A novel method for studying the buckling of nanotubes considering geometrical imperfections

A novel method for studying the buckling of nanotubesconsidering geometrical imperfections

N. M. Anoop Krishnan • Debraj Ghosh

Received: 13 March 2014 / Accepted: 1 May 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract Buckling of nanotubes has been studied using

many methods such as molecular dynamics (MD), molec-

ular mechanics, and continuum-based shell theories. In

MD, motion of the individual atoms is tracked under

applied temperature and pressure, ensuring a reliable esti-

mate of the material response. The response thus simulated

varies for individual nanotubes and is only as accurate as

the force field used to model the atomic interactions. On

the other hand, there exists a rich literature on the under-

standing of continuum mechanics-based shell theories.

Based on the observations on the behavior of nanotubes,

there have been a number of shell theory-based approaches

to study the buckling of nanotubes. Although some of these

methods yield a reasonable estimate of the buckling stress,

investigation and comparison of buckled mode shapes

obtained from continuum analysis and MD are sparse.

Previous studies show that the direct application of shell

theories to study nanotube buckling often leads to errone-

ous results. The present study reveals that a major source of

this error can be attributed to the departure of the shape of

the nanotube from a perfect cylindrical shell. Analogous to

the shell buckling in the macro-scale, in this work, the

nanotube is modeled as a thin-shell with initial imperfec-

tion. Then, a nonlinear buckling analysis is carried out

using the Riks method. It is observed that this proposed

approach yields significantly improved estimate of the

buckling stress and mode shapes. It is also shown that the

present method can account for the variation of buckling

stress as a function of the temperature considered. Hence,

this can prove to be a robust method for a continuum

analysis of nanosystems taking in the effect of variation of

temperature as well.

1 Introduction

Study of buckling of nanotubes has gained considerable

interest in the recent past. Owing to their slender structure,

nanotubes buckle and fail before reaching the ultimate

strength, often resulting in an early collapse of the struc-

ture. Hence, it is essential to study the buckling behavior of

nanotubes as the understanding of the theory of buckling

finds applications in nanoscale devices such as probe tips

[1–3], sensors, and nano-composites with nanotubes as

reinforcement to improve mechanical properties [4] or to

introduce piezoelectric properties [5]. Therefore, in addi-

tion to the studies on synthesis and mechanical properties

of nanotubes [6, 7], several experimental and simulation

studies have been carried out on the buckling of nanotubes

under axial-compressive loads [8–12]. Depending on the

length to diameter (L=D) ratio, nanotubes show three dif-

ferent buckling behavior. For instance, in [10], it was

reported that the carbon nanotubes (CNTs) show (1) wire-

like behavior for very long tubes, (2) shell buckling for

small aspect ratio (L=D� 1–20), and (3) Euler or column

buckling for intermediate L=D ratios. Computational

modeling of mechanics at nanoscale and buckling analysis

of nanotubes using such models have long been an

important area. There are ab-initio simulation methods

such as quantum mechanical or molecular dynamics (MD)

simulations—which are very accurate. But these simula-

tions are computationally expensive and hence cannot be

used for multiple simulations of nanotubes having different

N. M. Anoop Krishnan (&) � D. Ghosh

Department of Civil Engineering, Indian Institute of Science,

Bangalore 560012, India

e-mail: [email protected]

D. Ghosh

e-mail: [email protected]

123

Appl. Phys. A

DOI 10.1007/s00339-014-8489-x

dimensions and chiralities. There are simple continuum

mechanics-based models which are approximate yet more

economical. Such methods can make use of the rich and

vast literature and understanding of continuum mechanics

to predict the behavior of nanotubes individually and in a

system.

In MD simulations, the material is modeled at molecular

motion level and the computation amounts to solving a

large-coupled system of ordinary differential equations.

