Nonlinear vibration and buckling of functionally graded porous nanoscaled beams

  • Seyed Sajad Mirjavadi
  • Behzad Mohasel Afshari
  • Mohammad Khezel
  • Navvab ShafieiEmail author
  • Samira Rabby
  • Morteza Kordnejad
Technical Paper


Although many researchers have studied the vibration and buckling behavior of porous materials, the behavior of porous nanobeams is still a needed issue to be studied. This paper is focused on the buckling and nonlinear vibration of functionally graded (FG) porous nanobeam for the first time. Nonlinear Von Kármán strains are put into consideration to study the nonlinear behavior of nanobeam based on the Euler–Bernoulli beam theory. The nonlocal Eringen’s theory is used to study the size effects. The mechanical properties of ceramic and metal are used to model the functionally graded material through thickness, and the boundary conditions are considered as clamped–clamped (CC) and simply supported–simply supported (SS). The generalized differential quadrature method (GDQM) is used in conjunction with the iterative method to solve the equations. The parametric study is done to examine the effects of nonlinearity, porosity, sized effect, FG index, etc., on the vibration and buckling of porous nanobeam.


Nonlinear vibration Functionally graded Nonlocal nanobeam Porous GDQM 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Laiva AL et al (2014) Novel and simple methodology to fabricate porous and buckled fibrous structures for biomedical applications. Polymer 55(22):5837–5842CrossRefGoogle Scholar
  2. 2.
    Jiang G et al (2016) Characterization and investigation of the deformation behavior of porous magnesium scaffolds with entangled architectured pore channels. J Mech Behav Biomed Mater 64:139–150CrossRefGoogle Scholar
  3. 3.
    Bender S et al (2012) Mechanical characterization and modeling of graded porous stainless steel specimens for possible bone implant applications. Int J Eng Sci 53:67–73CrossRefGoogle Scholar
  4. 4.
    Li W et al (2015) Cell wall buckling mediated energy absorption in lotus-type porous copper. J Mater Sci Technol 31(10):1018–1026CrossRefGoogle Scholar
  5. 5.
    Joubaneh EF et al (2014) Thermal buckling analysis of porous circular plate with piezoelectric sensor–actuator layers under uniform thermal load. J Sandw Struct Mater 17(1):3–25MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pei-Sheng L (2010) Analyses of buckling failure mode for porous materials under compression. Acta Phys Sin 12:071Google Scholar
  7. 7.
    Chen D, Yang J, Kitipornchai S (2015) Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 133:54–61CrossRefGoogle Scholar
  8. 8.
    Magnucki K, Malinowski M, Kasprzak J (2006) Bending and buckling of a rectangular porous plate. Steel Compos Struct 6(4):319–333CrossRefGoogle Scholar
  9. 9.
    Magnucki K, Stasiewicz P (2004) Elastic buckling of a porous beam. J Theor Appl Mech 42(4):859–868zbMATHGoogle Scholar
  10. 10.
    Magnucka-Blandzi E (2008) Axi-symmetrical deflection and buckling of circular porous-cellular plate. Thin-Walled Struct 46(3):333–337CrossRefGoogle Scholar
  11. 11.
    Jabbari M et al (2013) Buckling analysis of a functionally graded thin circular plate made of saturated porous materials. J Eng Mech 140(2):287–295MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jabbari M et al (2014) Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials. J Therm Stresses 37(2):202–220CrossRefGoogle Scholar
  13. 13.
    Jabbari M et al (2013) Buckling analysis of porous circular plate with piezoelectric actuator layers under uniform radial compression. Int J Mech Sci 70:50–56CrossRefGoogle Scholar
  14. 14.
    Simone A, Gibson L (1997) The compressive behaviour of porous copper made by the GASAR process. J Mater Sci 32(2):451–457CrossRefGoogle Scholar
  15. 15.
    Jabbari M, Joubaneh EF, Mojahedin A (2014) Thermal buckling analysis of porous circular plate with piezoelectric actuators based on first order shear deformation theory. Int J Mech Sci 83:57–64CrossRefzbMATHGoogle Scholar
  16. 16.
    Leclaire P, Horoshenkov K, Cummings A (2001) Transverse vibrations of a thin rectangular porous plate saturated by a fluid. J Sound Vib 247(1):1–18CrossRefGoogle Scholar
  17. 17.
