a variation principle approach for buckling of carbon...

April 14, 2011 10:22 WSPC S1793-2920 S1793292011002676

NANO: Brief Reports and ReviewsVol. 6 (2011)c© World Scientific Publishing CompanyDOI: 10.1142/S1793292011002676

NANOArticle in Production

A Variation Principle Approach for Buckling of Carbon

Nanotubes based on nonlocal Timoshenko Beam Models

Yang Yanga,b and C.W. Limb,*

aDepartment of Engineering Mechanics, Kunming University of Science and

Technology, Kunming 650051, Yunnan, P.R. China

bDepartment of Building and Construction, City University of Hong Kong,

Tat Chee Avenue, Kowloon, Hong Kong, P.R. China

Abstract: Based on the nonlocal elastic theory and variation principle, new

Timoshenko beam models and analytical solutions for buckling of carbon nanotubes

considering nanoscale size effect and shear deformation are established. New

equilibrium equations and high-order boundary conditions are derived and the

buckling behavior of carbon nanotubes is numerical investigated. The numerical

solutions confirm that nanotube stiffness is enhanced by nanoscale size effect and

reduced by shear deformation. It is also concluded that nanotubes with different

boundary conditions show varying sensitivity to changes in nanoscale and dimension.

Comparison with molecular dynamics simulation results verifies the accuracy and

reliability of this new analytical nonlocal Timoshenko beam model.

Keywords: buckling; carbon nanotubes; nonlocal elastic theory; Timoshenko beam;

variation principle

*Corresponding author. E-mail: [email protected]

1

http://dx.doi.org/10.1142/S1793292011002676

April 14, 2011 10:22 WSPC S1793-2920 S1793292011002676

2 Y. Yang & C. W. Lim

1. Introduction

Since the discovery of carbon nanotubes (CNTs) in the early 1990’s,1 analysis

and experiment of the mechnical behaviors of CNTs have received much research

attention. The buckling behavior of CNTs has been a challenging research topic of

particular interest because CNTs are very much useful in nanobiological,

nanomechanical and nanochemical applications such as nano-fluid conveyance and

drug delivery.2,3 The research approaches on buckling behaviors of CNTs include

experiments as well as theoretical and numerical modeling and analyses. However,

experiments on CNT properties have been less conclusive due to the minute scales

and difficulties in manipulation and control.4 To complement the deficiencies and

other shortcomings of these approaches, theoretical and numerical modeling and

analyses have become effective alternatives with increasing importance in CNT

research.

The common approaches of theoretical and numerical methods to study the

mechanical behavior of CNTs include molecular dynamic (MD) simulations and

continuum mechanical modeling. While MD simulations are the most common

numerical methods for analyzing CNTs because every molecule within CNTs is

simulated and accounted for in the simulation program, the numerical models very

frequently ends up with billions of molecular entities for large scale systems and

hence tremendously amount of computational effort running into days is unavoidable.

The accuracy of the numerical solutions deteriorates due to numerical instability,

rounding off, truncation and other numerical tolerances. 5-7

As a result, elastic continuum models of CNTs received intensive attention

recently and they have been rigorously developed. Ru and his associates8 applied

the classical elastic models to study the bending and buckling behaviors of

single-walled carbon nanotubes (SWCNT) under high pressure and they concluded

that the critical buckling pressure was 1.8 GPa for SWCNTs with diameters around

1.3nm. The direct application of classical elastic models for CNTs may lead to

inaccurate solutions because the influence of nanoscale size effects on the mechanical

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A Variation Principle Approach for Buckling of Carbon Nanotubes Based on Nonloacl Timoshenko Beam Models 3

properties of CNTs cannot be captured by the classical models. 9 For these reasons,

new continuum elastic models have been established to study the mechanical

behaviors of CNTs and they include strain gradient models,9,10 couple stress

models,11-13 nonlocal stress models, etc.14-32

The nonlocal elastic stress theory was first developed by Eringen.14-18 It has

been regarded as an effective model which is sensitive to nanoscale size effects and it

is very suitable for modeling CNTs.18 Due to simplicity of the differential nonlocal

constitutive relation which is equivalent nonlocal stress in an integral form over the

entire elastic body,17 a large amount of research articles for bending and buckling of

CNTs based on this nonlocal models have been published since the beginning of the

new century.7,19-33 Using the nonlocal Euler-Bernoulli beam model, Sudak20 studied

column buckling of multiple-walled carbon nanotubes (MWCNT). The nonlocal

influence and van der Waals effect between adjacent tubes on buckling behaviors of

