Skip to main content
Log in

On bending and buckling responses of perforated nanobeams including surface energy for different beams theories

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Perforated beam is essential structural element of Nano-Electro-Mechanical-Systems (NEMS), whose design needs appropriate modelling of size of holes, hole numbers, and scale effects. The current manuscript presents a comprehensive study and develops non-classical closed form solutions to predict the static bending behavior and buckling stability of perforated nanobeams (PNBs) incorporating the surface energy for different four beams theories, for the first time. Equivalent geometrical models for both bulk and surface characteristics are presented. The Gurtin–Murdoch surface elasticity model is modified and adopted to include the perforation in surface characteristics. To consider the warping shear effect on bending as well as critical buckling loads with the presence of surface stress effects, four different beams theories with shear deformation are considered. The non-classical equilibrium equations relevant to each PNB theory are developed in detail. Closed-form solutions are developed considering the different classical and non-classical boundary conditions as well as loading conditions. The proposed methodology is verified by comparing the obtained results with the available analytical solutions for fully filled beams and excellent agreement is observed. Effects of perforation characteristics as well as the surface effects on bending and buckling behaviors are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Abdelrahman AA, Eltaher MA, Kabeel AM, Abdraboh AM, Hendi AA (2019) Free and forced analysis of perforated beams. Steel Compos Struct 31(5):489–502. https://doi.org/10.12989/scs.2019.31.5.489

    Article  Google Scholar 

  2. Agwa MA, Eltaher MA (2016) Vibration of a carbyne nanomechanical mass sensor with surface effect. Appl Phys A 122(4):335. https://doi.org/10.1007/s00339-016-9934-9

    Article  Google Scholar 

  3. Akbaş ŞD (2016) Static analysis of a nano plate by using generalized differential quadrature method. Int J Eng Appl Sci 8(2):30–39

    Google Scholar 

  4. Akbaş ŞD (2017) Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory. Int J Struct Stab Dyn 17(03):1750033. https://doi.org/10.1142/S021945541750033X

    Article  MathSciNet  Google Scholar 

  5. Akbaş ŞD (2018a) Forced vibration analysis of cracked nanobeams. J Braz Soc Mech Sci Eng 40(8):392. https://doi.org/10.1007/s40430-018-1315-1

    Article  Google Scholar 

  6. Akbaş ŞD (2018b) Bending of a cracked functionally graded nanobeam. Adv Nano Res 6(3):219. https://doi.org/10.12989/anr.2018.6.3.219

    Article  Google Scholar 

  7. Akbaş ŞD (2019a) Longitudinal forced vibration analysis of porous a nanorod. Mühendislik Bilimleri ve Tasarım Dergisi 7(4):736–743. https://doi.org/10.21923/jesd.553328

    Article  Google Scholar 

  8. Akbaş ŞD (2019b) Axially forced vibration analysis of cracked a nanorod. J Comput Appl Mech 50(1):63–68

    MathSciNet  Google Scholar 

  9. Akbas SD (2020) Modal analysis of viscoelastic nanorods under an axially harmonic load. Adv Nano Res 8(4):277–282. https://doi.org/10.12989/anr.2020.8.4.277

    Article  Google Scholar 

  10. Akgöz B, Civalek Ö (2016) Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut 119:1–12. https://doi.org/10.1016/j.actaastro.2015.10.021

    Article  Google Scholar 

  11. Allahyari E, Asgari M (2019) Thermo-mechanical vibration of double-layer graphene nanosheets in elastic medium considering surface effects; developing a nonlocal third order shear deformation theory. Eur J Mech A/Solids 75:307–321. https://doi.org/10.1016/j.euromechsol.2019.01.022

    Article  MathSciNet  MATH  Google Scholar 

  12. Almitani KH, Abdelrahman AA, Eltaher MA (2019) On forced and free vibrations of cutout squared beams. Steel Compos Struct 32(5):643–655. https://doi.org/10.12989/scs.2019.32.5.643

    Article  Google Scholar 

  13. Almitani KH, Abdelrahman AA, Eltaher MA (2020) Stability of PNBs incorporating surface energy effects. Steel Compos Struct 35(4):555–566. https://doi.org/10.12989/scs.2020.35.4.555

