. (2.24), the stress resultants are obtained as

Mb =−EI2
∂2wb
∂x2
−µEI2

∂4wb
∂x4
−ρµmI2

∂2ẅb
∂x2

Ms =−
EI2
84

∂2ws
∂x2
−µ
EI2
84

∂4ws
∂x4
−ρµm

I2
84

∂2ẅs
∂x2

Q=
5GA

6

(∂ws
∂x
+µ
∂3ws
∂x3

)
+ρµm

5A

6

∂ẅs
∂x

(2.31)

The equations ofmotion for the combined strain/inertia gradients constitutive relation, in terms
of generalized displacements, are

Î0(ẅb+ ẅs)− Î2
∂2ẅb
∂x2
+EI2

∂4wb
∂x4
+µEI2

∂6wb
∂x6
+ρµmI2

∂4ẅb
∂x4
+N0

∂2(wb+ws)

∂x2
=0

5GA

6

(∂2ws
∂x2
+µ
∂4ws
∂x4

)
+ρµm

5A

6

∂2ẅs
∂x2
−N0

∂2(wb+ws)

∂x2

= Î0(ẅb+ ẅs)−
Î2
84

∂2ẅs
∂x2
+
EI2
84

∂4ws
∂x4
+µ
EI2
84

∂6ws
∂x6
+ρµm

I2
84

∂4ẅs
∂x4

(2.32)

By setting the parameter µm equal to zero in Eqs. (2.32), we obtain the equation of motion for
strain gradient constitutive equations as

Î0(ẅb+ ẅs)− Î2
∂2ẅb
∂x2
+EI2

∂4wb
∂x4
+µEI2

∂6wb
∂x6
+N0

∂2(wb+ws)

∂x2
=0

5GA

6

(∂2ws
∂x2
+µ
∂4ws
∂x4

)
−N0

∂2(wb+ws)

∂x2

= Î0(ẅb+ ẅs)−
Î2
84

∂2ẅs
∂x2
+
EI2
84

∂4ws
∂x4
+µ
EI2
84

∂6ws
∂x6

(2.33)

The equation of motion of the local Huu-Tai beam theory can be obtained from Eq. (2.33) by
setting the nonlocal parameter µ equal to zero.

3. Analytical solutions of vibration and buckling of simply supported beams

3.1. The Reddy beam theory

In the present work, a simply supported beamwith lengthL, subjected to compressive axial
loading N0 is considered. The boundary conditions for the simply supported beam and the


226 D. Karličić et al.

Reddy beam theory are

w0(0, t) =Mxx(0, t) =Pxx(0, t)=w0(L,t)=Mxx(L,t)=Pxx(L,t)= 0 (3.1)

With theboundaryconditionsEq. (3.1), Eqs. (2.16), (2.19) and (2.20) can be solved byassuming
the solution in the following expansions of the generalized displacements w0 and φ(x), as

w0(x,t)=
∞∑

n=1

Wn sinαxe
iωnt φx(x,t) =

∞∑

n=1

Xncosαxe
iωnt (3.2)

where i=
√

−1, α=nπ/L, (Wn,Xn) are the amplitudes and ωn is the natural frequency.
Substituting the expansions for w0 and φx from Eqs. (3.2) into the equations of motion

for different gradient Reddy beam models for SWCNT’s, and solving the resulting eigenvalue
problem, the natural frequency and critical buckling load of SWCNT’s can be obtained

([
s11 s12
s21 s22

]

−ω2
n

[
m11 m12
m21 m22

]

−N0

[
n11 0
0 0

]){
Wn
Xn

}

=

{
0
0

}

(3.3)

where the coefficients for stress gradient theory are

s11 =Ec
2
1I6α

4+Gα2Â s12 = s21 =−Ec1(I4− c1I6)α
3+GαÂ

s22 =GÂ+E(I2− c1I4)α
2
−Ec1(I4−c1I6)α

2

m11 = Î0+α
2c21Î6+µα

2Î0+µα
4c21Î6 m12 =m21 =K1α+µK1α

3

m22 =K2+µK2α
2 n11 =α

2+µα4

(3.4)

The coefficients in the strain/inertia gradient theory are

s11 =Gα
2Â+ c21EI6α

4
−µGα4Â−µc21EI6α

6

s12 = s21 =GαÂ−µGα
3Â−Ec1(I4− c1I6)α

3+µEc1(I4− c1I6)α
5

s22 =GÂ−µGÂα
2+E(I2−c1I4)α

2
−Ec1(I4− c1I6)α

2

−Eµ(I2− c1I4)α
4+µEc1(I4−c1I6)α

4

m11 = Î0+α
2c21Î6+ρµmÂα

2+ρµmc
2
1I6α

4

m12 =m21 =K1α+ρµmÂα−ρµmc1(I4− c1I6)α
3

m22 =K2+ρµm(I2− c1I4)α
2
−ρµmc1(I4− c1I6)α

2+ρµmÂ

n11 =α
2

(3.5)