MWCNTs were analyzed in detail. Based on the nonlocal Timoshenko beam model

which accounts for shear deformation, Wang and his associates21 investigated

buckling of micro and nano rods and they reported that the buckling load was reduced

due to the effects of both nonlocal stress and shear deformation. There exist many

other research papers which at all concluded that, similar to that of Wang,21 the

buckling critical load nonlocal CNTs is reduced due to the presence of nonlocal

stress.20-32 In this respect, the authors herewith, based on the identical nonlocal

constitutive relation of Eringen,17 present a new analysis and interpretation of the

nonlocal stress effects that its presence should in fact induce strengthening of CNT

stiffness and thus enhancing CNT buckling loads.

Though the various nonlocal studies19-33 can simulate the small scale effect on

the mechanical behaviors of CNTs, contradictory predictions and surprising

conclusions based on the nonlocal models are observed. For example, the bending

behaviors of cantilever nanotube with a point load at the free end based on a nonlocal

beam model is not affected by the nonlocal effect, i.e. there is no nonlocal effect and

the “nonlocal solutions” is identical to the classical solutions.19 This implies the

bending response of a point-loaded cantilever nanotube is completely identical to the

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response of a classical tube without size effects. Another intriguing conclusion is

that virtually all nonlocal models predict decreased stiffness contributed by the small

scale effect, which contradicts the result of experiment and MD simulation.19-32

Having observed the contradictions and inconsistencies above, Lim and his

associates recently established a new approach for analytical nonlocal models (AN)

according to the variational principle.34-39 New higher-order nonlocal governing

equations and boundary conditions which contain higher-order nonlocal terms were

first derived. They also reported that the inconsistent and intriguing results of the

previous nonlocal studies were due to direct replacement of the classical stress terms

in the equilibrium equations or equations of motion by the corresponding nonlocal

quantities. The direct replacement has been without rigorous verification and hence

certain very important higher-order nonlocal terms have been inadvertently

neglected.34 For instance, the nonlocal bending moment derived from the nonlocal

constitutive equation was directly used to replace the classical bending moment, in

both the equilibrium conditions and boundary conditions, in the bending analysis of a

nonlocal nanobeam.19 These previous nonlocal models derived from direct

extension of the classical conditions are termed the “partial nonlocal models” (PN).

Based on the new AN model, Lim et al.34-39 examined the mechanical behaviors

of CNTs for bending, buckling, vibration and wave propagation are confirmed more

reasonable than PN models. The analyses adopted the Euler-Bernoulli beam theory

and no shear deformation was considered. To the authors’ knowledge, there has

been no study on buckling of shear deformable nanobeam or nanotubes based on the

new AN model and this subject is addressed in this paper. Here, buckling behavior

of SWCNTs applying the new analytical nonlocal Timoshenko beam model (ANT) is

investigated considering shear deformation effects. The buckling loads of SWCNTs

with four different boundary conditions are analyzed and discussed in detail.

2. ANT Model for Buckling of SWCNTs

Based on the nonlocal elasticity theory first proposed by Eringen,14-18 the

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stress at a point r in a nonlocal elastic continuum depends not only on the local

stress (classical stress) at that point but also on stresses at all other points in the body.

Thus, the nonlocal stress tensor r at point r is expressed as

VdVr r r r r (1)

where r is local classical stress at any particular points r within the domain

V . The nonlocal modulus ,r r depends on the Euclidean distance r r

between r and r and also a dimensionless length scale defined as

0e a

L (2)

where 0e is a material constant, a and L are the internal and external

characteristic lengths, respectively. In this context, L is taken as the length of

nanotube. Equation (1) above is an integral equation for which an analytical solution

may not be possible.

Instead of solving the integral nonlocal relation, it is also possible to solve an

equivalent second order differential constitutive relation as17

2 20e a (3)

where 2 2 2 2 2x z is the Laplace operator. Considering only normal

strain in the x direction, Eq. (3) is reduced to

22

0 2

d xx e a E x

dx (4)

where E is Young’s modulus and is the axial strain. The classical constitutive

equation denoted by Hooke’s law is obtained when 0 0e a in Eq. (4). For

convenience and generality, Eq. (4) is rearranged in a dimensionless form as

22

2

d xx x

dx (5)

where xx xx E and x x L are dimensionless quantities.

