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Buckling analysis of perforated nano/microbeams with deformable boundary conditions via nonlocal strain gradient elasticity

  • Ugur Kafkas (Faculty of Civil Engineering, Kutahya Dumlupinar University) ;
  • Yunus Unal (Faculty of Civil Engineering, Bursa Uludag University) ;
  • M. Ozgur Yayli (Faculty of Civil Engineering, Bursa Uludag University) ;
  • Busra Uzun (Faculty of Civil Engineering, Bursa Uludag University)
  • Received : 2022.11.28
  • Accepted : 2023.07.19
  • Published : 2023.10.25
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Abstract

This work aims to present a solution for the buckling behavior of perforated nano/microbeams with deformable boundary conditions using nonlocal strain gradient theory (NLSGT). For the first time, a solution that can provide buckling loads based on the non-local and strain gradient effects of perforated nanostructures on an elastic foundation, while taking into account both deformable and rigid boundary conditions. Stokes' transformation and Fourier series are used to realize this aim and determine the buckling loads under various boundary conditions. We employ the NLSGT to account for size-dependent effects and utilize the Winkler model to formulate the elastic foundation. The buckling behavior of the perforated nano/microbeams restrained with lateral springs at both ends is studied for various parameters such as the number of holes, the length and filling ratio of the perforated beam, the internal length, the nonlocal parameter and the dimensionless foundation parameter. Our results indicate that the number of holes and filling ratio significantly affect the buckling response of perforated nano/microbeams. Increasing the filling ratio increases buckling loads, while increasing the number of holes decreases buckling loads. The effects of the non-local and internal length parameters on the buckling behavior of the perforated nano/microbeams are also discussed. These material length parameters have opposite effects on the variation of buckling loads. This study presents an effective eigenvalue solution based on Stokes' transformation and Fourier series of the restrained nano/microbeams under the effects of elastic medium, perforation parameters, deformable boundaries and nonlocal strain gradient elasticity for the first time.

Keywords

References

  1. Abdelrahman, A.A. and Eltaher, M.A. (2022), "On bending and buckling responses of perforated nanobeams including surface energy for different beams theories", Eng. Comput., 38(3), 2385-2411. https://doi.org/10.1007/s00366-020-01211-8.
  2. Abdelrahman, A.A., Esen, I., O zarpa, C. and Eltaher, M.A. (2021a), "Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory", Appl. Math. Modell., 96, 215-235. https://doi.org/10.1016/j.apm.2021.03.008.
  3. Abdelrahman, A.A., Esen, I., O zarpa, C., Shaltout, R., Eltaher, M.A. and Assie, A.E. (2021b), "Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory", Smart Struct. Syst., 28(4), 515-553. https://doi.org/10.12989/sss.2021.28.4.515.
  4. Abdelrahman, A.A., Mohamed, N.A. and Eltaher, M.A. (2022), "Static bending of perforated nanobeams including surface energy and microstructure effects", Eng. Comput., 38(S1), 415-435. https://doi.org/10.1007/s00366-020-01149-x.
  5. Akbas, S., Ersoy, H., Akgoz, B. and Civalek, O . (2021), "Dynamic analysis of a fiber-reinforced composite beam under a moving load by the Ritz method", Mathematics, 9(9), 1048. https://doi.org/10.3390/math9091048
  6. Akgoz, B. and Civalek, O . (2011), "Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams", Int. J. Eng. Sci., 49(11), 1268-1280. https://doi.org/10.1016/j.ijengsci.2010.12.009.
  7. Alazwari, M.A., Esen, I., Abdelrahman, A.A., Abdraboh, A.M. and Eltaher, M.A. (2022), "Dynamic analysis of functionally graded (FG) nonlocal strain gradient nanobeams under thermomagnetic fields and moving load", Adv. Nano Res., 12(3), 231-251. https://doi.org/10.12989/anr.2022.12.3.231.
