Journal ofComputational and Theoretical Nanoscience
Vol. 7, 1–5, 2010
Buckling Temperature of a Single-Walled CarbonNanotube Using Nonlocal Timoshenko Beam Model
Win-Jin Chang and Haw-Long Lee∗
Department of Mechanical Engineering, Kun Shan University, Tainan 710, Taiwan
The thermal buckling of single-walled carbon nanotube (SWCNT) subjected to a uniform tempera-ture rise is studied using Timoshenko beam model, including the effects of transverse shear defor-mation and rotary inertia. The governing buckling equations of SWCNT are established on the basisof nonlocal continuum theory. An analytical solution to the equations is derived for the determina-tion of critical buckling temperature. The solution can be further reduced to obtain the results ofEuler beam model. According to the analysis, for mode 4, the Euler beam model overpredicts thecritical buckling temperature by 57%, 13% and 3.5% for L/d = 10, 30 and 60, respectively. TheTimoshenko beam model is able to predict the buckling temperature of the SWCNT at higher-ordermodes with small length-to-diameter ratios. In addition, the critical buckling temperature decreaseswith increasing the nonlocal value.
Carbon nanotubes (CNTs) have become the most signif-icant material in the coming nanotechnology age due totheir excellent physical and chemical properties.1–4 A greatnumber of researches have been made to investigate thediverse properties of them for a wide variety of applica-tions. CNTs subjected to compression or bending are espe-cially prone to buckling because of their high aspect ratios.The occurrence of buckling of CNTs can lead to potentialapplications as nanometer-sized tunnel barriers for electrontransport5 and fluid-flow control nano-valves.6–7
The theoretical methods, including atomistic simulationsand continuum mechanics, were often applied for studyingthe buckling behavior of CNTs because of the difficul-ties encountered by experiments on the nanoscale at thecurrent stage.8–12 For example, Wang et al.10 developed acontinuum mechanics models to investigate the torsionalbuckling of CNTs with different length-to-diameter ratiosand performed molecular dynamics simulations to verifythe buckling results. Kumar et al.12 studied the bucklingof carbon nanotubes, modeled as nonlocal one dimen-sional continuum within the framework of Euler–Bernoullibeams, and indicated significant dependence of nonlocalparameter on buckling load for particular types of bound-ary conditions.
∗Author to whom correspondence should be addressed.
Recently, Lee and Chang13 analyzed the buckling tem-perature of single-walled carbon nanotube (SWCNT) dueto thermal effect. Hsu et al.14 studied similar issue thatarose in the double-walled carbon nanotube. However, theeffects of nonlocal parameter and elastic medium wereignored in these analyses, which may influence accuracy ofthe results. In this study, the critical buckling temperatureof SWCNT subjected to a uniform temperature rise is ana-lyzed based on Euler beam and Timoshenko beam models.In the analysis, the SWCNT is embedded in an elasticmatrix, and the nonlocal continuum theory is introduced.In addition, the critical buckling temperature of SWCNTwith different modes and nonlocal parameters for differentlength-to-diameter ratios is investigated.
2. ANALYSIS
A SWCNT with length L is embedded in an elastic matrixand is assumed to be a cantilever beam fixed at X = 0and a stopper end at X = L as shown in Figure 1. Theaxial displacement is constrained by the stopper and thatinduces a thermal buckling load F due to temperaturerise T . The SWCNT has an equivalent Young’s modu-lus E, shear modulus G, length L, average diameter dand thickness t. The Timoshenko beam model, which istaken rotary inertia and transverse shear deformation intoaccount, is employed to analyze the buckling behavior
Buckling Temperature of a Single-Walled Carbon Nanotube Using Nonlocal Timoshenko Beam Model Chang and Lee
Fig. 1. An illustration of a SWCNT with length L considered as a can-tilever beam fixed at X = 0 and a stopper end at X = L subjected to athermal buckling load F due to temperature rise T .