A CNT modeled as a Timoshenko beam with external axial force xxN in

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Cartesian coordinate system is illustrated in Fig. 1, where x and z denote axial

and vertical coordinates, respectively, and w is the vertical deflection. The normal

strain in the axial direction is expressed as

d dz z

dx dx (6)

where z z L and denotes the rotation angle of beam cross section.

Using Eq. (6) and neglecting all pre-stresses and pre-strains which appear as

constants of integration, the solution to Eq. (5) can be expressed as33

2 1 2 12 1 2 1

1 1

n nn nxx xx

n n

z (7)

where n denotes the n th-order partial differentiation with respect to x . The

beam bending moment is defined as xx xxM z dA and the dimensionless bending

moment is xx xxM M L EI , where 2I z dA is the second moment of area over

the cross section A . Thus, the nonlocal dimensionless bending moment becomes

2 12 1

1

nnxx

n

M (8)

In the previous studies, it has been established that the classical governing

equations of motion and classical boundary conditions could not be directly applied in

nonlocal modeling even with the classical stress terms replaced by the corresponding

nonlocal quantities.34-39 The true moment equilibrium condition for a nonlocal stress

element should be instead replaced by an equivalent nonlocal bending moment

derived from the variational principle. Based on this new nonlocal modeling concept,

the buckling for a thick-walled, shear deformable nanostructure with nonlocal

constitutive relation for CNTs is developed here via the variational principle. The

unique feature is the history of nonlocal energy straining will be accounted for

throughout the processing of energy storing from a referenced unstrained state.

The strain energy density nu at a point in the nanostructure is the integral sum

of the nonlocal stress over the history of straining which can be expressed as

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0 0nu d E d (9)

Expanding Eq. (9) using the nonlocal constitutive relation in Eq. (7) yields37

1 2 3nu u u u (10)

where

21

21 22

1

1 2( 1)2 13

1 1

1

21

12

1

x

n nnx

n

nm m n mn

x xn m

u E

u E

u E

(11)

In addition, the energy density contributed by shear stress for a shear

deformable CNT should be considered and it is

21

2s xzu G (12)

where G is the shear modulus and xz is shear strain. Thus the strain energy in

the whole nonlocal CNT is

n s

v

U u u dV (13)

where V denotes the volume of CNT. On the other hand, the work done by

external axial force is

2 21

0 0

12 2

Lxx

xx

LNdw dwW N dx dx

dx dx (14)

According to the variational principle, variation of the energy functional should

be zero and it as

0F U W (15)

Considering Eqs. (10)-(14), Eq. (15) becomes

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212 2

1

1 2( 1)2 1 2

1 1

121

200

11 1 1 12 2 2

1

1 1 2 32 12

1 11

2 2

11

2

1

1 1

n nnx xV

n

nm m n mn

x x xzn m

xx xx

n n nn

Vn

m mm n mn

F U W

E E

E G dV

d w dwLN wdx LN w

dx dx

Ez Ez

Ez1 1 2 3

1 1

1 1 1 1

121

200

12 2 2 2

2

nm n m

n m

xx xx

G w w w w dV

d w dwLN wdx LN w

dx dx (16)

The shear force is defined as39

z xz

A A

w wQ G dA G dA AG

x x (17)

where is the shear correction factor, and hence

A

w wdA A

x x (18)

Substituting Eq. (18) into Eq. (16) yields

21 22 1

01

12 21 1 122 20

0 0 0

12 11 12 2 1 2 22 1

0 0

2 3

1

1 1

nn

n

nn n m n mn

xxn m

n mm i m im i n m i n m in

i

EI AG L wF n dx

L EI x

w w EIAG L LN wdx

x x x L

EI

L

111

1 1 0 0

11 1

0

0

n mm i

n m i

xxw GLA w LN w

(19)

Since and w do not vanish according to the variational principle, Eq. (19)

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yields the governing, higher-order nonlocal equilibrium equations

22 1

1

2 3 0nn

n

wa b n

x (20)

2 2

2 20xx

w wa LN

x x x (21)

where AG L a and EI

bL

. Further defining the dimensionless quantities as

2z

z

Q L a wQ

EI b x (22)

2 12 1

1

nnxx

n

M (23)

and effective nonlocal bending moment,34,37

2 2 12 12ef

1 1

2 2 3n nnnxx xx

n n

M M M n (24)

Eqs. (20) and (21) are reduced to

1ef 0zQ M (25)

2 0zxx

Qb LN w

x (26)

