Vibration Behaviour of Single Walled Carbon Nanotube using Finite Element

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ISSN : 2319 – 3182, Volume-2, Issue-4, 2013

129

Vibration Behaviour of Single Walled Carbon Nanotube using Finite Element

Ashirbad Swain¹, Tarapada Roy² & Bijoy Kumar Nanda³

^1&2Dept. of Mechanical Engineering, National Institute of Technology Rourkela, India

3VSSUT, Burla, India E-mail : [email protected]¹, [email protected]², [email protected]³

Abstract – The flexural vibration of single walled carbon nanotube has analyzed by finite element method.

Timoshenko beam element formulation has been used for this purpose. Axial deformation has also been taken into account apart from shear deformation for formulation of the element. Results from multi-scale modeling for free vibration analysis have been found to be in good agreement with the literatures available. Effects of chirality and aspect ratio on vibration characteristics are presented. More over effect of initial axial strain or stress on natural frequency have been analysed and found to have significant effect on the natural frequency of the nanotube.

Keywords – Free vibration analysis, Timoshenko beam element, Multi-scale modeling, Single walled carbon nanotube

INTRODUCTION

After the reported landmark discovery of Carbon Nanotube (CNT) by Iijima [1], lots of research work had been done for characterization of CNTs. It is now realized that CNTs have numerous extraordinary properties ranging from mechanical, thermal, electrical to magnetic. The comprehensive review of some of such property can be found from Ru et al. [2-5] and Lu et al.

[6,7]. Many studies had been conducted using experimental and molecular dynamics simulations to study the behaviors and properties of CNTs. As the experimental methods are quite difficult in nature in the scale of nanometers and molecular dynamics simulations are computationally expensive, continuum modeling found to be advantageous [8-10].

In continuum modeling the nanotubes which are discrete in nature, are represented by hollow beam. Here equivalent physical parameters like thickness, Young‟s modulus etc. is required to study such models. The same

kind of representation of a model with adequate equivalent property is needed for all types of nanotubes i.e. armchair, zigzag and chiral CNTs. In this present work only first two types of CNTs are considered.

Section 2 deals with the generalized Finite element (FE) formulation. In section 3 results and discussions has been shown as per the formulation. The method to find equivalent mechanical property is also discussed in this section. To be precise, all the required mechanical property for study has been shown as a function of CNT diameter [10].

The objective being to show the variation of first flexural natural frequency with respect to length to diameter ratio (L/D Ratio) for each CNT chiral index and secondly to show the sensitivity of natural frequency to initial load i.e. to show variation of first flexural natural frequency with respect to initial applied lode (T) for each CNT chiral index. Although the L/D ratio is the inverse of aspect ratio (D/L), in current studies L/D ratio is taken in to consideration as the prevailing literatures depicts the same. However trends could have also been plotted inking the aspect ratio. But for the sake of brevity, it is avoided.

II. FINITEELEMENTFORMULATIONOFCNT Transverse deformation of beam is a contribution of two parts. One is related to bending deformation and another related to shear deformation.

( , ) _b( , ) _s( , )

v x t v x t v x t (1)

Here b and s are abbreviations of bending and shear contribution respectively.

Hence it can be written that

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130

2

2 ( )

( )

b

s

y

EI v M x x A v

G S x

K x

 



 



(2)

Where E, G, M(x), S(x), A, K_yare the young‟s modulus, shear modulus, bending moment, shear force, cross sectional area and shear correction factor respectively.

Hence the strain energy of the beam are represented as follows with suitable shape functions denoted as „Ns‟

which are mentioned in the appendix.

6 2

( ) 0 1

6 2

( ) 0 1

1 2

l

b n n n

l

s n n

y n

U EI N v dx

G A N v dx

K



   

 

 

 

 

 

 

(3)

Hence the element of stiffness matrix can be written as

( ) ( )

0 0

( ) ( )

0 0

l l

ij xi xj b yi b yj

l l

s yi s yj yi yj

y

K EA N N dx EI N N dx

G A N N dx T N N dx k

   

 

   

 

 

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Where T is the initial loading. And the last term in the equation above represent elements of the geometric stiffness matrix. Hence the stiffness matrix can be shown as having two parts. One is elastic and other is geometric which are represented by subscripts e and g respectively.

[K_e][K_s] [ K_g] (5) Final governing equation of motion can be written as.

       

^M ^^ ^ ^K ^ ^ ^f ⁽⁶⁾

Where, [M] and [K] are the global mass, stiffness matrices respectively, vectors. {δ} and {f} are the vectors of nodal displacements and forces respectively.

Free vibration analysis of the complete system reveals the modal parameters such as mode shape and natural frequencies. Although mode shapes have been found but not represented in this work for brevity.

