Acta Mechanica

, Volume 217, Issue 1–2, pp 1–16

Multiscale modeling of the nonlinear response of nano-reinforced polymers



The present study uses a nonlinear representative volume element (RVE) to investigate the effective mechanical properties of a nano-reinforced polymer system. Here, the RVE represents the reinforcing carbon nanotube (CNT), the surrounding polymer matrix, and the CNT–polymer interface. Due to the inherent nanoscale involved in simulating CNT structures, an atomistic description is incorporated via the atomistic-based continuum multiscale modeling technique. In this way, the continuum constitutive relations are derived solely from atomistic formulations. The nonlinear response of armchair and zigzag nanotubes and their nano-reinforced polymer equivalents are considered and presented. The results reveal that reinforcing polymeric matrices with 1 to 10 vol% CNTs can result in upward of approximately 23- and 8-fold increases in the tensile and shear stiffness, respectively. These results have a direct bearing on the design and development of nano-reinforced composites.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Iijima S.: Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991)CrossRefGoogle Scholar
  2. 2.
    Endo M., Hayashi T., Kim Y.A., Terrones M., Dresselhaus M.S.: Applications of carbon nanotubes in the twenty-first century. Phil. Trans. R. Soc. Lond. A 362, 2223–2238 (2004)CrossRefGoogle Scholar
  3. 3.
    Kim B.C., Park S.W., Lee D.G.: Fracture toughness of the nano-particle reinforced epoxy composite. Compos. Struct. 86, 69–77 (2008)CrossRefGoogle Scholar
  4. 4.
    Zhai L.L., Ling G.P., Wang Y.W.: Effect of nano-Al2O3 on adhesion strength of epoxy adhesive and steel. Int. J. Adhes. Adhes. 28, 23–28 (2008)CrossRefGoogle Scholar
  5. 5.
    Salehi-Khojin A., Jana S., Wei-Hong Z.: Thermal-mechanical properties of a graphitic-nanofibers reinforced epoxy. J. Nanosci. Nanotechnol. 7, 898–906 (2007)CrossRefGoogle Scholar
  6. 6.
    Huang, C.K.: Prediction model of thermal conductivity for composite materials with nano particles. Technical Proceedings of the NSTI Nanotechnology Conference and Trade Show, NSTI, pp. 320–323 (2007)Google Scholar
  7. 7.
    Qinghua L., Jianhua Z.: Effects of nano fillers on the conductivity, adhesion strength, and reliability of isotropic conductive adhesives (ICAs). Key Eng. Mater. 353, 2879–2882 (2007)CrossRefGoogle Scholar
  8. 8.
    Qian D., Dickey E.C., Andrews R., Rantell T.: Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl. Phys. Lett. 76, 2868–2870 (2000)CrossRefGoogle Scholar
  9. 9.
    Schadler L.S., Giannaris S.C., Ajayan P.M.: Load transfer in carbon nanotube epoxy composites. Appl. Phys. Lett. 73, 3842–3844 (1998)CrossRefGoogle Scholar
  10. 10.
    Meguid S.A., Sun Y.: On the tensile and shear strength of nano-reinforced composite interfaces. Mater. Des. 25, 289–296 (2004)CrossRefGoogle Scholar
  11. 11.
    Chang T., Geng J., Guo X.: Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model. Proc. R. Soc. A 462, 2523–2540 (2006)CrossRefGoogle Scholar
  12. 12.
    Rudd R.E.: The atomic limit of finite element modeling in MEMS: Coupling of length scales. Analog. Integr. Circ. Signal Process. 29, 17–26 (2001)CrossRefGoogle Scholar
  13. 13.
    Abraham F.F., Walkup R., Gao H., Duchaineau M., DeLa Rubia T.D., Seager M.: Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture. Proc. Natl. Acad. Sci. USA 99, 5777–5782 (2002)CrossRefGoogle Scholar
  14. 14.
    Wernik J.M., Meguid S.A.: Coupling atomistics and continuum in solids: status, prospects, and challenges. Int. J. Mech. Mater. Des. 5, 79–110 (2009)CrossRefGoogle Scholar
  15. 15.
    Hyer, M.W.: Stress analysis of fiber-reinforced composite materials. McGraw-Hill, BostonGoogle Scholar
  16. 16.
    Nemat-Nasser S., Hori M., Denda M.: Micromechanics: overall properties of heterogeneous materials. Appl. Mech. Rev. 47, B24 (1998)Google Scholar
  17. 17.