Equations (25) and (26) above are the equilibrium equations for shear

deformable CNTs with external axial force. The remaining terms in Eq. (19)

constitute the higher-order nonlocal boundary conditions as

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ef 0,10,1

1 2 1 12 2

0,12 0,1

2 22 1

0,11 0,1

1 2 1 32 24

0,11 0,1

1 2 42 24

01 0,1

0 or 0

2 0 or 0

2 0 or 0

2 0 or 0

2 0 or

xx

nnxx xx

xn x

nnxx

xn x

nnxx xx

xn x

nnxx

xn x

M

M M

M

M M

M,1

1 2 1 52 34 6

0,11 0,1

0

2 0 or 0nnxx xx

xn x

M M

(27)

and

1

0,10,1 0 or 0z xx xxbQ LN w w (28)

In summary, Eqs. (25)-(28) constitute the higher-order equilibrium equations and

higher-order boundary conditions for buckling of nonlocal CNTs modeled as

Timoshenko nanobeams with axial force. The analytical model as expressed in Eqs.

(25)-(28) are termed as the analytical nonlocal Timoshenko nanobeam model (ANT)

for CNTs.

On the other hand, the classical equilibrium equations for Timoshenko beams

are

1c 0zQ M (29)

2 0zxx

Qb LN w

x (30)

where 1cM is the classical bending moment. In the previous studies on nanobeams

using the PN models,19-33 it is common to assume that the classical governing

equations of motion still hold if the stress and moment quantities are replaced by the

corresponding nonlocal quantities. Thus, for a PN Timoshenko nanobeam (PNT)

derived by directly replacing cM by xxM in Eqs. (29) and (30), the equilibrium

equations are21

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1 0z xxQ M (31)

2 0zxx

Qb LN w

x (32)

The corresponding boundary conditions for PNT models are treated in a similar

manner and they are

1

0,10,1 0 or 0z xx xxbQ LN w w (33)

0,10,1 0 or 0xx xx

M (34)

Note that comparing to Eqs. (27) and (28), the boundary conditions are without

higher-order nonlocal conditions.

In is interesting to note that, comparing Eqs. (25)-(28) for ANT models and Eqs.

(29)-(32) for PNT models, the equations are perfectly similar except that the former

are expressed in terms of the effective nonlocal bending moment efM while the

latter are in nonlocal bending momemt xxM . The two bending moment quantities

are related through Eq. (24). Furthermore, the higher-order boundary conditions in

Eq. (27) are not available based on the PNT models.

It has been established by Lim34-39 that the classical equilibrium conditions are

not in equilibrium through direct replacement of stress resultants with the

corresponding nonlocal quantities. This is due to the existence of the nonlinear

constitutive relation in Eq. (7) which relates the nonlocal stress with the higher-order

strain gradients. The nonlinear constitutive relation is a generalization of the

classical constitutive relation xx xx which could be obtained if only the first term

with 1n is retained. Here it is established that the use of the classical relations is

not appropriate because it will ends up with the classical strain energy 1

2u . In

Fig. 2(a), the linear stress/strain curve is shown while Fig. 2(b) illustrate a

representative nonlinear stress/strain curve. The strain energy in Figs. 2(a) and 2(b)

are denoted by the shadowed area under the stress/strain curves. In Fig. 2(a), the

shadowed area under linear curve is a triangle and its area is 1

2u , and this is the

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classical strain energy density. On the other hand, the shadowed area under the

nonlinear curve in Fig. 2(b) is 0

u d . It is obvious that 0

1

2d for the

nonlinear case. Using the classical strain energy density 1

2u but taking for

granted that is the nonlocal stress corresponding to Eqs. (1,3,4) will establish the

PN and including PNT models19-33. Obviously, the assumption of nonlocal stress

in the classical strain energy density 1

2u is inappropriate. The exact nonlocal

strain energy density is expressed in Eqs. (10) and (11) where the classical strain

energy density is only the first term in the nonlocal expression. For a true nonlocal

stress analysis, at least two terms in the nonlocal strain energy density in Eqs. (10) and

(11) should be taken into consideration.