    Article  Google Scholar 

  14. Ansari R, Sahmani S (2011) Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories. Int J Eng Sci 49(11):1244–1255. https://doi.org/10.1016/j.ijengsci.2011.01.007

    Article  Google Scholar 

  15. Apuzzo A, Barretta R, Faghidian SA, Luciano R, De Sciarra FM (2019) Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams. Compos B Eng 164:667–674. https://doi.org/10.1016/j.compositesb.2018.12.112

    Article  Google Scholar 

  16. Apuzzo A, Barretta R, Fabbrocino F, Faghidian SA, Luciano R, Marotti de Sciarra F (2019) Axial and torsional free vibrations of elastic nano-beams by stress-driven two-phase elasticity. J Appl Comput Mech 5(2):402–413. https://doi.org/10.22055/jacm.2018.26552.1338

    Article  Google Scholar 

  17. Attia MA, Rahman AAA (2018) On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int J Eng Sci 127:1–32. https://doi.org/10.1016/j.ijengsci.2018.02.005

    Article  MathSciNet  MATH  Google Scholar 

  18. Baghdadi H, Tounsi A, Zidour M, Benzair A (2015) Thermal effect on vibration characteristics of armchair and zigzag single-walled carbon nanotubes using nonlocal parabolic beam theory. Fullerenes Nanotubes Carbon Nanostruct 23(3):266–272. https://doi.org/10.1080/1536383X.2013.787605

    Article  Google Scholar 

  19. Barretta R, Čanađija M, Luciano R, de Sciarra FM (2018) Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams. Int J Eng Sci 126:53–67. https://doi.org/10.1016/j.ijengsci.2018.02.012

    Article  MathSciNet  MATH  Google Scholar 

  20. Barretta R, Luciano R, de Sciarra FM, Ruta G (2018) Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur J Mech A/Solids 72:275–286. https://doi.org/10.1016/j.euromechsol.2018.04.012

    Article  MathSciNet  MATH  Google Scholar 

  21. Barretta R, Faghidian SA, Luciano R, Medaglia CM, Penna R (2018) Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models. Compos B Eng 154:20–32. https://doi.org/10.1016/j.compositesb.2018.07.036

    Article  Google Scholar 

  22. Barretta R, de Sciarra FM, Vaccaro MS (2019) On nonlocal mechanics of curved elastic beams. Int J Eng Sci 144:103140. https://doi.org/10.1016/j.ijengsci.2019.103140

    Article  MathSciNet  MATH  Google Scholar 

  23. Barretta R, Faghidian SA, de Sciarra FM (2019) Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. Int J Eng Sci 136:38–52. https://doi.org/10.1016/j.ijengsci.2019.01.003

    Article  MathSciNet  MATH  Google Scholar 

  24. Barretta R, de Sciarra FM (2019) Variational nonlocal gradient elasticity for nano-beams. Int J Eng Sci 143:73–91. https://doi.org/10.1016/j.ijengsci.2019.06.016

    Article  MathSciNet  MATH  Google Scholar 

  25. Barretta R, Faghidian SA, de Sciarra FM, Pinnola FP (2020) On nonlocal Lam strain gradient mechanics of elastic rods. Int J Multiscale Comput Eng. https://doi.org/10.1615/IntJMultCompEng.2019030655

    Article  MATH  Google Scholar 

  26. Bedia WA, Houari MSA, Bessaim A, Bousahla AA, Tounsi A, Saeed T, Alhodaly MS (2019) A new hyperbolic two-unknown beam model for bending and buckling analysis of a nonlocal strain gradient nanobeams. J Nano Res 57:175–191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175

    Article  Google Scholar 

  27. Bendali A, Labedan R, Domingue F, Nerguizian V (2006) Holes effects on RF MEMS parallel membranes capacitors. In 2006 Canadian Conference on Electrical and Computer Engineering (pp. 2140–2143). IEEE. https://doi.org/10.1109/CCECE.2006.277600

  28. Bessaim A, Houari MSA, Bernard F, Tounsi A (2015) A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates. Struct Eng Mech 56(2):223–240. https://doi.org/10.12989/sem.2015.56.2.223