  8. Almitani, K.H., Abdelrahman, A.A., and Eltaher, M.A. (2020), "Stability of perforated nanobeams incorporating surface energy effects", Steel Compos. Struct., 35(4), 555-566. https://doi.org/https://doi.org/10.12989/scs.2020.35.4.555.
  9. Altenbach H and O chsner A. (2020), Encyclopedia of Continuum Mechanics (H. Altenbach and A. O chsner, Eds.), Springer, Berlin, Heidelberg.
  10. Apuzzo, A., Barretta, R., Faghidian, S. A., Luciano, R. and Marotti de Sciarra, F. (2018), "Free vibrations of elastic beams by modified nonlocal strain gradient theory", Int. J. Eng. Sci., 133, 99-108. https://doi.org/10.1016/j.ijengsci.2018.09.002.
  11. Arda, M. (2022), "Evaluation of optimum length scale parameters in longitudinal wave propagation on nonlocal strain gradient carbon nanotubes by lattice dynamics", Mech. Based Des. Struct., 50(12), 4363-4386. https://doi.org/10.1080/15397734.2020.1835488.
  12. Arefi, M. and Amabili, M. (2021), "A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory", Compos. Struct., 257, 113100. https://doi.org/10.1016/j.compstruct.2020.113100.
  13. Arefi, M., Mohammad-Rezaei Bidgoli, E., Dimitri, R., Bacciocchi, M. and Tornabene, F. (2019), "Nonlocal bending analysis of curved nanobeams reinforced by graphene nanoplatelets", Compos. Part B Eng., 166, 1-12. https://doi.org/10.1016/j.compositesb.2018.11.092.
  14. Asghari, M., Kahrobaiyan, M.H., Nikfar, M. and Ahmadian, M.T. (2012), "A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory", Acta Mechanica, 223(6), 1233-1249. https://doi.org/10.1007/s00707-012-0625-0.
  15. Barati, M.R. (2017), "Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities", Adv. Nano Res., 5(4), 393-414. https://doi.org/10.12989/anr.2017.5.4.393.
  16. Barretta, R., Ali Faghidian, S., de Sciarra, F.M. and Pinnola, F.P. (2021), "Timoshenko nonlocal strain gradient nanobeams: Variational consistency, exact solutions and carbon nanotube Young moduli", Mech. Adv. Mater. Struct., 28(15), 1523-1536. https://doi.org/10.1080/15376494.2019.1683660.
  17. Beni, Z.T. and Beni, Y.T. (2022), "Dynamic stability analysis of size-dependent viscoelastic/piezoelectric nano-beam", Int. J. Struct. Stabil. Dyn., 22(5). https://doi.org/10.1142/S021945542250050X
  18. Bensaid, I., Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279.
  19. Bourouina, H., Yahiaoui, R., Kerid, R., Ghoumid, K., Lajoie, I., Picaud, F. and Herlem, G. (2020), "The influence of hole networks on the adsorption-induced frequency shift of a perforated nanobeam using non-local elasticity theory", J. Phys. Chem. Solids, 136, 109201. https://doi.org/10.1016/j.jpcs.2019.109201.
  20. Bourouina, H., Yahiaoui, R., Sahar, A. and Benamar, M.E.A. (2016), "Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads", Physica E, 75, 163-168. https://doi.org/10.1016/j.physe.2015.09.014.
  21. Civalek, O ., Uzun, B. and Yayli, M.O . (2020), "Frequency, bending and buckling loads of nanobeams with different cross sections", Adv. Nano Res., 9(2), 91-104. https://doi.org/10.12989/anr.2020.9.2.091.
  22. Civalek, O ., Uzun, B. and Yayli, M.O . (2022a), "Size dependent torsional vibration of a restrained single walled carbon nanotube (SWCNT) via nonlocal strain gradient approach", Mater. Today Commun., 33, 104271. https://doi.org/10.1016/j.mtcomm.2022.104271.