The PNT approaches by direct substitution of nonlocal stress quantities into the

classical governing equations and boundary conditions have been proved by Lim and

his associates34-39 as unjustifiable and they yield rather surprising reduced stiffness

solutions. If the nonlocal quantities are directly substituted into the classical

Euler-Bernoulli beam model or the classical shell models, the partial nonlocal

Euler-Bernoulli beam model (PNE) and partial nonlocal shell models (PNS) are

established and several studies on the buckling of CNTs based on PNT, PNE and PNS

were published.19-33 All of these works predicted reduced stiffness and reduced free

vibration frequency for nanostructures, and the predictions are contradictory to the

solutions obtained by other approaches such as MD,5-7 the strain gradient models ,9,10

couple stress models,11-13 and experiment results.41

To analyze the buckling of a shear deformable CNTs using the ANT model, Eqs.

(25) and (26) should be expressed in terms of deflection w and rotation angle as

the independent variables. When higher-order terms of order 4O are omitted,

Eqs. (20) and (21) become

1 2 42 0a w b (35)

2 1 2 0xxa w LN w (36)

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The equations above can be reduced to one single equation when w is eliminated, as

5 3 12 0c (37)

where xx

xx

LNac

b a LN and it is a fifth-order homogeneous linear differential equation.

The characteristic equation of Eq. (37) is

2 5 3 0c (38)

Considering 21 4 0c as 0.3 for common CNTs, the solutions to Eq. (38)

are

1,2 3,4 5, , 0 (39)

where

122

2

1 1 4

2

c and

122

2

1 4 1

2

ci . Thus the general solution

to Eq. (37) is

1 2 3 4 5sin cosx xC e C e C x C x C (40)

where 1 2 5, , ,C C C are constants of integration to be determined form the boundary

conditions. Substituting Eq. (40) into Eqs. (35) and (36), w is expressed as

1 2 3 4 5 6

1 1 1 1cos sinx xa

w C e C e C x C x C x Ca LN

(41)

where 6C is also the constants of integration.

3. Result and Discussion

Numerical examples are presented in this section for (5,5) CNTs modeled as

Timoshenko nanobeams. The material constants and parameters are 5.5TPaE ,

2.3TPaG , 0.8 , 10nmL and thickness 0.066nmh . The constants 1C ,

2C , 3C , 4C , 5C , 6C in Eqs. (40) and (41) should be solved using the boundary

conditions in Eq. (27).

In the following examples, the mechanical responses of CNTs with various

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boundary conditions are analyzed. Because the CNTs are constructed based on a

nonlocal continuum model, the boundary conditions should not be viewed as

mechanical constraints on the end atoms/molecules. Instead, the boundary

conditions should be regarded as additional constraints imposed on the end

atoms/molecules in a collective manner. In molecular dynamics simulation, it is

generally acceptable to impose the boundary conditions by imposing additional

displacement constraints to the end atoms/molecules. The atomic/molecular

constraints can be physically implemented by nanomechanical effects such as electric

force, magnetic force, van der Waals interaction, hydrogen bond, etc.

3.1 Simply supported CNTs

For CNTs with simple supports at the ends, some of the atoms/molecules are

physically fixed while the rests are free without any constraints. Collectively, the

fixed atoms/molecules cannot move in the transverse direction but the end cross

section, which is simply supported, is allowed to rotate about a neutral axis.

However, axial movement of the atoms/molecules is permitted. In this example

where the mechanical responses are realized through micro effects in the axial

direction, the CNTs are either stretched or compressed. The compressed CNTs leads

to buckling that will be analyzed as follows.

According to Eq. (27), the simply supported boundary conditions at both ends of

the CNT are

22 1ef0,1 0,1

1 0,1

0 ; 0 ; 2 0nnxxx x

n x

w M M (42a-c)

Equations (42a,b) are the boundary conditions corresponding to the classical or PNT

solution21 when xxM is replaced by efM while Eq. (42c) is the new higher-order

boundary condition derived from the ANT models. The higher-order nonlocal

boundary conditions can be further simplified as

1 3

0,10,1 0,10 ; 0 ; 0

xx xw (43a-c)

Substituting Eqs. (40) and (41) into Eqs. (43a-c) yields

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1

3 3 32

3 3 3 33

4

5

6

0 0 0

cos sin 0 0

0 0 0

cos sin 0 00

1 1 10 0

1 1 1 1cos sin

Ce eC

Ce eCa LNCa

Ca LN a LNe e

a a

(44)

For nontrivial solutions of 1 2 3 4 5 6, , , , ,C C C C C C in Eq. (44), the determinant of matrix

should vanish. It leads to the following characteristic equation

3 3 3

3 3 3 3

0 0 0

cos sin 0 0

0 0 0

cos sin 0 00

1 1 10 0

1 1 1 1cos sin

xx

xx xx

e e

e e

a LN

a

a LN a LNe e

a a

(45)

The solution to Eq. (45) gives the relation between the nanoscale parameter

and the buckling load xxN . Referring to the classical buckling load of a simply

supported Timoshenko beam40

2

2 2CT

EIAGN

AG L EI (46)

the buckling load ratio xx CTN N is obtained. The relation of buckling loading ratio

for varying diameter ratio D L of CNT is presented in Fig. 3. In this case, the

diameter D of a (5,5) CNT is 1.3nm while the length L can be varied.