    Article  Google Scholar 

  29. Bourouina H, Yahiaoui R, Sahar A, Benamar MEA (2016) Analytical modeling for the determination of nonlocal resonance frequencies of PNBs subjected to temperature-induced loads. Phys E 75:163–168. https://doi.org/10.1016/j.physe.2015.09.014

    Article  Google Scholar 

  30. Bourouina H, Yahiaoui R, Kerid R, Ghoumid K, Lajoie I, Picaud F, Herlem G (2020) The influence of hole networks on the adsorption-induced frequency shift of a PNB using non-local elasticity theory. J Phys Chem Solids 136:109201. https://doi.org/10.1016/j.jpcs.2019.109201

    Article  Google Scholar 

  31. Civalek Ö, Demir Ç (2011) Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Appl Math Model 35(5):2053–2067. https://doi.org/10.1016/j.apm.2010.11.004

    Article  MathSciNet  MATH  Google Scholar 

  32. Daikh AA, Drai A, Houari MSA, Eltaher MA (2020) Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes. Steel Compos Struct 36(6):643–656. https://doi.org/10.12989/scs.2020.36.6.643

    Article  Google Scholar 

  33. Daikh AA, Bachiri A, Houari MSA, Tounsi A (2020) Size dependent free vibration and buckling of multilayered carbon nanotubes reinforced composite nanoplates in thermal environment. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1752232

    Article  Google Scholar 

  34. De Pasquale G, Veijola T, Somà A (2009) Modelling and validation of air damping in perforated gold and silicon MEMS plates. J Micromech Microeng 20(1):015010. https://doi.org/10.1088/0960-1317/20/1/015010

    Article  Google Scholar 

  35. Demir Ç, Civalek Ö (2017) On the analysis of microbeams. Int J Eng Sci 121:14–33. https://doi.org/10.1016/j.ijengsci.2017.08.016

    Article  MathSciNet  MATH  Google Scholar 

  36. Ebrahimi F, Barati MR (2017) A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Compos Struct 159:174–182. https://doi.org/10.1016/j.compstruct.2016.09.058

    Article  Google Scholar 

  37. Ebrahimi F, Barati MR (2018) Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory. Mech Adv Mater Struct 25(4):350–359. https://doi.org/10.1080/15376494.2016.1255830

    Article  Google Scholar 

  38. Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218(14):7406–7420. https://doi.org/10.1016/j.amc.2011.12.090

    Article  MathSciNet  MATH  Google Scholar 

  39. Eltaher MA, Emam SA, Mahmoud FF (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88. https://doi.org/10.1016/j.compstruct.2012.09.030

    Article  Google Scholar 

  40. Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201. https://doi.org/10.1016/j.compstruct.2012.11.039

    Article  Google Scholar 

  41. Eltaher MA, Mahmoud FF, Assie AE, Meletis EI (2013) Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl Math Comput 224:760–774. https://doi.org/10.1016/j.amc.2013.09.002

    Article  MathSciNet  MATH  Google Scholar 

  42. Eltaher MA, Hamed MA, Sadoun AM, Mansour A (2014) Mechanical analysis of higher order gradient nanobeams. Appl Math Comput 229:260–272. https://doi.org/10.1016/j.amc.2013.12.076

    Article  MathSciNet  MATH  Google Scholar 

  43. Eltaher MA, Khairy A, Sadoun AM, Omar FA (2014) Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl Math Comput 229:283–295. https://doi.org/10.1016/j.amc.2013.12.072

    Article  MathSciNet  MATH  Google Scholar 

  44. Eltaher MA, El-Borgi S, Reddy JN (2016) Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos Struct 153:902–913. https://doi.org/10.1016/j.compstruct.2016.07.013

    Article  Google Scholar 

  45. Eltaher MA, Agwa MA, Mahmoud FF (2016) Nanobeam sensor for measuring a zeptogram mass. Int J Mech Mater Des 12(2):211–221. https://doi.org/10.1007/s10999-015-9302-5

    Article  Google Scholar 

  46. Eltaher MA, Khater ME, Emam SA (2016) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5–6):4109–4128. https://doi.org/10.1016/j.apm.2015.11.026