of the SWCNT under axial compression due to temper-ature rise. The transverse deflection Y �X� and rotation��X� depend on the spatial coordinate X, respectively.The governing equations of Timoshenko beam consideringthe nonlocal continuum theory are two coupled differentialequations, which is expressed in terms of the flexural dis-placement and the angle of rotation due to bending. Thecoupled equations for the SWCNT are
KGA
(d2Y
dX2− d�
dX
)−F
d2Y
dX2+WY = 0 (1)
EId2�
dX2+KGA
(dY
dX−�
)
−�e0a�2
(Fd3Y
dX3−W
dY
dX
)= 0 (2)
where X is the distance along the center of the cantilever.A is cross-sectional area. I is the area moment of inertia.K is the shear factor, and its value is �2+2��/�4+3�� fora thin-walled tube,15 � is the Poisson’s ratio, W is a con-stant characterizing the Winkler model that represents theembedding medium, e0a is a nonlocal parameter reveal-ing the nano-scale effect on the response of structures andF denotes an axial compressive force which is depend
on temperature rise T and thermal expansion coefficient� of SWCNT. Therefore, the force can be expressed asF = �EAT .The corresponding boundary conditions are
��0�= 0 (3)
Y �0�= 0 (4)
EId��X�
dX−�e0a�
2
(Fd2Y
dX2−WY
)=0� at X=L (5)
KGA
[dY �X�
dX−��X�
]= F
dY �X�
dXat X = L (6)
The boundary conditions given by Eqs. (3) and (4) cor-respond to conditions of zero slope and zero displace-ment at X = 0. The boundary conditions are given byEqs. (5) and (6) correspond to zero moment and a forceinduced by temperature rise at X = L, respectively.The nondimensional parameters are defined as
x = X/L� y = Y /L� c = L/d
b = t/d� p = T
I/�AL2
s2 = EI/KGAL2 = �4+3���1+b2�/8c2
w = WL4
EI� �2 = �e0a/L�
2
(7)
where s, w, and � describe the effect of shear deforma-tion, the nondimensional Winkler constant and the nondi-mensional nonlocal parameter, respectively. Meanwhile,p denotes nondimensional temperature.Substituting Eqs. (7) into Eqs. (1)–(6), then the govern-
ing equations and the associated boundary conditions canbe simplified to the following dimensionless equations andboundary conditions
d2y
dx2− d�
dx− s2p
d2y
dx2− s2wy = 0 (8)
s2d2�
dx2− s2�2
(pd3y
dx3+w
dy
dx
)+(dy
dx−�
)= 0 (9)
��0�= 0 (10)
y�0�= 0 (11)
d��1�dx
−�2
[pd2y�1�dx2
+wy�1�]= 0 (12)
dy�1�dx
−��1�= s2pdy�1�dx
(13)
In order to solve the above differential system, we set thesolutions to be expressed as
y = Be�x (14)
� = Ze�x (15)
2 J. Comput. Theor. Nanosci. 7, 1–5, 2010
RESEARCH
ARTIC
LE
Chang and Lee Buckling Temperature of a Single-Walled Carbon Nanotube Using Nonlocal Timoshenko Beam Model
where B and Z are arbitrary constants. SubstitutingEqs. (14) and (15) into Eqs. (8) and (9), we find the fol-lowing associated auxiliary polynomial of function � as:
��� = �1−ps2−p�2��4
+ �p−ws2−w�2��2+w = 0 (16)
The general solutions of Eqs. (8) and (9) can be expressedas follows:
y =4∑
j=1
Bje�jx (17)
� =4∑
j=1
Zje�jx (18)
where Bj and Zj are constants. SubstitutingEqs. (17) and (18) into Eq. (9), we obtain the followingrelationship as
Bj = jZj� j = 1�2�3�4 (19)
where
j =s2�2
j −1
�j�ps2�2�2
j + s2�2w−1�(20)
Substituting Eqs. (17) and (18) into Eqs. (8) and (9), thenapplying the boundary conditions Eqs. (10)–(13), the fol-lowing equation can be yielded⎡
⎢⎢⎢⎢⎢⎣
1 1 1 1
1 2 3 4
�1 �2 �3 �4
�1 �2 �3 �4
⎤⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣
Z1
Z2
Z3
Z4
⎤⎥⎥⎥⎥⎥⎦= 0 (21)
where �j = e�j ��jj−ps2j−1�, �j = e�j ��j−p�2j�2j −
w�2j�, and j = 1�2�3�4. Hence, the nondimensionalcritical buckling temperature pC can be solved fromEq. (21).In case that w= 0 in Eq. (21) is assumed, the nondimen-
sional critical buckling temperature pCN for the SWCNTsubjected to a uniform temperature rise without an embed-ding medium is obtained.Moreover, when s = 0 is assumed in Eq. (21), it implies
that the effects of transverse shear deformation and rotaryinertia are not considered. Accordingly, the critical buck-ling temperature based on the Euler beam model could bealso obtained.
