1. 36th Dayton-Cincinnati Aerospace Sciences Symposium
Computational Study of Carbon Nanotubes under Compressive Loading
Quasi-static reduced-order general continuum method with
barycentric Interpolation
Yang Yang, William W. Liou
Computational Engineering Physics Lab
Western Michigan University
Kalamazoo, Michigan
36th Dayton-Cincinnati Aerospace Sciences Symposium
03/01/2011

2. Outline Introduction Numerical method Simulation results Conclusions
Properties of carbon nanotubes
Applications of carbon nanotubes
Definition of carbon nanotubes
Numerical method
Overview
Reduced-order general continuum method
Simulation results
Model setup
Buckling patterns
Buckling patterns after barycentric conversion
Loading-unloading stress-strain curves for CNTs of
different types
Conclusions

3. Properties of carbon nanotubes
Outline
Introduction
Properties of carbon nanotubes
Applications of carbon nanotubes
Definition of carbon nanotubes
Numerical method
Overview
Reduced-order general continuum method
Simulation results
Model setup
Buckling patterns
Buckling patterns after barycentric conversion
Loading-unloading stress-strain curves for CNTs of
different types
Conclusions

4. Introduction Properties of carbon nanotubes Average diameter of SWNT
Carbon bond length
Density
Thermal conductivity
Young’s modulus of SWNT
Max. tensile strength

5. Introduction Properties of carbon nanotubes
Composed of all-carbon molecules in shell-like cylindrical
structure formed by strong covalent bonding of atoms
Tend to undergo buckling with compression or bending loads
One of the strongest materials known, both in terms of tensile
strength and elastic modulus

6. Introduction Applications of carbon nanotubes
Carbon nanotubes enhanced composite materials
Efficient heat remover composed of aligned structures and ribbons of CNTs
Drug delivery to prevent medicine from damaging healthy cells
Intrinsic tubule character of CNTs attributing to their very high surface area leads to the applications in energy storage material
Used as electrical conducting additives to producing conductive plastics
Flat panel CNT field emission display

7. Introduction
Definition of carbon nanotubes

8. Properties of carbon nanotubes
Outline
Introduction
Properties of carbon nanotubes
Applications of carbon nanotubes
Definition of carbon nanotubes
Numerical method
Overview
Reduced-order general continuum method
Simulation results
Model setup
Buckling patterns
Buckling patterns after barycentric conversion
Loading-unloading stress-strain curves for CNTs of
different types
Conclusions

9. Numerical method Overview Classical molecular dynamics (MD)
Excels in modeling structural details of an atomic system by tracking
each atom
Computationally prohibitive for large systems; generally modeling a
system with the size up to a few hundred nanometers
Reduced-order general continuum method
Constitutive law is built based on an atomistic energy function by
intrinsic geometric quantities describing a deformation
No need for tracking individual atoms thus appropriate for modeling a
large system

10. Numerical method Reduced-order general continuum method General idea
Every point in the continuum body
is described by a representative atom
embedded in a crystallite of radius
Finite elements discretizing the
continuum body

11. Numerical method Reduced-order general continuum method
Cauchy-Born rule
Exponential map

12. Numerical method Reduced-order general continuum method
REBO potential function for CNT
The repulsive pair:
The attractive pair:
The bond order term:

13. Numerical method Reduced-order general continuum method
Lennard-Jones potential for long-range interaction

14. Numerical method Reduced-order general continuum method
Atomic potential energies expressed in continuum variables
Interatomic energy density
Total interatomic energy over the CNT surface
Long-range Lennard-Jones energy for the CNT
Total energy of the CNT
Equilibrium state of the CNT corresponds Min ( )

15. Properties of carbon nanotubes
Outline
Introduction
Properties of carbon nanotubes
Applications of carbon nanotubes
Definition of carbon nanotubes
Numerical method
Overview
Reduced-order general continuum method
Simulation results
Model setup
Buckling patterns
Buckling patterns after barycentric conversion
Loading-unloading stress-strain curves for CNTs of
different types
Conclusions

16. Simulation results Model setup
Fixed
end
Displacement
control B.C.
Buckling of different types of CNT under compressive loading
CNT cases studied
Case
Configuration number
Length (a)
Chiral angle 1
(14, 0) zigzag
40.3 2
(12, 3)
40.5
10.9 3
(10, 5)
36.8
19.1 4
(8, 8) armchair
40.9 30
Displacement control method is used to apply the loading

17. Simulation Results Buckling patterns Total energy vs. strain Case 1
Van der Waals energy vs. strain Case 1
buckling

18. Simulation Results Buckling patterns Case 1 Case 2 Case 3 Case 4
before
after

19. Simulation Results Buckling patterns Case Configuration number
Chiral angle (deg)
Strain at buckling 1
(14, 0) zigzag
0.056 2
(12, 3)
10.9
0.059 3
(10, 5)
19.1
0.069 4
(8, 8) armchair 30
0.08

20. Simulation Results Buckling patterns after barycentric conversion
Incipient state for Case 1
Buckled state for Case 1
Buckling events for Case 1
Bond direction
Un -deformed
Incipient
% of change
Buckled
Axial (b1,b4)
(1.45, 1.45)
(1.35,1.35)
(-6.9, -6.9)
(1.39,1.48)
Transverse (b2, b3)
(1.50,1.50)
(3.4, 3.4)
(1.51,1.48)
Transverse (b5, b6)
(1.50,1.51)
(3.4, 4.1)
(1.47,1.46)

21. Simulation Results Buckling patterns after barycentric conversion
Representative cells on the buckling surface of CNTs with different chiral angles.
(14, 0) CNT Case 1
(12, 3) CNT Case 2
(10, 5) CNT Case 3
(8, 8) CNT Case 4
The number of bonds that receives compressive load increases from Case 1 to Case 4
The bonds are compressed more uniformly in Case 4 than in Case 2 or Case 3

22. Simulation Results
Loading-unloading stress-strain curves for CNTs of different types
Case 2
Case 1
Case 4
Case 3

23. Properties of carbon nanotubes
Outline
Introduction
Properties of carbon nanotubes
Applications of carbon nanotubes
Definition of carbon nanotubes
Numerical method
Overview
Reduced-order general continuum method
Simulation results
Model setup
Buckling patterns
Buckling patterns after barycentric conversion
Loading-unloading stress-strain curves for CNTs of
different types
Conclusions

24. Conclusions
The reduced order general continuum method was used to study the behaviors of CNTs under compressive loading conditions.
Reverse mapping of the finite element results to the associated CNT lattice deformation using barycentric interpolation.
Different buckled configurations will be assumed by CNTs with different chiral angles.
The zigzag CNT has the most apparent buckling pattern.
The buckling strain increases with the increasing chiral angle.
The armchair CNT has the strongest resistance to the compressive loading.

25. Thank you!