    Article  MathSciNet  MATH  Google Scholar 

  47. Eltaher MA, Agwa M, Kabeel A (2018) Vibration analysis of material size-dependent CNTs using energy equivalent model. J Appl Comput Mech 4(2):75–86. https://doi.org/10.22055/JACM.2017.22579.1136

    Article  Google Scholar 

  48. Eltaher MA, Kabeel AM, Almitani KH, Abdraboh AM (2018) Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsyst Technol 24(12):4881–4893. https://doi.org/10.1007/s00542-018-3905-3

    Article  Google Scholar 

  49. Eltaher MA, Abdraboh AM, Almitani KH (2018) Resonance frequencies of size dependent perforated nonlocal nanobeam. Microsyst Technol 24(9):3925–3937. https://doi.org/10.1007/s00542-018-3910-6

    Article  Google Scholar 

  50. Eltaher MA, Omar FA, Abdalla WS, Gad EH (2019) Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity. Waves Random Complex Med 29(2):264–280. https://doi.org/10.1080/17455030.2018.1429693

    Article  MathSciNet  Google Scholar 

  51. Eltaher MA, Almalki TA, Ahmed KI, Almitani KH (2019) Characterization and behaviors of single walled carbon nanotube by equivalent-continuum mechanics approach. Adv Nano Res 7(1):39. https://doi.org/10.12989/anr.2019.7.1.039

    Article  Google Scholar 

  52. Eltaher MA, Mohamed N, Mohamed S, Seddek LF (2019) Postbuckling of curved carbon nanotubes using energy equivalent model. J Nano Res 57:136–157. https://doi.org/10.4028/www.scientific.net/JNanoR.57.136

    Article  Google Scholar 

  53. Eltaher MA, Almalki TA, Almitani KH, Ahmed KIE, Abdraboh AM (2019) Modal participation of fixed–fixed single-walled carbon nanotube with vacancies. Int J Adv Struct Eng 11(2):151–163. https://doi.org/10.1007/s40091-019-0222-8

    Article  Google Scholar 

  54. Eltaher MA, Mohamed N (2020a) Nonlinear stability and vibration of imperfect CNTs by doublet mechanics. Appl Math Comput 382:125311. https://doi.org/10.1016/j.amc.2020.125311

    Article  MathSciNet  MATH  Google Scholar 

  55. Eltaher MA, Mohamed NA (2020b) Vibration of nonlocal PNBs with general boundary conditions. Smart Struct Syst 25(4):501–514. https://doi.org/10.12989/sss.2020.25.4.501

    Article  Google Scholar 

  56. Eltaher MA, Mohamed N, Mohamed SA (2020) Nonlinear buckling and free vibration of curved CNTs by doublet mechanics. Smart Struct Syst 26(2):213–226. https://doi.org/10.12989/sss.2020.26.2.213

    Article  MATH  Google Scholar 

  57. Eltaher MA, Omar FA, Abdraboh AM, Abdalla WS, Alshorbagy AE (2020) Mechanical behaviors of piezoelectric nonlocal nanobeam with cutouts. Smart Struct Syst 25(2):219–228. https://doi.org/10.12989/sss.2020.25.2.219

    Article  Google Scholar 

  58. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16. https://doi.org/10.1016/0020-7225(72)90070-5

    Article  MathSciNet  MATH  Google Scholar 

  59. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710. https://doi.org/10.1063/1.332803

    Article  Google Scholar 

  60. Ferrari M, Granik VT, Imam A, Nadeau JC (1997) Advances in doublet mechanics, vol 45. Springer Science & Business Media, Berlin

    Book  Google Scholar 

  61. Gao G, Cagin T, Goddard WA III (1998) Energetics, structure, mechanical and vibrational properties of single-walled carbon nanotubes. Nanotechnology 9(3):184. https://doi.org/10.1088/0957-4484/9/3/007

    Article  Google Scholar 

  62. Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity—I. Theory. J Mech Phys Solids 47(6):1239–1263. https://doi.org/10.1016/S0022-5096(98)00103-3

    Article  MathSciNet  MATH  Google Scholar 

  63. Guha K, Kumar M, Agarwal S, Baishya S (2015) A modified capacitance model of RF MEMS shunt switch incorporating fringing field effects of perforated beam. Solid-State Electron 114:35–42. https://doi.org/10.1016/j.sse.2015.07.008

