13th International Conference on Fracture
June 16–21, 2013, Beijing, China
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Buckling Analysis of a Nanowire Lying on Winkler-Pasternak Elastic
Foundation
Tiankai Zhao
1
, Jun Luo
1,*
1
Department of Mechanics , Huazhong University of Science and Technology, Wuhan 430074, China
* Corresponding author: luojun.l@gmail.com
Abstract The Steigmann-ogden surface elasticity model and the Timoshenko beam theory are adopted to
study the axial buckling of a nanowire (NW) lying on Winkler-Pasternak substrate medium. Explicit
solutions of the critical buckling force and buckling mode are obtained analytically. The influences of the
surface stress effect, the geometry of the NW and the elastic foundation moduli are systematically discussed.
Keywords Surface effect, Nanowire, Timoshenko beam theory, Buckling
1. Introduction
Nanowire (NW) based devices have found important applications in various fields. Traditional
beams theories failed to interpret the size dependent mechanical behavior of these devices due to
their surface stress effect. The Gurtin-Murdoch model [1-2] has been widely accepted to study the
mechanical behavior of nanostructures and nano defects [3-7]. Some researchers found that the
effective elastic moduli of nanostructures under bending and tension are different[8-9], which could
not be explained by the Gurtin-Murdoch model. Chhapadia et al.[10] pointed out that the above
discrepancy could be explained with the modified framework proposed by Steigmann and Ogden
[11-12]. On the other hand, buckling has long been thought as an unwanted issue and should be
strictly avoided in structural designs. However, it was demonstrated by some researchers that
controlled buckling of slender structures such as thin films, nanowires and nanotubes on compliant
substrates could be utilized in the design of flexible electronics. This paper aims to study the axial
buckling of a simply supported NW lying on Winkler-Pasternak elastic foundation with the
Timoshenko beam model and Steigmann–Ogden theory. The influences of the surface stress effect,
the geometry of the NW and the elastic foundation moduli are systematically discussed.
2. Solution of the Problem
Consider a NW lying on a deformable substrate, which is subjected to distributed transverse load
and axial forces at both ends. We assume the NW has a circular cross section. The total energy of
the axially loaded nanobeam can be written as:
() () ()
Total Bulk Surface
UwUwU w
, (1)
where:
22
00
1
() ( ) ( )
22
LL
Bulk
AA
w
U w E y dAdx G dAdx
xx
, (2)
2
22
00 1
2
0
11
() [ ( ) ( )]
22
L
Surface
S
w
U w y C y C dSdx
xxx
. (3)
13th International Conference on Fracture
June 16–21, 2013, Beijing, China
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Here,
A
denotes the area of the cross section,
is the shear coefficient,
0
C and
1
C are the surface
elastic modulus and the Steigman-Ogden constant, respectively,
S
denotes the circumference of
the cross section. The potential energy of the lateral and axial load is given by:
2
00
1
() ()() ( )
2
LL
f
w
Uw fxwxdx N dx
x
, (4)
where
sf
f
xqxqxqx
. (5)
Here, ( )
s
qx stands for the equivalent lateral force due to the residual surface stress and ( )
f
qx
represents the distributed force arising from the substrate medium. To minimize the total potential
energy, we apply the variational theory,
() () 0
Total f
UwUw
. (6)
Finally, we get:
22
22
01
()() 0
()(( *))0
ww
GA f x N
xx x
w
GA EI C I C S
xx x
, (7)
where
2
A
I
ydA
,
2
*
S
I
ydS
,
2
*
y
S
SndS
. Take partial derivative of the second equation with
respect to
x
，and substitute the first equation into the second equation, we have:
42
42
()* ()* ()*
()* ( ) ()0
pwpw
EI EI w EI w
EI H N C N C H C C w x
GA GA x GA x
.
(8)
The general solution of Eq. (8) can be written as:
( )
mx
wx Ae . (9)
Substituting Eq. (9) into Eq. (8), we can solve
m .
The boundary conditions of a simply supported NW are given as follows:
(0) 0w
,
''
(0) 0w
, (10a)
() 0wL
,
''
() 0wL
. (10b)
Substitute Eqs.(10a,b) into Eq. (9), we can get the characteristic equation to solve the critical
buckling force. The corresponding buckling mode is given by Eq. (9).
