Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v9.i1.80
pages 97-118

DERIVATION OF THE YOUNG'S AND SHEAR MODULI OFSINGLE-WALLED CARBON NANOTUBES THROUGH A COMPUTATIONAL HOMOGENIZATION APPROACH

Elie El Khoury
Institut de Recherche en Genie Civil et Mecanique (UMR CNRS 6183), Ecole Centrale deNantes, BP 92101, 44321, Nantes Cedex 3, France
Tanguy Messager
Laboratoire de Mecanique de Lille (UMR CNRS 8107), Universite Lille 1, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France
Patrice Cartraud
Institut de Recherche en Genie Civil et Mecanique (UMR CNRS 6183), Ecole Centrale deNantes, BP 92101, 44321, Nantes Cedex 3, France

ABSTRACT

In this study, the computation of the traction-torsion-bending behavior of single-walled carbon nanotubes (SWCNTs) is investigated. A structural mechanics model is used to describe the response of the nanotube; the atomic interactions are represented with 3D beams. Nanotubes are slender structures, taking benefit from their axial periodicity or helical symmetry. Homogenization theory is used to obtain their overall beam behavior from the solution of basic cell problems. These problems are solved through a finite element approach and involve concise models, whatever the SWCNT type. The computed results show that the bending behavior appears to be decoupled from the axial one and independent of the moment direction. Young's and shear moduli are derived, and it is shown that the Young's moduli are very close in traction and bending. Comparisons with the data in the literature reveal good agreements. Finally, scale effects are studied, and the moduli of the SWCNTs are compared to those of the graphene, thus demonstrating mechanical sensitivity to curvature.

REFERENCES

  1. Agrawal, P. M., Sudalayandi, B. S., Raff, L. M., and Komanduri, R., A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations.

  2. Barros, E., Jorio, A., Samsonidze, G., Capaz, R., Souza Filho, A., Filho, J. M., Dresselhauss, G., and Dresselhauss, M. S., Review on the symmetry-related properties of carbon nanotubes. DOI: 10.1016/j.physrep.2006.05.007

  3. Bendsoe, M. P., Optimal shape design as a material distribution problem. DOI: 10.1007/BF01650949

  4. Boutin, C. and Hans, S., Homogenization of periodic discrete medium: Application to dynamics of framed structures. DOI: 10.1016/S0266-352X(03)00005-3

  5. Boutin, C. and Hans, S., Dynamics of discrete framed structures: A unified homogenized description.

  6. Buannic, N. and Cartraud, P., Higher-order effective modelling of periodic heterogeneous beams – Part I: Asymptotic expansion method. DOI: 10.1016/S0020-7683(00)00422-4

  7. Caillerie, D., Mourad, A., and Raoult, A., Discrete homogenization in graphene sheet modeling. DOI: 10.1007/s10659-006-9053-5

  8. Cartraud, P. and Messager, T., Computational homogenization of periodic beam-like structures. DOI: 10.1016/j.ijsolstr.2005.03.063

  9. Chang, T. and Gao, H., Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. DOI: 10.1016/S0022-5096(03)00006-1

  10. Chang, T., Geng, J., and Guo, X., Chirality- and size-dependent elastic properties of single-walled carbon nanotubes. DOI: 10.1063/1.2149216

  11. Cornell, W. D., Cieplak, P., Bayly, C. I., Gould, I. R., Merz, K. M., and Ferguson, D. M., A second generation force-field for the simulation of proteins, nucleic acids and organic molecules. DOI: 10.1021/ja00124a002

  12. Dallot, J., Sab, K., and Foret, G., Limit analysis of periodic beams. DOI: 10.1016/j.euromechsol.2008.04.001

  13. Dresselhaus, M. S., Dresselhaus, G., and Avouris, P., Carbon nanotubes (synthesis, structure properties and applications).