    Article  Google Scholar 

  64. Guha K, Laskar NM, Gogoi HJ, Borah AK, Baishnab KL, Baishya S (2017) Novel analytical model for optimizing the pull-in voltage in a flexured MEMS switch incorporating beam perforation effect. Solid-State Electron 137:85–94. https://doi.org/10.1016/j.sse.2017.08.007

    Article  Google Scholar 

  65. Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323. https://doi.org/10.1007/BF00261375

    Article  MathSciNet  MATH  Google Scholar 

  66. Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14(6):431–440

    Article  Google Scholar 

  67. Hamed MA, Eltaher MA, Sadoun AM, Almitani KH (2016) Free vibration of symmetric and sigmoid functionally graded nanobeams. Appl Phys A 122(9):829. https://doi.org/10.1007/s00339-016-0324-0

    Article  Google Scholar 

  68. Hamed MA, Sadoun AM, Eltaher MA (2019) Effects of porosity models on static behavior of size dependent functionally graded beam. Struct Eng Mech 71(1):89–98. https://doi.org/10.12989/sem.2019.71.1.089

    Article  Google Scholar 

  69. Hamed MA, Mohamed NA, Eltaher MA (2020) Stability buckling and bending of nanobeams including cutouts. Eng Comput. https://doi.org/10.1007/s00366-020-01063-2

    Article  Google Scholar 

  70. Hosseini-Hashemi S, Nazemnezhad R, Rokni H (2015) Nonlocal nonlinear free vibration of nanobeams with surface effects. Eur J Mech A/Solids 52:44–53. https://doi.org/10.1016/j.euromechsol.2014.12.012

    Article  MathSciNet  MATH  Google Scholar 

  71. Houari MSA, Bessaim A, Bernard F, Tounsi A, Mahmoud SR (2018) Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter. Steel Compos Struct 28(1):13–24. https://doi.org/10.12989/scs.2018.28.1.013

    Article  Google Scholar 

  72. Jalaei MH, Arani AG, Nguyen-Xuan H (2019) Investigation of thermal and magnetic field effects on the dynamic instability of FG Timoshenko nanobeam employing nonlocal strain gradient theory. Int J Mech Sci 161:105043. https://doi.org/10.1016/j.ijmecsci.2019.105043

    Article  Google Scholar 

  73. Jamalpoor A, Ahmadi-Savadkoohi A, Hosseini M, Hosseini-Hashemi S (2017) Free vibration and biaxial buckling analysis of double magneto-electro-elastic nanoplate-systems coupled by a visco-Pasternak medium via nonlocal elasticity theory. Eur J Mech A/Solids 63:84–98. https://doi.org/10.1016/j.euromechsol.2016.12.002

    Article  MathSciNet  MATH  Google Scholar 

  74. Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT (2011) A nonlinear strain gradient beam formulation. Int J Eng Sci 49(11):1256–1267. https://doi.org/10.1016/j.ijengsci.2011.01.006

    Article  MathSciNet  MATH  Google Scholar 

  75. Karami B, Janghorban M, Rabczuk T (2020) Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos B Eng 182:107622. https://doi.org/10.1016/j.compositesb.2019.107622

    Article  Google Scholar 

  76. Kerid R, Bourouina H, Yahiaoui R, Bounekhla M, Aissat A (2019) Magnetic field effect on nonlocal resonance frequencies of structure-based filter with periodic square holes network. Phys E 105:83–89. https://doi.org/10.1016/j.physe.2018.05.021

    Article  Google Scholar 

  77. Khater ME, Eltaher MA, Abdel-Rahman E, Yavuz M (2014) Surface and thermal load effects on the buckling of curved nanowires. Eng Sci Technol Int J 17(4):279–283. https://doi.org/10.1016/j.jestch.2014.07.003

    Article  Google Scholar 

  78. Khodabakhshi P, Reddy JN (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75. https://doi.org/10.1016/j.ijengsci.2015.06.006

    Article  MathSciNet  MATH  Google Scholar 

  79. Koiter WT (1964) Couple stresses in the theory of elasticity. Proc Koninklijke Nederl Akaad van Wetensch. https://hal.archives-ouvertes.fr/hal-00852443