    Jabbari M, Mojahedin A, Haghi M (2014) Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field. Thin-Walled Struct 85:50–56CrossRefGoogle Scholar
  18. 18.
    Liu P (2011) Failure by buckling mode of the pore-strut for isotropic three-dimensional reticulated porous metal foams under different compressive loads. Mater Des 32(6):3493–3498CrossRefGoogle Scholar
  19. 19.
    Amirkhani S, Bagheri R, Yazdi AZ (2012) Effect of pore geometry and loading direction on deformation mechanism of rapid prototyped scaffolds. Acta Mater 60(6):2778–2789CrossRefGoogle Scholar
  20. 20.
    Li F et al (2015) Fabrication, pore structure and compressive behavior of anisotropic porous titanium for human trabecular bone implant applications. J Mech Behav Biomed Mater 46:104–114CrossRefGoogle Scholar
  21. 21.
    Barati M, Sadr M, Zenkour A (2016) Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation. Int J Mech Sci 117:309–320CrossRefGoogle Scholar
  22. 22.
    Xia R et al (2011) Surface effects on the mechanical properties of nanoporous materials. Nanotechnology 22(26):265714CrossRefGoogle Scholar
  23. 23.
    Yu YJ et al (2016) Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity. Compos Struct 146:108–113CrossRefGoogle Scholar
  24. 24.
    Shen H-S, Xiang Y (2013) Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment. Compos B Eng 52:311–322CrossRefGoogle Scholar
  25. 25.
    Arani AG et al (2012) Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory. Compos B Eng 43(2):195–203MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mohammadabadi M, Daneshmehr A, Homayounfard M (2015) Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory. Int J Eng Sci 92:47–62MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kiani K (2016) Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy. Int J Eng Sci 106:57–76MathSciNetCrossRefGoogle Scholar
  28. 28.
    Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 41(9):1651–1655CrossRefGoogle Scholar
  29. 29.
    Thai H-T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang G-F, Feng X-Q (2009) Surface effects on buckling of nanowires under uniaxial compression. Appl Phys Lett 94(14):141913CrossRefGoogle Scholar
  31. 31.
    Paul A, Das D (2016) Non-linear thermal post-buckling analysis of FGM Timoshenko beam under non-uniform temperature rise across thickness. Eng Sci Technol Int J 19(3):1608–1625CrossRefGoogle Scholar
  32. 32.
    Ansari R et al (2014) Postbuckling analysis of Timoshenko nanobeams including surface stress effect. Int J Eng Sci 75:1–10CrossRefGoogle Scholar
  33. 33.
    Shafiei N, Mousavi A, Ghadiri M (2016) On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. Int J Eng Sci 106:42–56CrossRefGoogle Scholar
  34. 34.
    Wang Z-X, Shen H-S (2012) Nonlinear dynamic response of nanotube-reinforced composite plates resting on elastic foundations in thermal environments. Nonlinear Dyn 70(1):735–754MathSciNetCrossRefGoogle Scholar
  35. 35.
    He XQ, Rafiee M, Mareishi S (2015) Nonlinear dynamics of piezoelectric nanocomposite energy harvesters under parametric resonance. Nonlinear Dyn 79(3):1863–1880CrossRefzbMATHGoogle Scholar
  36. 36.
    Gholami R, Ansari R (2016) A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates. Nonlinear Dyn 84(4):2403–2422MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mashrouteh S et al (2016) Nonlinear vibration analysis of fluid-conveying microtubes. Nonlinear Dyn 85(2):1007–1021MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ansari R, Oskouie MF, Rouhi H (2017) Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/nano-beams using the strain gradient theory. Nonlinear Dyn 87(1):695–711CrossRefzbMATHGoogle Scholar
  39. 39.
    Nejad MZ, Hadi A (2016) Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams. Int J Eng Sci 106:1–9MathSciNetCrossRefGoogle Scholar
  40. 40.
    Reddy J, El-Borgi S (2014) Eringen’s nonlocal theories of beams accounting for moderate rotations. Int J Eng Sci 82:159–177MathSciNetCrossRefGoogle Scholar
  41. 41.
    Reddy J (2010) Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 48(11):1507–1518MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Fernández-Sáez J et al (2016) Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116MathSciNetCrossRefGoogle Scholar
  43. 43.