    Shan Z., Gokhale A.M.: Representative volume element for non-uniform micro-structure, Comput. Mater. Sci. 24, 361–379 (2002)Google Scholar
  18. 18.
    Sun C.T., Vaidya R.S.: Prediction of composite properties from a representative volume element. Compos. Sci. Technol. 56, 171–179 (1996)CrossRefGoogle Scholar
  19. 19.
    Bogetti T.A., Wang T., VanLandingham M.R., Gillespie J.W. Jr: Characterization of nanoscale property variations in polymer composite systems: 2. numerical modeling. Compos.: Part A 30, 85–94 (1999)CrossRefGoogle Scholar
  20. 20.
    Liu Y.J., Chen X.L.: Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mech. Mater. 35, 69–81 (2003)CrossRefGoogle Scholar
  21. 21.
    Hu N., Fukunaga H., Lu C., Kameyama M., Yan B.: Prediction of elastic properties of carbon nanotube reinforced composites. Proc. R. Soc. A 461, 1685–1710 (2005)CrossRefGoogle Scholar
  22. 22.
    Tserpes K.I., Papanikos P., Labeas G., Pantelakis S.G.: Multi-scale modeling of tensile behavior of carbon nanotube-reinforced composites. Theor. Appl. Fract. Mech. 49, 51–60 (2008)CrossRefGoogle Scholar
  23. 23.
    Li C., Chou T.: A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40, 2487–2499 (2003)MATHCrossRefGoogle Scholar
  24. 24.
    Li C., Chou T.: Multiscale modeling of compressive behavior of carbon nanotube/polymer composites. Compos. Sci. Technol. 66, 2409–2414 (2006)CrossRefGoogle Scholar
  25. 25.
    Shokrieh M.M., Rafiee R.: On the tensile behavior of an embedded carbon nanotube in polymer matrix with non-bonded interphase region. Compos. Struct. 92, 647–652 (2010)CrossRefGoogle Scholar
  26. 26.
    Belytschko T., Xiao S.P., Schatz G.C., Ruoff R.S.: Atomistic simulations of nanotube fracture. Phys. Rev. B 65, 1–8 (2002)CrossRefGoogle Scholar
  27. 27.
    Esfarjani K., Gorjizadeh N., Nasrollahi Z.: Molecular dynamics of single wall carbon nanotube growth on nickel surface. Comput. Mater. Sci. 3, 117–120 (2006)CrossRefGoogle Scholar
  28. 28.
    Liew K.M., Chen B.J., Xiao Z.M.: Analysis of fracture nucleation in carbon nanotubes through atomistic-based continuum theory. Phys. Rev. B 71, 235424-1–235424-7 (2005)Google Scholar
  29. 29.
    Sun X., Zhao W.: Prediction of stiffness and strength of single-walled carbon nanotubes by molecular mechanics based finite element approach. Mater. Sci. Eng. A 390, 366–371 (2005)CrossRefGoogle Scholar
  30. 30.
    Xiao J.R., Staniszewski J., Gillespie J.W. Jr: Fracture and progressive failure of defective graphene sheets and carbon nanotubes. Compos. Struct. 88, 602–609 (2009)CrossRefGoogle Scholar
  31. 31.
    Natsuki T., Endo M.: Structural dependence of nonlinear elastic properties for carbon nanotubes using continuum analysis. Appl. Phys. A 80, 1463–1468 (2005)CrossRefGoogle Scholar
  32. 32.
    Wernik J.M., Meguid S.A.: Atomistic-based continuum modeling of the nonlinear behavior of carbon nanotubes. Acta Mech. 212, 167–179 (2010)MATHCrossRefGoogle Scholar
  33. 33.
    Lu J.P.: Elastic properties of carbon nanotubes and nanoropes. Phys. Rev. Lett. 79, 1297–1300 (1997)CrossRefGoogle Scholar
  34. 34.
    Hernandez E., Goze C., Bernier P., Rubio A.: Elastic properties of C and B x C y N z composite nanotubes. Phys. Rev. Lett. 80, 4502–4505 (1998)CrossRefGoogle Scholar
  35. 35.
    Jin Y., Yuan F.G.: Simulation of elastic properties of single-walled carbon nanotubes. Compos. Sci. Technol. 63, 1507–1515 (2003)CrossRefGoogle Scholar
  36. 36.