  80. Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47(4):487–498. https://doi.org/10.1016/j.ijengsci.2008.08.008

    Article  MathSciNet  MATH  Google Scholar 

  81. Lam DC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508. https://doi.org/10.1016/S0022-5096(03)00053-X

    Article  MATH  Google Scholar 

  82. Levinson M (1981) A new rectangular beam theory. J Sound Vib 74(1):81–87. https://doi.org/10.1016/0022-460X(81)90493-4

    Article  MATH  Google Scholar 

  83. Li C, Chou TW (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40(10):2487–2499. https://doi.org/10.1016/S0020-7683(03)00056-8

    Article  MATH  Google Scholar 

  84. Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94. https://doi.org/10.1016/j.ijengsci.2015.08.013

    Article  MathSciNet  MATH  Google Scholar 

  85. Li L, Lin R, Ng TY (2020) Contribution of nonlocality to surface elasticity. Int J Eng Sci 152:103311. https://doi.org/10.1016/j.ijengsci.2020.103311

    Article  MathSciNet  MATH  Google Scholar 

  86. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313. https://doi.org/10.1016/j.jmps.2015.02.001

    Article  MathSciNet  MATH  Google Scholar 

  87. Liu H, Li B, Liu Y (2019) The inconsistency of nonlocal effect on carbon nanotube conveying fluid and a proposed solution based on local/nonlocal model. Eur J Mech A/Solids 78:103837. https://doi.org/10.1016/j.euromechsol.2019.103837

    Article  MathSciNet  MATH  Google Scholar 

  88. Liu Y, Wei Y (2020) Effect of surface energy on the indentation response of hard nanofilm/soft substrate composite structure. Int J Mech Sci. https://doi.org/10.1016/j.ijmecsci.2020.105759

    Article  Google Scholar 

  89. Lu L, Guo X, Zhao J (2018) On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy. Int J Eng Sci 124:24–40. https://doi.org/10.1016/j.ijengsci.2017.11.020

    Article  MathSciNet  MATH  Google Scholar 

  90. Luschi L, Pieri F (2014) An analytical model for the determination of resonance frequencies of perforated beams. J Micromech Microeng 24(5):055004. https://doi.org/10.1088/0960-1317/24/5/055004

    Article  Google Scholar 

  91. Luschi L, Pieri F (2016) An analytical model for the resonance frequency of square perforated Lamé-mode resonators. Sens Actuators B Chem 222:1233–1239. https://doi.org/10.1016/j.snb.2015.07.085

    Article  Google Scholar 

  92. Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56(12):3379–3391. https://doi.org/10.1016/j.jmps.2008.09.007

    Article  MathSciNet  MATH  Google Scholar 

  93. Mahmoud FF, Eltaher MA, Alshorbagy AE, Meletis EI (2012) Static analysis of nanobeams including surface effects by nonlocal finite element. J Mech Sci Technol 26(11):3555–3563. https://doi.org/10.1007/s12206-012-0871-z

    Article  Google Scholar 

  94. Malekzadeh P, Shojaee M (2013) Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams. Compos B Eng 52:84–92. https://doi.org/10.1016/j.compositesb.2013.03.046

    Article  Google Scholar 

  95. Malikan M (2019) On the buckling response of axially pressurized nanotubes based on a novel nonlocal beam theory. J Appl Comput Mech 5(1):103–112

    Google Scholar 

  96. Malikan M, Dimitri R, Tornabene F (2019) Transient response of oscillated carbon nanotubes with an internal and external damping. Compos B Eng 158:198–205. https://doi.org/10.1016/j.compositesb.2018.09.092

    Article  Google Scholar 

  97. Malikan M (2020) On the plastic buckling of curved carbon nanotubes. Theor Appl Mech Lett 10(1):46–56. https://doi.org/10.1016/j.taml.2020.01.004

    Article  Google Scholar 

  98. Malikan M, Eremeyev VA (2020a) On nonlinear bending study of a piezo-flexomagnetic nanobeam based on an analytical-numerical solution. Nanomaterials 10(9):1762. https://doi.org/10.3390/nano10091762

