Computational Study of Carbon Nanotubes under Compressive Loading

1. 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 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. 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. 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 correspondsMin ( )

15. 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 • Displacement control method is used to apply the loading

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

20. Simulation Results Buckling patterns after barycentric conversion • Buckled state for Case 1 • Incipient state for Case 1 • Buckling events for Case 1

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 (8, 8) CNT Case 4 (12, 3) CNT Case 2 (10, 5) CNT Case 3 • 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. 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

23. 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.

24. Thank you!