Skip to main content
SpringerLink
Log in

Mechanics of Carbon Nanotubes and Their Composites

  • Living reference work entry
  • First Online:
Handbook of Mechanics of Materials

  • 310 Accesses

Abstract

Carbon nanotubes (CNTs) display superior mechanical properties and have been used as reinforcements in polymer matrix composites. The first part of this chapter reviewed the finite-deformation shell theory, established directly from the atomic structure of CNT and the interatomic potential, for single-wall carbon nanotubes. This atomistic-based finite-deformation shell theory has been used to study the rigidity and instability of single-wall CNTs subject to tension, compression, internal and external pressure, and torsion. The second part of this chapter reviewed the cohesive law for interfaces between carbon nanotubes and polymers due to the van der Waals force. A micromechanics model for carbon nanotubes-reinforced composite with the incorporation of the nonlinear cohesive law has been established to investigate the mechanical behavior of nanocomposites.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Yakobson BI, Avouris P. Mechanical properties of carbon nanotubes. In: Dresselhaus MS, Dresselhaus G, Avouris P, editors. Carbon nanotubes. Berlin: Springer. Top Appl Phys. 2001;80:287.

    Google Scholar 

  2. Lee CG, Wei XD, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science. 2008;321:385.

    Article  Google Scholar 

  3. Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: Instabilities beyond linear response. Phys Rev Lett. 1996;76:2511.

    Article  Google Scholar 

  4. Belytschko T, Xiao SP, Schatz GC, Ruoff RS. Atomistic simulations of nanotube fracture. Phys Rev B. 2002;65:235430.

    Article  Google Scholar 

  5. Ge G, Samsonidze G, Yakobson BI. Energetics of Stone-Wales defects in deformations of monoatomic hexagonal layers. Comput Mater Sci. 2002;23:62.

    Article  Google Scholar 

  6. Liu B, Huang Y, Jiang H, Qu S, Hwang KC. The atomic-scale finite element method. Comput Methods Appl Mech Eng. 2004;193:1849.

    Article  Google Scholar 

  7. Liu B, Jiang H, Huang Y, Qu S, Yu MF, Hwang KC. Atomic-scale finite element method in multiscale computation with applications to carbon nanotubes. Phys Rev B. 2005;72:035435.

    Article  Google Scholar 

  8. Govindjee S, Sackman JL. On the use of continuum mechanics to estimate the properties of nanotubes. Solid State Commun. 1999;110:227.

    Article  Google Scholar 

  9. Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent-continuum modeling of nano-structured materials. Compos Sci Technol. 1869, 2002;62

    Google Scholar 

  10. Gao XL, Li K. Finite deformation continuum model for single-walled carbon nanotubes. Int J Solids Struct. 2003;40:7329.

    Article  Google Scholar 

  11. Sears A, Batra RC. Macroscopic properties of carbon nanotubes from molecular-mechanics simulations. Phys Rev B. 2004;69:235406.

    Article  Google Scholar 

  12. Wang M, Zhang X, Lu MW. Nonlinear membrane-spring model for carbon nanotubes. Phys Rev B. 2005;72:205403.

    Article  Google Scholar 

  13. Chen X, Cao G. A structural mechanics study of single-walled carbon nanotubes generalized from atomistic simulation. Nanotechnol. 2006;17:1004.

    Article  Google Scholar 

  14. Arroyo M, Belytschko T. An atomistic-based finite deformation membrane for single layer crystalline films. J Mech Phys Solids. 1941, 2002;50

    Google Scholar 

  15. Arroyo M, Belytschko T. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys Rev B. 2004;69:115415.

    Article  Google Scholar 

  16. Zhang P, Huang Y, Gao H, Hwang KC. Fracture nucleation in single-wall carbon nanotubes under tension: a continuum analysis incorporating interatomic potentials. J Appl Mech-T ASME. 2002;69:454.

    Article  Google Scholar 

  17. Zhang P, Huang Y, Geubelle PH, Klein PA, Hwang KC. The elastic modulus of single-walled carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int J Solids Struct. 2002;39:3893.

    Article  Google Scholar 

  18. Zhang P, Jiang H, Huang Y, Geubelle PH, Hwang KC. An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. J Mech Phys Solids. 2004;52:977.