    Article  Google Scholar 

  99. Malikan M, Eremeyev VA (2020b) On the dynamics of a visco–piezo–flexoelectric nanobeam. Symmetry 12(4):643. https://doi.org/10.3390/sym12040643

    Article  Google Scholar 

  100. Malikan M, Eremeyev VA (2020c) Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method. Mater Res Exp 7(2):025005. https://doi.org/10.1088/2053-1591/ab691c

    Article  Google Scholar 

  101. Merzouki T, Houari MSA, Haboussi M, Bessaim A, Ganapathi M (2020) Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory. Eng Comput. https://doi.org/10.1007/s00366-020-01156-y

    Article  Google Scholar 

  102. Malikan M, Eremeyev VA, Sedighi HM (2020) Buckling analysis of a non-concentric double-walled carbon nanotube. Acta Mech. https://doi.org/10.1007/s00707-020-02784-7

    Article  MathSciNet  Google Scholar 

  103. Malikan M, Uglov NS, Eremeyev VA (2020) On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures. Int J Eng Sci 157:103395. https://doi.org/10.1016/j.ijengsci.2020.103395

    Article  MathSciNet  MATH  Google Scholar 

  104. Mindlin RD (1962) Influence of couple-stresses on stress concentrations. Columbia University, New york

    Google Scholar 

  105. Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438. https://doi.org/10.1016/0020-7683(65)90006-5

    Article  Google Scholar 

  106. Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3):139. https://doi.org/10.1088/0957-4484/11/3/301

    Article  Google Scholar 

  107. Mirjavadi SS, Afshari BM, Barati MR, Hamouda AMS (2019) Transient response of porous FG nanoplates subjected to various pulse loads based on nonlocal stress-strain gradient theory. Eur J Mech A/Solids 74:210–220. https://doi.org/10.1016/j.euromechsol.2018.11.004

    Article  MathSciNet  MATH  Google Scholar 

  108. Mohamed N, Eltaher MA, Mohamed SA, Seddek LF (2019) Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation. Struct Eng Mech 70(6):737–750. https://doi.org/10.12989/sem.2019.70.6.737

    Article  Google Scholar 

  109. Mohamed N, Mohamed SA, Eltaher MA (2020) Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng Comput. https://doi.org/10.1007/s00366-020-00976-2

    Article  Google Scholar 

  110. Mohammadi M, Hosseini M, Shishesaz M, Hadi A, Rastgoo A (2019) Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads. Eur J Mech A/Solids 77:103793. https://doi.org/10.1016/j.euromechsol.2019.05.008

    Article  MathSciNet  MATH  Google Scholar 

  111. Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46(3):411–425. https://doi.org/10.1016/S0022-5096(97)00086-0

    Article  MATH  Google Scholar 

  112. Park SK, Gao XL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16(11):2355. https://doi.org/10.1088/0960-1317/16/11/015

    Article  Google Scholar 

  113. Pei YL, Geng PS, Li LX (2018) A modified higher-order theory for FG beams. Eur J Mech A/Solids 72:186–197. https://doi.org/10.1016/j.euromechsol.2018.05.008

    Article  MathSciNet  MATH  Google Scholar 

  114. Phung-Van P, Thai CH, Nguyen-Xuan H, Abdel-Wahab M (2019) An isogeometric approach of static and free vibration analyses for porous FG nanoplates. Eur J Mech A/Solids 78:103851. https://doi.org/10.1016/j.euromechsol.2019.103851

    Article  MathSciNet  MATH  Google Scholar 

  115. Pinnola FP, Faghidian SA, Barretta R, de Sciarra FM (2020) Variationally consistent dynamics of nonlocal gradient elastic beams. Int J Eng Sci 149:103220. https://doi.org/10.1016/j.ijengsci.2020.103220

    Article  MathSciNet  MATH  Google Scholar 

  116. Rabhi M, Benrahou KH, Kaci A, Houari MSA, Bourada F, Bousahla AA, Tounsi A (2020) A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Geomech Eng 22(2):119. https://doi.org/10.12989/gae.2020.22.2.119

