Advertisement

Acta Mechanica

, Volume 225, Issue 7, pp 1945–1953 | Cite as

Small-scale effects in nanorods

  • Raffaele Barretta
  • Francesco Marotti de Sciarra
  • Marina Diaco
Article

Abstract

The Eringen model of nonlocal elasticity provides an effective theoretical tool to assess small-scale effects in carbon nanotubes (CNTs). A variational formulation of the nonlocal elastostatic problem is proposed in this paper. The merit of the variational treatment, over standard ones based on integration of a second-order differential equation, consists in revealing a simple basic analogy. According to this analogy, the nonlocality effect is simulated by prescribing an axial distortion linearly depending on the first derivative of the axial load intensity. The nonlocal elastostatic problem can then be solved by standard tools of structural analysis with prescription of equivalent imposed distortions. Examples of nanorods with one fixed and one free end and with fixed ends are explicitly carried out.

Keywords

Axial Force Beam Theory Axial Displacement Nonlocal Elasticity Nonlocality Effect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Marotti de Sciarra F.: Nonlocal and gradient rate plasticity. Int. J. Solids Struct. 41(26), 7329–7349 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Marotti de Sciarra F.: Variational formulations, convergence and stability properties in nonlocal elastoplasticity. Int. J. Solids Struct. 45, 2322–2354 (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Marotti de Sciarra, F.: A finite element for nonlocal elastic analysis. In: Coupled Problems, IV International Conference on Computational Methods for Coupled Problems in Science and Engineering, 496505, Greek (2011)Google Scholar
  4. 4.
    Marotti de Sciarra F.: Hardening plasticity with nonlocal strain damage. Int. J. Plast. 34, 114–138 (2012)CrossRefGoogle Scholar
  5. 5.
    Marotti de Sciarra F.: Novel variational formulations for nonlocal plasticity. Int. J. Plast. 25, 302–331 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Marotti de Sciarra F.: On non-local and non-homogeneous elastic continua. Int. J. Solids Struct. 46, 651–676 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Marotti de Sciarra F.: A nonlocal model with strain-based damage. Int. J. Solids Struct. 46, 4107–4122 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Aydogdu M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 41, 1651–1655 (2009)CrossRefGoogle Scholar
  10. 10.
    Phadikar J.K., Pradhan S.C.: Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comp. Mat. Sci. 49, 492–499 (2010)CrossRefGoogle Scholar
  11. 11.
    Demir Ç., Demir Ç.: Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Appl. Math. Model. 35, 2053–2067 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Roque C.M.C., Ferreira A.J.M., Reddy J.N.: Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. Int. J. Eng. Sci. 49, 976–984 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Aydogdu M.: Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity. Mech. Res. Comm. 43, 34–40 (2012)CrossRefGoogle Scholar
  14. 14.
    Arash B., Wang Q.: A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comp. Mat. Sci. 51, 303–313 (2012)CrossRefGoogle Scholar
  15. 15.
    Kazemi-Lari M.A., Fazelzadeh S.A., Ghavanloo E.: Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation. Physica E 44, 1623–1630 (2012)CrossRefGoogle Scholar
  16. 16.
    Lim C.W., Xu R.: Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size effects. Acta Mech. 223, 789–809 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mahmoud F.F., Eltaher M.A., Alshorbagy A.E., Meletis E.I.: Static analysis of nanobeams including surface effects by nonlocal finite elements. J. Mech. Sci. Tech. 26(11), 3555–3563 (2012)CrossRefGoogle Scholar
  18. 18.
    Pradhan S.C.: Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finite Elem. Anal. Des. 50, 8–20 (2012)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefMathSciNetGoogle Scholar
  20. 20.
