Stability of Compressed Carbon Nanotubes
Using Shell Models
N. Silvestre and D. Camotim1
Abstract. This paper presents some remarks on the use of shell models to analyse
the stability behaviour of single-walled NTs under compression. It is shown that
there are three different categories of critical buckling modes of NTs under compression: while the axi-symmetric mode is critical for very short NTs, the flexural
buckling mode is critical for long tubes. While the former exhibits cross-section
contour deformation but no warping deformation, the later is characterised by the
opposite situation (warping deformation but no contour deformation). Additionally, a third category exists (distortional buckling): it takes place for NTs with
moderate length, it is related to the transitional buckling behaviour between the
shell (axi-symmetric mode) and the rod (flexural mode) and it is characterised by
both cross-section contour deformation and warping deformation. Concerning the
distortional buckling behaviour of moderately long NTs, it is also shown that the
well known Donnell-type theory of shells leads to erroneous results.
1 Introduction
After the seminal work of Iijima [5], much research has been done on carbon
nanotubes (NTs). Since then, most of the studies performed were based on molecular dynamics approach to simulate the NT non-linear behaviour. Nevertheless,
it is known that molecular dynamics analyses are computationally expensive and
time consuming and are often limited to a maximum number of atoms. Yakobson
et al. [11] analysed the buckling behaviour of single-walled NTs under compression, bending and torsion and compared the results (critical measures and buckling
modes) obtained by molecular dynamics simulations with those determined by
continuum shell models. They showed that shell model results agreed fairly well
with those obtained from molecular dynamics simulations. Since then, a large
N. Silvestre and D. Camotim
Department of Civil Engineering and Architecture, ICIST/IST, Technical University of
Lisbon, Lisboa, Portugal
e-mail: [email protected]
358
N. Silvestre and D. Camotim
amount of investigations has been carried out, most of them confirming the accuracy of continuum shell analyses. However, the use of shell models to analyse the
NT buckling behaviour should be carefully addressed, as different shell theories
lead to very dissimilar results (Silvestre [9]).
2 Buckling Modes, Critical Strains and Shell Theories
The local buckling behaviour of cylindrical shells and tubes, which involve deformation of the circular section contour, can be classified in two categories:
• Local buckling (axi-symmetric) modes: they occur for very short lengths and are
characterised by null warping deformation of the circular section (Fig. 1(a)).
• Distortional buckling (diamond-shape) modes: they occur for short lengths and
are characterised by significant warping deformation of the circular section.
Figure 1(a) depicts the in-plane deformed configuration of a distortional buckling mode with two circumferential half-waves (m=2) and its warping displacement profile (in perspective).
εc
0.20
z
z
y
y
0.18
Donnell Theory
0.16
Love Theory
0.14
0.12
0.10
0.08
z
0.06
y
x
0.04
4
3
0.02
5
4
3
m=1
m=1
0.00
1
(a)
m=2
m=2
10
L (Å)
100
1000
(b)
Fig. 1 (a) Local mode (m=0) and two-wave distortional mode (m=2) configuration and
warping displacement profile, (b) Variation of the critical strain εc with the length L of
NT(23,0)
In the theoretical investigations of NTs under compression, it is always assumed that the critical strain εc and the buckling half-wavelength values of both
local and distortional buckling modes are based on the Donnell theory of shells
and are given by
Stability of Compressed Carbon Nanotubes Using Shell Models
εc =
h r
3(1 − ν )
2
Lc =
nπ h r
4
12(1 − ν 2 )
359
(1)
where h and r are the shell thickness and radius, respectively, and n is the number of
longitudinal half-waves. Less well known than Donnell theory is the Love theory of
shells. For the local buckling modes, both Donnell and Love theories lead to the same
expressions (Eq. (1)). However, for the distortional modes, the Love theory of shells
gives the expressions for the critical strain εc and the buckling half-wavelength Lc,
εc =
m2 − 1
2
3(1 − ν 2 ) m + 1
h r
L c = nπ r
2
r 4 12(1 − ν )
h m m2 − 1
(2)
For illustration purposes, figure 1(b) depicts the variation of the critical strain εc
with L and m, for the NT(23,0) with a single longitudinal half-wave (n=1). In this
analysis, one adopted ν=0.19, r=9Å and h=0.66Å. It is seen that the two curves
exhibit several local minima. The critical strain obtained from the Donnell theory
(εc=0.043) does not depend on the NT length L. Unlike the Donnell-type theory,
the critical strain εc obtained from Love theory for the distortional buckling modes
(m>1) decreases with increasing lengths. The number of circumferential halfwaves of the critical buckling mode (m) also decreases. From this example, it is
concluded that the use of Donnell theory is exact for local buckling of very short
NTs (axi-symmetric buckling mode with m=0) but the results for long and moderately long tubes (distortional buckling modes) become unsafe, with differences in
εc values reaching up to 40%. The Donnell-type theory is unable to predict accurately both the warping and tangential displacement profiles due to the omission of
some important terms in the kinematic relations. The non accurate estimation of
warping and tangential displacements (u and v) has far reaching implications (errors) in the results obtained from the Donnell-type theory for the distortional buckling modes of moderately long NTs. For more detailed information, see Ref. [9].
