Multiscale Evaluation of the Nonlinear Elastic Properties of Carbon Nanotubes Under Finite Deformation

Document Type: Research Paper

Authors

Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran.

Abstract

This paper deals with the calculation of the elastic properties for single-walled carbon nanotubes (SWCNTs) under axial deformation and hydrostatic pressure using the atomistic-based continuum approach and the deformation mapping technique. A hyperelastic model based on the higher-order Cauchy-Born (HCB) rule being applicable at finite strains and accounting for the chirality and material nonlinearity is presented. Mechanical properties of several carbon nanotubes (CNTs) are computed and compared with the existing theoretical results and a good agreement is observed. Moreover, by comparison with atomistic calculations, it is found that the present model can reproduce the energetics of axially deformed CNTs. The model is then adopted to study the dependence of the elastic properties on chirality, radius and strain which yields an upper bound on the stability limit of axially and circumferentially stretched nanotubes. The influence of chirality is found to be more prominent for smaller tubes and as the diameter increases, the anisotropy induced by finite deformations gets nullified. It is discerned that the constitutive properties of the CNT can vary with deformation in a nonlinear manner. It is also found that the CNT displays a martial softening behavior at finite tensile strains and a hardening behavior at slightly compressive strains.

Keywords


1. Iijima S. Helical microtubules of graphitic carbon. Nature. 1991;354(6348):56.

2. Bethune DS, Klang CH, De Vries MS, Gorman G, Savoy R, Vazquez J, Beyers R. Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls. Nature. 1993;363(6430):605-7.

3. Zakeri M, Shayanmehr M. On the mechanical properties of chiral carbon nanotubes. Journal of Ultrafine Grained and Nanostructured Materials. 2013;46(1):1-9.

4. Chandraseker K, Mukherjee S. Atomistic-continuum and ab initio estimation of the elastic moduli of single-walled carbon nanotubes. Computational Materials Science. 2007;40(1):147-58.

5. Treacy MJ, Ebbesen TW, Gibson JM. Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature. 1996;381(6584):678.

6. Krishnan A, Dujardin E, Ebbesen TW, Yianilos PN, Treacy MM. Young’s modulus of single-walled nanotubes. Physical Review B. 1998;58(20):14013.

7. Lourie O, Wagner HD. Evaluation of Young's modulus of carbon nanotubes by micro-Raman spectroscopy. Journal of Materials Research. 1998;13(09):2418-22.

8. Muster J, Burghard M, Roth S, Duesberg GS, Hernandez E, Rubio A. Scanning force microscopy characterization of individual carbon nanotubes on electrode arrays. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena. 1998;16(5):2796-801.

9. Salvetat JP, Briggs GA, Bonard JM, Bacsa RR, Kulik AJ, Stöckli T, Burnham NA, Forró L. Elastic and shear moduli of single-walled carbon nanotube ropes. Physical Review Letters. 1999;82(5):944.

10. Yu MF, Files BS, Arepalli S, Ruoff RS. Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Physical Review Letters. 2000;84(24):5552.

11. Hernandez E, Goze C, Bernier P, Rubio A. Elastic properties of single-wall nanotubes. Applied Physics A: Materials Science & Processing. 1999;68(3):287-92.

12. Goze C, Vaccarini L, Henrard L, Bernier P, Hemandez E, Rubio A. Elastic and mechanical properties of carbon nanotubes. Synthetic Metals. 1999;103(1-3):2500-1.

13. Vaccarini L, Goze C, Henrard L, Hernandez E, Bernier P, Rubio A. Mechanical and electronic properties of carbon and boron–nitride nanotubes. Carbon. 2000;38(11):1681-90.

14. Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejón P. Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Physical Review B. 1999;59(19):12678-88.

15. Kudin KN, Scuseria GE, Yakobson BI. C 2 F, BN, and C nanoshell elasticity from ab initio computations. Physical Review B. 2001;64(23):235406.

16. Van Lier G, Van Alsenoy C, Van Doren V, Geerlings P. Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene. Chemical Physics Letters. 2000;326(1):181-5.

17. Robertson DH, Brenner DW, Mintmire JW. Energetics of nanoscale graphitic tubules. Physical Review B. 1992;45(21):12592.

18. Overney G, Zhong W, Tomanek D. Structural rigidity and low frequency vibrational modes of long carbon tubules. Zeitschrift für Physik D Atoms, Molecules and Clusters. 1993;27(1):93-6.

19. Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: instabilities beyond linear response. Physical Review Letters. 1996;76(14):2511.

20. Cornwell CF, Wille LT. Elastic properties of single-walled carbon nanotubes in compression. Solid State Communications. 1997;101(8):555-8.

