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Nonlinear bending and postbuckling analysis of FG nanoscale beams using the two-phase fractional nonlocal continuum mechanics

  • M. Faraji Oskouie
  • R. Ansari
  • H. RouhiEmail author
Regular Article
  • 22 Downloads

Abstract.

The geometrically nonlinear bending and postbuckling of nanoscale beams are investigated herein according to the two-phase fractional nonlocal continuum model. It is considered that the beams have been made from functionally graded materials (FGMs), and the Bernoulli-Euler beam model is employed for their modeling. The variational form of the governing fractional equation is obtained first by means of an energy approach. Thereafter, a novel numerical solution method is proposed named as fractional variational differential quadrature method (FVDQM). In FVDQM, which is applied to the variational statement of the problem in a direct way, a combination of the differential quadrature method and matrix operators is utilized. The efficiency of the proposed fractional nonlocal model is evaluated by molecular dynamics (MD) simulations. Selected numerical results are given to explore the influences of fractional order, nonlocality, length-to-thickness ratio and FG index on the nonlinear bending and postbuckling responses of FG nanobeams with various types of boundary conditions.

References

  1. 1.
    U. Andreaus, L. Placidi, G. Rega, Commun. Nonlinear Sci. Numer. Simul. 15, 2603 (2010)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    U. Andreaus, L. Placidi, G. Rega, Proc. IMechE, Part C 225, 2444 (2011)CrossRefGoogle Scholar
  3. 3.
    U. Andreaus, B. Chiaia, L. Placidi, Continuum Mech. Thermodyn. 25, 375 (2013)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    U. Andreaus, L. Placidi, G. Rega, J. Appl. Phys. 113, 224302 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    U. Andreaus, B. Baragatti, L. Placidi, Int. J. Non-Linear Mech. 80, 96 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    E. Kröner, Int. J. Solids Struct. 3, 731 (1967)CrossRefGoogle Scholar
  7. 7.
    J. Krumhansl, IUTAM Symposia (Springer, Berlin, Heidelberg, 1968) pp. 298--311Google Scholar
  8. 8.
    I.A. Kunin (Editor), The theory of elastic media with microstructure and the theory of dislocations, in IUTAM Symposia (Springer, Berlin, Heidelberg, 1968) pp. 321--329Google Scholar
  9. 9.
    A.C. Eringen, Int. J. Eng. Sci. 10, 1 (1972)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.C. Eringen, D.G.B. Edelen, Int. J. Eng. Sci. 10, 233 (1972)CrossRefGoogle Scholar
  11. 11.
    H. Rouhi, R. Ansari, Nano 7, 1250018 (2012)CrossRefGoogle Scholar
  12. 12.
    R. Ansari, A. Shahabodini, H. Rouhi, Curr. Appl. Phys. 15, 1062 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    H.S. Shen, Y.M. Xu, C.L. Zhang, Comput. Meth. Appl. Mech. Eng. 267, 458 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    R. Ansari, H. Rouhi, S. Sahmani, Int. J. Mech. Sci. 53, 786 (2011)CrossRefGoogle Scholar
  15. 15.
    Y. Liang, Q. Han, Eur. J. Mech. - A 45, 153 (2014)CrossRefGoogle Scholar
  16. 16.
    H.T. Thai, T.P. Vo, Int. J. Eng. Sci. 54, 58 (2012)CrossRefGoogle Scholar
  17. 17.
    Y.S. Li, P. Ma, W. Wang, J. Intell. Mater. Sys. Struct. 27, 1139 (2016)CrossRefGoogle Scholar
  18. 18.
    M.A. Maneshi, E. GHavanloo, S.A. Fazelzadeh, Eur. Phys. J. Plus 133, 256 (2018)CrossRefGoogle Scholar
  19. 19.
    Y.Z. Wang, F.M. Li, K. Kishimoto, ASME J. Vib. Acoust. 134, 031011 (2012)CrossRefGoogle Scholar
  20. 20.
    R. Ansari, R. Gholami, H. Rouhi, Compos. Struct. 126, 216 (2015)CrossRefGoogle Scholar
  21. 21.
    M. Arefi, P. Pourjamshidian, A. Ghorbanpour Arani, Eur. Phys. J. Plus 133, 193 (2018)CrossRefGoogle Scholar
  22. 22.
    R. Ansari, M. Faraji Oskouie, R. Gholami, Physica E 75, 266 (2016)ADSCrossRefGoogle Scholar
  23. 23.