3 Shell Models and Results for NTs under Compression
The critical value of the compressive force (Pc) applied to the NT corresponds to
critical strain by means of ε c = Pc / EA , where EA is the NT axial stiffness. With
the objective of investigating the buckling behaviour of NTs under compression,
let us first consider the NT(7,7) with armchair helicity under uniform compression
(r=4.75Å). This NT was investigated by Yakobson et al. [11] using both molecular
dynamics simulations and analytical formulae derived from shell models. The NT
has fix-ended rigid supports (cross-section deformation and global rotation is restrained at both supports and the axial translation of one end section was left free,
in order to allow the axial shortening of the NT under large compression) and, like
Yakobson et al. [11], the following properties were adopted: ν=0.19, E=5.5TPa,
h=0.66Å. Using a shell model based on finite element simulation, the results shown in
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N. Silvestre and D. Camotim
figure 2 were obtained, where it is depicted the variation of the critical strain εc and the
buckling mode configuration with L. From the observation of figure 2, the following
remarks can be drawn:
• For very short NTs (L<3Å), buckling takes place in local modes (m=0) – note
that the length is only 30% of the NT diameter and, thus, its buckling behaviour
is more similar to that of a short (shallow) cylindrical shell. The critical strain
εc=0.082 at the local minimum and the corresponding half-wavelength (Lc=3Å)
are in perfect agreement with the values obtained from Eq. (1).
• For small to moderate lengths (3Å<L<100Å), the NT buckles in distortional
modes with two circumferential waves (m=2). The εc vs. L variation exhibits
several local minima, corresponding to an increasing number n of longitudinal
half-waves (from 1 to 5). The critical strain at the local minima (εc=0.049) and
the corresponding half-wavelength (Lc=21.0Å) are in perfect agreement with
the values obtained from Eq. (2). Notice that εDc=0.049=0.6×0.082=0.6εLc and
LDc=21Å=3Å×4.75/0.66=LLc×r/h (superscripts L and D denote local and distortional modes).
0.12
εc
0.1
0.08
0.06
m=0
0.04
m=2
0.02
L (Å)
0
1
10
100
300
Fig. 2 Variation of the critical strain εc with the NT length L
• Using molecular dynamics simulations, Yakobson et al. [11] studied a NT(7,7)
with L=60Å and obtained a critical strain εc=0.050 (see the top white dot in figure 2), which matches virtually with the numerical value obtained here
(εc=0.049). Moreover, the critical mode shape determined by these authors also
agrees very well with the two-wave distortional mode obtained herein. Using
Eq. (1), Yakobson et al. [11] obtained εc=0.077 (the small difference with respect to εc=0.082 is due to the fact that they used r=5Å, instead of r=4.75Å) and
Stability of Compressed Carbon Nanotubes Using Shell Models
361
argued that this value is “close to the value obtained in MD simulations”, i.e.,
εc=0.050. However, this is not the case, since there is a 50% difference between
those values (εc=0.077 and εc=0.049). This discrepancy is due to the fact that
the value εc=0.077 is valid only for local (axi-symmetric) mode and not valid
for the mode they found (pure distortional), exhibiting identical “… flattenings
perpendicular to each other…” − see figure 1 in Yakobson et al. [11]. The
main difference between these two buckling modes is that the local (axisymmetric) mode does not exhibit warping (axial) displacements while the distortional one displays warped cross-sections at x=L/3 and x=2L/3 (see white
lines in figure 2). These warped cross-sections are qualitatively similar to those
represented in figure 1(a).