One of the pioneering work on MD simulations on nano-

tube buckling is [8], where the observations were explained

using a continuum shell theory. In [13], MD was used for

buckling and post-buckling analysis of nanotubes. How-

ever, studying more complicated and bigger systems using

an MD simulation may not be realistic. Thus, using the

knowledge that nanotube buckles similar to that of a shell,

researchers started developing theories to model the

nanotube from a continuum shell perspective. Several

attempts have been reported in this direction, such as

Timoshenko beam theory [14], thin-shell theories [11, 12,

15–18], a thin-shell theory followed by finite element

discretization [19], structural mechanics at atomistic level

[20–23], molecular statics [24–28], nonlocal elasticity [29,

30]. Based on these studies, it is found that the thin-shell

models [31] can be very useful in studying nanotube

buckling, primarily due to the similarity in shapes and

mechanical behavior between continuum shells and nano-

tubes. The effectiveness of various continuum-based

methods to analyze CNT buckling was studied in [32]. The

study revealed a variation in the estimation of buckling

strains among different methods. Hence, selection of an

appropriate shell theory and associated parameters is cru-

cial for accuracy of the estimates. The limitations of the

thin-shell theories in nanotube buckling estimation are

critically studied in [11, 16, 17], underlining the need for a

better method.

It will be demonstrated in this paper that for a nanotube

under axial compression, the classical thin-shell theory

overestimates the buckling stress and predicts the buckled

mode shape incorrectly. Similar observations were made in

the macroscopic engineering level for many years where

the experimental observations of buckling load of thin-

shells remained consistently lower than the estimates from

classical thin-shell theory [33]. Later on, the researchers

attributed this difference to the unavoidable presence of

initial imperfections in the structures, particularly the out-

of-roundness of the cylinder, locally or along the length of

the tube. This issue was resolved to a large extent in the

seminal thesis [34] by Koiter. Through an asymptotic

analysis of the post-buckling behavior he demonstrated that

the buckling load reduces dramatically with the increase in

initial imperfection. This work found numerous applica-

tions in the follow-up works on estimating a realistic

buckling load for various types of structures under various

loading conditions [15, 35–37]. Led by this success, Ko-

iter’s theory [34, 38, 39]—and more generally the theory of

thin-shell buckling considering initial imperfections (see

[33]), has been widely accepted as very reliable approaches

for accurate prediction buckling of thin-shells. A nanotube,

although modeled as a thin-shell, is essentially a collection

of finite number of atoms. Hence, it is not truly a contin-

uum. Furthermore, while equilibration, one or more

atom(s) can move out of the ideal cylindrical surface

leading to an initial structural imperfection in the form of

out-of-roundness along the length of the tube. Note that the

equilibration is carried out at a specific temperature and

hence, the degree of imperfection depends on the temper-

ature as well. Thus, usage of shell theories with a geometric

imperfection could serve as a potential direction for

improving the continuum mechanics-based analysis of

nanotube buckling incorporating the effect of temperature

on the nanotube buckling. There are a few continuum-

based models such as nonlocal elasticity used to study the

effect of temperature on the behavior of nanotubes[40–43].

However, to the best of authors’ knowledge, the effect of

out-of-roundness on the buckling stress of nanotube has not

yet been explored thus far by any previous studies, and

therefore the present work is aimed at this direction. Note

that the geometric imperfections considered here refer to

the out-of-roundness of the CNTs from that of the perfect

structure and are not related to the other imperfections or

defects such as missing-atom [44], Stone-Wales [45].

This paper is organized as follows. The theory of thin-

shells and the effect of initial imperfections are discussed

in Sect. 2. The theory of analysis of nanotubes using finite

element (FE) method is also discussed there. In Sect. 3, the

modeling and implementation of MD and FE methods to

study the buckling behavior of nanotubes are explained.

The major results obtained from thin-shell, shell with

imperfections, FE analysis and MD are discussed in Sect. 4

followed by concluding remarks in Sect. 5.

2 Theory

Shell theories have been used extensively to study the

buckling of nanotubes. Few commonly used theories

include thin-shell, first order shear deformation, and

nonlocal elasticity. Here, the theory of thin-shell is

explained followed by the Koiter’s theory of imperfection

in shells. The computational modeling of the nanotubes

with and without imperfection, respectively, to study shell

buckling using linear and nonlinear methods using the FE

analysis is also explained. The numerical implementation

of the FE method is discussed in detail in the next

section.