III. RESULTSANDDISCUSSION

As per the work of previous researchers, the thickness of the nanotubes reported to be between 0.066

nm to 0.34 nm [11-13]. To overcome these ambiguities, Lee et al. proposed a technique to determine equivalent structural property of a nanotube which only require diameter of nanotube to evaluate them. The equivalent properties of armchair and zigzag nanotube are listed below. [10]

Structural Properties For arm chair nanotubes (n,n)

2

3 2

2 3 2

24

2 25

1122.80 1.04 (kg·nm / s ) 962.83 23.4 (kg·nm / s )

428.48 397.08 109.24 (kg·nm / s ) 2.4 10 (kg / nm)

(9.12 8.47 2.33) 10 (kg·nm)

EA d

GA d

EI d d

A d

I d d





 

  

 

   

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Structural Properties For zigzag nanotubes (n,0)

2

3 2

2 3 2

24

2 25

(kg·nm)

1053.76 12.3 (kg·nm / s ) 974.51 35.73 (kg·nm / s )

249.94 146.49 26.67 (kg·nm / s ) 2.39 10 (kg / nm)

(5.62 3.3 .559) 10

EA d

GA d

EI d d

A d

I d d





 

  

 

   

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Although mode shapes have been found but not represented in this work for brevity. Focus has been concentrated to represent the variation of first natural frequency with respect to length to diameter ratio for nanotubes. There are three boundary conditions taken in to account for arm chair and zigzag nanotubes. They are cantilever, simply supported and fixed-fixed. The first flexural natural frequency of armchair nanotubes have been found for (3,3), (4,4), (5,5), (6,6), (7,7), (10,10), (15,15), (20,20) nanotubes with length to diameter ratio varying from 1 to 20. Similarly, natural frequency of zigzag nanotubes (5,0), (6,0), (7,0), (8,0), (9,0), (10,0), (15,0), (20,0), (25,0) have been taken in to consideration.

A. Result and discussion for armchair CNT:

Fig.1 : Variation of first flexural frequency with respect to L/D ratio for Armchair CNT in fixed-free condition (cantilever)

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131

Fig.2 : Variation of first flexural frequency with respect to L/D ratio for Armchair CNT in simply supported condition

Fig. 3 :Variation of first flexural frequency with respect to L/D ratio for Armchair CNT in simply supported condition

It may be seen from the above fig. 1, 2 and 3 that the natural frequency declines with increase in aspect ratio. It can also be observed that with increase with increase in the value of „n‟ the natural frequency decreases. The variations of parameters are shown in the table no. 1 below.

Table 1. Variations of response parameters in case of armchair CNT

Variation in (n,n)

Variation in L/D

ratio (nm)

Variation in natural frequency

(GHz)

Boundary Condition

(3,3) to

(20,20) 6 to 60

13302.54 to

2.42387 Cantilever 10920.14 to

1.08569

Simply supported 26614.19 to

6.784235 Fixed-fixed The shows that of an armchair CNT, when the „n‟ of the chiral index change from 3 to 20, changing the L/D ratio from 6 to 60 will shift the first flexural natural frequency from 13302.54 to 2.42387 GHz in case of cantilever type boundary condition. Similarly, in simply supported condition it changes from 10920.14 to

1.08569 GHz and in fixed-fixed condition it changes from 26614.19 to 6.784235 GHz.

B. Result and discussion for armchair CNT (n, 0):

Fig.4 : Variation of first flexural frequency with respect to L/D ratio for Zigzag CNT in fixed-free condition (cantilever)

Fig.5 : Variation of first flexural frequency with respect to L/D ratio for Zigzag CNT in simply supported condition

Fig. 6 : Variation of first flexural frequency with respect to L/D ratio for Zigzag CNT in fixed-fixed condition From above fig. 4, 5 and 6, same kind of variation or trend has been observed in natural frequency in case of zigzag nanotube i.e. the natural frequency decreases

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132 with increase in L/D and index „n‟. The variations are represented in the table No. 2 below.

Table 2. Variations of response parameters in case of zigzag CNT

Variation in (n,0)

Variation in L/D

ratio (nm)

Variation in natural frequency

(GHz)

Boundary Condition

(5,0) to (25,0)

6 to 60

13626.12 to

3.102569 Cantilever 13194.02 to

1.38969

Simply supported 29844.72 to

8.683864 Fixed-fixed The shows that of an zigzag CNT, when the „n‟ of the chiral index change from 5 to 25, changing the L/D ratio from 6 to 60, for each „n‟, will shift the first flexural natural frequency from 13626.12 to 3.102569 GHz in case of cantilever type boundary condition. Similarly, in simply supported condition it changes from 13194.02 to 1.38969 GHz and in fixed-fixed condition it changes from 29844.72 to 8.683864 GHz.