  14. Dumitrica, T. and James, R. D., Objective molecular dynamics. DOI: 10.1016/j.jmps.2007.03.001

  15. Geers, M. G. D., Coenen, E. W. C., and Kouznetsova, V. G., Multi-scale computational homogenization of structured thin sheets. DOI: 10.1088/0965-0393/15/4/S06

  16. Giannopoulos, G. I., Kakavas, P. A., and Anifantis, N. K., Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach. DOI: 10.1016/j.commatsci.2007.05.016

  17. Gupta, S., Dharamvir, K., and Jindal, V. K., Elastic moduli of single-walled carbon nanotubes and their ropes. DOI: 10.1103/PhysRevB.72.165428

  18. Gupta, S. S. and Batra, R. C., Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes.

  19. Hernandez, E., Goze, C., Bernier, P., and Rubio, A., Elastic properties of C and B<sub>x</sub>C<sub>y</sub>N<sub>z</sub> composite nanotubes. DOI: 10.1103/PhysRevLett.80.4502

  20. Huang, Y.,Wu, J., and Hwang, K. C., Thickness of graphene and single-wall carbon nanotubes. DOI: 10.1103/PhysRevB.74.245413

  21. Kalamkarov, A. L., Georgiades, A. V., Rokkam, S. K., Veedu, V. P., and Ghasemi-Nejhad, M. N., Analytical and numerical techniques to predict carbon nanotubes properties. DOI: 10.1016/j.ijsolstr.2006.02.009

  22. Kasti, N., Zigzag carbon nanotubes&#8212;Molecular/structural mechanics and the finite element method. DOI: 10.1016/j.ijsolstr.2007.03.017

  23. Krishnan, A., Dujardin, E., Ebbessen, T. W., and Yianilos et Treacy, M., Young’s modulus of single-walled nanotubes. DOI: 10.1103/PhysRevB.58.14013

  24. L&#233;n&#233;, F., Contribution &#224; l’&#233;tude des Mat&#233;riaux Composites et de Leur Endommagement.

  25. Li, C. and Chou, T. W., A structural mechanics approach for the analysis of carbon nanotubes. DOI: 10.1016/S0020-7683(03)00056-8

  26. Liu, W. K., Karpov, E. G., and Park, H. S., Nano Mechanics and Materials; Theory Multiscale Methods and Applications.

  27. Lu, J. P., Elastic properties of carbon nanotubes and nanoropes. DOI: 10.1103/PhysRevLett.79.1297

  28. Meo, M. and Rossi, M., Prediction of Young’s modulus of single wall carbon nanotubes by molecular-mechanics based finite element modelling. DOI: 10.1016/j.compscitech.2005.11.015

  29. Messager, T. and Cartraud, P., Homogenization of helical beam-like structures: Application to single-walled carbon nanotubes. DOI: 10.1007/s00466-007-0189-3

  30. Mintmire, J. W. and White, C. T., Electronic and structural properties of carbon nanotubes.

  31. Moustaghfir, N., Daya, E. M., Braikat, B., Damil, N., and Potier-Ferry, M., Evaluation of continuous modelings for the modulated vibration modes of long repetitive structures. DOI: 10.1016/j.ijsolstr.2007.03.023

  32. Natsuki, T., Tantrakarn, K., and Endo, M., Effect of carbon nanotube structures on mechanical properties. DOI: 10.1007/s00339-003-2492-y

  33. Odegard, G. M., Gates, T. S., Nicholson, L. M., and Wise, K. E., Equivalent-continuum modeling of nano-structured materials. DOI: 10.1016/S0266-3538(02)00113-6

  34. Pantano, A., Parks, D. M., and Boyce, M. C., Mechanics of deformation of single- and multi-wall carbon nanotubes. DOI: 10.1016/j.jmps.2003.08.004

  35. Qian, D., Wagner, G. J., Liu, W. K., Yu, M. F., and Ruoff, R. S., Mechanics of carbon nanotubes. DOI: 10.1115/1.1490129

  36. Rafii-Tabar, H., Computational modelling of thermo-mechanical and transport properties of carbon nanotubes. DOI: 10.1016/j.physrep.2003.10.012