N. M. A. Krishnan, D. Ghosh

123

2.1 Buckling theory for cylindrical thin-shells

Cylindrical shells under uniform compressive loading in

the axial direction buckle at loads much lower than the

crushing load. The stress corresponding to the minimum of

such loads is defined as the critical buckling stress. The

buckled shape of the shell may be symmetrical or asym-

metrical to the axis of the cylinder. In the case of a sym-

metrical buckling, the radial displacement w during

buckling can be assumed as

w ¼ �A sinmpx

lð1Þ

where l is the length of the cylinder along the X axis—the

axial direction, x is the coordinate along the X axis, A is the

amplitude of the radial displacement and m is an integer.

Assuming that the load during the buckling remains con-

stant, the buckling load per unit length immediately before

buckling, Ncr ¼ �0Eh is equated to the buckling load per

unit length immediately after buckling,

Ncr ¼ ½Eh=ð1� m2Þ�ð�1 þ m�2Þ, where m is the Poisson’s

ratio and E is the modulus of elasticity and h is the shell

thickness. Thus, the axial strain before buckling �0 is

related to the axial strain �1 and circumferential strain �2

after buckling as

�1 þ m�2 ¼ ð1� m2Þ�0 ð2Þ

Here, �2 can be written �2 ¼ �m�0 � ðw=aÞ ¼ �m�0 þðA=aÞ sin mpx=lð Þ where a is the radius of the cylinder. The

change in curvature in the axial plane is given as

vx ¼ Am2p2

l2sin

mpx

lð3Þ

Also, owing to the symmetrical buckling,

c ¼ vy ¼ vxy ¼ 0. Thus, the increase of strain energy dur-

ing buckling can be written as

DU ¼ �2phEm�0

Z l

0

A sinmpx

ldxþ pA2Ehl

2aþ A2 p4m4

2l2palD

ð4Þ

where D ¼ Eh3=12ð1� m2Þ is the flexural rigidity of the

cylinder. Now, the work done by the compressive force

during buckling is

DW ¼ 2pNcr mZ l

0

A sinmpx

ldxþ a

4A2 m2p2

l

0@

1A ð5Þ

This work done can be equated to the change in strain

energy of the cylinder to obtain the expression for the

buckling stress as

rcr ¼ Dm2p2

hl2þ El2

a2Dm2p2

� �ð6Þ

Since m can take any integer value, the wavelength of

deformation can be lower than the length of the cylinder.

Thus, the minimum value of rcr with respect to ðmp=lÞ can

be given as

rcr ¼Eh

affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1� m2Þ

p ð7Þ

The location of this critical stress along the shell length is

at ðmp=lÞ ¼ Eh=a2Dð Þ1=4. Hence, it can be seen that the

critical buckling stress is independent of the length of the

cylinder. It can also be shown that the magnitude of critical

stress in the case of asymmetrical buckling is same as that

of symmetrical buckling [31].