The coefficients in the strain gradient theory (µm =0) are

s11 =Gα
2Â+ c21EI6α

4
−µGα4Â−µc21EI6α

6

s12 = s21 =GαÂ−µGα
3Â−Ec1(I4− c1I6)α

3+µEc1(I4− c1I6)α
5

s22 =GÂ−µGÂα
2+E(I2−c1I4)α

2
−Ec1(I4− c1I6)α

2
−Eµ(I2− c1I4)α

4

+µEc1(I4− c1I6)α
4

m11 = Î0+α
2c21Î6 m12 =m21 =K1α m22 =K2 n11 =α

2

(3.6)

The buckling load is obtained from Eq. (3.3) by setting ωn to zero

N0 =
s11s22−s

2
12

n11s22
(3.7)

The natural frequency is obtained from Eq. (3.3) by setting N0 to zero

ω2
n
=

1

2(m11m22−m
2
12)

[
m22s11−2m12s12+m11s22

−

√
(−m22s11+2m12s12−m11s22)2−4(−m

2
12+m11m22)(−s

2
12+s11s22)

] (3.8)


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 227

3.2. The Huu-Tai beam theory

The simply supported beam boundary conditions for the Huu-Tai beam theory are

wb(0, t) =ws(0, t) =Mb(0, t)=Ms(0, t) =wb(L,t)=ws(L,t)=Mb(L,t)

=Ms(L,t)= 0
(3.9)

With the governing boundary conditions Eq. (3.9), Eqs. (2.30), (2.32) and (2.33) can be solved
by assuming the solution in the following expansions of the generalized displacementswb andws,
as

wb(x,t)=
∞∑

n=1

Wbn sinαxe
iωnt ws(x,t) =

∞∑

n=1

Wsn sinαxe
iωnt (3.10)

where α=nπ/L,Wbn andWsn are the amplitudes and ωn is the natural frequency.
The closed-form solutions for the natural frequencies and buckling load of SWCNTusing the

Huu-Tai beam theory (Huu-Tai, 2012) for different gradients elasticity theories can be obtained
by substituting the expansions of wb and ws into equations of motion Eqs. (2.30), (2.32) and
(2.33), from the following equations

([
s11 0
0 s22

]

−ω2
n

[
m11 m12
m21 m22

]

−N0n

[
1 1
1 1

]){
Wbn
Wsn

}

=

{
0
0

}

(3.11)

where the coefficients in the stress gradient theory are

s11 =EI2α
4 s22 =

EI2
84
α4+

5GA

6
α2 λ=1+µα2

m11 =λ(Î0+α
2Î2) m12 =m21 =λÎ0 m22 =λ

(
Î0+α

2 Î2
84

)

n=λα2

(3.12)

Also, with the strain/inertia gradient theory one can obtain the coefficients

s11 =EI2α
4λ s22 =

EI2
84
α4λ+

5GA

6
α2λ λ=1−µα2

m11 = Î0+α
2Î2+ρµmI2α

4 m12 =m21 = Î0

m22 = Î0+α
2 Î2
84
+ρµm

I2
84
α4+ρµm

5A

6
α2 n=α2

(3.13)

and in the strain gradient theory (µm =0)

s11 =EI2α
4λ s22 =

EI2
84
α4λ+

5GA

6
α2λ λ=1−µα2

m11 = Î0+α
2Î2 m12 =m21 = Î0 m22 = Î0+α

2 Î2
84

n=α2

(3.14)

The buckling load is obtained from Eq. (3.11) by setting ωn to zero

N0 =
s11s22

n(s11+s22)
(3.15)

The natural frequency is obtained from Eq. (3.11) by settingN0 to zero

ω2
n
=
m22s11+m11s22−

√
(m22s11+m11s22)

2
−4(−m212+m11m22)s11s22

2(−m212+m11m22)
(3.16)


228 D. Karličić et al.

4. Numerical results and discussion

In thefirstpart of this Section,we compare theobtainedanalytical results for bothbeamtheories
with the results previously published in literature. An excellent agreement is shown with the
results obtained for the Timoshenkomodel of the nanobeam and also with the results obtained
by MD simulation, proposed by Ansari et al. (2012). To investigate the effect of the nonlocal
parameter and the aspect ratio on vibration and stability behavior of the Reddy and Huu-
Tai beam for different constitutive relations, numerical analyses of the frequency and buckling
load are performed and discussed. In what follows, we consider the armchair (8,8) single-walled
carbon nanotube (SWCNT) with thickness (h = 0.34nm), Poisson’s ratio (ν = 0.3), mass
density (ρ=2300kg/m3),Young’smodulus (E =1.1TPa) andnonlocal parameters (l=0.6nm,
lm/l=0.3).Numerical results for thenatural frequencyandbuckling load ofReddyandHuu-Tai
beam theories are presented in form of graphs and a table, using different constitutive relations,
where influences of high order rotary inertias are also taken into account only to compute the
natural frequencies.