It is shown in Fig. 3 that xx CTN N increases when varies from 0 (the

classical solution) to 0.15. It implies the ANT buckling load xxN is enhanced by

increasing nanoscale effect. The increase in buckling load and higher CNT stiffness

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due to nanoscale effect is confirmed here. Furthermore, larger D L corresponds to

higher xx CTN N , which means CNTs with larger D L are stiffer.

The response of xx CTN N of different nonlocal models for varying with

10nmL is illustrated in Fig. 4 where PNE denotes the solution based on the partial

nonlocal Euler-Bernoulli beam model,20 PNT for the partial nonlocal Timoshenko

beam model,21 ANE for the analytical nonlocal Euler-Bernoulli beam model38 and

ANT the new solutions in this paper. The buckling load ratio for the ANE and PNE

models are xx CEN N , where CEN is classical buckling load for a simply supported

Euler-Bernoulli beam as40

2

2CE

EIN

L (47)

From Fig. 4, it is clear that the buckling load ratios xx CEN N and xx CTN N ,

respectively, for PNE and PNT models decrease with increasing , which indicates

higher nanoscale effect leads to lower stiffness. In the contrary, ANT and ANE

models predict higher stiffness for CNTs when increases. It is also obvious that

shear deformation effect tends to caused reduced CNT stiffness where the buckling

load ratios for ANT and PNT are lower than the corresponding values for ANE and

PNE, respectively, in Fig. 4.

3.2 Clamped-Clamped CNTs

For CNTs fully clamped at the ends, the atoms/molecules are physically fixed

by micro effects. Collectively, the atoms/molecules are restricted from any

transverse movement and end rotation while axial movement is permitted. In this

example where CNTs are compressed through micro effects in the axial direction,

buckling will be induced similar to the simply supported CNTs. The buckling

behaviors of fully clamped CNTs are investigated as follows.

From Eq. (27), the nonlocal boundary conditions at clamped ends are

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1 2 12 2

0,1 0,12 0,1

0 ; 0 ; 2 0nnxx xxx x

n x

w M M (48a-c)

Eqs. (48a,b) above correspond to the classical solutions while Eq. (48c) is the new

higher-order nonlocal boundary condition for ANT. The boundary conditions in Eqs.

(48a-c) can be further simplified as

2

0,1 0,1 0,10 ; 0 ; 0

x x xw (49a-c)

Substituting Eqs. (40) and (41) into Eqs. (49a-c) yields

1

2 2 22

2 2 2 23

4

5

6

1 1 0 1 1 0

sin cos 1 0

0 0 0

sin cos 0 00

1 1 10 0

1 1 1 1cos sin

xx

xx xx

Ce eC

Ce eCa LNCa

Ca LN a LNe e

a a

(50)

Similar to Eq. (44), for nontrivial solutions Eq. (50) also leads to the following

characteristic equation

2 2 2

2 2 2 2

1 1 0 1 1 0

sin cos 1 0

0 0 0

sin cos 0 00

1 1 10 0

1 1 1 1cos sin

xx

xx xx

e e

e e

a LN

a

a LN a LNe e

a a

(51)

The solution to Eq. (51) shows the relation between and the buckling load xxN

for clamped-clamped CNTs.

The classical buckling load for a clamped-clamped Timoshenko beam is40

2

2 2

4

4CT

EIAGN

AG L EI (52)

Based on Eqs. (51) and (52), the relation of xx CTN N and /D L for a

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clamped-clamped CNT is presented in Fig. 5.

In Fig. 5, xx CTN N increases with increasing form 0 (classical solution) to

0.15. The stiffness enhancement contributed by nanoscale effect is again confirmed

here for a clamped-clamped CNT based on ANT. Furthermore, CNT with higher

D L also leads to higher xx CTN N , which implies longer CNT has lower buckling

resistance.

