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Nonlinear bending analysis of nanoplates made of FGMs based on the most general strain gradient model and 3D elasticity theory

  • Y. Gholami
  • R. AnsariEmail author
  • R. Gholami
  • H. Rouhi
Regular Article
  • 30 Downloads

Abstract.

In this article, the nonlinear bending behavior of rectangular nanoplates made of functionally graded materials (FGMs) is studied in the context of a variational formulation. To capture size effects, the most general form of strain gradient theory is employed. The three-dimensional (3D) elasticity theory is used for modeling the nanoplate. The governing equations are also derived in the discretized weak form using the variational differential quadrature (VDQ) method. Finally, the solution of the nonlinear bending problem is obtained by the pseudo arc-length continuation algorithm. In the numerical results, the effects of thickness-to-length scale ratio, side length-to-thickness ratio and material gradient index on the nonlinear bending response of nanoplates subject to different types of boundary conditions are analyzed. Moreover, a comparison is provided between the predictions of various strain gradient-based theories.

References

  1. 1.
    M. Birkholz, K.E. Ehwald, P. Kulse, J. Drews, M. Fröhlich, U. Haak et al., Adv. Funct. Mater. 21, 1652 (2011)CrossRefGoogle Scholar
  2. 2.
    E. Hosseinian, O.N. Pierron, Nanoscale 5, 12532 (2013)ADSCrossRefGoogle Scholar
  3. 3.
    J.S. Bunch, A.M. Van Der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbaum, J.M. Parpia et al., Science 315, 490 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    R. Barretta, R. Luciano, F. Marotti de Sciarra, Math. Probl. Eng. 2015, 495095 (2015)CrossRefGoogle Scholar
  5. 5.
    M. Čanadija, R. Barretta, F.M. De Sciarra, Eur. J. Mech. A-Solid 55, 243 (2016)CrossRefGoogle Scholar
  6. 6.
    R. Barretta, M. Brčić, M. Čanadija, R. Luciano, F.M. de Sciarra, Eur. J. Mech. A-Solid 65, 1 (2017)ADSCrossRefGoogle Scholar
  7. 7.
    N. Fleck, G. Muller, M. Ashby, J. Hutchinson, Acta Metall. Mater. 42, 475 (1994)CrossRefGoogle Scholar
  8. 8.
    J.S. Stölken, A.G. Evans, Acta Mater. 46, 5109 (1998)CrossRefGoogle Scholar
  9. 9.
    A.W. McFarland, J.S. Colton, J. Micromech. Microeng. 15, 1060 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    R.D. Mindlin, H.F. Tiersten, Arch. Ration. Mech. Anal. 11, 415 (1962)CrossRefGoogle Scholar
  11. 11.
    W.T. Koiter, Nederl. Akad. Wetensch. Proc. Ser. B 67, 17 (1964)MathSciNetGoogle Scholar
  12. 12.
    R.D. Mindlin, Arch. Ration. Mech. Anal. 16, 51 (1964)CrossRefGoogle Scholar
  13. 13.
    R.D. Mindlin, Int. J. Solids Struct. 1, 417 (1965)CrossRefGoogle Scholar
  14. 14.
    F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Int. J. Solids Struct. 39, 2731 (2002)CrossRefGoogle Scholar
  15. 15.
    D. Lam, F. Yang, A. Chong, J. Wang, P. Tong, J. Mech. Phys. Solids 51, 1477 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    R. Gholami, A. Darvizeh, R. Ansari, F. Sadeghi, Eur. J. Mech. A-Solid 58, 76 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    R. Ansari, M.F. Shojaei, F. Ebrahimi, H. Rouhi, Arch. Appl. Mech. 85, 937 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    B.R. Goncalves, A. Karttunen, J. Romanoff, J. Reddy, Compos. Struct. 165, 233 (2017)CrossRefGoogle Scholar
  19. 19.
    M. Tang, Q. Ni, L. Wang, Y. Luo, Y. Wang, Int. J. Eng. Sci. 85, 20 (2014)CrossRefGoogle Scholar
  20. 20.
    R. Ansari, R. Gholami, H. Rouhi, J. Vib. Control 19, 708 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M.M. Adeli, A. Hadi, M. Hosseini, H.H. Gorgani, Eur. Phys. J. Plus 132, 393 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    M.H. Ghayesh, M. Amabili, H. Farokhi, Int. J. Eng. Sci. 63, 52 (2013)CrossRefGoogle Scholar
  23. 23.
    R. Ansari, R. Gholami, H. Rouhi, Compos. B: Eng. 43, 2985 (2012)CrossRefGoogle Scholar
  24. 24.
    B. Akgöz, Ö. Civalek, Compos. Struct. 98, 314 (2013)CrossRefGoogle Scholar
  25. 25.
    R. Ansari, M.F. Shojaei, H. Rouhi, Eur. J. Mech. A-Solid 53, 19 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    F. Alinaghizadeh, M. Shariati, J. Fish, Appl. Math. Model. 44, 540 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. Ansari, M.F. Shojaei, V. Mohammadi, M. Bazdid-Vahdati, H. Rouhi, Comput. Methods Appl. Mech. Eng. 295, 56 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    M. Simşek, Compos. Struct. 112, 264 (2014)CrossRefGoogle Scholar
  29. 29.
    R. Ansari, M.F. Shojaei, A. Shakouri, H. Rouhi, J. Comput. Nonlin. Dyn. 11, 051014 (2016)CrossRefGoogle Scholar
  30. 30.
    M. Faghih Shojaei, R. Ansari, Appl. Math. Model. 49, 705 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory, Proc. Advanced Sem., Univ. Wisconsin, Madison (Academic Press, New York, 1977) pp. 359--384Google Scholar
  32. 32.
    M. Fares, M.K. Elmarghany, D. Atta, Compos. Struct. 91, 296 (2009)CrossRefGoogle Scholar
  33. 33.
    J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics (John Wiley & Sons, 2017)Google Scholar
  34. 34.
    J.N. Reddy, Theory and Analysis of Elastic Plates and Shells (CRC Press, 2006)Google Scholar
  35. 35.
    C. Shu, Differential Quadrature and Its Application in Engineering (Springer Science & Business Media, London, 2012)Google Scholar
  36. 36.
    M. Ganapathi, Compos. Struct. 79, 338 (2007)CrossRefGoogle Scholar
  37. 37.
    H. Matsunaga, Compos. Struct. 82, 499 (2008)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran
  2. 2.Department of Mechanical Engineering, Lahijan BranchIslamic Azad UniversityLahijanIran
  3. 3.Department of Engineering Science, Faculty of Technology and Engineering, East of GuilanUniversity of GuilanRudsar-VajargahIran

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