Bending Instabilities of Carbon Nanotubes
N. Silvestre and D. Camotim1
Abstract. This paper presents an investigation on the buckling behaviour of single-walled carbon nanotubes (NTs) under bending and unveils several aspects
concerning the dependence of critical bending curvature on the NT length.
The buckling results are obtained by means of non-linear shell finite element
analyses using ABAQUS code. It is shown that eigenvalue analyses do not give a
correct prediction of the critical curvature of NTs under bending. Conversely,
incremental-iterative non-linear analyses provide a better approximation to the
molecular dynamics results due to the progressive ovalization of the NT crosssection under bending. For short NTs, the limit curvature drops with the increasing
length mostly due to the decreasing influence of end effects. For moderate to long
tubes, the limit curvature remains practically constant and independent on the tube
length. An approximate formula based on the Brazier expression is proposed to
predict the limit curvature.
1 Bifurcation (Eigenvalue) Analysis of Carbon Nanotubes
Let us start by consider the NT(13,0) under uniform bending, previously investigated by Yakobson et al. [8]. The NT is modelled with rigid end sections (the
flexural rotations at both supports are free and the axial translation of one end section was left free, in order to enable the axial shortening of the tube under large
bending displacements) and the following properties were adopted: ν=0.19,
E=5.5TPa, h=0.66Å, r=5.09Å. Performing bifurcation (eigenvalue) analyses, it is
found that the NT buckling is triggered by a local mode, characterized by the deformation of the top compressed zone. This local mode is very similar to the axisymmetric mode of NTs under compression, both displaying a large number of
-1
half-waves. The critical curvature at the local minimum is κc=0.0160Å and the
corresponding half-wavelength is Lc=3.4Å. These values are close to the ones
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(κc=0.0150Å and Lc=3.1Å) obtained from
N. Silvestre and D. Camotim
Department of Civil Engineering and Architecture, ICIST/IST, Technical University of
Lisbon, Lisboa, Portugal
e-mail: [email protected]
366
N. Silvestre and D. Camotim
κc =
h r2
(1)
3(1 − ν 2 )
where h and r are the shell thickness and radius, respectively. The minor discrepancy
is due to the fact that the top compressed zone of the NT under bending is partially
restrained by its bottom tensioned zone. Using molecular dynamics simulations
and adopting Brenner’s empirical potential for the atomic interactions, Yakobson
et al. [8] studied the NT(13,0) with L=80Å and obtained a critical curvature
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κc=0.0155Å , which is close to the numerical value obtained (κc=0.0160Å ). Nevertheless, it seems fair to say that the critical mode obtained by the current analysis looks very different to that identified by Yakobson et al. [8]. The later displays
only one kink at the middle of the tube and the deformed configuration of the midspan section is very similar to the two half-wave (m=2) distortional mode.
In order to shed light on this subject, several data on NTs under bending obtained by other authors were collected. Figure 1 shows the variation of the critical
curvature with the NT radius and includes several dots corresponding to the mentioned data. The solid curve is obtained from Eq. (1), which is related to the critical local mode of the top compressed zone. The first remark is that the critical
curvature decreases with the NT radius. However, the decreasing rate of the solid
curve is much more pronounced than that of the data. The carefull observation of
this figure shows that, apart from the Yakobson’s result (black dot), all data (white
dots) are located below and far from the solid curve. This evidence leads to us to
question the accuracy of Eq. (1), which is extensively used in NT analysis (Iijima
et al. [4], Buehler et al. [1]), and to have some reservations on the performance of
pure bifurcation (eigenvalue) analysis of NTs under bending.
κc (Å-1)
0.045
Yakobson et al. [8]
Cao and Chen [2]
Guo et al. [3]
Shibutani and Ogata [5]
0.040
0.035
Bifurcation − Eq. (1)
Brazier − Eq. (3)
Modified Brazier − Eq. (4)
0.030
0.025
0.020
0.015
0.010
ξ=0.15
0.005
ξ=0.10
0.000
3
5
r (Å)
7
9
11
Fig. 1 Variation of the critical curvature with the NT radius
13
15
Bending Instabilities of Carbon Nanotubes
367
With the aim of investigating the above mentioned difference between the critical curvatures, one is aimed to perform fully non-linear analysis. Details of the
numerical model can be found in the recent work by the author [6]. First, the nonlinear analysis of the NT(13,0) with L=80Å was performed. Figure 2 shows (i) the
non-linear equilibrium path M(θ) (or M(κ)) obtained from the incrementaliterative analysis (solid curve) and (ii) the linear equilibrium path (dashed line).