    Odegard G.M., Gates T.S., Nicholson L.M., Wise K.E.: Equivalent-continuum modeling of nano-structured materials . Compos. Sci. Technol. 62, 1869–1880 (2002)CrossRefGoogle Scholar
  37. 37.
    Natsuki T., Tantrakan K., Endo M.: Effects of carbon nanotubes structures on mechanical properties. Appl. Phys. A 79, 117–124 (2004)CrossRefGoogle Scholar
  38. 38.
    Keller T., De Castro J., Schollmayer M.: Adhesively bonded and translucent glass fiber reinforced polymer sandwich girders. J. Compos. Constr. 8, 461–470 (2004)CrossRefGoogle Scholar
  39. 39.
    Lordi V., Yao N.: Molecular mechanics of binding in carbon-nanotube-polymer composites. J. Mater. Res. 15, 2770–2779 (2000)CrossRefGoogle Scholar
  40. 40.
    Fiedler B., Gojny F.H., Wichmann M.H.G., Nolte M.C.M., Schulte K.: Fundamental aspects of nano-reinforced composites. Compos. Sci. Technol. 66, 3115–3125 (2006)CrossRefGoogle Scholar
  41. 41.
    Hu Y., Shenderova O.A., Zushou H., Padgett C.W., Brenner D.W.: Carbon nanostructures for advanced composites. Rep. Prog. Phys. 69, 1847–1895 (2006)CrossRefGoogle Scholar
  42. 42.
    Battezzati L., Pisani C., Ricca F.J.: Equilibrium conformation and surface motion of hydrocarbon molecules physisorbed on graphite. Chem. Soc., Faraday Trans. 71, 1629–1639 (1975)CrossRefGoogle Scholar
  43. 43.
    Montazeri, A., Naghdabadi, R.: Investigation the stability of SWCNT-polymer composites in the presence of CNT geometrical defects using multiscale modeling. Proc. Fourth Int. Conf. Multiscale Mater. Model., pp. 163–166 (2008)Google Scholar
  44. 44.
    Yu M.F., Files B.S., Arepalli S., Ruoff R.S.: Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys. Rev. Lett. 84, 5552–5555 (2000)CrossRefGoogle Scholar
  45. 45.
    Meo M., Rossi M.: Tensile failure prediction of single wall carbon nanotube. Eng. Fract. Mech. 73, 2589–2599 (2006)CrossRefGoogle Scholar
  46. 46.
    Giannopoulos G.I., Kakavas P.A., Anifantis N.K.: Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach. Comput. Mater. Sci. 41, 561–569 (2008)CrossRefGoogle Scholar
  47. 47.
    Srivastava D., Wei C.: Nanomechanics of carbon nanotubes and composites. Appl. Mech. Rev. 56, 215–230 (2003)CrossRefGoogle Scholar
  48. 48.
    Gupta S., Dharamvir K., Jindal V.K.: Elastic moduli of single-walled carbon nanotubes and their ropes. Phys. Rev. B 72, 165428-1–165428-16 (2005)Google Scholar
  49. 49.
    Xiao J.R., Gama B.A., Gillespie J.W. Jr: An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int. J. Solids Struct. 42, 3075–3092 (2005)MATHCrossRefGoogle Scholar
  50. 50.
    Yeh M., Hsieh T., Tai N.: Fabrication and mechanical properties of multi-walled carbon nanotubes/epoxy nanocomposites. Mater. Sci. Eng. A 483, 289–292 (2008)CrossRefGoogle Scholar
  51. 51.
    To C.W.S.: Bending and shear moduli of single-walled carbon nanotubes. Finite Elements Anal. Des. 42, 404–413 (2006)CrossRefGoogle Scholar
  52. 52.
    Krishnan A., Dujardin E., Ebbesen T.W., Yianilos P.N., Treacy M.M.J.: Young’s modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998)CrossRefGoogle Scholar
  53. 53.
    Tombler T.W., Zhou C., Kong J., Dai H., Liu L., Jayanthi C.S., Tang M., Wu S.Y.: Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation. Nature 405, 769–772 (2000)CrossRefGoogle Scholar
  54. 54.
    Hall A.R., An L., Liu J., Vicci L., Falvo M.R., Superfine R., Washburn S.: Experimental measurement of single-wall carbon nanotubes torsional properties. Phys. Rev. Lett. 96, 256102-1–256102-4 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

Personalised recommendations