  23. Civalek, O ., Uzun, B., and Yayli, M. O . (2022b), "A Fourier sine series solution of static and dynamic response of nano/micro micro-scaled FG rod under torsional effect", Adv. Nano Res., 12(5), 467-482. https://doi.org/10.12989/anr.2022.12.5.467
  24. Civalek, O ., Uzun, B., and Yayli, M. O . (2022c), "Thermal buckling analysis of a saturated porous thick nanobeam with arbitrary boundary conditions", J. Therm Stress., 1-21. https://doi.org/10.1080/01495739.2022.2145401
  25. Daghigh, H., Daghigh, V., Milani, A., Tannant, D., Lacy, T. E. and Reddy, J. N. (2020), "Nonlocal bending and buckling of agglomerated CNT-Reinforced composite nanoplates", Compos. Part B Eng., 183, 107716. https://doi.org/10.1016/j.compositesb.2019.107716.
  26. Demir, C . and Civalek, O . (2017), "On the analysis of microbeams", Int. J. Eng. Sci., 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  27. Ebnali Samani, M.S. and Beni, Y.T. (2018), "Size dependent thermo-mechanical buckling of the flexoelectric nanobeam", Mater. Res. Express, 5(8), 085018. https://doi.org/10.1088/2053-1591/aad2ca.
  28. Ebrahimi, F. and Barati, M.R. (2017), "Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 231(23), 4457-4469. https://doi.org/10.1177/0954406216668912.
  29. Ebrahimi, F. and Barati, M.R. (2018a), "Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232(11), 2067-2078. https://doi.org/10.1177/0954406217713518.
  30. Ebrahimi, F. and Barati, M.R. (2018b), "Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory", Mech. Adv. Mater. Struct., 25(11), 953-963. https://doi.org/10.1080/15376494.2017.1329467.
  31. Ebrahimi, F., Karimiasl, M. and Mahesh, V. (2019), "Vibration analysis of magneto-flexo-electrically actuated porous rotary nanobeams considering thermal effects via nonlocal strain gradient elasticity theory", Adv. Nano Res., 7(4), 223-231. https://doi.org/10.12989/ANR.2019.7.4.223.
  32. Eltaher, M.A., Abdraboh, A.M. and Almitani, K.H. (2018a), "Resonance frequencies of size dependent perforated nonlocal nanobeam", Microsyst. Technol., 24(9), 3925-3937. https://doi.org/10.1007/s00542-018-3910-6.
  33. Eltaher, M.A., Kabeel, A.M., Almitani, K.H. and Abdraboh, A.M. (2018b), "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.
  34. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.A. (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.
  35. Eltaher, M.A., Khater, M.E. and Emam, S.A. (2016), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Modell., 40(5-6), 4109-4128. https://doi.org/10.1016/J.APM.2015.11.026.
  36. Eltaher, M.A. and Mohamed, N.A. (2020), "Vibration of nonlocal perforated nanobeams with general boundary conditions", Smart Struct. Syst., 25(4), 501-514. https://doi.org/10.12989/SSS.2020.25.4.501.
  37. Eltaher, M.A., Omar, F.A., Abdalla, W.S., Kabeel, A.M. and Alshorbagy, A.E. (2020a), "Mechanical analysis of cutout piezoelectric nonlocal nanobeam including surface energy effects", Struct. Eng. Mech., 76(1), 141-151. https://doi.org/10.12989/sem.2020.76.1.141.
  38. Eltaher, M.A., Omar, F.A., Abdraboh, A.M., Abdalla, W.S. and Alshorbagy, A.E. (2020b), "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.
  39. Eltaher, M.A., Shanab, R.A. and Mohamed, N.A. (2023), "Analytical solution of free vibration of viscoelastic perforated nanobeam", Arch. Appl. Mech., 93(1), 221-243. https://doi.org/10.1007/s00419-022-02184-4.
  40. Eringen, A.C. (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.
  41. Eringen, A.C. (1987), "Theory of nonlocal elasticity and some applications", Res Mechanica, 21(4), 313-342. https://doi.org/10.21236/ada145201.