2.2 Theory of shells with imperfection

Though the thin-shell equation gives a theoretical estimate

of the buckling stresses, the actual value obtained in

experiments varies from this. Such a variation from the

theoretical estimate is due to the presence of imperfections

in the initial conditions or the inaccuracies in ensuring the

conditions while conducting the experiments, for example,

imperfection in loading, shape, structure, and boundary

conditions. For imperfection sensitive structures, the vari-

ation of the shape of a shell from that of a perfect cylinder

can reduce the buckling stress. The imperfection sensitivity

of a structure was analyzed by Koiter by studying the

asymptotic form of a smooth potential energy function,

Pðq; k; aÞ, given by,

Pðq; k; aÞ ¼ ðc1 � a1aÞqþ ðc2 � a2aÞq2

þ ðcn � anaÞqn � kuð8Þ

where u ¼ b2ðq2 � a2Þ þ b4ðq4 � a4Þ is the generalized

displacement and ku corresponds to work done under

loading. c1; c2; cn; a1; a2; an and n are constants, q is the

parameter of the asymptotic expansion, k is the load

parameter characterizing either a single load or a system of

loads, and a is the small imperfection parameter. It should

be noted that n ¼ 3 for asymmetric bifurcation and n ¼ 4

for symmetric bifurcation. The equilibrium condition

oP=oqð Þ ¼ 0 yields,

c1 � a1aþ 2ðc2 � a2a� b2kÞqþ nðcn � anaÞqn�1

�4b4kq3 ¼ 0ð9Þ

Similarly, the critical load condition for the structure given

by o2P=oq2� �

¼ 0 yields,

A novel method for studying the buckling of nanotubes

123

2ðc2 � a2a� b2kÞ þ nðn� 1Þðcn � anaÞqn�2

�12b4kq2 ¼ 0ð10Þ

Applying the equilibrium and bifurcation conditions, and

neglecting higher-order terms, Eqs. (9) and (10) reduce to

k ¼ 1� k1

aq

� �� ðk2 þ k3Þqn�2 ð11Þ

k ¼ 1� ½ðn� 1Þk2 þ 3k3�qn�2 ð12Þ

respectively, where k1 ¼ a1=2c2, k2 ¼ ncn=2c2, k3 ¼ 0 for

n ¼ 3 and k3 ¼ 2b4=c2 6¼ 0 for other values of n. Using

elimination and substitution, k can be obtained as

k ¼ 1� aa0

� �m

ð13Þ

where a0 ¼ ½ðn� 2Þk2 þ 2k3�k�11 ½ðn� 1Þk2 þ 3k3��1=m

is a

constant, the exponent m is given by m ¼ ½ðn� 2Þ=ðn� 1Þ�and k ¼ Pmax=Pcr for a single load. Here, Pmax is the

maximum load at which the cylinder with imperfection

buckles and Pcr is the critical buckling load of a perfect

cylinder. The imperfection sensitivity is stronger when the

exponent m is smaller. Hence, for unstable symmetric

bifurcation (n ¼ 4 and m ¼ 2=3), the imperfection sensi-

tivity is more compared to asymmetric bifurcation (n ¼ 3

and m ¼ 1=2).

In the case of a shell with imperfection, the Eq. (13) can

be written as [39]

1� rrcr

� �¼ ad

rrcr

ð14Þ

Here, e ¼ d=h is the imperfection of the cylinder of

thickness h, with maximum amplitude of imperfection

being d. r is the buckling stress obtained for shell with

imperfection while rcr refers to the buckling stress of a

shell without imperfection. From Eq. (14), the buckling

stress r of a cylinder with imperfection, e, can be derived

as follows

r ¼ rcr

�1þ 3

4effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1� m2Þ

p

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3


p2þ 3


p� �s � ð15Þ

The effect of the imperfection e on the buckling stress of

a CNT is shown in Fig. 1. Here, the thickness of CNT is

assumed as 0.34 nm and Poisson’s ratio as 0.19. It is noted

that as the imperfection e increases from 0 to 0.5, the ratio

of buckling stress of imperfect nanotube to the buckling

stress of perfect nanotube reduces monotonically from 1 to

0.3. Hence, it is observed that the initial imperfection plays

a critical role in determining the buckling stress of a

nanotube.

2.3 Finite element analysis

The buckling behavior of shells can also be studied by FE

analysis, where the governing differential equation of shell

deformation is discretized using a set of FE bases followed

by a Galerkin projection. The resulting system of equations

is solved numerically. Both linear eigenvalue and nonlinear

analyses can be used to study buckling in FE. While

deriving the analytical solutions such as Eq. (7), the solu-

tion field was approximated using simple sinusoids such as

Eq. (1). However, in FE analysis, the search space for the

solution is much larger. Therefore, FE analysis can yield

more accurate results compared to analytical estimates

such as Eq. (7). In this work, both linear and nonlinear FE

analyses are carried out using Abaqus FEA [46] to estimate

the buckling stresses of nanotubes.

