Compressive behavior and buckling response of carbon nanotubes (CNTs)

Aswath Narayanan R

Dianyun Zhang

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• Introduction– Buckling problem of carbon nanotube– Literature review

• Approach– Mathematical model– Simulation

• GULP• Abaqus

• Future work• Conclusion

Outline

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What’s carbon nanotubes (CNTs)

Building blocks – beyond molecules

ME 599 (Nanomaufecturing) lecture notes, Fall 2009, Intstructor: A.J. Hart, University of Michigan

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Exceptional properties of CNTs

National Academy of Sciences report (2005), http://www.nap.edu/catalog/11268.html and many other sources

High Young’s modulus~1 TPa

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CNTs kink like straws

Yakobson et al., Physical Review B 76 (14), 1996.

High recoverable strains and reversible kinking

Kink shape develops!Seiji et al., Japan Society of Applied Physics, 45 (6B): 5586-9, 2006.

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• Types of buckling of CNTs– Euler‐type buckling

• general case

– hollow cylinder– shell buckling

• short or large‐diameter CNTs

• We are interested in Euler-type buckling

Buckling problem of CNTs

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From a recent research paper…

Seiji et al., Japan Journal of Applied Physics, 44(34): L1097-9, 2005.

E ~ 0.8 TPa

(a) 20 Shellsdouter = 14.7 nmdinner = 1.3 nmL = 1.19 µmFcr = 24.5 nN (b) 6 Shellsdouter = 14.7 nmdinner = 10.3 nmL = 1.07 µmFcr = 24.0 nN

Euler-type buckling!

Boundary Condition:Clamp – free

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Something interesting…

Motoyuki et al., Mater. Res. Symp. Proc. 1081:13-05, 2008 Poncharal et al., 283:1513, 1999.

Ripple – like distortions

Outer wall

Inner wall

Multi-wall carbon nanotubes (MWCNTs)

Two-DOF model

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0

0

0

0

P

Kt1

Kt1

Kr1Kt2

L/2

L/2

(L- R) cos(θ)

u

θ

P

Initial Configuration Deformed Configuration

2 2

L R

Inner wall:kt1, k1

Outer wall:kt2

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• Total potential energy

• Non-dimensional form

where

• Equilibrium condition

Two-DOF model cont.

2 2 21 2

1 1 1(2 ) ( ) 2 [ ( ) ( )]

2 2 2 2r t t

Rk k k L L R Cos

2 2 21 2

12 4 [1 (1 ) ( )] (1 ) ( )

2k r k r Cos p r Cos p

4 rk

R

rL

4 r

Pp

k

21

1 32t

r

k Lk

k

22

2 32t

r

k Lk

k

0r

0

Inner wall Outer wall

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Force – displacement curve

0.2 0.4 0.6 0.8 1.0Displacement

1

2

3

4Force

k1 = 1, k2 = 0(no outer wall)

Trifurcation

θ = 0

0 .2 0 .4 0 .6 0 .8 1 .0D is p la c eme nt

2

4

6

8

1 0

1 2F o r c e

k1 = 1, k2 = 1

Outer wall increases the slope of post-buckling curve

Snapback behavior

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0 .2 0 .4 0 .6 0 .8 1 .0D is p la c eme nt

2

4

6

8

1 0

1 2F o r c e

k2 = 1k2 = 0.8

k2 = 1.2

k2 = 1.5

k2 = 0.5

Force – displacement curve cont.

k1 = 1, vary k2

• Initial slope = 4 (k1 +2 k2)

• Snapback behaviors are observed when k1 = 1

• Trifurcation point is based on both k1 and k2

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Compared with the experimental data

0.2 0.4 0.6 0.8 1.0Displacement

2

4

6

8

10

12

Force

Experimental data

k1 = 0.99, k2 = 1.1

Trifurcation

Snapback

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GULP simulation of 6,6 CNT (Armchair)

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• Minimization of the potential of the multi atom system

• Takes into account various multi body potentials

• NON LOCAL interactions (twisting, three body moments)

General Utility Lattice Program (GULP)

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What are non local interactions?

Ref. C. Li et al, Int J Sol & Str

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• Force –displacement curve for 6,6 CNT

Force – displacement curve

INTERNAL ENERGY - DISPLACEMENT

-37 -36 -35 -34 -33 -32 -31 -30 -29-2.54

-2.52

-2.5

-2.48

-2.46

-2.44

-2.42

-2.4

-2.38

-2.36x 10

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FORCE - DISPLACEMENT

1 2 3 4 5 6 7-6000

-4000

-2000

0

2000

4000

6000

8000

XX

FE

F=dE/dX

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• Potential – it decides the way atoms interact with each other

• Tersoff Potential is used for this simulation

• It is a multi body potential, consisting of terms which depend on the angles between the atoms as well as on the distances between the corresponding atoms (bond order potential)

• Selected due to its applicability to covalent molecules and faster speed of computation compared to other potentials

Parameters used in simulation

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• Frame-like structure

• Primary bonds between two nearest-neighboring atoms act like load-bearing beam members

• Individual atom acts as the joint of the related load-bearing beam members

FEA using Abaqus

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Buckling mode

1 2

3 4

5

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• Mathematical model– Imperfection sensitivity– Non-linear springs

• Post-buckling analysis using Abaqus– Figure out parameters in the model– Implement rotational springs in the joints

Future work

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• 2-DOF model represents the Euler-type buckling of CNTs– Trifurcation– Snapback

• GULP simulation– Minimization of potential energy– Force – displacement curve

• Buckling analysis using Abaqus– Frame-like structure

Conclusion

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NASA Video on MWCNTs

Thank You!

Questions?

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