21. Halicioglu T. Stress calculations for carbon nanotubes. Thin Solid Films. 1998;312(1):11-4.

22. Lu JP. Elastic properties of carbon nanotubes and nanoropes. Physical Review Letters. 1997;79(7):1297.

23. Prylutskyy YI, Durov SS, Ogloblya OV, Buzaneva EV, Scharff P. Molecular dynamics simulation of mechanical, vibrational and electronic properties of carbon nanotubes. Computational Materials Science. 2000;17(2):352-5.

24. Jin Y, Yuan FG. Simulation of elastic properties of single-walled carbon nanotubes. Composites Science and Technology. 2003;63(11):1507-15.

25. Kwon YK, Berber S, Tománek D. Thermal contraction of carbon fullerenes and nanotubes. Physical Review Letters. 2004;92(1):015901.

26. Agrawal PM, Sudalayandi BS, Raff LM, Komanduri R. A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations. Computational Materials Science. 2006;38(2):271-81.

27. DiBiasio CM, Cullinan MA, Culpepper ML. Difference between bending and stretching moduli of single-walled carbon nanotubes that are modeled as an elastic tube. Applied Physics Letters. 2007;90(20):203116.

28. Hu N, Jia B, Arai M, Yan C, Li J, Liu Y, Atobe S, Fukunaga H. Prediction of thermal expansion properties of carbon nanotubes using molecular dynamics simulations. Computational Materials Science. 2012;54:249-54.

29. Bian L, Zhao H. Elastic properties of a single-walled carbon nanotube under a thermal environment. Composite Structures. 2015;121:337-43.

30. Li T, Tang Z, Huang Z, Yu J. A comparison between the mechanical and thermal properties of single-walled carbon nanotubes and boron nitride nanotubes. Physica E: Low-dimensional Systems and Nanostructures. 2017;85:137-42.

31. Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics. 1983;54(9):4703-10.

32. Eringen AC. Nonlocal continuum field theories. Springer Science & Business Media; 2002.

33. Mindlin RD, Tiersten HF. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and analysis. 1962;11(1):415-48.

34. Toupin RA. Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis. 1964;17(2):85-112.

35. Aifantis EC. Gradient deformation models at nano, micro, and macro scales. Journal of Engineering Materials and Technology. 1999;121(2):189-202.

36. Lam DC, Yang F, Chong AC, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids. 2003;51(8):1477-508.

37. Wang CM, Zhang YY, He XQ. Vibration of nonlocal Timoshenko beams. Nanotechnology. 2007;18(10):105401.

38. Reddy JN, Pang SD. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics. 2008;103(2):023511.

39. Hu YG, Liew KM, Wang Q, He XQ, Yakobson BI. Nonlocal shell model for elastic wave propagation in single-and double-walled carbon nanotubes. Journal of the Mechanics and Physics of Solids. 2008;56(12):3475-85.

40. Murmu T, Pradhan SC. Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Computational Materials Science. 2009;46(4):854-9.

41. Civalek Ö, Akgöz B. Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. International Journal of Engineering and Applied Sciences. 2009;1(2):47-56.

42. Yang J, Ke LL, Kitipornchai S. Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Physica E: Low-dimensional Systems and Nanostructures. 2010;42(5):1727-35.

43. Lim CW, Yang Y. New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. Journal of Computational and Theoretical Nanoscience. 2010;7(6):988-95.

44. Akgöz B, Civalek Ö. Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories. Journal of Computational and Theoretical Nanoscience. 2011;8(9):1821-7.

45. Yang Y, Zhang L, Lim CW. Wave propagation in fluid-filled single-walled carbon nanotube on analytically nonlocal Euler–Bernoulli beam model. Journal of Sound and Vibration. 2012;331(7):1567-79.

46. Demir Ç, Civalek Ö. Nonlocal finite element formulation for vibration. International Journal of Engineering and Applied Sciences. 2016;8:109-17.

47. Civalek Ö, Demir C. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation. 2016;289:335-52.

48. Akgöz B, Civalek Ö. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica. 2016;119:1-2.

49. Shahabodini A, Ansari R, Darvizeh M. Multiscale modeling of embedded graphene sheets based on the higher-order Cauchy-Born rule: Nonlinear static analysis. Composite Structures. 2017;165:25-43.

50. Arroyo M, Belytschko T. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Physical Review B. 2004;69(11):115415.

51. Zhang P, Huang Y, Geubelle PH, Klein PA, Hwang KC. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids and Structures. 2002;39(13):3893-906.

52. Arroyo M, Belytschko T. Large deformation atomistic-based continuum analysis of carbon nanotubes. In 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2002.

53. Arroyo M. Finite crystal elasticity for curved single layer lattices: applications to carbon nanotubes. PhD dissertation, Northwestern University, USA, 2003.

54. Jiang H, Zhang P, Liu B, Huang Y, Geubelle PH, Gao H, Hwang KC. The effect of nanotube radius on the constitutive model for carbon nanotubes. Computational Materials Science. 2003;28(3):429-42.

55. Jiang H, Huang Y, Hwang KC. A finite-temperature continuum theory based on interatomic potentials. Journal of Engineering Materials and Technology. 2005;127(4):408-16.