    Article  Google Scholar 

  117. Rapaport DC, Rapaport DCR (2004) The art of molecular dynamics simulation. Cambridge University Press, Cambridge

    Book  Google Scholar 

  118. Rappé AK, Casewit CJ, Colwell KS, Goddard WA III, Skiff WM (1992) UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J Am Chem Soc 114(25):10024–10035. https://doi.org/10.1021/ja00051a040

    Article  Google Scholar 

  119. Rebeiz GM (2004) RF MEMS: theory, design, and technology. John Wiley & Sons, New York

    Google Scholar 

  120. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307. https://doi.org/10.1016/j.ijengsci.2007.04.004

    Article  MATH  Google Scholar 

  121. Reddy JN (2010) Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int J Eng Sci 48(11):1507–1518. https://doi.org/10.1016/j.ijengsci.2010.09.020

    Article  MathSciNet  MATH  Google Scholar 

  122. Sedighi HM (2014a) Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut 95:111–123. https://doi.org/10.1016/j.actaastro.2013.10.020

    Article  Google Scholar 

  123. Sedighi HM (2014b) The influence of small scale on the pull-in behavior of nonlocal nanobridges considering surface effect, Casimir and Van der Waals attractions. Int J Appl Mech 6(03):1450030. https://doi.org/10.1142/S1758825114500306

    Article  Google Scholar 

  124. Shenoy VB (2005) Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys Rev B 71(9):094104. https://doi.org/10.1103/PhysRevB.71.094104

    Article  Google Scholar 

  125. She GL, Yuan FG, Karami B, Ren YR, Xiao WS (2019) On nonlinear bending behavior of FG porous curved nanotubes. Int J Eng Sci 135:58–74. https://doi.org/10.1016/j.ijengsci.2018.11.005

    Article  MathSciNet  MATH  Google Scholar 

  126. Sheng GG, Wang X (2018) Nonlinear vibration of FG beams subjected to parametric and external excitations. Eur J Mech A/Solids 71:224–234. https://doi.org/10.1016/j.euromechsol.2018.04.003

    Article  MathSciNet  MATH  Google Scholar 

  127. Shodja HM, Delfani MR (2011) A novel nonlinear constitutive relation for graphene and its consequence for developing closed-form expressions for Young’s modulus and critical buckling strain of single-walled carbon nanotubes. Acta Mech 222(1–2):91. https://doi.org/10.1007/s00707-011-0528-5

    Article  MATH  Google Scholar 

  128. Shokrieh MM, Rafiee R (2010) Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater Des 31(2):790–795. https://doi.org/10.1016/j.matdes.2009.07.058

    Article  Google Scholar 

  129. Şimşek M, Kocatürk T, Akbaş ŞD (2013) Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory. Compos Struct 95:740–747. https://doi.org/10.1016/j.compstruct.2012.08.036

    Article  Google Scholar 

  130. Şimşek M, Yurtcu HH (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 97:378–386. https://doi.org/10.1016/j.compstruct.2012.10.038

    Article  Google Scholar 

  131. Thai HT, Vo TP, Nguyen TK, Lee J (2015) Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Compos Struct 123:337–349. https://doi.org/10.1016/j.compstruct.2014.11.065

    Article  Google Scholar 

  132. Thai S, Thai HT, Vo TP, Patel VI (2018) A simple shear deformation theory for nonlocal beams. Compos Struct 183:262–270. https://doi.org/10.1016/j.compstruct.2017.03.022

    Article  Google Scholar 

  133. Toupin RA (1964) Theories of elasticity with couple-stress. Arch Ration Mech Anal 17:85–112. https://doi.org/10.1007/BF00253050

    Article  MathSciNet  MATH  Google Scholar 

  134. Wang GF, Feng XQ (2007) Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl Phys Lett 90(23):231904. https://doi.org/10.1063/1.2746950

    Article  Google Scholar 

  135. Wu Y, Zhang X, Leung AYT, Zhong W (2006) An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes. Thin-walled Struct 44(6):667–676. https://doi.org/10.1016/j.tws.2006.05.003

    Article  Google Scholar 

  136. Yang FACM, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. A. Eltaher.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abdelrahman, A.A., Eltaher, M.A. On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Engineering with Computers 38, 2385–2411 (2022). https://doi.org/10.1007/s00366-020-01211-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01211-8

Keywords