Static bending and buckling of perforated nonlocal size-dependent nanobeams

  • M. A. Eltaher
  • A. M. Kabeel
  • K. H. Almitani
  • A. M. Abdraboh
Technical Paper


Due to their superior electronic, mechanical and thermal properties of nanomaterials, they have been widely used in nano-electromechanical systems (NEMS). Nanobeams are common elements in these systems. In some situations perforation is necessary in nanostructures for manufacturing process and technological reasons. Therefore, studying of static bending and buckling behavior of perforated nanobeams is essential for accurate and reliable design of nanostructures. Because of the importance of size effects on the mechanical performance of these structures the size-scale effect should be considered in their design. In this paper an analytical model capable of predicting bending response and critical buckling load of perforated nanobeams is presented for the first time. The model is formulated based on both Timoshenko and Euler–Bernoulli beam theories with nonlocal differential form of Eringen model. The present model is validated by comparing the obtained predictions with that of the available published results in literature for fully beam. Effects of size-scale and perforation parameters (perforation size and number of cutouts) on the static and buckling behavior of the perforated nanobeams are analyzed.



  1. Alshorbagy AE, Eltaher MA, Mahmoud FF (2013) Static analysis of nanobeams using nonlocal FEM. J Mech Sci Technol 27:2035–2041CrossRefGoogle Scholar
  2. Bourouina H, Yahiaoui R, Sahar A, Benamar MEA (2016) Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads. Physica E 75:163–168CrossRefGoogle Scholar
  3. Deng CS, Peng HG, Gao YS, Zhong JX (2014) Ultrahigh-Q photonic crystal nanobeam cavities with H-shaped holes. Phys E 63:8–13CrossRefGoogle Scholar
  4. Ebrahimi F, Barati MR (2018a) Axial magnetic field effects on dynamic characteristics of embedded multiphase nanocrystalline nanobeams. Microsystem Technologies, pp 1–16Google Scholar
  5. Ebrahimi F, Barati MR (2018b) Magnetic field effects on buckling characteristics of smart flexoelectrically actuated piezoelectric nanobeams based on nonlocal and surface elasticity theories. Microsystem Technologies, pp 1–11Google Scholar
  6. Ebrahimi F, Dabbagh A (2018) Wave dispersion characteristics of orthotropic double-nanoplate-system subjected to a longitudinal magnetic field. Microsystem Technologies, pp 1–11Google Scholar
  7. El-Sinawi AH, Bakri-Kassem M, Landolsi T, Awad O (2015) A novel comprehensive approach to feedback control of membrane displacement in radio frequency micro-electromechanical switches. Sens Actuators, A 221:123–130CrossRefGoogle Scholar
  8. Eltaher MA, Emam SA, Mahmoud FF (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88CrossRefGoogle Scholar
  9. Eltaher MA, Khairy A, Sadoun AM, Omar FA (2014a) Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl Math Comput 229:283–295MathSciNetzbMATHGoogle Scholar
  10. Eltaher MA, Hamed MA, Sadoun AM, Mansour A (2014b) Mechanical analysis of higher order gradient nanobeams. Appl Math Comput 229:260–272MathSciNetzbMATHGoogle Scholar
  11. Eltaher MA, Khater ME, Emam SA (2016a) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5):4109–4128MathSciNetCrossRefGoogle Scholar
  12. Eltaher MA, El-Borgi S, Reddy JN (2016b) Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos Struct 153:902–913CrossRefGoogle Scholar
  13. Eltaher MA, Khater MA, Abdel-Rahman E, Yavuz M (2016c) On the static stability of nonlocal nanobeams using higher-order beam theories. Adv Nano Res 4(1):51–64CrossRefGoogle Scholar
  14. Eltaher MA, Omar FA, Abdalla WS, Gad EH (2018a) Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity. Waves in Random and Complex Media, pp 1–17Google Scholar
  15. Eltaher MA, Fouda N, El-midany T, Sadoun AM (2018b) Modified porosity model in analysis of functionally graded porous nanobeams. J Braz Soc Mech Sci Eng 40(3):141CrossRefGoogle Scholar
  16. Emam SA (2013) A general nonlocal nonlinear model for buckling of nanobeams. Appl Math Model 37:6929–6939MathSciNetCrossRefGoogle Scholar
  17. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710CrossRefGoogle Scholar
  18. Eringen AC (2002) Nonlocal continuum field theories. Springer, BerlinGoogle Scholar
  19. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248MathSciNetCrossRefzbMATHGoogle Scholar
  20. Fang DM, Li XH, Yuan Q, Zhang HX (2010) Effect of etch holes on the capacitance and pull-in voltage in MEMS tunable capacitors. Int J Electron 97(12):1439–1448CrossRefGoogle Scholar
