Advertisement

Stress-driven nonlocal and strain gradient formulations of Timoshenko nanobeams

  • M. Faraji Oskouie
  • R. AnsariEmail author
  • H. Rouhi
Regular Article

Abstract.

In this paper, three size-dependent formulations are developed for the analysis of Timoshenko nanobeams with various end conditions based on the nonlocal and strain gradient theories. The nonlocal governing equations are presented based on the stress-driven model of Eringen’s theory. First, a strain gradient Timoshenko beam model is developed. The governing equations of the integral stress-driven model, and then those of differential stress-driven model together with associated constitutive boundary conditions are obtained in the next step. With the aim of addressing the static bending and free vibration problems, the nonlocal governing equations in integral form are directly solved by constructing matrix differential and integral operators. Furthermore, the governing equations in differential form together with constitutive boundary conditions are discretized and solved via the mentioned operators. It is shown that there is a good agreement between the results obtained from solving the integral and differential governing equations of stress-driven nonlocal models. Several comparative studies are also conducted for the bending and vibration analyses of nanobeams based on the strain gradient and stress-driven nonlocal models. The results reveal that in both models, increasing the nonlocal/length scale parameter has a stiffening effect on the response of the system. However, the stiffening effect corresponding to the strain gradient model is more pronounced than that corresponding to the stress-driven nonlocal model.