In the case of a linear analysis, the buckling load of a

cylinder is obtained by solving the eigenvalue problem

Ku ¼ 0 ð16Þ

where K ¼ KE þ vKG is the tangent stiffness, KE is the

elastic stiffness, KG is the geometric stiffness, u represents

the nontrivial displacement solutions and v denotes the

eigenvalues. The eigenanalysis yields different mode

shapes corresponding to different eigenvalues. The lowest

eigenvalue provides an upper bound for the buckling load.

The corresponding mode shape denotes the mode of failure

of the shell for the applied load.

In nonlinear analysis, the imperfection is imposed on the

cylinder as a superposition of a few possible mode shapes

obtained from the linear analysis with desired amplitudes.

This imperfection leads to a non-zero radial displacement

0 0.1 0.2 0.3 0.4 0.5

0.4

0.5

0.6

0.7

0.8

0.9

1

Imperfection, ε

Rat

io o

f Buc

klin

g st

ress

es

Fig. 1 The ratio of buckling stresses of imperfect nanotube to that of

a perfect nanotube is plotted as a function of initial imperfection, e. It

should be noted that the buckling stress of a nanotube reduces to

almost 30 % of its actual buckling stress for an imperfection, e, value

of 0.5


123

in initial conditions. Hence, the load–displacement relation

varies from that of a pure bifurcation to a smooth curve

without discontinuities before buckling. The maximum

applicable load is obtained from eigenanalysis as it gives

the upper bound for the buckling load. The modified Riks

method is used to analyze the effect of geometric nonlin-

earity due to the initial imperfection in the structure of the

cylinder. In this method, the increment at each step is

governed by a parameter called arc-length, w, which pro-

gresses along the static equilibrium path in the load–dis-

placement space. The load increments are optimally chosen

using a load proportionality factor, the value of which is

estimated at each step using w. The load proportionality

factor j estimates the load Pi at each incremental step i as

Pi ¼ P0 þ jðPmax � P0Þ ð17Þ

where Pmax is the maximum load that can be applied in the

analysis, and P0 is the initial load present, if any, before the

application of the external load. Hence, at any step during

the analysis, the tangent stiffness, the displacement, and the

loading are used to obtain the static equilibrium path. Then,

the optimal load proportionality factor along with the dis-

placements is estimated. Riks method yields solution for

both stable and unstable responses.

The buckling load obtained from a linear analysis will be

greater than that obtained from a nonlinear analysis. This is

due to the fact that the eigenanalysis assumes very little

deformation before buckling and the entire FE analysis

leading to buckling failure is carried out in a single loading

step. Hence, the effect of incremental loading on the geo-

metric stiffness matrix is not included in eigenanalysis. Thus,

for better results, the nonlinear analysis needs to be used.

3 Modeling and implementation

Here, the buckling simulations are conducted on CNTs.

The nanotubes are modeled with varying diameters. The

L=D ratio is kept close to 5 for all the CNTs to ensure shell

buckling. The thickness of each of the nanotubes is

assumed as 0.34 nm, Poisson’s ratio as 0.19 and elastic

modulus as 1 TPa. Fixed–fixed boundary condition is

applied on both the boundaries of the nanotube.

3.1 Molecular dynamics

The open-source package visual molecular dynamics

[47] (VMD) is used to model the structural configuration

of nanotubes. The MD simulations are carried out using

the open-source package large-scale atomic/molecular

massively parallel simulator [48] (LAMMPS). Interac-

tions between the carbon atoms are modeled using

adaptive intermolecular reactive empirical bond order

[49] (AIREBO) potential. The AIREBO potential is

defined by

E ¼ 1

2

Xi

Xj6¼i

EREBOij þ ELJ

ij þXk 6¼i;j

Xl 6¼i;j;k

ETORSIONkijl

!

ð18Þ

Here, all the indices run from 1 to N atoms in the neighbor

list. The first term, EREBO, represents the second generation

REBO potential [50]. The second term in the Eq. (18) is the

12–6 Lennard-Jones potential that takes into account the

long-range interactions among atoms with distance beyond

2 A. The effect of long-range force becomes prominent for

smaller tubes throughout and in the post-buckling behavior

for larger CNTs. The third term, ETORSION, represents the

torsional interactions about single bonds.