    Du H, Lim M, Lin R (1994) Application of generalized differential quadrature method to structural problems. Int J Numer Methods Eng 37(11):1881–1896MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49(1):1–28CrossRefGoogle Scholar
  45. 45.
    Khorshidi MA, Shariati M (2016) Free vibration analysis of sigmoid functionally graded nanobeams based on a modified couple stress theory with general shear deformation theory. J Braz Soc Mech Sci Eng 38(8):2607–2619CrossRefGoogle Scholar
  46. 46.
    Alinaghizadeh F, Shariati M (2015) Static analysis of variable thickness two-directional functionally graded annular sector plates fully or partially resting on elastic foundations by the GDQ method. J Braz Soc Mech Sci Eng 37(6):1819–1838CrossRefGoogle Scholar
  47. 47.
    Maarefdoust M, Kadkhodayan M (2015) Elastic/plastic buckling analysis of skew plates under in-plane shear loading with incremental and deformation theories of plasticity by GDQ method. J Braz Soc Mech Sci Eng 37(2):761–776CrossRefGoogle Scholar
  48. 48.
    Shokrani MH et al (2016) Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method. J Braz Soc Mech Sci Eng 38(8):2589–2606CrossRefGoogle Scholar
  49. 49.
    Shokrollahi H, Kargarnovin MH, Fallah F (2015) Deformation and stress analysis of sandwich cylindrical shells with a flexible core using harmonic differential quadrature method. J Braz Soc Mech Sci Eng 37(1):325–337CrossRefGoogle Scholar
  50. 50.
    Shafiei N et al (2017) Vibration analysis of Nano-Rotor’s Blade applying Eringen nonlocal elasticity and generalized differential quadrature method. Appl Math Model 43:191–206MathSciNetCrossRefGoogle Scholar
  51. 51.
    Shafiei N, Kazemi M (2017) Buckling analysis on the bi-dimensional functionally graded porous tapered nano-/micro-scale beams. Aerosp Sci Technol 66:1–11CrossRefGoogle Scholar
  52. 52.
    Shafiei N, Kazemi M, Ghadiri M (2016) Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM. Appl Phys A 122(8):728CrossRefGoogle Scholar
  53. 53.
    Shafiei N et al (2017) Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput Methods Appl Mech Eng 322:615–632MathSciNetCrossRefGoogle Scholar
  54. 54.
    Ebrahimi F, Shafiei N (2017) Influence of initial shear stress on the vibration behavior of single-layered graphene sheets embedded in an elastic medium based on Reddy’s higher-order shear deformation plate theory. Mech Adv Mater Struct 24(9):761–772CrossRefGoogle Scholar
  55. 55.
    Shafiei N et al (2016) Nonlinear vibration of axially functionally graded non-uniform nanobeams. Int J Eng Sci 106:77–94MathSciNetCrossRefGoogle Scholar
  56. 56.
    Yang J, Shen H-S (2002) Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J Sound Vib 255(3):579–602CrossRefGoogle Scholar
  57. 57.
    Lu P et al (2006) Dynamic properties of flexural beams using a nonlocal elasticity model. J Appl Phys 99(7):073510CrossRefGoogle Scholar
  58. 58.
    Malekzadeh P, Shojaee M (2013) Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams. Compos B Eng 52:84–92CrossRefGoogle Scholar
  59. 59.
    Lestari W, Hanagud S (2001) Nonlinear vibration of buckled beams: some exact solutions. Int J Solids Struct 38(26–27):4741–4757CrossRefzbMATHGoogle Scholar
  60. 60.
    Singh G, Sharma AK, Rao GV (1990) Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions. J Sound Vib 142(1):77–85CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.School of Mechanical Engineering, College of EngineeringUniversity of TehranTehranIran
  2. 2.School of Mechanical Engineering, College of EngineeringSharif University of TechnologyTehranIran
  3. 3.Department of MathematicPayame Noor University (PNU)TehranIran
  4. 4.Department of Mechanical EngineeringPayame Noor University (PNU)TehranIran
  5. 5.Department of Information Technology, College of EngineeringPayame Noor University (PNU)TehranIran
  6. 6.Catalysis and Nano-Structured Materials Laboratory, School of Chemical Engineering, College of EngineeringUniversity of TehranTehranIran

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