Determination of rigidities, stiffness coefficients and elastic constants of multi-layer graphene sheets by an asymptotic homogenization method

  • Masoud TahaniEmail author
  • Sobhan Safarian
Technical Paper


This work is concerned with the analysis of bending and torsional rigidities of multi-layer graphene sheets (MLGSs) based on the Kalamkarov’s general asymptotic homogenization composite shell model. Also, the effective stiffness coefficients and elastic constants of MLGSs are estimated with this analytical method. The unit cell with both in-plane and out-of-plane interactions is assumed in this model. A MLGS as a homogeneous honeycomb network sheet with the periodic hexagonal unit cell is considered here in which the layers are held together by different densities of van der Waals interactions. The stiffness coefficients, elastic constants and rigidities of MLGS are found by considering different densities of the van der Waals forces and different number of layers. The results show good agreements in comparison with other experiments and numerical solutions. It is found that the homogenization method gives the ability to create promise analytical approach that can be used for other nanostructures.


Asymptotic homogenization method Multi-layer graphene sheet Bending and torsional rigidities Stiffness coefficients Elastic constants 



This research did not receive any specific grant from funding agencies in the public, commercial or not-for profit sectors.


  1. 1.
    Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56–58Google Scholar
  2. 2.
    Krishnan A, Dujardin E, Ebbesen TW, Yianilos PN, Treacy MMJ (1998) Young’s modulus of single-walled nanotubes. Phys Rev B 58(20):14013–14019Google Scholar
  3. 3.
    Vadukumpully S, Paul J, Mahanta N, Valiyaveettil S (2011) Flexible conductive graphene/poly(vinyl chloride) composite thin films with high mechanical strength and thermal stability. Carbon 49(1):198–205Google Scholar
  4. 4.
    Tsai J-L, Tu J-F (2010) Characterizing mechanical properties of graphite using molecular dynamics simulation. Mater Des 31(1):194–199Google Scholar
  5. 5.
    WenXing B, ChangChun Z, WanZhao C (2004) Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. Phys B Condens Matter 352(1–4):156–163Google Scholar
  6. 6.
    Zhao H, Min K, Aluru NR (2009) Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension. Nano Lett 9(8):3012–3015Google Scholar
  7. 7.
    Zhang YY, Wang CM, Cheng Y, Xiang Y (2011) Mechanical properties of bilayer graphene sheets coupled by sp3 bonding. Carbon 49(13):4511–4517Google Scholar
  8. 8.
    Van Lier G, Van Alsenoy C, Van Doren V, Geerlings P (2000) Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene. Chem Phys Lett 326(1):181–185Google Scholar
  9. 9.
    Jiao MD, Wang L, Wang CY, Zhang Q, Ye SY, Wang FY (2016) Molecular dynamics simulations on deformation and fracture of bi-layer graphene with different stacking pattern under tension. Phys Lett A 380(4):609–613Google Scholar
  10. 10.
    Odegard GM, Gates TS, Nicholson LM, Wise KE (2002) Equivalent-continuum modeling of nano-structured materials. Compos Sci Technol 62(14):1869–1880Google Scholar
  11. 11.
    Scarpa F, Adhikari S, Phani AS (2009) Effective elastic mechanical properties of single layer graphene sheets. Nanotechnology 20(6):065709Google Scholar
  12. 12.
    Golkarian AR, Jabbarzadeh M (2013) The density effect of van der Waals forces on the elastic modules in graphite layers. Comput Mater Sci 74:138–142Google Scholar
  13. 13.
    Cho J, Luo JJ, Daniel IM (2007) Mechanical characterization of graphite/epoxy nanocomposites by multi-scale analysis. Compos Sci Technol 67(11):2399–2407Google Scholar
  14. 14.
    Behfar K, Seifi P, Naghdabadi R, Ghanbari J (2006) An analytical approach to determination of bending modulus of a multi-layered graphene sheet. Thin Solid Films 496(2):475–480Google Scholar
  15. 15.
    Lu Q, Arroyo M, Huang R (2009) Elastic bending modulus of monolayer graphene. J Phys D Appl Phys 42:(10)Google Scholar
  16. 16.
    Kudin KN, Scuseria GE, Yakobson BI (2001) C 2 F, BN, and C nanoshell elasticity from ab initio computations. Phys Rev B 64(23):235406Google Scholar
  17. 17.
    Kalamkarov AL, Veedu VP, Ghasemi-Nejhad MN (2005) Mechanical properties modeling of carbon single-walled nanotubes: an asymptotic homogenization method. J Comput Theor Nanosci 2(1):124–131Google Scholar