56. Guo X, Wang JB, Zhang HW. Mechanical properties of single-walled carbon nanotubes based on higher order Cauchy–Born rule. International Journal of Solids and Structures. 2006;43(5):1276-90.

57. Wang JB, Guo X, Zhang HW, Wang L, Liao J. Energy and mechanical properties of single-walled carbon nanotubes predicted using the higher order Cauchy-Born rule. Physical Review B. 2006;73(11):115428.

58. Guo X, Zhang T. A study on the bending stiffness of single-walled carbon nanotubes and related issues. Journal of the Mechanics and Physics of Solids. 2010;58(3):428-43.

59. Guo X, Liao J, Wang X. Investigation of the thermo-mechanical properties of single-walled carbon nanotubes based on the temperature-related higher order Cauchy–Born rule. Computational Materials Science. 2012;51(1):445-54.

60. Ansari R, Shahabodini A, Alipour A, Rouhi H. Stability of a single-layer graphene sheet with various edge conditions: a non-local plate model including interatomic potentials. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems. 2012;226(2):51-60.

61. Ansari R, Shahabodini A, Rouhi H. Prediction of the biaxial buckling and vibration behavior of graphene via a nonlocal atomistic-based plate theory. Composite Structures. 2013;95:88-94.

62. Ansari R, Shahabodini A, Rouhi H. A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Current Applied Physics. 2015;15(9):1062-9.

63. Ansari R, Shahabodini A, Rouhi H. A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression. Composite Structures. 2013;100:323-31.

64. Ansari R, Shahabodini A, Rouhi H, Alipour A. Thermal buckling analysis of multi-walled carbon nanotubes through a nonlocal shell theory incorporating interatomic potentials. Journal of Thermal Stresses. 2013;36(1):56-70.

65. Giannopoulos GI, Kakavas PA, Anifantis NK. Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach. Computational Materials Science. 2008;41(4):561-9.

66. Ghavamian A, Rahmandoust M, Öchsner A. On the determination of the shear modulus of carbon nanotubes. Composites Part B: Engineering. 2013;44(1):52-9.

67. Mohammadpour E, Awang M. Predicting the nonlinear tensile behavior of carbon nanotubes using finite element simulation. Applied Physics A: Materials Science & Processing. 2011;104(2):609-14.

68. Sakharova NA, Pereira AF, Antunes JM, Fernandes JV. Numerical simulation study of the elastic properties of single-walled carbon nanotubes containing vacancy defects. Composites Part B: Engineering. 2016;89:155-68.

69. Singh S, Patel BP. Nonlinear elastic properties of graphene sheet under finite deformation. Composite Structures. 2015;119:412-21.

70. Zhou J, Huang R. Internal lattice relaxation of single-layer graphene under in-plane deformation. Journal of the Mechanics and Physics of Solids. 2008;56(4):1609-23.

71. Lu Q, Huang R. Nonlinear mechanics of single-atomic-layer graphene sheets. International Journal of Applied Mechanics. 2009;1(03):443-67.

72. Lu Q, Arroyo M, Huang R. Elastic bending modulus of monolayer graphene. Journal of Physics D: Applied Physics. 2009;42(10):102002.

73. Tersoff J. New empirical approach for the structure and energy of covalent systems. Physical Review B. 1988;37(12):6991.

74. Brenner DW. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Physical Review B. 1990;42(15):9458.

75. Wang X, Guo X. Numerical simulation for finite deformation of single-walled carbon nanotubes at finite temperature using temperature-related higher order Cauchy-Born rule based quasi-continuum model. Computational Materials Science. 2012;55:273-83.

76. Belytschko T, Liu WK, Moran B, Elkhodary K. Nonlinear finite elements for continua and structures. John wiley & sons; 2013.

77. Peng J, Wu J, Hwang KC, Song J, Huang Y. Can a single-wall carbon nanotube be modeled as a thin shell?. Journal of the Mechanics and Physics of Solids. 2008;56(6):2213-24.

78. Belytschko T, Xiao SP, Schatz GC, Ruoff RS. Atomistic simulations of nanotube fracture. Physical Review B. 2002;65(23):235430.

79. Zhang P, Jiang H, Huang Y, Geubelle PH, Hwang KC. An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. Journal of the Mechanics and Physics of Solids. 2004;52(5):977-98.

80. Hill R. On the elasticity and stability of perfect crystals at finite strain. In Mathematical Proceedings of the Cambridge Philosophical Society. 1975;77(1):225-40.

81. Song J, Wu J, Huang Y, Hwang KC. Continuum modeling of boron nitride nanotubes. Nanotechnology. 2008;19(44):445705.

82. Yakobson BI, Campbell MP, Brabec CJ, Bernholc J. High strain rate fracture and C-chain unraveling in carbon nanotubes. Computational Materials Science. 1997;8(4):341-8.