Skip to main content
SpringerLink
Log in

Study on the effect of an eccentric hole on the vibrational behavior of a graphene sheet using an analytical approach

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Because of the production process and constraint conditions, a circular graphene sheet may be opposed to structural defect and pin hole, respectively. Some of the defects and pin hole on a circular graphene sheet can be considered as an eccentric hole. So, analyzing the behavior of a circular graphene sheet with an eccentric hole is important. Free vibration of an eccentric annular graphene sheet, as the basis of any dynamical analysis, is analytically studied in this paper. Nonlocal thin plate theory is used to model the problem. The translational addition theorem for cylindrical vector wave functions is employed to solve the equation of motion for various boundary conditions. Results are compared with the literature, and their accuracy is approved. Effects of boundary conditions, geometrical properties and nonlocal parameter changes on symmetric and antisymmetric vibrational modes are investigated. It is approved that the eccentricity has a significant effect on the natural frequencies. Also, symmetric and antisymmetric modes of an annular graphene sheet have different behavior when geometrical and nonlocal parameters change.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Singh V., Joung D., Zhai L., Das S., Khondaker S.I., Seal S.: Graphene based materials: past, present and future. Prog. Mater. Sci. 56, 1178–1271 (2011). doi:10.1016/j.pmatsci.2011.03.003

    Article  Google Scholar 

  2. Eringen A.C.: Nonlocal Continuum Field Theories. Springer, Berlin (2002)

    MATH  Google Scholar 

  3. Sudak L.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)

    Article  Google Scholar 

  4. Wang Q., Wang C.: The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18, 075702 (2007)

    Article  Google Scholar 

  5. Aghababaei R., Reddy J.: Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289 (2009)

    Article  Google Scholar 

  6. Moosavi H., Mohammadi M., Farajpour A., Shahidi S.: Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory. Phys. E 44, 135–140 (2011)

    Article  Google Scholar 

  7. Danesh M., Farajpour A., Mohammadi M.: Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mech. Res. Commun. 39, 23–27 (2012)

    Article  MATH  Google Scholar 

  8. Pradhan S.: Buckling analysis and small scale effect of biaxially compressed graphene sheets using non-local elasticity theory. Sadhana 37, 461–480 (2012)

    Article  Google Scholar 

  9. Akgöz B., Civalek Ö.: Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory. Mater. Des. 42, 164–171 (2012)

    Article  Google Scholar 

  10. Ma H., Gao X.-L., Reddy J.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ma H., Gao X.-L., Reddy J.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)

    Article  MATH  Google Scholar 

  12. Arash B., Wang Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012). doi:10.1016/j.commatsci.2011.07.040

    Article  Google Scholar 

  13. Rafiee R., Moghadam R.M.: On the modeling of carbon nanotubes: a critical review. Compos. Part B 56, 435–449 (2014)

    Article  Google Scholar 

  14. Duan W., Wang C.: Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory. Nanotechnology 18, 385704 (2007)

    Article  Google Scholar 

  15. Gürses M., Akgöz B., Civalek Ö.: Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl. Math. Comput. 219, 3226–3240 (2012)

    Article  MathSciNet  Google Scholar 

  16. Farajpour A., Dehghany M., Shahidi A.R.: Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment. Compos. Part B 50, 333–343 (2013). doi:10.1016/j.compositesb.2013.02.026

    Article  Google Scholar 

  17. Malekzadeh P., Farajpour A.: Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium. Acta Mech. 223, 2311–2330 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gheshlaghi B., Hasheminejad S.M.: Size dependent damping in axisymmetric vibrations of circular nanoplates. Thin Solid Films 537, 212–216 (2013). doi:10.1016/j.tsf.2013.04.014

    Article  Google Scholar 

  19. Mahmoudinezhad E., Ansari R.: Vibration analysis of circular and square single-layered graphene sheets: an accurate spring mass model. Phys. E 47, 12–16 (2013). doi:10.1016/j.physe.2012.09.029

    Article  Google Scholar 

  20. Mohammadi M., Goodarzi M., Ghayour M., Farajpour A.: Influence of in-plane preload on the vibration frequency of a circular graphene sheet via nonlocal continuum theory. Compos. Part B 51, 121–129 (2013). doi:10.1016/j.compositesb.2013.02.044

    Article  Google Scholar 

  21. Ravari M.K., Shahidi A.: Axisymmetric buckling of the circular annular nanoplates using finite difference method. Meccanica 48, 135–144 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Bedroud M., Hosseini-Hashemi S., Nazemnezhad R.: Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity. Acta Mech. 224, 2663–2676 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hosseini-Hashemi S., Bedroud M., Nazemnezhad R.: An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity. Compos. Struct. 103, 108–118 (2013). doi:10.1016/j.compstruct.2013.02.022

    Article  Google Scholar 

  24. Zhou S.-M., Sheng L.-P., Shen Z.-B.: Transverse vibration of circular graphene sheet-based mass sensor via nonlocal Kirchhoff plate theory. Comput. Mater. Sci. 86, 73–78 (2014). doi:10.1016/j.commatsci.2014.01.031

    Article  Google Scholar 

  25. Asemi, S.R., Farajpour, A., Borghei, M., Hassani, A.H.: Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics. Lat. Am. J. Solids Struct. 11, 704–724 (2014)

  26. Gurtin M.E., Murdoch A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. Asemi S.R., Farajpour A.: Decoupling the nonlocal elasticity equations for thermo-mechanical vibration of circular graphene sheets including surface effects. Phys. E 60, 80–90 (2014). doi:10.1016/j.physe.2014.02.002

    Article  Google Scholar 

  28. Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V., Sahmani, S.: Surface stress effect on the pull-in instability of circular nanoplates. Acta Astronaut. 102, 140–150 (2014). doi:10.1016/j.actaastro.2014.05.020

  29. Ansari R., Gholami R., Shojaei M.F., Mohammadi V., Sahmani S.: Surface stress effect on the vibrational response of circular nanoplates with various edge supports. J. Appl. Mech. 80, 021021 (2013)

    Article  Google Scholar 

  30. Ansari R., Mohammadi V., Faghih Shojaei M., Gholami R., Sahmani S.: Surface stress effect on the postbuckling and free vibrations of axisymmetric circular Mindlin nanoplates subject to various edge supports. Compos. Struct. 112, 358–367 (2014). doi:10.1016/j.compstruct.2014.02.028

    Article  Google Scholar 

  31. Watson G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, MA (1995)

    MATH  Google Scholar 

  32. Alsahlani A., Mukherjee R.: Dynamics of a circular membrane with an eccentric circular areal constraint: analysis and accurate simulations. Simul. Modell. Pract. Theory 31, 149–168 (2013). doi:10.1016/j.simpat.2012.10.008

    Article  Google Scholar 

  33. Lin W.: Free transverse vibrations of uniform circular plates and membranes with eccentric holes. J. Sound Vib. 81, 425–435 (1982)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, Qom University of Technology, Qom, Iran

    M. Fadaee

  2. Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    M. R. Ilkhani

Authors
  1. M. Fadaee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Ilkhani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fadaee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fadaee, M., Ilkhani, M.R. Study on the effect of an eccentric hole on the vibrational behavior of a graphene sheet using an analytical approach. Acta Mech 226, 1395–1407 (2015). https://doi.org/10.1007/s00707-014-1259-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-014-1259-1

Keywords

Search

Navigation