  42. Eringen, A.C. (2002), Nonlocal Continuum Field Theories. Springer New York, NY. https://doi.org/10.1007/b97697.
  43. Esen, I., Abdelrahman, A.A. and Eltaher, M.A. (2021a), "On vibration of sigmoid/symmetric functionally graded nonlocal strain gradient nanobeams under moving load", Int. J. Mech. Mater. Des., 17(3), 721-742. https://doi.org/10.1007/s10999-021-09555-9.
  44. Esen, I., Daikh, A.A. and Eltaher, M.A. (2021b), "Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load", Eur. Phys. J. Plus, 136(4), 458. https://doi.org/10.1140/epjp/s13360-021-01419-7.
  45. Esen, I., Abdelrahman, A.A. and Eltaher, M.A. (2022a), "Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields", Eng. Comput., 38(4), 3463-3482. https://doi.org/10.1007/s00366-021-01389-5.
  46. Esen, I., Abdelrahman, A.A. and Eltaher, M.A. (2022b), "Dynamics analysis of timoshenko perforated microbeams under moving loads", Eng. Comput., 38(3), 2413-2429. https://doi.org/10.1007/s00366-020-01212-7.
  47. Esen, I. and O zmen, R. (2022), "Thermal vibration and buckling of magneto-electro-elastic functionally graded porous nanoplates using nonlocal strain gradient elasticity", Compos. Struct., 296, 115878. https://doi.org/10.1016/j.compstruct.2022.115878.
  48. Faghidian, S.A. (2020a), "Two-phase local/nonlocal gradient mechanics of elastic torsion", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6877
  49. Faghidian, S.A. (2020b), "Higher order mixture nonlocal gradient theory of wave propagation", Math. Method Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.6885
  50. Faghidian, S.A. (2021a), "Contribution of nonlocal integral elasticity to modified strain gradient theory", Eur. Phys. J. Plus, 136(5), 559. https://doi.org/10.1140/epjp/s13360-021-01520-x
  51. Faghidian, S.A. (2021b), "Flexure mechanics of nonlocal modified gradient nano-beams", J. Comput. Des. Eng., 8(3), 949-959. https://doi.org/10.1093/jcde/qwab027
  52. Faghidian, S.A., Zur, K.K. and Reddy, J.N. (2022a), "A mixed variational framework for higher-order unified gradient elasticity", Int. J. Eng. Sci., 170, 103603. https://doi.org/10.1016/j.ijengsci.2021.103603
  53. Faghidian, S.A., Zur, K.K., Reddy, J.N. and Ferreira, A.J.M. (2022b), "On the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams based on the higher-order nonlocal gradient elasticity", Compos. Struct., 279, 114819. https://doi.org/10.1016/J.COMPSTRUCT.2021.114819
  54. Faghidian, S.A., Zur, K.K., Pan, E. and Kim, J. (2022c), "On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension", Eng. Anal. Bound. Elem., 134, 571-580. https://doi.org/10.1016/J.ENGANABOUND.2021.11.010
  55. Faghidian, S.A., Zur, K.K. and Rabczuk, T. (2022d), "Mixture unified gradient theory: A consistent approach for mechanics of nanobars", Appl. Phys. A, 128(11), 996. https://doi.org/10.1007/s00339-022-06130-7
  56. Faghidian, S.A., Zur, K.K. and Pan, E. (2023a), "Stationary variational principle of mixture unified gradient elasticity", Int. J. Eng. Sci., 182, 103786. https://doi.org/10.1016/J.IJENGSCI.2022.103786
  57. Faghidian, S.A., Zur, K.K. and Elishakoff, I. (2023b), "Nonlinear flexure mechanics of mixture unified gradient nanobeams", Commun. Nonlinear Sci. Numer. Simul., 117, 106928. https://doi.org/10.1016/J.CNSNS.2022.106928
  58. Fakher, M. and Hosseini-Hashemi, S. (2022), "On the vibration of nanobeams with consistent two-phase nonlocal strain gradient theory: exact solution and integral nonlocal finite-element model", Eng. Comput., 38(3), 2361-2384. https://doi.org/10.1007/s00366-020-01206-5.