  21. Gul U, Aydogdu M (2017) Structural modelling of nanorods and nanobeams using doublet mechanics theory. Int J Mech Mater Des: 1–18Google Scholar
  22. Jeong KH, Amabili M (2006) Bending vibration of perforated beams in contact with a liquid. J Sound Vib 298(1):404–419CrossRefGoogle Scholar
  23. Jia N, Yao Y, Yang Y, Chen S (2017) Size effect in the bending of a Timoshenko nanobeams. Acta Mech 228:2363–2375MathSciNetCrossRefzbMATHGoogle Scholar
  24. Joshi AY, Sharma SC, Harsha SP (2011) Zeptogram scale mass sensing using single walled carbon nanotube based biosensors. Sens Actuators, A 168(2):275–280CrossRefGoogle Scholar
  25. Khadem SE, Rasekh M, Toghraee A (2012) Design and simulation of a carbon nanotube-based adjustable nano-electromechanical shock switch. Appl Math Model 36(6):2329–2339CrossRefGoogle Scholar
  26. Luschi L, Pieri F (2012) A simple analytical model for the resonance frequency of perforated beams. Procedia Eng 47:1093–1096CrossRefGoogle Scholar
  27. Luschi L, Pieri F (2014) Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity. J Micromech Microeng 24(5):055004CrossRefGoogle Scholar
  28. Luschi L, Pieri F (2016) An analytical model for the resonance frequency of square perforated Lamé-mode resonators. Sens Actuators B: Chem 222:1233–1239CrossRefGoogle Scholar
  29. Luschi L, Iannaccone G, Pieri F (2017) Temperature compensation of silicon Lamé resonators using etch holes: theory and design methodology. IEEE Trans Ultrason Ferroelectr Freq Control 64(5):879–887CrossRefGoogle Scholar
  30. Mitin VV, Sementsov DI, Vagidov NZ (2010) Quantum mechanics for nanostructures, 1st edn. Cambridge University Press, New YorkCrossRefGoogle Scholar
  31. Miura R, Imamura S, Ohta R, Ishii A, Liu X, Shimada T, Kato YK (2014) Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters. Nature Commun 5:5580CrossRefGoogle Scholar
  32. Mohite SS, Sonti VR, Pratap R (2008) A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures. J Microelectromech Syst 17(3):709–723CrossRefGoogle Scholar
  33. Nagase T, Kawamura J, Pahlovy SA, Miyamoto I (2010) Ion beam fabrication of natural single crystal diamond nano-tips for potential use in atomic force microscopy. Microelectron Eng 87(5):1494–1496CrossRefGoogle Scholar
  34. Rasekh M, Khadem SE (2011) Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia. Int J Mech Sci 53(2):108–115CrossRefGoogle Scholar
  35. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307CrossRefzbMATHGoogle Scholar
  36. Reddy JN, Pang SD (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103:023511CrossRefGoogle Scholar
  37. Refaeinejad V, Rahmani O, Hosseini SAH (2017) An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories. Sci Iran, Trans F: Nanotechnol 24:1635–1653Google Scholar
  38. Shao L, Palaniapan M (2008) Effect of etch holes on quality factor of bulk-mode micromechanical resonators. Electron Lett 44(15):938–939CrossRefGoogle Scholar
  39. Sharma JN, Grover D (2011) Thermoelastic vibrations in micro-/nano-scale beam resonators with voids. J Sound Vib 330(12):2964–2977CrossRefGoogle Scholar
  40. Shaterzadeh AR, Rezaei R, Abolghasemi S (2015) Thermal buckling analysis of perforated functionally graded plates. J Therm Stress 38(11):1248–1266CrossRefGoogle Scholar
  41. Sun C, Zhang H (2003) Size-dependent elastic moduli of platelike nanomaterials. Appl Phys 93(2):1212–1218CrossRefGoogle Scholar
  42. Tounsi A, Semmah A, Bousahla AA (2013) Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. J Nanomech Micromech 3:37–42CrossRefGoogle Scholar
  43. Tu C, Lee JEY (2012) Increased dissipation from distributed etch holes in a lateral breathing mode silicon micromechanical resonator. Appl Phys Lett 101(2):023504CrossRefGoogle Scholar
  44. Wang KF, Wang BL, Kitamura T (2016) A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech Sin 32(1):83–100MathSciNetCrossRefzbMATHGoogle Scholar
  45. Zhang X, Wang X, Kong W, Yi G, Jia J (2011) Tribological behavior of micro/nano-patterned surfaces in contact with AFM colloidal probe. Appl Surf Sci 258(1):113–119CrossRefGoogle Scholar
  46. Zohoor H, Kakavand F (2013) Timoshenko versus Euler–Bernoulli beam theories for high speed two-link manipulator. Sci Iran 20(1):172–178Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • M. A. Eltaher
    • 1
    • 2
  • A. M. Kabeel
    • 2
  • K. H. Almitani
    • 1
  • A. M. Abdraboh
    • 3
  1. 1.Mechanical Engineering Department, Faculty of EngineeringKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Mechanical Design and Production Department, Faculty of EngineeringZagazig UniversityZagazigEgypt
  3. 3.Physics Department, Faculty of ScienceBanha UniversityBanhaEgypt

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