Skip to main content
SpringerLink
Log in

Free torsional vibration of cracked nanobeams incorporating surface energy effects

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton’s principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.

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. Fei, P., Yeh, P. H., Zhou, J., Xu, S., Gao, Y., Song, J., Gu, Y., Huang, Y., and Wang, Z. L. Piezoelectric potential gated field-effect transistor based on a free-standing ZnO wire. Nano Lett., 9(10), 3435–3439 (2009)

    Article  Google Scholar 

  2. He, J. H., Hsin, C. L., Liu, J., Chen, L. J., and Wang, Z. L. Piezoelectric gated diode of a single ZnO nanowire. Adv. Mater., 19(6), 781–784 (2007)

    Google Scholar 

  3. Wang, Z. L. and Song, J. Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science, 312(5771), 242–246 (2006)

    Article  Google Scholar 

  4. Zhong, Z., Wang, D., Cui, Y., Bockrath, M. W., and Lieber, C. M. Nanowire crossbar arrays as address decoders for integrated nanosystems. Science, 302(5649), 1377–1379 (2003)

    Article  Google Scholar 

  5. Bai, X., Gao, P., Wang, Z. L., and Wang, E. Dual-mode mechanical resonance of individual ZnO nanobelts. Appl. Phys. Lett., 82(26), 4806–4808 (2003)

    Google Scholar 

  6. Nazemnezhad, R. and Hosseini-Hashemi, S. Nonlinear free vibration analysis of Timoshenko nanobeams with surface energy. Meccanica, 50(4), 1027–1044 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hosseini-Hashemi, S., Nazemnezhad, R., and Bedroud, M. Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity. Appl. Math. Model., 38(14), 3538–3553 (2014)

    MathSciNet  Google Scholar 

  8. Hosseini-Hashemi, S., Nahas, I., Fakher, M., and Nazemnezhad, R. Nonlinear free vibration of piezoelectric nanobeams incorporating surface effects. Smart. Mater. Struct., 23(3), 035012 (2014)

    MATH  Google Scholar 

  9. Hosseini-Hashemi, S., Nahas, I., Fakher, M., and Nazemnezhad, R. Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity. Acta Mech., 225(6), 1555–1564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, C. and Rajapakse, R. Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Trans. Nanotechnol., 9(4), 422–431 (2010)

    Google Scholar 

  11. Gurtin, M. E. and Murdoch, A. I. A continuum theory of elastic material surfaces. Arch. Ration. Mech. An., 57(4), 291–323 (1975)

  12. Gurtin, M. E. and Murdoch, A. I. Surface stress in solids. Int. J. Solids. Struct., 14(6), 431–440 (1978)

    MATH  Google Scholar 

  13. Hosseini-Hashemi, S., Nazemnezhad, R., and Rokni, H. Nonlocal nonlinear free vibration of nanobeams with surface effects. Eur. J. Mech. A-Solids., 52, 44–53 (2015)

    Article  MathSciNet  Google Scholar 

  14. Ansari, R., Mohammadi, V., Shojaei, M. F, Gholami, R., and Sahmani, S. On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory. Compos. Part BEng, 60, 158–166 (2014)

    Article  Google Scholar 

  15. Wang, G. F. and Feng, X. Q. Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D. Appl. Phys., 42(15), 155411 (2009)

    Google Scholar 

  16. Li, Y., Chen, C., Fang, B., Zhang, J., and Song, J. Postbuckling of piezoelectric nanobeams with surface effects. Int. J. Appl. Mech., 4(2), 1250018 (2012)

    Google Scholar 

  17. Guo, J. G. and Zhao, Y. P. The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology, 18(29), 295701 (2007)

    Article  Google Scholar 

  18. Assadi, A. and Farshi, B. Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects. Acta Mech., 222(1/2), 27–39 (2011)

    Article  MATH  Google Scholar 

  19. Hosseini-Hashemi, S., Fakher, M., Nazemnezhad, R., and Haghighi, M. H. S. Dynamic behavior of thin and thick cracked nanobeams incorporating surface effects. Compos. Part B-Eng., 61, 66–72 (2014)

    Article  Google Scholar 

  20. Wang, K. and Wang, B. Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. J. Vib. Control., 21(12), 2452–2464 (2015)

    MathSciNet  Google Scholar 

  21. Hasheminejad, S. M., Gheshlaghi, B., Mirzaei, Y., and Abbasion, S. Free transverse vibrations of cracked nanobeams with surface effects. Thin. Solid. Films, 519(8), 2477–2482 (2011)

    Article  Google Scholar 

  22. Wang, G. F. and Feng, X. Q. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett., 90(23), 231904 (2007)

    Google Scholar 

  23. Nazemnezhad, R., Salimi, M., Hashemi, S. H., and Sharabiani, P. A. An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. Compos. Part B-Eng., 43, 2893–2897 (2012)

    Article  Google Scholar 

  24. Fennimore, A., Yuzvinsky, T., Han, W. Q., Fuhrer, M., Cumings, J., and Zettl, A. Rotational actuators based on carbon nanotubes. nature, 424(6947), 408–410 (2003)

    Article  Google Scholar 

  25. Witkamp, B., Poot, M., Pathangi, H., H¨ uttel, A., Van, D., and Zant, H. Self-detecting gatetunable nanotube paddle resonators. Appl. Phys. Lett., 93(11), 111909 (2008)

    Google Scholar 

  26. Meyer, J. C., Paillet, M., and Roth, S. Single-molecule torsional pendulum. Science, 309(5740), 1539–1541 (2005)

    Article  Google Scholar 

  27. Dong, L., Nelson, B. J., Fukuda, T., and Arai, F. Towards nanotube linear servomotors. IEEE Trans. Autom. Sci. Eng, 3(3), 228–235 (2006)

    Google Scholar 

  28. Williams, P., Papadakis, S., Patel, A., Falvo, M., Washburn, S., and Superfine, R. Torsional response and stiffening of individual multiwalled carbon nanotubes. Phys. Rev. Lett., 89(25), 255502 (2002)

    Article  Google Scholar 

  29. Gheshlaghi, B. and Hasheminejad, S. M. Size dependent torsional vibration of nanotubes. Physica E, 43(1), 45–48 (2010)

    Article  Google Scholar 

  30. Murmu, T., Adhikari, S., and Wang, C. Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory. Physica E, 43(6), 1276–1280 (2011)

    Article  Google Scholar 

  31. Lim, C. W., Li, C., and Yu, J. Free torsional vibration of nanotubes based on nonlocal stress theory. J. Sound. Vib., 331(12), 2798–2808 (2012)

    Article  Google Scholar 

  32. Loya, J., Aranda-Ruiz, J., and Fernández-Sáez, J. Torsion of cracked nanorods using a nonlocal elasticity model. J. Phys. D Appl. Phys., 47(11), 115304 (2014)

    MathSciNet  Google Scholar 

  33. Arda, M. and Aydogdu, M. Torsional statics and dynamics of nanotubes embedded in an elastic medium. Compos. Struct., 114, 80–91 (2014)

    Google Scholar 

  34. Rao, S. S. Vibration of Continuous Systems, John Wiley and Sons, Hoboken (2007)

    Google Scholar 

  35. Freund, L. and Herrmann, G. Dynamic fracture of a beam or plate in plane bending. J. Appl. Mech., 43(1), 112–116 (1976)

    Google Scholar 

  36. Miller, R. E. and Shenoy, V. B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 11(3), 139–147 (2000)

    Article  Google Scholar 

  37. Assadi, A. and Farshi, B. Vibration characteristics of circular nanoplates. J. Appl. Phys., 108(7), 074312 (2010)

    Google Scholar 

  38. Hosseini-Hashemi, S. and Nazemnezhad, R. An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Compos. Part B-Eng., 52, 199–206 (2013)

    Article  Google Scholar 

  39. Gheshlaghi, B. and Hasheminejad, S. M. Surface effects on nonlinear free vibration of nanobeams. Compos. Part B-Eng., 42, 934–937 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering, Damghan University, Damghan, 36716-41167, Iran

    R. Nazemnezhad

  2. School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran

    P. Fahimi

  3. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, 14399-57131, Iran

    P. Fahimi

Authors
  1. R. Nazemnezhad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Fahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Nazemnezhad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nazemnezhad, R., Fahimi, P. Free torsional vibration of cracked nanobeams incorporating surface energy effects. Appl. Math. Mech.-Engl. Ed. 38, 217–230 (2017). https://doi.org/10.1007/s10483-017-2167-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-017-2167-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation