Compressive behavior and buckling
response of carbon nanotubes (CNTs)
Aswath Narayanan R
Dianyun Zhang
Outline
• Introduction
– Buckling problem of carbon nanotube
– Literature review
• Approach
– Mathematical model
– Simulation
• GULP
• Abaqus
• Future work
• Conclusion
2
What’s carbon nanotubes (CNTs)
Building blocks – beyond molecules
ME 599 (Nanomaufecturing) lecture notes, Fall 2009,
Intstructor: A.J. Hart, University of Michigan
3
Exceptional properties of CNTs
High Young’s modulus
~1 TPa
National Academy of Sciences report (2005), http://www.nap.edu/catalog/11268.html and
many other sources
4
CNTs kink like straws
High recoverable strains and reversible kinking
Kink shape develops!
Yakobson et al., Physical Review B 76 (14), 1996.
Seiji et al., Japan Society of Applied Physics,
45 (6B): 5586-9, 2006.
5
Buckling problem of CNTs
• 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
6
From a recent research paper…
E ~ 0.8 TPa
(a) 20 Shells
douter = 14.7 nm
dinner = 1.3 nm
L = 1.19 µm
Fcr = 24.5 nN
(b) 6 Shells
douter = 14.7 nm
dinner = 10.3 nm
L = 1.07 µm
Fcr = 24.0 nN
Boundary Condition:
Clamp – free
Seiji et al., Japan Journal of Applied Physics,
44(34): L1097-9, 2005.
Euler-type buckling!
7
Something interesting…
Multi-wall carbon nanotubes (MWCNTs)
Outer wall
Inner wall
Ripple – like distortions
Motoyuki et al., Mater. Res. Symp. Proc.
1081:13-05, 2008
Poncharal et al., 283:1513, 1999.
8
Two-DOF model
P
P
u
Outer wall:
kt2
Kt1
θ
L/2
Kr1
Kt1
Kt2
0
Inner wall:
kt1, k1
0
(L- R) cos(θ)
L/2
L
2
Initial Configuration

R
2
Deformed Configuration
9
Two-DOF model cont.
• Total potential energy
 
1
2
k r (2 ) 
2
1
2
k t1 (
R
) 2 
2
2
1
2
k t 2 [ L  ( L  R )  C os ( )]
2
• Non-dimensional form
 
1
2
  2 k1  r  4 k 2  [1  (1  r )  C os ( )]  p (1  r )  C os ( )  p
2
where
 
2

2
r 
R
p 
L
4kr
P
4kr
• Equilibrium condition

r
0
k1 
k t1 L
2
32 k r
Inner wall


k2 
kt 2 L
2
32 k r
Outer wall
0
10
Force – displacement curve
k1 = 1, k2 = 0
(no outer wall)
k1 = 1, k2 = 1
Force
4
Force
12
10
3
Trifurcation
8
Snapback
behavior
2
6
4
1
θ=0
0.2
0.4
0.6
Outer wall increases the
slope of post-buckling
curve
2
0.8
1.0
Displacement
0.2
0.4
0.6
0.8
1.0
Displacement
11
Force – displacement curve cont.
Force
12
k1 = 1, vary k2
• Initial slope = 4 (k1 +2 k2)
10
• Snapback behaviors are
observed when k1 = 1
8
6
4
k2 = 1.5
• Trifurcation point is based
on both k1 and k2
k2 = 1.2
k2 = 0.5
k2 = 0.8
k2 = 1
2
0.2
0.4
0.6
0.8
1.0
Displacement
12
Compared with the experimental data
Force
k1 = 0.99, k2 = 1.1
12
10
Trifurcation
8
Snapback
6
4
Experimental data
2
0.2
0.4
0.6
0.8
1.0
Displacement
13
GULP simulation of 6,6 CNT (Armchair)
14
General Utility Lattice Program (GULP)
• Minimization of the potential of the multi
atom system
• Takes into account various multi body
potentials
• NON LOCAL interactions (twisting, three
body moments)
15
What are non local interactions?
Ref. C. Li et al, Int J
16
Sol & Str
Force – displacement curve
• Force –displacement curve for 6,6 CNT
INTERNAL ENERGY - DISPLACEMENT
FORCE - DISPLACEMENT
5
-2.36
x 10
8000
-2.38
6000
-2.4
4000
-2.42
2000
-2.44
E
-2.46
F
0
-2.48
-2000
-2.5
-4000
-2.52
-2.54
-37
-36
-35
-34
-33
X
-32
-31
-30
-29
-6000
1
2
F=dE/dX
3
4
5
6
7
X
17
Parameters used in simulation
• 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
18
FEA using Abaqus
• Frame-like structure
• Primary bonds between two nearestneighboring atoms act like load-bearing
beam members
• Individual atom acts as the joint of the
related load-bearing beam members
19
Buckling mode
1
2
3
4
5
20
Future work
• Mathematical model
– Imperfection sensitivity
– Non-linear springs
• Post-buckling analysis using Abaqus
– Figure out parameters in the model
– Implement rotational springs in the joints
21
Conclusion
• 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
22
NASA Video on MWCNTs
23
Thank You!
Questions?
24