  37. Rappe, A. K., Casemit, C. J., Colwell, K. S., Goddard,W. A., and Skiff,W. M., UFF, A full periodic-table force field for molecular mechanics and molecular dynamics simulations. DOI: 10.1021/ja00051a040

  38. Reddy, C. D., Rajendran, S., and Liew, K. M., Equilibrium configuration and continuum elastic properties of finite sized graphene. DOI: 10.1088/0957-4484/17/3/042

  39. Salvetat, J. P., Briggs, A. D., Bonard, J. M., Basca, R. R., Kulik, A. J., Stockli, T., Burnham, N. A., and Forro, L., Elastic and shear moduli of single-walled carbon nanotube ropes. DOI: 10.1103/PhysRevLett.82.944

  40. Srivastava, D., Wei, C., and Cho, K., Computational nanomechanics of carbon nanotubes and composites. DOI: 10.1115/1.1538625

  41. To, W. S., Bending and shear moduli of single-walled carbon nanotubes. DOI: 10.1016/j.finel.2005.08.004

  42. Trabucho, L. and Viano, J. M., Mathematical Modelling of Rods.

  43. Tserpes, K. I. and Papanikos, P., Finite element modelling of single-walled carbon nanotubes. DOI: 10.1016/j.compositesb.2004.10.003

  44. Tu, Z. C. and Ou-Yang, Z. C., Elastic theory of low-dimensional continua and its applications in bio- and nano-structures. DOI: 10.1166/jctn.2008.002

  45. Van Lier, G., Van Alsenoy, C., Van Doren, V., and Geerlings, P., Ab initio study of the elastic properties of single-walled carbon nanotubes and grapheme. DOI: 10.1016/S0009-2614(00)00764-8

  46. Wang, Q. and Varadan, V. K., Wave characteristics of carbon nanotubes. DOI: 10.1016/j.ijsolstr.2005.02.047

  47. Wenxing, B., Changchun, Z., and Wanzhao, C., Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. DOI: 10.1016/j.physb.2004.07.005

  48. Wu, Y., Zhang, X., Leung, A. Y. T., and Zhong, W., An energy-equivalent model on studying the mechanical properties of singlewalled carbon nanotubes. DOI: 10.1016/j.tws.2006.05.003

  49. Xiao, J. R., Lopatnikov, S. L., Gama, B. A., and Gillespie, J. W., Nanomechanics of the deformation of single- and multi-walled carbon nanotubes under radial pressure. DOI: 10.1016/j.msea.2005.09.105

  50. Zhang, D.-B. and Dumitrica, T., Elasticity of ideal single-walled carbon nanotubes via symmetry-adapted tight-binding objective modeling. DOI: 10.1063/1.2965465


Articles with similar content:

Mathematical Modeling of Nonlinearly Elastic Deformation of Orthotropic Composite Shells
Journal of Automation and Information Sciences, Vol.31, 1999, issue 1-3
V. A. Maksimyuk
Deformation and Stability of Copper Nanowires under Bending
International Journal for Multiscale Computational Engineering, Vol.7, 2009, issue 3
Hongwu Zhang, Shan Jiang, Yonggang Zheng, Zhen Chen
FULL COUPLING RESPONSE OF SINGLE-WALLED CARBON NANOTUBES
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 1
Hongwu Zhang, Jin Bao Wang, Xu Guo
STATIC FLEXURE OF CROSS-PLY LAMINATED CANTILEVER BEAMS
Composites: Mechanics, Computations, Applications: An International Journal, Vol.5, 2014, issue 3
Sangita B. Shinde, Yuwaraj M. Ghugal
FLEXURAL ANALYSIS OF FIBROUS COMPOSITE BEAMS UNDER VARIOUS MECHANICAL LOADINGS USING REFINED SHEAR DEFORMATION THEORIES
Composites: Mechanics, Computations, Applications: An International Journal, Vol.5, 2014, issue 1
Atteshamuddin Shamshuddin Sayyad, Yuwaraj M. Ghugal, R. R. Borkar