  59. Farajpour, A., Shahidi, A.R., Mohammadi, M. and Mahzoon, M. (2012), "Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics", Compos. Struct., 94(5), 1605-1615. https://doi.org/10.1016/J.COMPSTRUCT.2011.12.032.
  60. Farrahi, G.H., Faghidian, S.A. and Smith, D.J. (2009), "Reconstruction of residual stresses in autofrettaged thick-walled tubes from limited measurements", Int. J. Press. Vessels Pip., 86(11), 777-784. https://doi.org/10.1016/J.IJPVP.2009.03.010
  61. Farrahi, G.H., Faghidian, S.A. and Smith, D.J. (2010), "an inverse method for reconstruction of the residual stress field in welded plates", J. Press. Vessel Technol., 132(6). https://doi.org/10.1115/1.4001268
  62. Fleck, N.A. and Hutchinson, J.W. (1993), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Solids, 41(12), 1825-1857. https://doi.org/10.1016/0022-5096(93)90072-N.
  63. Fleck, N.A. and Hutchinson, J.W. (2001), "A reformulation of strain gradient plasticity", J. Mech. Phys. Solids, 49(10), 2245-2271. https://doi.org/10.1016/S0022-5096(01)00049-7.
  64. Giannuzzi, L.A. and Stevie, F.A. (1999), "A review of focused ion beam milling techniques for TEM specimen preparation", Micron, 30(3), 197-204. https://doi.org/10.1016/S0968-4328(99)00005-0.
  65. Guha, K., Kumar, M., Agarwal, S. and Baishya, S. (2015), "A modified capacitance model of RF MEMS shunt switch incorporating fringing field effects of perforated beam", Solid-State Electronics, 114, 35-42. https://doi.org/10.1016/J.SSE.2015.07.008.
  66. Guclu, G. and Artan, R. (2020), "Large elastic deflections of bars based on nonlocal elasticity", ZAMM J. Appl. Math. Mech., 100(4), e201900108. https://doi.org/10.1002/zamm.201900108.
  67. Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (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.
  68. Harik, V.M. (2002), "Mechanics of carbon nanotubes: applicability of the continuum-beam models", Comput. Mater. Sci., 24(3), 328-342. https://doi.org/10.1016/S0927-0256(01)00255-5.
  69. Hutchinson, J. and Fleck, N. (1997), "Strain gradient plasticity", Adv. Appl. Mech., 33, 295-361. https://doi.org/10.1016/S0065-2156(08)70388-0
  70. Jena, S. K., Chakraverty, S. and Tornabene, F. (2019), "Dynamical behavior of nanobeam embedded in constant, linear, parabolic, and sinusoidal types of Winkler elastic foundation using first-Order nonlocal strain gradient model", Mater. Res. Express, 6(8), 0850f2. https://doi.org/10.1088/2053-1591/ab2779.
  71. Jeong, K.H. and Amabili, M. (2006), "Bending vibration of perforated beams in contact with a liquid", J. Sound Vib., 298(1-2), 404-419. https://doi.org/10.1016/J.JSV.2006.05.029
  72. Kar, V.R. and Panda, S.K. (2017), "Postbuckling analysis of shear deformable FG shallow spherical shell panel under nonuniform thermal environment", J. Therm. Stress., 40(1), 25-39. https://doi.org/10.1080/01495739.2016.1207118
  73. Katariya, P.V., Panda, S.K., Hirwani, C.K., Mehar, K. and Thakare, O. (2017), "Enhancement of thermal buckling strength of laminated sandwich composite panel structure embedded with shape memory alloy fibre", Smart Struct. Syst., 20(5), 595-605. https://doi.org/10.12989/sss.2017.20.5.595
  74. Katariya, P.V. and Panda, S.K. (2016), "Thermal buckling and vibration analysis of laminated composite curved shell panel", Aircr. Eng. Aerosp. Technol., 88(1), 97-107. https://doi.org/10.1108/AEAT-11-2013-0202
  75. Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Naddaf Oskouei, A. (2017), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin Wall. Struct., 113, 162-169. https://doi.org/10.1016/J.TWS.2017.01.016.