Moreover, while several points (A to M) are located along the non-linear path,
point P is located on the top of the linear equilibrium path and corresponds to the
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NT bifurcation in the local mode (the values κc=0.0150Å and θc=0.0150Å
1
×80Å=1.20rad are obtained by means of Eq. (1), with linear pre-buckling path).
The observation of figure 2 deserves the following comments:
• Until point D is achieved, the equilibrium path is almost linear. After point D,
the non-linear path starts to deviate from the linear one and reaches a limit
point at G. After that, the bending moment always decreases and the bending
angle (or curvature) increases after a small snapback. In the descending branch
of the M(θ) curve, the increase in the bending angle leads to a final deformed
configuration (point M) with three kinks in the mid-span zone. From this nonlinear analysis, it is seen that the NT under bending never reaches the critical
bending moment value Mc=22.5TPaÅ3, which is associated with the local
mode (point P) determined from the bifurcation analysis. It should be stressed
that the same non-linear trend of the ascending branch of the M(θ) curve was
also unveiled by Yakobson et al. [8] and Vodenitcharova and Zhang [7].
κ (×10-3Å-1)
0
6.25
25
10.25
12.5
18.75
22.5
20
θ
15.0
12.06
P
M
M
Non-Linear Analysis
Bifurcation Analysis
Brazier Analysis
15
M
Q
12.2
(TPaÅ3) 11.8
10
E
F
H
G
J
D
A
I
K
B
C
D
L
M
C
5
E
F
G
B
0
0.83
A
0
0.5
θ (rad)
0.96
1.0
1.20
1.5
H
I
J
K
L
M
Fig. 2 Non-linear behaviour and progressive ovalization and collapse of NT(13,0) with
L=80Å under uniform bending
368
N. Silvestre and D. Camotim
• The roundness of the non-linear path is related to the well known Brazier effect, which is due to the action of normal (longitudinal) stresses on the curvature of the bended NT, thus resulting in transverse (vertical) pressure directed
towards the NT neutral axis. This pressure leads the top (compressive) and bottom (tensioned) zones to move towards the neutral axis, thus resulting in the
NT ovalization. It is obvious that the ovalized shape of the cross-section coincides perfectly with the two-wave distortional mode represented in figure 1(b).
The NT flattening (ovalization) is responsible to a decrease in the cross-section
second moment of area (I(κ)), which depends on the curvature. Consequently,
it also leads to a drop in the actual bending moment (M=EI(κ)κ). The Brazier
equilibrium path is shown in figure 2 (dotted curve) and is given by
r4
M = EIκ(1 − 32 ξ)
ξ = κ 2 2 (1 − ν 2 )
(2)
h
3
where I=πr h is the second moment of area of the circular section and ξ is the
ovalisation parameter (oval minor axis width / 2r). The Brazier curve exhibits a
local maximum (point Q) given by
2π 2 Erh 2
2 h / r2
κ BR =
(3)
9
3 1 − ν2
1− ν2
• The Brazier curve (dotted line) approximates very well the non-linear path until
the limit point G is reached. While the limit point G is associated with
3
-1
Mlim=11.8TPaÅ and κlim=0.01025Å , the Brazier curve local maximum point Q
3
-1
is characterised by MBR=12.2TPaÅ and κBR=0.01206Å . It is fair to say that
both points lead to similar values of the bending moment M and to a much
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lower curvature κ than the critical local mode one (κc=0.01500Å ).
• From the Brazier analysis, the limit value of the bending moment
3
(Mlim=11.8TPaÅ ) is reached for an ovalization parameter ξ=0.157, which
means that the circular section vertical axis width decreased about 16%. Vodenitcharova and Zhang [7] found a value of the ovalization parameter ξ=0.14,
very close to the one obtained here. Moreover, the picture in the right of figure
10 shows the variation of the ovalization parameter ξ with the bending angle θ.
3
For the maximum Brazier bending moment (MBR=12.2TPaÅ – point Q), the
circular cross-section exhibits an ovalization parameter equal to 21%.
• For the several coloured points (A to M), figure 2 also shows the corresponding
deformed configurations of the NT mid-span section. While A is the underformed circular section configuration, the configurations B, C and D remain
almost circular after bending. Nevertheless, the configurations E, F and G exhibit clearly visible ovalized configurations, where the two-wave distortional
mode is prevalent. In particular, it should be noticed that the (black) deformed
configuration G, corresponding to the limit situation, exhibits an ovalization
parameter equal to 19%. This value is located between those mentioned before
in the context of the Brazier analysis (16% and 21%), thus reflecting the relative accuracy and usefulness of the later.