    De Rosa M.A., Franciosi C.: A simple approach to detect the nonlocal effects in the static analysis of Euler–Bernoulli and Timoshenko beams. Mech. Res. Commun. 48, 66–69 (2013)CrossRefGoogle Scholar
  21. 21.
    Eltaher M.A., Alshorbagy A.E., Mahmoud F.F.: Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37, 4787–4797 (2013)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Eltaher M.A., Emam S.A, Mahmoud F.F.: Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88 (2013)CrossRefGoogle Scholar
  23. 23.
    Emam, S.A.: A general nonlocal nonlinear model for buckling of nanobeams. Appl. Math. Modelling. 37(10–11), 6929–6939 (2013)Google Scholar
  24. 24.
    Fang B., Zhen Y-X., Zhang C-P., Tang Y.: Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Appl. Math. Model. 37, 1096–1107 (2013)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Ghannadpour S.A.M., Mohammadi B., Fazilati J.: Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Compos. Struct. 96, 584–589 (2013)CrossRefGoogle Scholar
  26. 26.
    Şimşek M., Yurtcu H.H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013)CrossRefGoogle Scholar
  27. 27.
    Wang B.L., Wang K.F.: Vibration analysis of embedded nanotubes using nonlocal continuum theory. Composites: Part B 47, 96–101 (2013)CrossRefGoogle Scholar
  28. 28.
    Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)CrossRefGoogle Scholar
  29. 29.
    Reddy J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)CrossRefzbMATHGoogle Scholar
  30. 30.
    Romano G., Diaco M., Barretta R.: Variational formulation of the first principle of continuum thermodynamics. Continuum Mech. Thermodyn. 22(3), 177–187 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Barretta R.: Analogies between Kirchhoff plates and Saint–Venant beams under torsion. Acta Mech. 224, 2955–2964 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Romano G., Barretta R., Barretta A.: On Maupertuis principle in dynamics. Rep. Math. Phys. 63(3), 331–346 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Romano G., Barretta R., Diaco M.: Algorithmic tangent stiffness in elastoplasticity and elastoviscoplasticity: a geometric insight. Mech. Res. Comm. 37(3), 289–292 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Romano G., Barretta R.: Covariant hypo-elasticity. Eur. J. Mech. A-Solids 30(6), 1012–1023 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Romano G., Barretta R.: On Euler’s stretching formula in continuum mechanics. Acta Mech. 224, 211–230 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Romano G., Barretta R.: Geometric constitutive theory and frame invariance. Int. J. Non-linear Mech. 51, 75–86 (2013)CrossRefGoogle Scholar
  37. 37.
    Barretta R.: On the relative position of twist and shear centres in the orthotropic and fiberw homogeneous SaintVenant beam theory. Int. J. Solids Struct. 49, 3038–3046 (2012)CrossRefGoogle Scholar
  38. 38.
    Barretta R.: On Cesàro–Volterra method in orthotropic Saint-Venant beam. J. Elast. 112, 233–253 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Irschik H.: Analogies between bending of plates and torsion problem. J. Eng. Mech. 117(11), 2503–2508 (1991)CrossRefGoogle Scholar
  40. 40.
    Irschik H.: Analogy between refined beam theories and the Bernoulli–Euler theory. Int. J. Solids Struct. 28(9), 1105–1112 (1991)CrossRefzbMATHGoogle Scholar
  41. 41.
    Furukawa T., Irschik H.: Body-force analogy for one-dimensional coupled dynamic problems of thermoelasticity. J. Therm. Stresses 28(4), 455–464 (2005)CrossRefGoogle Scholar
  42. 42.
    Irschik H., Gusenbauer M.: Body force analogy for transient thermal stresses. J. Therm. Stresses 30(9-10), 965–975 (2007)CrossRefGoogle Scholar
  43. 43.
    Irschik H., Krommer M., Zehetner C.: A generalized body force analogy for the dynamic theory of thermoelasticity. J. Therm. Stresses 35(1–3), 235–247 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  • Raffaele Barretta
    • 1
  • Francesco Marotti de Sciarra
    • 1
  • Marina Diaco
    • 1
  1. 1.Department of Structures for Engineering and ArchitectureUniversity of Naples Federico IINaplesItaly

Personalised recommendations