Table 3. Variations of response parameters in case of CNT for a particular variation in „n

Variation if „n‟ of chiral index

Vitiation of natural frequency

(GHz) for Boundary

condition Armchair Zigzag

5 to 20

7981.821 to 10.096

13626.12 to 18.169

Cantilever 1993.698 to

2.424

3364.979 to 4.1573 7346.347 to

4.522

13194.02 to

8.1384 simply supported 1816.603 to

1.086

3201.03 to 1.8621 16617.486 to

28.258

29844.72 to

50.855 fixed- fixed 4109.633 to

6.784

7242.297 to 11.636

Comparing the natural frequency for same index „n‟ for both the type of CNT, It can be found that zigzag CNTs shows higher natural frequency than armchair CNT which is shown in the above table No. 3.

C. Result and discussion for sensitivity to initial load:

It can be seen from fig.1 to 6 the natural frequency change rapidly with respect to L/D ratio from 6 to 15 for all considered nanotubes. So a particular L/D ratio of 9 which is in between this rang for the study of sensitively to initial load has been taken. The initial load has been varied between 0.1 Kg to 1.0 Kg, to find the variation of

natural frequency with the initial load for all type of nanotube taken in to account.

Fig.7 : Variation of first flexural frequency with respect to initial loading for Armchair CNT in cantilever condition

Fig. 8 : Variation of first flexural frequency with respect to initial loading for Zigzag CNT in cantilever condition

From Fig. 7 and 8 it may be seen that Zigzag nanotubes are more sensitive to preloading. CNT with chiral index (5,0), (6,0), (7,0), (8,0), (9,0) are the most sensitive to preloading as the rate of change of change of Natural frequency with respect to preload is higher than the other zigzag nanotube. In case of armchair CNTs the most sensitive one are (3,3), (4,4), (5,5) and (6,6) than other armchair CNTs.

IV. CONCLUSIONS

The following conclusions have been drawn from the above results and discussions.

1. The natural frequency decreases with increase in L/D and „n‟ in the chiral index. Hence as per this study the chiral index have effect on vibration characteristic.

2. With increase in the diameter of CNT, the natural frequency increases.

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133 3. Zigzag nanotubes are more sensitive to preloading

than Armchair CNT.

V. APPENDIX

The shape function that satisfy the above two relations are as follows.

1

4

2 3

2

2 3

3

2 3

4

6

2

2 1

1 1 3 2 1

1 2 1

1 2

1 3 2

1

x

y

N x l N x

l

x x x

N

l l l

l x x x

N

l l l

x x x

N

l l l

N l

x x

l l

x x

l l

 



 



    



   



  



 

 

       

          

 

 

     

        

             

   

   

 

     

     

   

 

 

  

   

2 3

( ) 2

2 3 2

( )3

( )5

( )6

3 2

2 3

1 1 3 2

1 2 1

1 2

1 1

1

1 2

3 2

b

x x

N

l l

l x x x x x

N

l l l l l

N N l

x x

l l

x x

l l



 



  



    



 

      

         

      

 

 

     

       

 

 

         

 

     

         

   

     

  

       

 

 



( ) 2

( )3

( )5

2 3 2

1 1

1 2

1

1 2

s

N x

l N x

N x

l

x x x

l l l



 



 



  

 

       

   

       

     

 

 

 

 

 

  

  

  

2

( )6 12

1 2

y

s EI

k GAl

N x





  

  

 

VI. REFERENCES

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1533–1542, 2003

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[6] J.P. Lu, “Elastic properties of carbon nanotubes and nanoropes,” Physical Review Letters vol. 79, pp. 1297–300, 1997

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7, pp. 1123–1126, 1995

[8] L. Wang and Q .Ni, “On vibration and instability of carbon nanotubes conveying fluid”, Computational Materials Science, vol. 43,pp.

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[9] S. Govindjee and J.L. Sackman, “On the use of continuum mechanics to estimate the properties of nanotubes”, Solid State Communications, vol.

110,pp. 227–230, 1999

[10] Usik Lee and Hyukjin Oh, “Evaluation of the Structural Properties of Single-Walled Carbon Nanotubes Using a Dynamic Continuum Modeling Method”, Mechanics of Advanced Materials and Structures, vol. 15, no. 2, pp. 79–

87, Feb. 2008

[11] E. Hernandez, C. Goze, P. Bernier, and A. Rubio,

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4505, 1998

[12] B. I. Yakobson, C. J. Brabec, and J. Bernholc,

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[13] C. F. Cornwell, and L. T. Wille, “Elastic properties of single-walled carbon nanotubes in compression,” Solid State Communications, vol.

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