• For long NTs (L>100Å), buckling takes place in a bending mode (m=1) with
one longitudinal wave (n=1), and the strain always decreases with L.
Finally, the figure 3 shows the variation of the critical strain with the NT radius, for the local (m=0 – top curve) and distortional (m=2, 3, 4 – bottom curves)
buckling modes. Also depicted are the results obtained by several researchers,
who carried out atomistic simulations of NTs with different radius and lengths.
Very rencently, Cao and Chen [2] performed an extensive study on the buckling
behaviour of NTs using molecular mechanics simulations. They investigated NTs
with rather small (but similar) radii (NT(5,5), NT(6,4), NT(7,2), NT(8,2) and
NT(9,0)) but with different lengths (L=50, 100, 190, 390Å). In order to study the
diameter effects, these authors also performed analyses for NT(10,10), NT(14,6),
NT(17,0) with equal length (L=50Å) and moderate radii. Using the molecular dynamics method, Liew et al. [6] also carried out extensive numerical invetigations
on the buckling behaviour of compressed NTs. They determined the buckling
loads of 32 zig-zag NTs and 32 armchair NTs with small, moderate and large radii. Conversely, Cornwell and Wille [3] investigated the buckling behaviour
of NTs with larger radii. Additionally, also shown are the results by Yakobson
et al. [11], Buehler et al. [4], Arroyo and Belytschko [1], Srivastava et al. [10] and
Ni et al. [7].
From the observation of figure 3, it is possible to mention that the great majority of the molecular dynamics estimates and atomistic simulation results are very
close to the curve corresponding to the local (axi-symmetric) buckling mode (top
curve). This is the case of the NT(8,0) studied by Srivastava et al. [10], which
undoubtedly buckled in a local (axi-symmetric) mode. However, there is a non
negligible number of results that fall well below this local buckling curve. These
results correspond to several points located close to the curves with m=2, 3, 4,
which suggests that the corresponding NTs buckle preferably in distortional buckling modes. This clearly occured in the case of the NTs studied by Yakobson et al.
[11], Buehler et al. [4], Arroyo and Belytschko [1] and Ni et al. [7]. Pantano et al.
[8] also simulated numerically (using continuum shell finite elements) the behaviour of the NT(10,10) investigated by Ni et al. [7] and they found that it buckled in
362
N. Silvestre and D. Camotim
εc
0.18
Cao and Chen [2] – L=50Å + varying chirality
Cao and Chen [2] – L=100Å
Cao and Chen [2] – L=190Å
Cao and Chen [2] – L=390Å
Cao and Chen [2] – L=50Å + varying radius
Liew et al. [6] – zig-zag NTs
Liew et al. [6] – armchair NTs
Cornwell and Wille [3]
Srivastava et al. [10]
Yakobson et al. [11]
Ni et al. [7]
Arroyo and Belytschko [1]
Buehler et al. [4]
0.16
0.14
0.12
0.10
0.08
0.06
m=4
m=0
0.04
m=3
0.02
m=2
r (Å)
0.00
1
3
5
7
9
11
13
15
17
19
21
23
25
Fig. 3 NT data and variation of the critical strain with the radius
a clear distortional mode (m=2) configuration similar to that depicted in figure
1(a). Moreover, figure 3 shows that several points are very close to the curve
corresponding to the distortional buckling curve (m=2, bottom curve). Finally, it
becomes visible that the absolute difference between the local (top curve) and distortional (m=2, bottom curve) critical strains increases substantially for diminishing radii. However, remember that the relative difference is always 40%.
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