The buckling load ratio versus with 10nmL for a clamped-clamped

CNT based on different models is presented in Fig. 6. In this case, the buckling load

of a classical Euler-Bernoulli beam is40

2

2

4CE

EIN

L(53)

As is shown in Fig. 6, CNT stiffness is again enhanced by the ANT and ANE

models while PNE and PNT predict otherwise. Comparing with Fig. 4, the buckling

load ratios indicate that xx CTN N or xx CEN N ranges from 0.25 to 1.75 in Fig. 6

while the range in Fig. 4 is from 0.75 to 1.3. Hence, the clamped-clamped CNTs are

more sensitive to changes in with respect to the simply supported CNTs.

3.3 Cantilever CNTs

For cantilever CNTs clamped at 0x and free at 1x , atoms/molecules at

the clamped end are physically fixed in the transverse direction but axial movement is

allowed. The buckling behaviors for cantilever CNTs are investigated as follows.

From Eq. (27), the nonlocal boundary conditions for a cantilever CNT are

ef 10

1

10

1 1

0 1

2 2

0 1

0 ; 0

0 ; 0

0 ; 0

0 ; 0

xx

z xx xx

x x

x x

M w

bQ LN w (54)

The conditions in Eqs. (54) are further reduced to

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1

10

2 2

0 1

1

10

0 ; 0

0 ; 0

0 ; 0

xx

x x

xx xxa LN w a w

(55)

Substituting Eqs. (40) and (41) into Eqs. (55) yields

1

22 2 2

32 2 2 2

4

5

6

0 0 0

sin cos 1 0

0 0 00

sin cos 0 0

0 0 0 0 0

0 0 0 0 0xx

xx

C

e e C

C

e e C

LN C

LN C

(56)

The characteristic equation for Eq. (56) is

2 2 2

2 2 2 2

0 0 0

sin cos 1 0

0 0 00

sin cos 0 0

0 0 0 0 0

0 0 0 0 0xx

xx

e e

e e

LN

LN

(57)

The classical buckling load of cantilever a Timoshenko beam is40

2

2 24CT

EIAGN

AG L EI(58)

Using Eqs. (57) and (58), the relation of xx CTN N and D L is illustrated in

Fig. 7. Similar to Fig. 3 and Fig. 5, xx CTN N increases with increasing D L and

. However, xx CTN N in Fig. 7 is lower than 1.15, which is much smaller than the

corresponding value in Fig. 3 and Fig. 5. Hence the buckling load for a cantilever

CNT is less sensitive to changes in D L and .

The relation of buckling load ratios xx CTN N or xx CEN N with respect to

increasing based on different models in presented in Fig. 8 where CEN is defined

as40

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2

24CE

EIN

L (59)

In this figure, again stiffness and buckling loads are enhanced by increasing based

on ANT and ANE. The contradictory results of PNE and PNT are also indicated in

Fig. 8. The buckling load ratios in Fig. 8 by different models range from 0.9 to 1.1,

which are smaller than the corresponding values in Fig. 4 and Fig. 6. Thus a

cantilever CNT is less sensitive to changes in .

3.4 Clamped-Simply supported CNTs

For CNTs clamped at 0x and simply supported at 1x , the constraints of

atoms/molecules at the clamped end follow that of Sec. 3.2 while the atoms/molecules

at the simply supported end follow that of Sec. 3.1. The buckling behaviors of such

CNTs are investigated as follows.

From Eqs. (43a-c) and (49a-c), the nonlocal boundary conditions for a

clamped-simply supported CNT are

0 1

1

10

3 2

0 1

0 ; 0

0 ; 0

0 ; 0

x x

xx

x x

w w

(60)

Substituting Eqs. (40) and (41) into Eqs. (60) yields

1

3 3 32

2 2 2 23

4

5

6

0 1 0

sin cos 1 0

0 0 0

sin cos 0 00

1 1 10 0

1 1 1 1cos sin

Ce eC

Ce eCa LNCa

Ca LN a LNe e

a a

(61)

For nontrivial solution, the characteristic equation for Eq. (61) is then obtained as

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3 3 3

2 2 2 2

0 1 0

sin cos 1 0

0 0 0

sin cos 0 00

1 1 10 0

1 1 1 1cos sin

e e

e e

a LN

a

a LN a LNe e

a a

(62)

The classical buckling load of a clamped-simply supported Timoshenko beam is40

2

2 20.49 1.1CT

EIAGN

AG L EI (63)

The relation of xx CTN N and D L obtained from Eqs. (62) and (63) are illustrated

in Fig. 9. Comparing to Figs. 3, 5 and 7, it is also observed in Fig. 9 the similar

trend for xx CTN N versus D L . The range of xx CTN N in Fig. 9 is from 1.0 to

2.0, which is small than the ranges for clamped-clamped CNTs in Fig. 5 and larger

than the values of simply supported and cantilever CNTs in Fig. 3 and Fig. 7,

respectively.