    Article  Google Scholar 

  19. 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. Comput Mater Sci. 2003;28:429.

    Article  Google Scholar 

  20. Born M, Huang K. Dynamical theory of crystal lattices. Oxford: Oxford University Press; 1959.

    MATH  Google Scholar 

  21. Khang DY, Xiao J, Kocabas C, MacLaren S, Banks T, Jiang H, Huang Y, Rogers J. Molecular scale buckling mechanics in individual, aligned single-wall carbon nanotubes on elastomeric substrates. Nano Lett. 2008;8:124.

    Article  Google Scholar 

  22. Tombler TW, Zhou CW, Alexseyev L, Kong J, Dai HJ, Lei L, Jayanthi CS, Tang MJ, Wu SY. Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation. Nature. 2000;405:769.

    Article  Google Scholar 

  23. Jiang H, Yu M-F, Liu B, Huang Y. Intrinsic energy loss mechanisms in a cantilevered carbon nanotube beam oscillator. Phys Rev Lett. 2004;93:185501.

    Article  Google Scholar 

  24. Zhou W, Huang Y, Liu B, Wu J, Hwang KC, Wei BQ. Adhesion between carbon nanotubes and substrate: mimicking the gecko foothair. Nano. 2007;2:175.

    Article  Google Scholar 

  25. Shi DL, Feng XQ, Huang Y, Hwang KC, Gao H, Eng J. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. Mater-T ASME. 2004;126:250.

    Article  Google Scholar 

  26. Tan H, Jiang LY, Huang Y, Liu B, Hwang KC. The effect of van der Waals-based interface cohesive law on carbon nanotube-reinforced composite materials. Compos Sci Technol. 2007;67:2941.

    Article  Google Scholar 

  27. Wu J, Hwang KC, Huang Y. An atomstic-based finite-deformation shell theory for single-wall carbon nanotubes. J Mech Phys Solids. 2008;56:279.

    Article  MathSciNet  Google Scholar 

  28. Wu J, Hwang KC, Huang Y. A shell theory for carbon nanotubes based on the interatomic potential and atomic structure. Adv Appl Mech. 2009;43:1.

    Article  Google Scholar 

  29. Sanders JL. Nonlinear theories for thin shells. Q Appl Math. 1963;21:21.

    Article  MathSciNet  Google Scholar 

  30. Koiter WT. On the nonlinear theory of thin elastic shells, I, II, III. In: Proceedings of the Koninklijke Nederlandse Akademie Van Wetenschappen. Series B, Amsterdam, Vol. 69; 1966. p. 1.

    Google Scholar 

  31. Niordson F. Shell theoriy. North-Holland series in applied mathematics and mechanics, vol. 29. New York/Oxford: Norh-holland-Amsterdam; 1985.

    Google Scholar 

  32. Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J Phys-Condens Mat. 2002;14:783.

    Article  Google Scholar 

  33. Brenner DW. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B. 1990;42:9458.

    Article  Google Scholar 

  34. Huang Y, Wu J, Hwang KC. Thickness of graphene and single-wall carbon nanotubes. Phys Rev B. 2006;74:245413.

    Article  Google Scholar 

  35. Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. New York: McGraw-Hill; 1959.

    MATH  Google Scholar 

  36. Eisenhart LP. An introduction to differential geometry : with use of the tensor calculus. Princeton/London: Princeton University Press/Geoffrey Cumberlege, Oxford University Press; 1947.

    MATH  Google Scholar 

  37. Wu J, Hwang KC, Huang Y, Song J. A finite-deformation shell theory for carbon nanotubes based on the interatomic potential. Part I: basic theory. J Appl Mech-T ASME. 2008;75:061006.

    Article  Google Scholar 

  38. Wu J, Hwang KC, Huang Y, Song J. A finite-deformation shell theory for carbon nanotubes based on the interatomic potential. Part II: instability analysis. J Appl Mech-T ASME. 2008;75:061007.

    Article  Google Scholar 

  39. Peng J, Wu J, Hwang KC, Song J, Huang Y. Can a single-wall carbon nanotube be modeled as a thin shell? J Mech Phys Solids. 2008;56:2213.