    L.L. Ke, Y.S. Wang, Physica E 63, 52 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    F. Ebrahimi, M.R. Barati, Arab. J. Sci. Eng. 41, 1679 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    R. Ansari, M. Faraji Oskouie, R. Gholami, F. Sadeghi, Composites Part B 89, 316 (2016)CrossRefGoogle Scholar
  26. 26.
    K. Kiani, Appl. Math. Model. 37, 1836 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    R. Ansari, M. Faghih Shojaei, V. Mohammadi, R. Gholami, H. Rouhi, Z. Angew. Math. Mech. 95, 939 (2015)CrossRefGoogle Scholar
  28. 28.
    S. Sahmani, R. Ansari, J. Mech. Sci. Technol. 25, 1 (2011)CrossRefGoogle Scholar
  29. 29.
    A. Ghorbanpour Arani, M. Ghaffari, A. Jalilvand, R. Kolahchi, Acta Mech. 224, 3005 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M.T. Ahmadian, A. Pasharavesh, A. Fallah, in Proceedings of the 5th International Conference on Micro Nanosystems (ASME, 2011) Paper No. DETC2011-48862, pp. 255--261,  https://doi.org/10.1115/DETC2011-48862
  31. 31.
    Y.-G. Wang, H.-F. Song, W.-H. Lin, J. Mech. 32, 737 (2016)CrossRefGoogle Scholar
  32. 32.
    J. Fernández-Sáez, R. Zaera, J.A. Loya, J.N. Reddy, Int. J. Eng. Sci. 99, 107 (2016)CrossRefGoogle Scholar
  33. 33.
    M. Faraji Oskouie, A. Norouzzadeh, R. Ansari, H. Rouhi, Appl. Math. Mech. 40, 767 (2019)CrossRefGoogle Scholar
  34. 34.
    G. Romano, R. Barretta, Int. J. Eng. Sci. 115, 14 (2017)CrossRefGoogle Scholar
  35. 35.
    G. Romano, R. Barretta, Compos. Part B 114, 184 (2017)CrossRefGoogle Scholar
  36. 36.
    G. Romano, R. Barretta, M. Diaco, Int. J. Mech. Sci. 131-132, 490 (2017)CrossRefGoogle Scholar
  37. 37.
    R. Barretta, M. Canajija, R. Luciano, F. Marottide Sciarra, Int. J. Eng. Sci. 126, 53 (2018)CrossRefGoogle Scholar
  38. 38.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Eur. Phys. J. Plus 133, 336 (2018)CrossRefGoogle Scholar
  39. 39.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Acta Mech. Sin. 34, 871 (2018)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Int. J. Comput. Mater. Sci. Eng. 7, 1850016 (2018)Google Scholar
  41. 41.
    R. Barretta, F. Fabbrocino, R. Luciano, F. Marotti de Sciarra, G. Ruta, Mech. Adv. Mater. Struct. (2019)  https://doi.org/10.1080/15376494.2018.1501523
  42. 42.
    R. Barretta, S.A. Faghidian, F. Marotti de Sciarra, Int. J. Eng. Sci. 136, 38 (2019)CrossRefGoogle Scholar
  43. 43.
    R. Barretta, A. Caporale, S.A. Faghidian, R. Luciano, F. Marotti de Sciarra, C.M. Medaglia, Compos. Part B 164, 590 (2019)CrossRefGoogle Scholar
  44. 44.
    A. Apuzzo, R. Barretta, F. Fabbrocino, S.A. Faghidian, R. Luciano, F. Marotti de Sciarra, J. Appl. Comput. Mech. 5, 402 (2019)Google Scholar
  45. 45.
    N. Heymans, J. Vib. Control 14, 1587 (2008)CrossRefGoogle Scholar
  46. 46.
    M.P. Lazarević, Mech. Res. Commun. 33, 269 (2006)CrossRefGoogle Scholar
  47. 47.
    D. Sierociuk, D. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, T. Skovranek, Philos. Trans. R. Soc. A 371, 20130146 (2013)CrossRefGoogle Scholar
  48. 48.
    R. Ansari, M. Faraji Oskouie, F. Sadeghi, M. Bazdid-Vahdati, Physica E 74, 318 (2015)ADSCrossRefGoogle Scholar
  49. 49.
    M. Faraji Oskouie, R. Ansari, F. Sadeghi, Acta Mech. Solida Sin. 30, 416 (2017)CrossRefGoogle Scholar
  50. 50.
    W.M. Ahmad, R. El-Khazali, Chaos, Solitons Fractals 33, 1367 (2007)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    G.S. Frederico, D.F. Torres, Nonlinear Dyn. 53, 215 (2008)CrossRefGoogle Scholar
  52. 52.
    V.S. Gubenko, J. Appl. Math. Mech. 21, 279 (1957)MathSciNetGoogle Scholar
  53. 53.
    N.A. Rostovtsev, J. Appl. Math. Mech. 23, 1143 (1959)MathSciNetCrossRefGoogle Scholar
  54. 54.
    K.A. Lazopoulos, Mech. Res. Commun. 33, 753 (2006)CrossRefGoogle Scholar
  55. 55.
    G. Cottone, M. Di Paola, M. Zingales, in Proceedings of the 6th WESEAS International Conference, Cairo, Egypt, Dec. 29--31 (2007) pp. 81--89Google Scholar