In each of the MD simulations carried out, the nanotube is

equilibrated initially at a temperature of 300 K. After suffi-

cient equilibration, both the ends of the nanotube are fixed by

arresting the motion of a few layers of the end atoms. A

constant displacement rate of 0.1 A/ps is applied to the lower

end of the nanotube in the axial-compressive direction. The

time step is fixed as 1 fs. The CNT is compressed until it failed

by buckling, characterized by a sudden drop in the stress–

strain curve. The buckled mode shapes are examined using

VMD, which revealed that all the CNTs underwent shell

buckling.

The virial definition of stress at the atomic scale is used

to find out the equivalent continuum stress, given by

rij ¼1

V

XN

a¼1

�mava;iva;j þ1

2

XN

a¼1

XN

b¼1;b 6¼a

o/ðrÞor

rirj

rjr¼rab

!

ð19Þ

Since the virial stress is found out for individual atoms and

summed up during the simulation, the resultant stress has to

be divided by the total volume of the carbon nanotube to

obtain the equivalent Cauchy stress. The volume of the

nanotube is calculated assuming wall thickness of 0.34 nm

[51, 52]. The thermodynamic definition using the change in

total energy is also used to estimate stress as

r ¼ 1

V

oU

o�xx

ð20Þ

Here, r is the stress, V represents the volume, U is the total

energy, and �xx is the axial strain of the nanotube. Using

Eqs. (19) and (20) for estimating the buckling stress

ensures consistency in the value of stress obtained.


The FE analysis is conducted using two methods—linear

and nonlinear. The inputs for the nonlinear analysis


123

considering imperfections in the initial shape are obtained

from the linear analysis. Details of implementation of the

linear and nonlinear methods are explained below.

3.2.1 Linear analysis

The admissible diameter for each of the nanotubes is

obtained using

D ¼ 0:783ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn2 þ nmþ m2Þ

pð21Þ

where n and m are the chiral indices of the nanotube. The

4-noded S4RS conventional shell element, available in

Abaqus FEA, is used for meshing the structure. The ele-

ment has three displacements and three rotational degrees

of freedom at each node and the thickness is defined as the

section property. The formulation of this element allows

thickness change as a function of in-plane deformation as

well. Hence, this element is suitable for modeling buckling

problems because it yields accurate results for extreme in-

plane bending and rotational behavior in case of both thick-

and thin-shells. Both the ends of the nanotube are fixed

against rotational and translational motion. Axial motion of

the loading end is allowed for the buckling to happen. The

linear eigenanalysis is carried out to obtain the eigenvalues

and corresponding mode shapes. The buckling stress is

then calculated from the obtained buckling load. The mode

shapes and the buckling loads are stored to carry out the

nonlinear analysis.

3.2.2 Nonlinear analysis

The meshing and the boundary conditions used in nonlin-

ear analysis are the same as the linear analysis. However,

the initial condition is varied by introducing imperfection

in the initial shape of the nanotube in the form of out-of-

roundness. The imperfection is modeled using the buckled

mode shapes obtained from the linear analysis. The effect

of the number of modes to model the initial imperfection is

studied by modeling the imperfection as the superposition

of multiple modes. The maximum displacement allowed

for the initial imperfection is restricted to 0.17 nm, which

is half of the wall thickness. This value is obtained from the

maximum out-of-roundness of the CNT after equilibration

in a MD simulation at 300 K. Hence, depending on the

temperature of the simulation, the maximum displacement

varies accounting for the change in buckling stress as a

function of temperature. This will be demonstrated later.

The amplitude for modes other than the first mode is

always restricted to half the amplitude of the first mode,

0.09 nm in this case. Since the failure due to the first mode

is more probable than the failure due higher modes, the first

mode is given maximum weight. For instance, if only one

mode is used, the imperfect initial shape of the nanotube

will be similar to the first buckling mode with amplitude as

0.17 nm. If three modes are used, the imperfect shape will

be a linear superposition of first, second and third modes

with amplitudes 0.17, 0.09, 0.09 nm, respectively. The

inclusion of more than one mode could ensure a more

realistic initial structure. The nonlinear analysis using the

Riks algorithm is carried out for each of the nanotubes. The

buckling stresses obtained in each of these cases are

compared with the results obtained from MD.