    Article  MathSciNet  Google Scholar 

  40. Thostenson ET, Ren ZF, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol. 1899, 2001;61

    Google Scholar 

  41. Coleman JN, Cadek M, Ryan KP, Fonseca A, Nagy JB, Blau WJ, Ferreira MS. Reinforcement of polymers with carbon nanotubes. The role of an ordered polymer interfacial region. Experiment and modeling, Polymer 2006;47:8556–8561

    Article  Google Scholar 

  42. Maruyama B, Alam H. Carbon nanotubes and nanofibers in composite materials. SAMPE J. 2002;38:59.

    Google Scholar 

  43. Deepak S, Wei C, Cho K. Nanomechanics of carbon nanotube and composites. Appl Mech Rev. 2003;56:215.

    Article  Google Scholar 

  44. Breuer O, Sundararaj U. Big returns from small fibers: A review of polymer/carbon nanotube composites. Polym Compos. 2004;25:630.

    Article  Google Scholar 

  45. Harris PJF. Carbon nanotube composites. Int Mater Rev. 2004;49:31.

    Article  Google Scholar 

  46. Schadler LS, Giannaris SC, Ajayan PM. Load transfer in carbon nanotube epoxy composites. Appl Phys Lett. 1998;73:3842.

    Article  Google Scholar 

  47. Ajayan PM, Schadler LS, Giannaris C, Rubio A. Single-walled carbon nanotube-polymer composites: strength and weakness. Adv Mater. 2000;12:750.

    Article  Google Scholar 

  48. Lau KT, Shi SQ. Failure mechanisms of carbon nanotube/epoxy composites pretreated in different temperature environments. Carbon. 2002;40:2965.

    Article  Google Scholar 

  49. Liao K, Li S. Interfacial characteristics of a carbon nanotube-polystyrene composite system. Appl Phys Lett. 2001;79:4225.

    Article  Google Scholar 

  50. Frankland SJV, Caglar A, Brenner DW, Griebel M. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube-polymer interfaces. J Phys Chem B. 2002;106:3046.

    Article  Google Scholar 

  51. Li CY, Chou TW. Multiscale modeling of carbon nanotube reinforced polymer composites. J Nanosci Nanotechnol. 2003;3:423.

    Article  Google Scholar 

  52. Wong M, Paramsothy M, Xu XJ, Ren Y, Li S, Liao K. Physical interactions at carbon nanotube-polymer interface. Polymer. 2003;44:7757.

    Article  Google Scholar 

  53. Gou J, Minaie B, Wang B, Liang Z, Zhang C. Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites. Comput Mater Sci. 2004;31:225.

    Article  Google Scholar 

  54. Odegard GM, Gates TS, Nicholson LM, Wise KE. Equivalent continuum modeling of nano-structured materials. Compos Sci Technol. 1869, 2002;62

    Google Scholar 

  55. Odegard GM, Gates TS, Wise KE, Park C, Siochi EJ. Constitutive modeling of nanotube-reinforced polymer composites. Compos Sci Technol. 2003;63:1671.

    Article  Google Scholar 

  56. Liu YJ, Chen XL. Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mech Mater. 2003;35:69.

    Article  Google Scholar 

  57. Thostenson ET, Chou TW. On the elastic properties of carbon nanotube-based composites: modelling and characterization. J Phys D-Appl Phys. 2003;36:573.

    Article  Google Scholar 

  58. Shi DL, Feng XQ, Huang Y, Hwang KC, Gao H. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J Eng Mater Technol. 2004;126:250.

    Article  Google Scholar 

  59. Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech. 1987;54:525.

    Article  Google Scholar 

  60. Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. Int J Solids Struct. 1996;33:2899.

    Article  Google Scholar 

  61. Geubelle PH, Baylor JS. Impact-induced delamination of composites: a 2D simulation. Compos Part B. 1998;29:589.

    Article  Google Scholar 

  62. Huang Y, Gao H. Intersonic crack propagation - Part I: The fundamental solution. J Appl Mech. 2001;68:169.

    Article  Google Scholar 

  63. Zhang P, Klein PA, Huang Y, Gao H, Wu PD. Numerical simulation of cohesive fracture by the virtual-internal-bond model. Comput Model Eng Sci. 2002;3:263.

