Compressive behavior and buckling response of carbon nanotubes (CNTs) Aswath Narayanan R Dianyun...
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Compressive behavior and buckling response of carbon nanotubes (CNTs)
Aswath Narayanan R
Dianyun Zhang
2
• Introduction– Buckling problem of carbon nanotube– Literature review
• Approach– Mathematical model– Simulation
• GULP• Abaqus
• Future work• Conclusion
Outline
3
What’s carbon nanotubes (CNTs)
Building blocks – beyond molecules
ME 599 (Nanomaufecturing) lecture notes, Fall 2009, Intstructor: A.J. Hart, University of Michigan
4
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
5
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.
6
• 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
7
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
8
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
9
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
12
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
13
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
15
• 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)
17
• 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
5
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
21
• 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
22
• 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