  18. 18.
    Kalamkarov AL, Georgiades AV, Rokkam SK, Veedu VP, Ghasemi-Nejhad MN (2006) Analytical and numerical techniques to predict carbon nanotubes properties. Int J Solids Struct 43(22):6832–6854zbMATHGoogle Scholar
  19. 19.
    Safarian S, Tahani M (2018) Evaluation of tension, bending and twisting rigidities of single-layer graphene sheets by an analytical asymptotic homogenization model. Mechanics 24(2):161–168Google Scholar
  20. 20.
    Hassani B, Hinton E (1998) A review of homogenization and topology optimization I—homogenization theory for media with periodic structure. Comput Struct 69(6):707–717zbMATHGoogle Scholar
  21. 21.
    Hassani B, Hinton E (1998) A review of homogenization and topology optimization II—analytical and numerical solution of homogenization equations. Comput Struct 69(6):719–738zbMATHGoogle Scholar
  22. 22.
    Hassani B, Hinton E (2012) Homogenization and structural topology optimization: theory, practice and software. Springer, BerlinzbMATHGoogle Scholar
  23. 23.
    Challagulla K, Georgiades A, Kalamkarov A (2007) Asymptotic homogenization model for three-dimensional network reinforced composite structures. J Mech Mater Struct 2(4):613–632Google Scholar
  24. 24.
    Kalamkarov AL, Savi MA (2012) Micromechanical modeling and effective properties of the smart grid-reinforced composites. J Braz Soc Mech Sci Eng 34:(SPE):343–351Google Scholar
  25. 25.
    Mantic V (2014) Mathematical methods and models in composites. Imperial College Press, LondonzbMATHGoogle Scholar
  26. 26.
    Kalamkarov AL, Kolpakov AG (1997) Analysis, design and optimization of composite structures. Wiley, ChichesterzbMATHGoogle Scholar
  27. 27.
    Kalamkarov A, Georgiades A (2002) Micromechanical modeling of smart composite structures. Smart Mater Struct 11(3):423Google Scholar
  28. 28.
    Kalamkarov A (1987) On the determination of effective characteristics of cellular plates and shells of periodic structure. Mech Solids 22:175–179Google Scholar
  29. 29.
    Kalamkarov AL (1992) Composite and reinforced elements of constructions. Wiley, ChichesterzbMATHGoogle Scholar
  30. 30.
    Kalamkarov AL, Georgiades AV (2002) Micromechanical modeling of smart composite structures. Smart Mater Struct 11(3):423Google Scholar
  31. 31.
    Giannopoulos G, Kakavas P, Anifantis N (2008) Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach. Comput Mater Sci 41(4):561–569Google Scholar
  32. 32.
    Jorgensen WL, Severance DL (1990) Aromatic-aromatic interactions: free energy profiles for the benzene dimer in water, chloroform, and liquid benzene. J Am Chem Soc 112(12):4768–4774Google Scholar
  33. 33.
    Li C, Chou T-W (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40(10):2487–2499zbMATHGoogle Scholar
  34. 34.
    Georgantzinos SK, Giannopoulos GI, Anifantis NK (2010) Numerical investigation of elastic mechanical properties of graphene structures. Mater Des 31(10):4646–4654Google Scholar
  35. 35.
    Reddy JN (2004) Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, Boca RatonzbMATHGoogle Scholar
  36. 36.
    Blakslee OL, Proctor DG, Seldin EJ, Spence GB, Weng T (1970) Elastic constants of compression-annealed pyrolytic graphite. J Appl Phys 41(8):3373–3382Google Scholar
  37. 37.
    Shabana YM, Wang G-T (2013) Thermomechanical modeling of polymer nanocomposites by the asymptotic homogenization method. Acta Mech 224(6):1213–1224zbMATHGoogle Scholar
  38. 38.
    Xiao JR, Gama BA, Gillespie JW Jr (2005) An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int J Solids Struct 42(11–12):3075–3092zbMATHGoogle Scholar
  39. 39.
    Arroyo M, Belytschko T (2004) Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy–Born rule. Phys Rev B 69(11):115415Google Scholar
  40. 40.
    Lu Q, Huang R (2009) Nonlinear mechanics of single-atomic-layer graphene sheets. Int J Appl Mech 1(03):443–467Google Scholar
  41. 41.
    Hosseini Kordkheili S, Moshrefzadeh-Sani H (2013) Mechanical properties of double-layered graphene sheets. Comput Mater Sci 69:335–343Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Faculty of EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.Department of Mechanical Engineering, Faculty of EngineeringKhayyam UniversityMashhadIran

Personalised recommendations