    MATH  Google Scholar 

  64. Kubair DV, Geubelle PH, Huang Y. Intersonic crack propagation in homogeneous media under shear-dominated loading: theoretical analysis. J Mech Phys Solids. 2002;50:1547.

    Article  Google Scholar 

  65. Kubair DV, Geubelle PH, Huang Y. Analysis of a rate-dependent cohesive model for dynamic crack propagation. Eng Fract Mech. 2003;70:685.

    Article  Google Scholar 

  66. Samudrala O, Huang Y, Rosakis AJ. Subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone. J Mech Phys Solids. 2002;50:1231.

    Article  MathSciNet  Google Scholar 

  67. Song SH, Paulino GH, Buttlar WG. A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk, Eng Fract Mech. 2006;73:2829–2848.

    Google Scholar 

  68. Thiagarajan G, Hsia KJ, Huang Y. Finite element implementation of virtual internal bond model for simulating crack behavior. Eng Fract Mech. 2004;71:401.

    Article  Google Scholar 

  69. Thiagarajan G, Huang Y, Hsia KJ. Fracture simulation using an elasto-viscoplastic virtual internal bond model with finite elements. J Appl Mech. 2004;71:796.

    Article  Google Scholar 

  70. Tan H, Huang Y, Liu C, Geubelle PH. The Mori-Tanaka method for composite materials with nonlinear interface debonding. Int J Plast. 1890, 2005;21

    Google Scholar 

  71. Tan H, Liu C, Huang Y, Geubelle PH. The cohesive law for the particle/matrix interfaces in high explosives. J Mech Phys Solids. 2005;53:1892.

    Article  Google Scholar 

  72. Li VC, Chan CM, Leung KY. Experimental-determination of the tension-softening relations for cementitious composites. Cem Concr Res. 1987;17:441.

    Article  Google Scholar 

  73. Guo ZK, Kobayashi AS, Hay JC, White KW. Fracture process zone modeling of monolithic Al2O3. Eng Fract Mech. 1999;63:115.

    Article  Google Scholar 

  74. Mohammed I, Liechti KM. Cohesive zone modeling of crack nucleation at bimaterial corners. J Mech Phys Solids. 2000;48:735.

    Article  Google Scholar 

  75. Bazant ZP. Concrete fracture models: testing and practice. Eng Fract Mech. 2002;69:165.

    Article  Google Scholar 

  76. Elices M, Guinea GV, Gomez J, Planas J. The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech. 2002;69:137.

    Article  Google Scholar 

  77. Hong SS, Kim KS. Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method. J Mech Phys Solids. 2003;51:1267.

    Article  Google Scholar 

  78. Jiang LY, Huang Y, Jiang H, Ravichandran G, Gao H, Hwang KC, Liu B. A cohesive law for carbon nanotube/polymer interfaces based on the van der Waals force. J Mech Phys Solids. 2006;54:2436.

    Article  Google Scholar 

  79. Namilae S, Chandra N. Multiscale model to study the effect of interfaces in carbon nanotube-based composites. J Eng Mater Technol. 2005;127:222.

    Article  Google Scholar 

  80. Thostenson ET, Li CY, Chou TW. Nanocomposites in context. Sci Technol. 2005;65:491.

    Google Scholar 

  81. Samudrala O, Rosakis AJ. Effect of loading and geometry on the subsonic/intersonic transition of a bimaterial interface crack. Eng Fract Mech. 2003;70:309.

    Article  Google Scholar 

  82. Pantano A, Boyce MC, Parks DM. Nonlinear structural mechanics based modeling of carbon nanotube deformation. Phys Rev Lett. 2003;91:145504.

    Article  Google Scholar 

  83. Pantano A, Parks DM, Boyce MC. Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids. 2004;52:789.

    Article  Google Scholar 

  84. Lu WB, Wu J, Jiang LY, Huang Y, Hwang KC, Liu B. A cohesive law for multi-wall carbon nanotubes. Philos Mag. 2007;87:2221.

    Article  Google Scholar 

  85. Lu WB, Wu J, Song J, Hwang KC, Jiang LY, Huang Y. A cohesive law for interfaces between multi-wall carbon nanotubes and polymers due to the van der Waals interactions. Comput Methods Appl Mech Eng. 2008;197:3261.