4.1. Validation study

The results of fundamental natural frequencies of the simply supported SWCNT’s for different
constitutive relations are presented inTable 1. In thepresentedanalysis, the ratio of thedynamic
and static length scale parameter lm/l is adopted fromAnsari et al. (2012), which has obtained
via MD simulations.The results obtained by presented analyses for different gradient theories
for fundamental frequency are found to be in excellent agreementwith the one obtained from
MD simulation and for the Timoshenko beam model shown by Ansari et al. (2012). From this
table it can be seen that the fundamental frequencies decrease with the increasing beam aspect
ratio for all cases of gradient elasticity and beam theories. Also, it is observed that the results
obtained by applying theHuu-Tai andReddybeammodels lead to the identical results obtained
for values of aspect ratios higher than 17.3. It is indicated that the influence of the nonlocal
parameteris reflected in reduction of the fundamental frequency, but that it has no effect on the
classical theory of elasticity. When the nonlocal parameter is sufficiently small, the discrepancy
between various constitutive relations decreases so that the fundamental frequencies tend to
converge at the fundamental frequencies in the classical elasticity theory. The comparison of the
fundamental frequencies from the Reddy and Huu-Tai beam theory shows lower values in case
of the combined strain/inertia gradient beam theory than for other gradient elasticity theories.
Also, comparing the results for the combined strain/inertia gradient with the results obtained
fromMD simulation (Ansari et al., 2012), one can observe excellent agreement.

4.2. Numerical examples

Figure 1 shows the natural frequency from the Reddy and Huu-Tai beam theories as a
function of the aspect ratio (L/D) for various constitutive relations. It can be seen that the
natural frequency decreases as the beam aspect ratio increases. When the beam aspect ratio is
sufficiently larger, the discrepancy between various constitutive relations decreases so that the
natural frequencies tend to converge to the aspect ratio. However, for small values of the aspect
ratio, differences in the fundamental frequencies for different constitutive relations are much
greater than for larger values. It should be noted that the natural frequency of the combined
strain/inertia gradient beam theory is lower than those of other gradient elasticity theories and,
therefore, this beam theory can accommodate the results of other ones for both cases. From the
physical point of view, we can say that the influence of the aspect ratio (L/D) on the dynamic
behavior has damping effects.


F
le
x
u
ra
l
v
ib
ra
tio
n
a
n
d
b
u
c
k
lin
g
a
n
a
ly
sis
o
f
sin
g
le
-w
a
lled
ca
r
bo
n
n
a
n
o
tu
be
s...

2
2
9

Table 1. Comparison of fundamental frequencies [THz] from different beam and gradient elasticity theories for (8,8)

armchair SWCNT’s (R/l=0.6nm, lm/l=8)

Timoshenko beam theory
Reddy beam theory Huu-Tai beam theory

Ref. Ansari et al. (2012)

L/D
MD
simu-
lation

Classical
theory

Stress
gra-
dient
theory

Strain
gra-
dient
theory

Strain/

Classical
theory

Stress
gra-
dient
theory

Strain
gra-
dient
theory

Strain/

Classical
theory

Stress
gra-
dient
theory

Strain
gra-
dient
theory

Strain/

inertia inertia inertia

gradient gradient gradient

theory theory theory

8.3 0.5299 0.5306 0.5302 0.5302 0.5299 0.5403 0.5393 0.5393 0.5387 0.5398 0.5387 0.5387 0.5381

10.1 0.3618 0.3606 0.3604 0.3604 0.3603 0.3669 0.3664 0.3664 0.3662 0.3666 0.3661 0.3661 0.3659

13.7 0.1931 0.1972 0.1971 0.1971 0.1971 0.2004 0.2003 0.2003 0.2002 0.2004 0.2002 0.2002 0.2002

17.3 0.1103 0.1240 0.1240 0.1240 0.1240 0.1260 0.1259 0.1259 0.1259 0.1259 0.1259 0.1259 0.1259

20.9 0.0724 0.0851 0.0851 0.0851 0.0851 0.0864 0.0864 0.0864 0.0864 0.0864 0.0864 0.0864 0.0864

24.5 0.0519 0.0620 0.0620 0.0620 0.0620 0.0629 0.0629 0.0629 0.0629 0.0629 0.0629 0.0629 0.0629

28.1 0.0425 0.0471 0.0471 0.0471 0.0471 0.0478 0.0478 0.0478 0.0478 0.0478 0.0478 0.0478 0.0478

31.6 0.0358 0.0373 0.0373 0.0373 0.0373 0.0378 0.0378 0.0378 0.0378 0.0378 0.0378 0.0378 0.0378