To study the effect of variation in temperature, the

buckling analysis is carried out at five different tempera-

tures—50, 100, 150, 200, and 300 K, on a (10,10) CNT.

The CNT is first equilibrated at the mentioned temperatures

separately using MD simulation. The maximum out-of-

roundness of the CNT is estimated considering the dis-

placement of the atom undergoing maximum displacement

from its original undeformed configuration. This dis-

placement is used to model the initial imperfection of the

CNT in FE analysis.

4 Results and discussions

4.1 Shell theory

The results obtained from MD are compared with the

thin-shell theory in Fig. 2. It is verified that the results from

MD simulations match closely with previous studies [8–

11]. Equation (7) is used to obtain the critical buckling

stresses according to the thin-shell theory. It can be seen

that even if the buckling stress decreases monotonically as

a function of the diameter of nanotubes in both the cases,

the classical thin-shell theory overpredicts the buckling

stresses. Hence, to obtain realistic buckling stresses which

will be comparable to MD estimates, the effect of imper-

fections needs to be considered. Furthermore, the buckled

mode shape for thin-shell buckling is a symmetric sinusoid

as given in Eq. (1), which is different from the MD esti-

mate shown later in Fig. 6.

The theory of imperfections is then employed to obtain

the buckling stresses of nanotubes. To this end, the Koiter’s

formula given by Eq. (15) is invoked. The critical buckling

stress rcr is obtained from the thin-shell equation. A real-

istic value of the imperfection e—0.5—is chosen for the

analysis. This value is obtained from MD, considering the

maximum displacement of the equilibrated nanotube from

its perfect structure. Using Eq. (15), the buckling stress

according to Koiter’s theory as a function of the diameter is

plotted in Fig. 3. This buckling stress is compared to that

obtained from thin-shell theory and MD. It is observed that

the Koiter’s theory gives a better estimate of the buckling

stress compared to thin-shell theory. It is also observed that


123

as the diameter of the nanotubes increases, the Koiter’s

estimate is closer to MD. But for smaller diameters, the

Koiter’s theory still overpredicts the buckling stress con-

siderably. Another drawback of the Koiter’s theory is its

inability to predict the buckled mode shape correctly.


The buckling stresses obtained using linear and nonlin-

ear FE analysis are plotted in Fig. 4. The results thus

obtained are compared with the buckling stresses obtained

from MD simulations. It is observed from Fig. 4 that the

linear analysis gives maximum estimate for the buckling

stress. This results from the fact that the linear analysis

does not take into account the variation in geometric

stiffness due to the loading. The nonlinear analysis is

carried out with an initial imperfection. The buckling

stresses obtained considering one, two, three, four and five

eigenmodes, respectively, in the initial imperfection reveal

that as the number of eigenmodes increases, the buckling

stress decreases monotonically. The buckling stresses

obtained considering three buckling modes matche well

with the buckling stress obtained using MD. In some cases,

an analysis considering four or five modes gives a mar-

ginally closer match with MD. However, the difference in

buckling stresses estimated using four and five modes are

not significantly different compared to the results obtained

from three modes. Hence, it can be inferred that the

superposition of first three buckling modes gives a realistic

initial imperfection for the nonlinear FE analysis.