3. RESULTS AND DISCUSSION
In order to know the effect of relative parameters onthe critical buckling temperature of SWCNT, we con-sidered the geometric and material parameters as fol-lows: � = 1 1× 10−6/�C,13 L = 30 �m, � = 0 2, t =0 34 nm.16 In addition, the nonlocal parmeter is assumed
Mode numberN
ondi
men
sion
al c
ritic
al b
uckl
ing
tem
pera
ture
0 1 2 3 4 5 6 7 8 9 10
0
200
400
600
800
Euler
L/d = 60
L/d = 30
L/d = 10
Fig. 2. Comparison of critical temperature buckling of SWCNT withdifferent length-to-diameter ratios at e0a= 0 5 for different modes basedon Euler beam and Timoshenko beam models.
to be e0a = 0 5 nm, and the Winkler constant of w =0 035 which is equivalent to an elastic medium constantof W = 0 2 MPa. The comparison of nondimensional criti-cal buckling temperature of SWCNT with different length-to-diameter ratios and the nonlocal value of e0a = 0 5for the first ten modes based on Euler beam and Tim-oshenko beam models is shown in Figure 2. It can beobserved that for the higher-order modes, the critical buck-ling temperature of the Timoshenko beam model are sig-nificantly lower than that of the Euler beam model dueto the effects of shear deformation and rotary inertia. Thediscrepancy is large, especially at relatively small length-to-diameter ratios �L/d < 30� and higher-order modes.For example, for mode 4, the Euler beam model over-predicts the critical buckling temperature of SWCNT by57%, 13% and 3.5% for L/d = 10, 30 and 60, respec-tively. The relative percentage difference between the twobeam models increases with respect to increasing modenumbers.Figure 3 depicts the critical buckling temperature of
SWCNT with different nonlocal values and L/d = 30 fordifferent modes based on Timoshenko beam model. Theparameter value of e0a= 0 implies that the nonlocal effectis neglected. It can be seen that the effect of nonlocalparameter e0a on the critical buckling temperature is sig-nificant, especially at higher-order modes. Increasing thenonlocal effect decreases the critical buckling temperature.This can be seen from Eq. (21).
J. Comput. Theor. Nanosci. 7, 1–5, 2010 3
RESEARCH
ARTIC
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Buckling Temperature of a Single-Walled Carbon Nanotube Using Nonlocal Timoshenko Beam Model Chang and Lee
Mode number
Non
dim
ensi
onal
cri
tical
buc
klin
g te
mpe
ratu
re
0 1 2 3 4 5 6 7 8 9 10
0
150
300
450
e0a = 0 nm
e0a = 0.5 nm
e0a = 1 nm
Fig. 3. Critical buckling temperature of SWCNT with different nonlocalvalues and L/d = 30 for different modes based on Timoshenko beammodel.
Winkler constant, W (MPa)
Buc
klin
g te
mpe
ratu
re r
atio
, pC
/pC
N
0.1 0.2 0.3 0.4 0.5
1
1.1
1.2
Mode 1
Mode 2
Mode 3
Fig. 4. The ratio of buckling temperature of SWCNT with embed-ding medium (pC ) to that without embedding medium (pCN ) for dif-ferent modes at L/d = 30 and e0a = 0 5 based on Timoshenko beammodel.
For comparison purposes, the variation in the ratio ofbuckling temperature of SWCNT with embedding medium(pC) to that without embedding medium (pCN ) with dif-ferent Winkler constants, for the first three modes atL/d = 30 and e0a = 0 5 based on Timoshenko beammodel has been evaluated as shown in Figure 4. It canbe seen that the buckling temperature of SWCNT withembedding medium is larger than that without embeddingmedium due to a stiffer structure. In addition, the temper-ature ratio, pC/pCN , increases with increasing the Winklerconstant for the modes. The trend is more obvious in thelower-order modes.
4. CONCLUSIONS
The nonlocal Timoshenko beam model was applied toanalyze the critical buckling temperature of SWCNTsubjected to axial compression due to temperature rise.According to the analysis, the following results wereobtained:1. The effects of rotary inertia and shear deformation onthe critical buckling temperature of SWCNT increasedwith decreasing the length-to-diameter ratio, especially athigher-order modes.2. Increasing the value of nonlocal parameter decreasedthe critical buckling temperature, especially at higher-order modes.3. The ratio of buckling temperature of SWCNT withembedding medium to that without embedding mediumincreased with increasing the Winkler constant. The trendwas more obvious at lower-order modes.
Acknowledgment: The authors wish to thank theNational Science Council of the Republic of China in Tai-wan for providing financial support for this study underProjects NSC 98-2221-E-168-019 and NSC 97-2221-E-168-019-MY2.
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Received: 5 February 2010. Accepted: 30 March 2010.