35.3 0.0287 0.0299 0.0299 0.0299 0.0299 0.0303 0.0303 0.0303 0.0303 0.0303 0.0303 0.0303 0.0303

39.1 0.0259 0.0244 0.0244 0.0244 0.0244 0.0247 0.0247 0.0247 0.0247 0.0247 0.0247 0.0247 0.0247


230 D. Karličić et al.

Fig. 1. Influence of the beam aspect ratio (L/D) on various fundamental frequencies for (8,8) armchair
SWCNT’s based on the (a) Reddy beam theory, (b) Huu-Tai beam theory; l=0.6nm, l

m
/l=3

The influence of the beam aspect ratio on the critical buckling load for various constitutive
relations are shown in Fig. 2. It can be seen that the buckling load decreases with the increase
in the beam aspect ratio for both beam theories. One can notice that small values of the beam
aspect ratio (L/D) cause significant differences in the buckling load for different constitutive
relations.Also, the effect of the beamaspect ratio (L/D) on the buckling load decreaseswith the
increase in the aspect ratio. It can be concluded that the influences of µ on the critical buckling
load are significant for small values of the aspect ratio and not for larger values. This conclusion
also applies to the fundamental frequency. The buckling load for the combined strain/inertia
gradients and strain gradients theory is the same for both beam theories, because µm does not
affect the value of the buckling load.

Fig. 2. Influence of the beam aspect ratio (L/d) on various buckling loads for (8,8) armchair SWCNT’s
based on the (a) Reddy beam theory, (b) Huu-Tai beam theory; l=0.6nm, l

m
/l=3

In order to investigate the effect of the nonlocal parameter on the natural frequency and
buckling load of the SWCNT’s, curves have been plotted as a function of the nonlocal parameter
for four constitutive equations andbothbeamtheories (Figs. 3 and 4). FromFig. 3 it can be seen
that the increase in the nonlocal parameter causes a decrease in the natural frequency for all
casesof constitutive relations. This implies that the overall stiffness of SWCNTs is significantly
reduced due to the effects of the nonlocal parameter. From the physical point of view,we can say
that the nonlocal parameter has a damping effect on the dynamic behavior of SWCNTs. Also,
it is interesting to note that the values of the natural frequencyin the case of the strain/inertia
gradient theory have the lowest value, which is in line with the previous results byAnsari et al.
(2012). By comparing the values of the natural frequency from theReddy (Fig. 3a) andHuu-Tai
(Fig. 3a) beam theory, it can be concluded that the nonlocal parameter has a similar influence
in both cases. Considering the influence of the nonlocal parameter on the buckling load (Fig. 4),


Flexural vibration and buckling analysis of single-walled carbon nanotubes... 231

a similar conclusion can be drawn as in the case of natural frequencies. Moreover, it can be
seen that the strain/inertia gradient theory is reduced to the strain gradient theory, so it can be
concluded that the inertia gradient length scale factors µm have no effect on the buckling load.

Fig. 3. Influence of the nonlocal parameter on various fundamental frequencies for (8,8) armchair
SWCNT’s based on the (a) Reddy beam theory, (b) Huu-Tai beam theory;L/d=5

Fig. 4. Influence of the nonlocal parameter on various buckling loads for (8,8) armchair SWCNT’s based
on the (a) Reddy beam theory and (b) Huu-Tai beam theory;L/d=5

5. Conclusions

In this paper, the influence of the aspect ratio and the nonlocal parameter on free vibration and
buckling of SWCNT’s based on different gradient elasticity theories is discussed.TheReddy and
Huu-Tai beam theories have been utilized for modeling SWCNT’s. The closed form solutions of
natural frequencies and buckling load, which include the effect of the nonlocal parameter, shear
deformation and rotary inertia, have been obtained for different constitutive relations. From
the numerical results, it can be observed that the nonlocal parameter reduces the fundamental
frequency and buckling load of simply supported SWCNT’s. Also, it can be concluded that
the aspect ratio increases the fundamental frequency but the buckling load decreases in both
beam theories. Comparing these two beam theories, it is found that the fundamental frequency
and buckling load predicted by the stress gradient, classical elasticity theories, combined stra-
in/inertia and strain gradient elasticity are in good agreement. Also, the influence of nonlocal
parametersµ is greater on the buckling load than on the fundamental frequencies. The dynamic
nonlocal parameter µm does not affect the buckling load. The obtained analytical results have
been compared with the results found in literature for the Timoshenko beam model and MD
simulation, and excellent agreement is shown.


232 D. Karličić et al.

Acknowledgments

This investigation has been supported by the research grants of the Serbian Ministry of Education,

Science and Technological Development, number OI 174001 and OI 174011.

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Manuscript received September 6, 2013; accepted for print September 26, 2014