  56. 56.
    M. Di Paola, M. Zingales, Int. J. Solids Struct. 45, 5642 (2008)CrossRefGoogle Scholar
  57. 57.
    G. Alotta, M. Di Paola, G. Failla, F.P. Pinnola, Compos. Part B 137, 102 (2018)CrossRefGoogle Scholar
  58. 58.
    G. Alotta, G. Failla, F. Paolo Pinnola, J. Risk Uncertain. Eng. Syst. B 3, 030904 (2017)CrossRefGoogle Scholar
  59. 59.
    A. Carpinteri, P. Cornetti, A. Sapora, M. Di Paola, M. Zingales, Phys. Scr. T136, 014003 (2009)ADSCrossRefGoogle Scholar
  60. 60.
    T.M. Atanackovic, B. Stankovic, Acta Mech. 208, 1 (2009)CrossRefGoogle Scholar
  61. 61.
    A. Carpinteri, P. Cornetti, A. Sapora, Eur. Phys. J. ST 193, 193 (2011)CrossRefGoogle Scholar
  62. 62.
    C.S. Drapaca, S. Sivaloganathan, J. Elast. 107, 105 (2012)CrossRefGoogle Scholar
  63. 63.
    A. Sapora, P. Cornetti, A. Carpinteri, Commun. Nonlinear Sci. Numer. Simul. 18, 63 (2013)ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    N. Challamel, D. Zorica, T.M. Atanackovic, D.T. Spasic, C. R. Mec. 341, 298 (2013)ADSCrossRefGoogle Scholar
  65. 65.
    A. Carpinteri, P. Cornetti, A. Sapora, Meccanica 49, 2551 (2014)MathSciNetCrossRefGoogle Scholar
  66. 66.
    W. Sumelka, Acta Mech. 225, 3247 (2014)MathSciNetCrossRefGoogle Scholar
  67. 67.
    W. Sumelka, Arc. Civ. Mech. Eng. 15, 231 (2015)CrossRefGoogle Scholar
  68. 68.
    V.E. Tarasov, J. King Saud Univ. Sci. 28, 33 (2016)CrossRefGoogle Scholar
  69. 69.
    V.E. Tarasov, Fractional Nonlocal Continuum Mechanics and Microstructural Models, in Handbook of Nonlocal Continuum Mechanics for Materials and Structures, edited by G. Voyiadjis (Springer, Cham, 2017)Google Scholar
  70. 70.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Meccanica 53, 1115 (2018)MathSciNetCrossRefGoogle Scholar
  71. 71.
    M. Faraji Oskouie, R. Ansari, H. Rouhi, Microsyst. Technol. 24, 2775 (2018)CrossRefGoogle Scholar
  72. 72.
    A.H. Bhrawy, T.M. Taha, J.A. Tenreiro Machado, Nonlinear Dyn. 81, 1023 (2015)CrossRefGoogle Scholar
  73. 73.
    A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)MathSciNetCrossRefGoogle Scholar
  74. 74.
    E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011)MathSciNetCrossRefGoogle Scholar
  75. 75.
    M. Lakestani, M. Dehghan, S. Irandoust-pakchin, Commun. Nonlinear Sci. Numer. Simul. 17, 1149 (2012)ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    M.H. Akrami, M.H. Atabakzadeh, G.H. Erjaee, Int. J. Sci. Techol. 37A4, 439 (2013)Google Scholar
  77. 77.
    A. Saadatmandi, Appl. Math. Model. 38, 1365 (2014)MathSciNetCrossRefGoogle Scholar
  78. 78.
    L. Wang, Y. Ma, Z. Meng, Appl. Math. Comput. 227, 66 (2014)MathSciNetGoogle Scholar
  79. 79.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)Google Scholar
  80. 80.
    G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach, Amsterdam, 1993)Google Scholar
  81. 81.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  82. 82.
    O.P. Agrawal, J. Phys. A 40, 6287 (2007)ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    M. Faghih Shojaei, R. Ansari, Appl. Math. Model. 49, 705 (2017)MathSciNetCrossRefGoogle Scholar
  84. 84.
    R. Ansari, M. Faghih Shojaei, R. Gholami, Compos. Struct. 136, 669 (2016)CrossRefGoogle Scholar
  85. 85.
    R. Ansari, M. Faghih Shojaei, A. Shahabodini, M. Bazdid-Vahdati, Compos. Struct. 131, 753 (2015)CrossRefGoogle Scholar
  86. 86.
    R. Ansari, A. Shahabodini, M. Faghih Shojaei, Compos. Struct. 139, 167 (2016)CrossRefGoogle Scholar
  87. 87.
    R. Ansari, J. Torabi, M. Faghih Shojaei, Eur. J. Mech. - A 60, 166 (2016)CrossRefGoogle Scholar
  88. 88.
    R. Ansari, S. Sahmani, H. Rouhi, Phys. Lett. A 375, 1255 (2011)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

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

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