Furthermore, the buckled mode shape obtained using

nonlinear FE analysis, shown in Fig. 5, is compared with

the buckled shape obtained from MD as in Fig. 6. It can be

seen that the buckled shape predicted by MD and nonlinear

FEM match well. This similarity in the failure mode vali-

dates the assumption that the inclusion of initial imper-

fection improves the estimate of both the buckling stress

and buckling shape. The FE method is computationally

much cheaper than the MD simulation. As the nanotube

size increases, MD simulation becomes more expensive

due to the force calculation involved at each time step for

each of the atoms. Hence, the proposed method for the

modeling of nanotubes considering the initial

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8x 1011

Radius (nm)

Buc

klin

g S

tres

s (N

/m2 )

MDShell

Fig. 2 The buckling stresses of nanotubes obtained using MD and

thin-shell theory are plotted as a function of the diameter of the

nanotubes. The thin-shell is observed to overestimate the buckling

stress of the nanotubes

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8x 1011

Radius (nm)

Buc

klin

g S

tres

s (N

/m2 )

MD

Shell

Koiter

Fig. 3 The buckling stresses obtained from MD, thin-shell theory and

Koiter’s theory of imperfection are plotted as a function of diameter.

It is observed that the Koiter’s theory gives a better estimate

0 0.5 1 1.5 20

0.5

1

1.5

2x 1011

Radius (nm)

Buc

klin

g S

tres

s (N

/m2 )

MDLinearNL − 1 modeNL − 2 modesNL − 3 modesNL − 4 modesNL − 5 modes

Fig. 4 The buckling stresses obtained from MD and FE analysis are

plotted as a function of the tube diameter. FE analysis includes both

linear and nonlinear analysis with initial imperfection modeled as the

superposition one, two, three, four and five modes, respectively


123

imperfections using FE would be very useful especially for

larger nanotubes.

The effect of temperature on the prediction of buckling

stress is studied next. Three modes are considered for

modeling the out-of-roundness of the CNT as they gave a

good estimate of buckling stress. The results obtained from

the FE analysis as a function of the temperature are plotted

in Fig. 7. It is observed that the buckling stresses obtained

using FE analysis matches closely with the MD simulation

result for a range of temperatures varying from 50 to

300 K. However, as we go to even higher temperatures, the

out-of-roundness due to thermal vibrations becomes very

large. This phenomenon results in an unrealistic FE model

leading to erroneous estimation of the buckling stress.

5 Concluding remarks

This study reveals the importance of considering the

imperfections in the analysis of nanostructures, nanotubes

in particular. The imperfections in the initial shape of the

nanotube affect the buckling load and the buckling mode

shapes. It is demonstrated here that while the classical thin-

shell theory overpredicts the buckling load, Koiter’s theory

of shells with imperfection gives a better estimate. More

importantly, it is found out that the nonlinear FE analysis,

considering the imperfections in the initial structure, can be

used to explain the shell-like behavior of nanotubes

observed in MD simulations. The FE analysis predicts the

buckling load of the nanotubes accurately and is compu-

tationally cheaper compared to MD simulations by more

than an order of magnitude. The buckling mode shapes are

also predicted accurately by the nonlinear FE method.

Prediction of buckling modes would be useful to carry out

post-buckling analysis. The present method can accurately

take into account the effect of temperature in the form

Fig. 5 The mode shape of the nanotube after buckling obtained from

a nonlinear FE analysis is shown here. This is the mode shape of a

nanotube with initial imperfection as a superposition of three modes

Fig. 6 The buckled shape of a nanotube obtained from MD, with the

same diameter as that of the FE analysis, is shown. Note that the

buckled shape is very similar to that obtained from FE analysis

50 100 150 200 250 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 1011

Temperature (K)

Buc

klin

g S

tres

s (N

/m2 )

LinearNL − 1 modeNL − 2 modesNL − 3 modesMD

Fig. 7 The buckling stresses of CNTs obtained from MD and FE are

plotted as a function of temperature for (10,10) CNT. Note that NLFE

with three modes matches well with MD at different temperatures


123

initial imperfection. This ability promises a novel approach

to incorporate the effect of temperature into continuum

buckling analysis. The effectiveness and usefulness of the

present method to study post-buckling behavior need to be

further studied and understood. This method can be further

extended to study the effect of imperfections in other

behaviors such as torsional buckling, fracture.

Acknowledgements The authors would like to thank the Board of

Research in Nuclear Sciences (BRNS) grant no. 2012/36/37-BRNS/

1683 for financial support.

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A novel method for studying the buckling of nanotubes considering geometrical imperfections