M BR =
Figure 3(a) shows the non-linear M(θ) equilibrium paths obtained for the
NT(15,0), for several length values (10Å<L<120Å). Obviously, the inclination
Bending Instabilities of Carbon Nanotubes
369
values of the initial branches (M/θ) are proportional to the NT bending stiffness
values (EI/L). It is also visible that shorter NTs possess almost linear (straight)
equilibrium paths until the limit moment Mlim is reached. Conversely, the equilibrium path of the longer NTs is more rounded near the point of limit bending moment Mlim. This fact proves that the ovalization phenomenon (Brazier effect) is
more evident in the longer NTs. Moreover, it is also seen that the limit bending
moment Mlim decreases abruptly for the shorter NTs but remains nearly constant
for longer lengths. A possible explanation for this Mlim decrease in the shorter NT
behaviour resides in the influence of the boundary conditions, which is absent in the
longer NT behaviour: the mid-span section of longer NTs, where collapse takes place,
is too distant from their end supports, a fact that does not occur in the shorter NTs. The
transition between these two (shorter and longer NT) behaviours is different in the case
3
of NT(15,0). It is clear that the NT(15,0) length-independence of Mlim (≈13.7 TPaÅ )
occurs for L>40Å. As a first approach, one can state that this limit length depends on
the NT radius (r) and is relatively well approximated by the NT perimeter (p):
L>p=36.9Å for the NT(15,0).
NT(15,0)
25
0.012
L=10Å
0.010
20
L=20Å
0.008
L=40
L=60
L=80
L=100
κ lim (Å-1)
M (TPaÅ3)
L=30Å
15
L=120Å
10
0.006
0.004
Non-Linear Analysis
Bifurcation − Eq. (1)
Brazier − Eq. (3)
0.002
5
Modified Brazier (ξ=0.15) − Eq. (4)
Guo et al. [3]
0.000
0
0
0
0.2
0.4
0.6
θ (rad)
(a)
0.8
1
30
60
90
120
L (Å)
(b)
Fig. 3 Buckling behaviour of NT(15,0) under bending: (a) M-θ equilibrium paths and (b)
variation of limit curvature with the length
The figure 3(b) shows the variation of the limit curvature (κlim) with the length
of the NT(15,0). The limit bending curvature κlim also decreases for the shorter
NTs, exhibits a local minimum and then augments sligthly for increasing lengths.
It is interesting to mention that Cao and Chen [2], using molecular dynamics
to simulate NTs under bending, also found that the curve κ(L/d) exhibited a
“kind” of local minimum for very short NTs (i.e., with very low aspect ratio L/d).
Moreover, the black dots in figure 3(b) represents the critical curvatures (κc) obtained by Guo et al. [3], using the atomist-scale finite element method, for the
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NT(15,0) with L=83.5 Å. The NT(15,0) critical curvature (κc=0.0077 Å ) is rather
370
N. Silvestre and D. Camotim
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close to the limit curvature calculated from non-linear analysis (κlim=0.0070 Å ).
Moreover, figure 3(b) also shows four horizontal lines corresponding to the curvature values obtained from (i) bifurcation analysis (dashed line − Eq. (1)), (ii) Brazier
analysis (dashed-doted line − Eq. (3b)) and (iii) modified Brazier analysis (dotted
line). This modified Brazier analysis is based on the following expression,
h / r2
(4)
κ= ξ
1 − ν2
This expression can be used to evaluate the limit curvature as a function of the
ovalization parameter ξ. While the dotted line in figure 3(b) correspond to the
value ξ=0.15, the bifurcation and Brazier lines correspond to the adoption of
ξ=3/9 and ξ=2/9, respectively. From the observation of figure 3(b), it is possible to
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-1
conclude that both bifurcation (κ=0.0113 Å ) and Brazier (κ=0.0092 Å ) curvature
estimates are too high, in comparison with the non-linear values (white dots).
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However, the modified Brazier curvature values with ξ=0.15 (κ=0.0075 Å ) lead
to lower and more accurate estimates of the critical curvature. Finally, let us look
at figure 1, where the bifurcation (ξ=3/9), Brazier (ξ=2/9) and modified Brazier
(ξ=0.15) curves are represented and compared with available data. Despite the
modified Brazier with ξ=0.15 curve is the one that gives more accurate estimates,
it is also obvious that it does not fit well with available data. Therefore, one proposes the use of a modified Brazier analysis with ξ=0.10, which leads to very accurate results for all data (bottom dotted curve), with the exception of Yakobson’s
result. For a more detailed discussion of the results, the reader is referred to a recent work by the author [6].
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