3. Results and Discussions
13th International Conference on Fracture
June 16–21, 2013, Beijing, China
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The buckling force of the NW is normalized with the critical buckling force of a classic simply
supported Euler-Bernoulli beam. To see qualitatively how the surface stress effect and the
Steigmann-Ogden correction influence the buckling behavior of the NW, we adopt the following
material parameters in this study:
76GpaE ,
0.3
,
0
0.65 N/m
,
0
1.39N/mC
,
1
153.6eVC
.
In Figure 1., we ignore the transverse shear effect of the substrate. The diameter and
Steigmann-Ogden constant are chosen to be 3.5nmand 153.6eV respectively. All the results are
normalized with the critical buckling force of a classic simply supported Euler-Bernoulli beam at
each aspect ratio. Figure 1. shows that the surface effect has significant influences on the
normalized critical buckling forces. When the Steigmann-Ogden constant is positive, the critical
buckling force is greater than that predicted by the Gurtin-Murdoch theory, which implies that the
NW is stiffened. It is also noted that the influence of the surface effect (predicted either by the
Gurtin-Murdoch model or the Steigmann-Ogden model) on the critical buckling force decreases
when the diameter of the NW increases. The difference between the result predicted by the
Gurtin-Murdoch theory and that by the Steigmann-Ogden theory is also largely decreased as the
diameter of the NW increases. Fig. 1 also shows that the critical buckling force for a finitely long
NW is always larger than that for an infinitely long NW.
10 12 14 16 18 20 22 24 26 28 30
1
2
3
4
5
6
7
8
9
10
11
L/D
an infinite nanowire with curvature-dependent surface energy
a finite nanowire with curvature-dependent surface energy
a classical Timoshenko beam of infinite length
a classical Timoshenko beam of finite length
an infinite nanowire based on the Gurtin-Murdoch theory
a finite nanowire based on the Gurtin-Murdoch theory
Figure 1. The normalized critical buckling force of a simply support NW lying on Winkler substrate medium.
In Figure 2., the influences of the Winkler and Pasternak moduli on the critical buckling force are
studied with the Timoshenko beam model. The material parameters are the same with those in
Figure 1.. The diameter of the NW is set to
3.5nm. From Figure 2., we find that both the Winkler
modulus and the Pasternak modulus tend to stiffen the NW. As a result, the buckling force for a NW
lying on substrate medium is always larger than that of a free NW. It is also noticed that the critical
buckling force is more sensitive to the Pasternak modulus.
13th International Conference on Fracture
June 16–21, 2013, Beijing, China
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10 12 14 16 18 20 22 24 26 28 30
2
3
4
5
6
7
8
9
10
11
12
wp
wp
wp
ˆ
ˆ
C =30, C =3
ˆˆ
C =30, C =0
ˆ
ˆ
C=0,C=0
L/D
Figure 2. Variation of the normalized critical buckling force of a simply supported NW lying on
Winkler-Pasternak substrate medium with respect to its aspect ratio. The diameter of the NW is
3.5nm .
4. Conclusions
In this paper, the Steigmann-Ogden model is adopted to characterize the surface effect of the NW.
Explicit solutions are obtained for the critical buckling force and buckling mode of a simply
supported NW lying on Winkler-Pasternak substrate medium with the Timoshenko beam theory.
The following conclusions are drawn through this study:
(1) The shear effect, the surface stress effect and curvature dependent surface energy all have
influences on the critical buckling force of the NW. The importance of these influences is highly
dependent on the diameter and aspect ratio of the NW.
(2) The Steigmann-Ogden correction can stiffen or soften the NW, depending on the sign of the
Steigmann-Ogden constant.
(3) Both the Winkler modulus and Pasternak modulus tend to stiffen the NW. The critical
buckling force of the NW is more sensitive to the Pasternak modulus.
Acknowledgement
The support of National Natural Science Foundation of China under contract No. 10802032 is most
sincerely appreciated.
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June 16–21, 2013, Beijing, China
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