Figure 10 illustrates the buckling load ratios xx CTN N or xx CEN N versus

based on different models where CEN is defined as40

2

20.49CE

EIN

L (64)

Similar to Figs. 4, 6 and 8, stiffness is enhanced for clamped-simply supported CNTs

based on ANT and ANE as indicated in Fig. 10. The buckling load ratio by different

models in Fig. 10 are between 0.35 to 1.65, which are smaller than the corresponding

values in Fig. 6 and larger than the ranges in Fig. 4 and Fig. 8. Therefore, in terms

of buckling response, clamped-clamped CNTs are most sensitive to changes in

and length, follows by clamped-simply supported CNTs and simply supported CNTs

while cantilever CNTs are least sensitive to changes in . It is also shown from the

results that shear deformable CNTs have lower buckling load than the ones without

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shear deformation. Therefore, shear effect tend to reduce the stiffness of CNTs.

3.5 Comparison with MD simulation

To further verify and confirm the reliability as well as accuracy of ANT in Eqs

(25)-(28), MD simulation solution7 for a clamped-clamped CNT is compared with the

nonlocal solutions using ANT, ANE, 38 PNE7 and the classical Euler-Bernoulli beam

model (E).7 The comparative results are presented in Fig. 11 where the buckling

strain is defined as xxN EA . The material constants and parameters are

121 10 PaE , 120.42 10 PaG , 1.1 , 0.004 , 0.066nmh and

0.678nmD .7

As is shown in Fig. 11, the buckling strain obtained by ANT, ANE and MD are

at all higher than the classical Euler-Bernoulli beam solution. Therefore the stiffness

enhancement of CNT due to nanoscale effect predicted by ANT and ANE are justified

with respect to MD solutions. In the contrary, PNE not only fails to predict this

trend, but instead predict lower buckling capacity due to nanoscale size effect. In

addition, the buckling strain by ANT is not only lower than ANE solution but it is

closer to the MD solution. The results of PNT and the classical Timoshenko beam

solution are not shown in Fig. 11 because these solutions are lower than the PNE and

E solutions and hence they are further away with respect to the MD solutions. In

summary, the ANT model is the most accurate model, as compared to MD simulation,

for buckling analysis of CNTs based on the nonlocal elastic stress theory.

4. Conclusion

A new analytical Timoshenko beam models for buckling of carbon nanotubes

considering nanoscale effect and shear deformation are established based on the

nonlocal elastic stress theory. The higher-order nonlocal equilibrium equations and

higher-order nonlocal boundary conditions are obtained via the variational principle.

The analytical solutions confirm that the stiffness of nanotubes is enhanced by the

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presence and increase in nanoscale effect and it is reduced by shear deformation.

Nonlocal nanotubes with different boundary conditions show different sensitivities to

nanoscale effect and changing dimension. The accuracy of this analytical nonlocal

Timoshenko CNT model is verified and confirmed by comparing with MD simulation

solutions.

Acknowledgment

This project is supported by a grant from City University of Hong Kong (No.

9667036).

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Figures and captions

Fig.1. CNT with axial force in Cartesian coordinate system

Fig. 2. The stress/strain curve: (a) Linear; (b) nonlinear

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Fig. 3. Buckling load ratio versus diameter ratio for a simply supported CNT

Fig. 4. Buckling load ratio versus nonlocal parameter for a simply supported CNT.

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Fig. 5. Buckling load ratio versus diameter ratio for a clamped-clamped CNT

Fig. 6. Buckling load ratio versus nonlocal parameter for a clamped-clamped CNT.

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Fig. 7. Buckling load ratio versus diameter ratio for a cantilever CNT.

Fig. 8. Buckling load ratio versus nonlocal parameter for a cantilever CNT.

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Fig. 9. Buckling load ratio versus diameter ratio for a clamped-simply supported

CNT.

Fig. 10. Buckling load ratio versus nonlocal parameter for a clamped-simply

supported CNT.

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Fig. 11. Buckling strain obtained by different approaches.

a variation principle approach for buckling of carbon...

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