    Article  Google Scholar 

  86. Tan H, Jiang LY, Huang Y, Liu B, Hwang KC. The effect of van der Waals-based interface cohesive law on carbon nanotube-reinforced composite materials. Compos Sci Technol. 2007;67:2941.

    Article  Google Scholar 

  87. Girifalco LA, Hodak M, Lee RS. Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential. Phys Rev B. 2000;62:13104.

    Article  Google Scholar 

  88. Frankland SJV, Harik VM, Odegard GM, Brenner DW, Gates TS. The stress-strain behavior of polymer-nanotube composites from molecular dynamics simulation. Compos Sci Technol. 2003;63:1655.

    Article  Google Scholar 

  89. Lordi V, Yao N. Molecular mechanics of binding in carbon-nanotube-polymer composites. J Mater Res. 2000;15:2770.

    Article  Google Scholar 

  90. Frankland SJV, Caglar A, Brenner DW, Griebel M. Molecular simulation of the influence of chemical cross-links on the shear strength of carbon nanotube-polymer interfaces, J. Phys. Chem. B 2002;106:3046–3048.

    Article  Google Scholar 

  91. Rapaport DC. The art of molecular dynamics simulation. Cambridge: Cambridge University Press; 2004.

    Book  Google Scholar 

  92. Chen B, Gao M, Zuo JM, Qu S, Liu B, Huang Y. Binding energy of parallel carbon nanotubes. Appl Phys Lett. 2003;83:3570.

    Article  Google Scholar 

  93. 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. Comput Mater Sci. 2003;28:429.

    Article  Google Scholar 

  94. Jiang LY, Tan HL, Wu J, Huang YG, Hwang KC. Continuum modeling of interfaces in polymer matrix composites reinforced by carbon nanotubes. Nano. 2007;2:139.

    Article  Google Scholar 

  95. Benveniste Y, Aboudi J. A continuum model for fiber reinforced materials with debonding. Int J Solids Struct. 1984;20:935.

    Article  Google Scholar 

  96. Timoshenko S, Goodier JN. Theory of elasticity. 3rd ed. New York: McGraw- Hill; 1970.

    Google Scholar 

  97. Reuss A. Berechung der Fliessgrenze von Mischkristallenx auf Grund der Plastizita¨tsbedingung fu¨ r Einkristalle. Z Angew Math Mech. 1929;9:49.

    Article  Google Scholar 

  98. Huang Y, Wu J, Hwang KC. Thickness of graphene and single-wall carbon nanotubes. Phys Rev B. 2006;74:245413.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AML, Department of Engineering Mechanics, Tsinghua University, Beijing, China

    Jian Wu & Keh-Chih Hwang

  2. Center for Mechanics and Materials, Tsinghua University, Beijing, China

    Jian Wu & Keh-Chih Hwang

  3. Department of Engineering Mechanics, Soft Matter Research Center and Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou, China

    Chenxi Zhang & Jizhou Song

Authors
  1. Jian Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chenxi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jizhou Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Keh-Chih Hwang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jizhou Song .

Editor information

Editors and Affiliations

  1. Materials Science and Strength of Materials, University of Stuttgart Institute for Materials Testing, Stuttgart, Germany

    Siegfried Schmauder

  2. Department of Civil Engineering, National Taiwan University Department of Civil Engineering, Taipei, T´ai-pei, Taiwan

    Chuin-Shan Chen

  3. BEC 254, University of Alabama at Birmingham, Birmingham, Alabama, USA

    Krishan K. Chawla

  4. Fulton School of Engineering, Arizona State University, Tempe, Arizona, USA

    Nikhilesh Chawla

  5. Engineering Mechanics, Zhejiang University Engineering Mechanics, Hangzhou, China

    Weiqiu Chen

  6. Katayanagi Advanced Research Laboratorie, Tokyo University of Technology Katayanagi Advanced Research Laboratorie, Tokyo, Japan

    Yutaka Kagawa

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Wu, J., Zhang, C., Song, J., Hwang, KC. (2018). Mechanics of Carbon Nanotubes and Their Composites. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_17-1

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-6855-3_17-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-6855-3

  • Online ISBN: 978-981-10-6855-3

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation