Skip to main content
SpringerLink
Log in

Large deflection analysis of nanowires based on nonlocal theory using total Lagrangian finite element method

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A model based on a nonlocal theory is developed for the large deflection analysis of nanowires for the first time. The nonlocal differential equation of Eringen is used as the nonlocal constitutive equation. Shear deformation is introduced by using the Timoshenko beam theory. The governing equations are derived by the variational formulation. The total Lagrangian finite element formulation is applied as a numerical method for the solution of the problem. In addition, a novel approach is used to overcome the difficulty in modeling concentrated loads on a nanowire with nonlocal theory. Several examples are presented for both of the small and large deflections. The numerical results for small deflections are verified with the results reported in the literature. The numerical results for large deflections can be used as benchmark.

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. Wang, Q., Varadan, V.K., Quek, S.T.: Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Phys. Lett. A 357, 130–135 (2006)

    Article  Google Scholar 

  2. Wang, Q., Varadan, V.K.: Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater. Struct. 16, 178–190 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  4. Arash, B., Wang, Q., Liew, K.M.: Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation. Comput. Methods Appl. Mech. Eng. 223–224, 1–9 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang, Yang: Zhang, L.W., Liew, K.M., Yu, J.L.: Transient analysis of single-layered graphene sheet using the kp-Ritz method and nonlocal elasticity theory. Appl. Math. Comput. 258, 489–501 (2015)

    MathSciNet  Google Scholar 

  6. Zhang, Yang: Lei, Z.X., Zhang, L.W., Liew, K.M., Yu, J.L.: Nonlocal continuum model for vibration of single-layered graphene sheets based on the element-free kp-Ritz method. Eng. Anal. Bound. Elem. 56, 90–97 (2015)

    Article  Google Scholar 

  7. Zhang, Yang, Zhang, L.W., Liew, K.M., Yu, J.L.: Nonlocal continuum model for large deformation analysis of SLGSs using the kp-Ritz element-free method. Int. J. Nonlinear Mech. 79, 1–9 (2016)

  8. Zhang, Yang: Zhang, L.W., Liew, K.M., Yu, J.L.: Free vibration analysis of bilayer graphene sheets subjected to in-plane magnetic fields. Compos. Struct. 144, 86–95 (2016)

    Article  Google Scholar 

  9. Liew, K.M., Zhang, Yang, Zhang, L.W.: Nonlocal elasticity theory for graphene modeling and simulation: prospects and challenges. J. Model. Mech. Mater. 20160159 (2017)

  10. Craighead, H.G.: Nanoelectromechanical systems. Science 290, 1532–1535 (2000)

    Article  Google Scholar 

  11. Patolsky, F., Lieber, C.M.: Nanowire nanosensors. Mater. Today 8, 20–28 (2005)

    Article  Google Scholar 

  12. Feng, X.L., He, R., Yang, P., Roukes, M.L.: Very high frequency silicon nanowire electromechanical resonators. Nano Lett. 7, 1953–1959 (2007)

    Article  Google Scholar 

  13. Jensen, K., Kim, K., Zettl, A.: An atomic-resolution nanomechanical mass sensor. Nat. Nanotechnol. 3, 533–537 (2008)

    Article  Google Scholar 

  14. Yakoboson, B.I., Brabed, C.J., Bernholc, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (1996)

    Article  Google Scholar 

  15. Liu, B., Jiang, H., Huang, Y., Qu, S., Yu, M.F., Hwang, K.C.: Atomic-scale finite element method in multiscale computation with applications to carbon nanotubes. Phys. Rev. B 72, 035435 (2005)

    Article  Google Scholar 

  16. Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)

    Article  Google Scholar 

  17. Taghipour, Y., Baradaran, G.H.: A finite element modeling for large deflection analysis of uniform and tapered nanowires with good interpretation of experimental results. Int. J. Mech. Sci. 114, 111–119 (2016)

    Article  Google Scholar 

  18. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  20. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  21. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peddieson, J., Buchanan, G.G., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)

    Article  Google Scholar 

  23. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    Article  MATH  Google Scholar 

  24. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E 41, 1651–1655 (2009)

    Article  Google Scholar 

  25. Civalek, O., Demir, C.: Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Appl. Math. Model. 35, 2053–2067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thai, H.T.: A nonlocal beam theory for bending, buckling, and of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)

    Article  MathSciNet  Google Scholar 

  27. Thai, H.T., Vo, T.P.: A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54, 58–66 (2012)

    Article  MathSciNet  Google Scholar 

  28. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Static analysis of nanobeams using nonlocal FEM. J. Mech. Sci. Technol. 27, 2035–2041 (2013)

    Article  Google Scholar 

  29. Nguyen, N.T., Kim, N.I., Lee, J.: Mixed finite element analysis of nonlocal Euler-Bernoulli nanobeams. Finite Elem. Anal. Des. 106, 65–72 (2015)

    Article  Google Scholar 

  30. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008)

    Article  Google Scholar 

  31. Zhang, Y.Y., Wang, C.M., Challamel, N.: Bending, buckling, and vibration of micro-nanobeams by hybrid nonlocal beam model. J. Eng. Mech. 136, 562–574 (2010)

    Article  Google Scholar 

  32. He, J., Lilley, C.M.: The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation. Comput. Mech. 44, 395–403 (2009)

    Article  MATH  Google Scholar 

  33. Felippa, C.A.: Nonlinear Finite Element Methods. University of Colorado, Boulder, Colorado (2014)

    Google Scholar 

  34. Tauchert, T.R.: Energy Principles in Structural Mechanics. McGraw-Hill, New York (1974)

    Google Scholar 

  35. Wang, Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 124301 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shahid Bahonar University of Kerman, Kerman, Iran

    Yasser Taghipour & Gholam Hossein Baradaran

Authors
  1. Yasser Taghipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gholam Hossein Baradaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gholam Hossein Baradaran.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taghipour, Y., Baradaran, G.H. Large deflection analysis of nanowires based on nonlocal theory using total Lagrangian finite element method. Acta Mech 228, 2429–2442 (2017). https://doi.org/10.1007/s00707-017-1837-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1837-0

Search

Navigation