Skip to main content
SpringerLink
Log in

Structural modelling of nanorods and nanobeams using doublet mechanics theory

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

Abstract

In this study, statics and dynamics of nanorods and nanobeams are investigated by using doublet mechanics. Classical rod theory and Euler–Bernoulli beam theory is used in the formulation. After deriving governing equations static deformation, buckling, vibration and wave propagation problems in nanorods and nanobeams are investigated in detail. The results obtained by using of doublet mechanics are compared to that of the classical elasticity theory. The importance of the size dependent mechanical behavior at the nano scale is shown in the considered problems. In doublet mechanics, bond length of atoms of the considered solid are used as an intrinsic length scale.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  • Aifantis, E.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)

    Article  MATH  Google Scholar 

  • Aizawa, T., Souda, R., Otani, S., Ishizawa, Y., Oshima, C.: Bond softening in monolayer graphite formed on transition-metal carbide surfaces. Phys. Rev. B 43, 12060 (1991)

    Article  Google Scholar 

  • Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mat. 8, 231–282 (1997)

    Google Scholar 

  • Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51(1), 303–313 (2012)

    Article  Google Scholar 

  • Askes, H., Suiker, A., Sluys, L.: A classification of higher-order strain gradient models–linear analysis. Arch. Appl. Mech. 72, 171–188 (2002)

    Article  MATH  Google Scholar 

  • Aydogdu, M.: Axial vibration of the nanorods with the nonlocal continuum rod model. Phys. E 41, 861–864 (2009a)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Beskou, P.S., Polyzos, D., Beskos, D.E.: Dynamic analysis of gradient elastic flexural beams. Struct. Eng. Mech. 15, 705–716 (2003a)

    Article  MATH  Google Scholar 

  • Beskou, P.S., Tsepoura, K.G., Polyzos, D., Beskos, D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40, 385–400 (2003b)

    Article  MATH  Google Scholar 

  • Challamel, N., Wang, C.M., Elishakoff, I.: Nonlocal or gradient elasticity macroscopic models: a question of concentrated or distributed microstructure. Mech. Res. Commun. 71, 25–31 (2016)

    Article  Google Scholar 

  • Cosserat, E., Cosserat, F.: Sur la théorie des corps déformables. Herman, Paris (1909). (in French)

    MATH  Google Scholar 

  • Dresselhaus, M.S., Dresselhaus, G., Saito, R.: Physics of carbon nanotubes. Carbon 33(7), 883–891 (1995)

    Article  Google Scholar 

  • Ece, M.C., Aydogdu, M.: Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mech. 190, 185–195 (2007)

    Article  MATH  Google Scholar 

  • Eringen, A.C.: Nonlocal Polar Field Models. Academic Press, New York (1976)

    Google Scholar 

  • 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 

  • Ferrari, M., Granik, V.T., Imam, A., Nadeau, J.: Advances in Doublet Mechanics. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  • Feynman, R.P.: There is plenty of room in the bottom. Caltech Eng. Sci. 23(5), 22–36 (1960)

    Google Scholar 

  • Gentile, F., Sakamoto, J., Righetti, R., Decuzzi, P., Ferrari, M.: A doublet mechanics model for the ultrasound characterization of malignant tissues. J. Biomed. Sci. Eng. 4, 362–374 (2011)

    Article  Google Scholar 

  • Granik, V.T.: Microstructural mechanics of granular media. Technique report IM/MGU, Institute of Mechanics of Moscow State University 78–241 (1978)

  • Granik, V.T., Ferrari, M.: Microstructural mechanics of granular media. Mech. Mater. 15, 301–322 (1993)

    Article  Google Scholar 

  • Gul, U., Aydogdu, M., Gaygusuzoglu, G.: Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics. Compos. Struct. 160, 1268–1278 (2017)

    Article  Google Scholar 

  • Kojic, M., Vlastelica, I., Decuzzi, P., Granik, V.T., Ferrari, M.: A finite element formulation for the doublet mechanics modeling of microstructural materials. Comput. Methods Appl. Mech. Eng. 200, 1446–1454 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, J., Ferrari, M.: Mechanical spectral signatures of malignant disease? A small-sample, comparative study of continuum vs. nano-biomechanical data analyses. Dis. Markers 18, 175–183 (2002)

    Article  Google Scholar 

  • Lin, S.S., Shen, Y.C.: Stress fields of a half-plane caused by moving loads-resolved using doublet mechanics. Soil Dyn. Earthq. Eng. 25, 893–904 (2005)

    Article  Google Scholar 

  • Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, New York (2002)

    Google Scholar 

  • Sadd, M.H., Dai, Q.: A comparison of micro-mechanical modeling of asphalt materials using finite elements and doublet mechanics. Mech. Mater. 37, 641–662 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Thostenson, E.T., Ren, Z., Chou, T.W.: Advances in the science and technology of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 61, 1899–1912 (2001)

    Article  Google Scholar 

  • Tsepoura, K.G., Beskou, S.P., Polyzos, D., Beskos, D.E.: Static and dynamic analysis of a gradient-elastic bar in tension. Arch. Appl. Mech. 72, 483–497 (2002)

    Article  MATH  Google Scholar 

  • Vajari, A.F., Imam, A.: Analysis of radial breathing mode of vibration of single-walled carbon nanotubes via doublet mechanics. ZAMM-J Math. Mech. (2016a). doi:10.1002/zamm.201500160

    MATH  Google Scholar 

  • Vajari, A.F., Imam, A.: Axial vibration of single-walled carbon nanotubes using doublet mechanics. Indian J. Phys. 90(4), 447–455 (2016b)

    Article  MATH  Google Scholar 

  • Wu, J., Layman, C., Liu, J.: Wave equations, dispersion relations and van Hove singularities for applications of doublet mechanics to ultrasound propagation in bio and nanomaterials. J. Acoust. Soc. Am. 115(2), 893–900 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Trakya University, 22030, Edirne, Turkey

    Ufuk Gul & Metin Aydogdu

Authors
  1. Ufuk Gul
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Metin Aydogdu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Metin Aydogdu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gul, U., Aydogdu, M. Structural modelling of nanorods and nanobeams using doublet mechanics theory. Int J Mech Mater Des 14, 195–212 (2018). https://doi.org/10.1007/s10999-017-9371-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-017-9371-8

Keywords

Search

Navigation