References

  1. 1.
    A.C. Eringen, Int. J. Eng. Sci. 10, 1 (1972)CrossRefGoogle Scholar
  2. 2.
    A.C. Eringen, D.G.B. Edelen, Int. J. Eng. Sci. 10, 233 (1972)CrossRefGoogle Scholar
  3. 3.
    A.C. Eringen, Int. J. Eng. Sci. 10, 425 (1972)CrossRefGoogle Scholar
  4. 4.
    A.C. Eringen, J. Appl. Phys. 54, 4703 (1983)ADSCrossRefGoogle Scholar
  5. 5.
    A.C. Eringen, Int. J. Eng. Sci. 30, 1551 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Engelbrecht, M. Braun, Appl. Mech. Rev. 51, 475 (1998)ADSCrossRefGoogle Scholar
  7. 7.
    A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002)Google Scholar
  8. 8.
    J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41, 305 (2003)CrossRefGoogle Scholar
  9. 9.
    A.E. Alshorbagy, M.A. Eltaher, F.F. Mahmoud, J. Mech. Sci. Technol. 27, 2035 (2013)CrossRefGoogle Scholar
  10. 10.
    R. Ansari, R. Gholami, H. Rouhi, Compos. Struct. 126, 216 (2015)CrossRefGoogle Scholar
  11. 11.
    F. Ebrahimi, M.R. Barati, Eur. Phys. J. Plus 131, 279 (2016)CrossRefGoogle Scholar
  12. 12.
    R. Ansari, M. Faghih Shojaei, V. Mohammadi, R. Gholami, H. Rouhi, Z. Angew. Math. Mech. 95, 939 (2014)CrossRefGoogle Scholar
  13. 13.
    C. Liu, L.L. Ke, J. Yang, S. Kitipornchai, Y.S. Wang, Mech. Adv. Mater. Struct. (2016)  https://doi.org/10.1080/15376494.2016.1149648
  14. 14.
    R. Ansari, A. Shahabodini, H. Rouhi, Curr. Appl. Phys. 15, 1062 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    E. Khanmirza, A. Jamalpoor, A. Kiani, Eur. Phys. J. Plus 132, 422 (2017)CrossRefGoogle Scholar
  16. 16.
    R. Ansari, B. Arash, H. Rouhi, Compos. Struct. 93, 2419 (2011)CrossRefGoogle Scholar
  17. 17.
    S. Sarrami-Foroushani, M. Azhari, Acta Mech. 227, 721 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Ansari, M. Faghih Shojaei, A. Shahabodini, M. Bazdid-Vahdati, Compos. Struct. 131, 753 (2015)CrossRefGoogle Scholar
  19. 19.
    A. Farajpour, M.R. Hairi Yazdi, A. Rastgoo, M. Loghmani, M. Mohammadi, Compos. Struct. 140, 323 (2016)CrossRefGoogle Scholar
  20. 20.
    R. Ansari, H. Rouhi, Solid State Commun. 152, 56 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    R. Ansari, A. Shahabodini, H. Rouhi, Compos. Struct. 95, 88 (2013)CrossRefGoogle Scholar
  22. 22.
    A. Shakouri, R.M. Lin, T.Y. Ng, J. Appl. Phys. 106, 094307 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    R. Ansari, H. Rouhi, M. Mirnezhad, Curr. Appl. Phys. 14, 1360 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    K. Kiani, Appl. Math. Model. 37, 1836 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Natsuki, N. Matsuyama, Q.Q. Ni, Appl. Phys. A 120, 1309 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    H. Rouhi, R. Ansari, NANO 7, 1250018 (2012)CrossRefGoogle Scholar
  27. 27.
    H.S. Shen, C.L. Zhang, Compos. Struct. 92, 1073 (2010)CrossRefGoogle Scholar
  28. 28.
    R. Ansari, A. Shahabodini, H. Rouhi, A. Alipour, J. Therm. Stresses 36, 56 (2013)CrossRefGoogle Scholar
  29. 29.
    B. Wang, Z. Deng, K. Zhang, J. Zhou, Multidiscip. Model. Mater. Struct. 8, 432 (2012)CrossRefGoogle Scholar
  30. 30.
    R. Ansari, H. Rouhi, B. Arash, Iran. J. Sci. Technol. Trans. Mech. Eng. 37, 91 (2013)Google Scholar
  31. 31.
    A. Ghorbanpour Arani, R. Kolahchi, Proc. Inst. Mech. Eng., Part C 228, 366 (2014)CrossRefGoogle Scholar
  32. 32.
    Q. Wang, K.M. Liew, Phys. Lett. A 363, 236 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    N. Challamel, C. Wang, Nanotechnology 19, 345703 (2008)CrossRefGoogle Scholar
  34. 34.
    N. Challamel, Z. Zhang, C. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, Arch. Appl. Mech. 84, 1275 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    J. Fernández-Sáez, R. Zaera, J. Loya, J.N. Reddy, Int. J. Eng. Sci. 99, 107 (2016)CrossRefGoogle Scholar
  36. 36.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Meccanica 53, 1115 (2018)MathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Microsyst. Technol. 24, 2775 (2018)CrossRefGoogle Scholar
  38. 38.
    G. Romano, R. Barretta, M. Diaco, F. Marotti de Sciarra, Int. J. Mech. Sci. 121, 151 (2017)CrossRefGoogle Scholar
  39. 39.
    J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41, 305 (2003)CrossRefGoogle Scholar
  40. 40.
    G. Romano, R. Barretta, Int. J. Eng. Sci. 115, 14 (2017)CrossRefGoogle Scholar
  41. 41.
    G. Romano, R. Barretta, Compos. Part B 114, 184 (2017)CrossRefGoogle Scholar
  42. 42.
    G. Romano, R. Barretta, M. Diaco, Int. J. Mech. Sci. 131-132, 490 (2017)CrossRefGoogle Scholar
  43. 43.
    R. Barretta, L. Feo, R. Luciano, F. Marotti de Sciarra, R. Penna, PSU Res. Rev. 1, 164 (2017)CrossRefGoogle Scholar
  44. 44.
    A. Apuzzo, R. Barretta, R. Luciano, F. Marotti de Sciarra, R. Penna, Compos. Part B: Eng. 123, 105 (2017)CrossRefGoogle Scholar
  45. 45.
    R.D. Mindlin, Arch. Ration. Mech. Anal. 6, 51 (1964)CrossRefGoogle Scholar
  46. 46.
    R.D. Mindlin, Int. J. Solids Struct. 1, 417 (1965)CrossRefGoogle Scholar
  47. 47.
    D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, J. Mech. Phys. Solids 51, 1477 (2003)ADSCrossRefGoogle Scholar
  48. 48.
    K.A. Lazopoulos, A.K. Lazopoulos, Eur. J. Mech. - A/Solids 29, 837 (2010)ADSCrossRefGoogle Scholar
  49. 49.
    R. Ansari, R. Gholami, S. Sahmani, Compos. Struct. 94, 221 (2011)CrossRefGoogle Scholar
  50. 50.
    M.H. Kahrobaiyan, M. Asghari, M. Rahaeifard, M.T. Ahmadian, Int. J. Eng. Sci. 49, 1256 (2011)CrossRefGoogle Scholar
  51. 51.
    N. Challamel, Compos. Struct. 105, 351 (2013)CrossRefGoogle Scholar
  52. 52.
    R. Ansari, M. Faraji Oskouie, H. Rouhi, Nonlinear Dyn. 87, 695 (2017)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran
  2. 2.Department of Engineering Science, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran

Personalised recommendations