  76. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and 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.
  77. Langford, R.M., Nellen, P.M., Gierak, J., and Fu, Y. (2007), "Focused ion beam micro- and nanoengineering", MRS Bulletin, 32(5), 417-423. https://doi.org/10.1557/mrs2007.65.
  78. Li, L. and Hu, Y. (2016), "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 107, 77-97. https://doi.org/10.1016/j.ijengsci.2016.07.011.
  79. Li, Y.D., Bao, R. and Chen, W. (2018), "Buckling of a piezoelectric nanobeam with interfacial imperfection and van der Waals force: Is nonlocal effect really always dominant?", Compos. Struct., 194, 357-364. https://doi.org/10.1016/j.compstruct.2018.04.031.
  80. Lim, C.W., Zhang, G. and Reddy, J.N. (2015a), "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.
  81. Lim, C.W., Islam, M.Z. and Zhang, G. (2015b), "A nonlocal finite element method for torsional statics and dynamics of circular nanostructures", Int. J. Mech. Sci., 94-95, 232-243. https://doi.org/10.1016/J.IJMECSCI.2015.03.002.
  82. Lu, L., Guo, X. and Zhao, J. (2017), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms", Int. J. Eng. Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024.
  83. Luschi, L. and 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.
  84. Melaibari, A., Abdelrahman, A.A., Hamed, M.A., Abdalla, A.W. and Eltaher, M.A. (2022), "Dynamic analysis of a piezoelectrically layered perforated nonlocal strain gradient nanobeam with flexoelectricity", Mathematics, 10(15), 2614. https://doi.org/10.3390/math10152614.
  85. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., 16, 51-78. https://doi.org/10.1007/BF00248490.
  86. Mindlin, R.D. (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.
  87. Mirjavadi, S.S., Forsat, M., Nia, A.F., Badnava, S. and Hamouda, A.M.S. (2020), "Nonlocal strain gradient effects on forced vibrations of porous FG cylindrical nanoshells", Adv. Nano Res., 8(2), 149-156. https://doi.org/10.12989/anr.2020.8.2.149.
  88. Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., 7(3), 181-190. https://doi.org/https://doi.org/10.12989/anr.2019.7.3.181
  89. Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324
  90. Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216, 406-414. https://doi.org/10.1016/J.COMPSTRUCT.2019.03.002
  91. Murmu, T. and Pradhan, S.C. (2009), "Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Physica E, 41(7), 1232-1239. https://doi.org/10.1016/J.PHYSE.2009.02.004.
  92. Najafzadeh, M., Adeli, M.M., Zarezadeh, E. and Hadi, A. (2020), "Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field", Mech. Based Des. Struct. Mach., 50(2), 726-740. https://doi.org/10.1080/15397734.2020.1733602.
  93. Nematollahi, M.S., Mohammadi, H. and Nematollahi, M.A. (2017), "Thermal vibration analysis of nanoplates based on the higher-order nonlocal strain gradient theory by an analytical approach", Superlatt. Microstruct., 111, 944-959. https://doi.org/10.1016/j.spmi.2017.07.055.
  94. Noroozi, R., Barati, A., Kazemi, A., Norouzi, S. and Hadi, A. (2020), "Torsional vibration analysis of bi-directional FG nanocone with arbitrary cross-section based on nonlocal strain gradient elasticity", Adv. Nano Res., 8(1), 13-24. https://doi.org/10.12989/anr.2020.8.1.013.
  95. Numanoglu, H.M., Akgoz, B. and Civalek, O . (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  96. Panda, S.K. and Singh, B.N. (2010), "Nonlinear free vibration analysis of thermally post-buckled composite spherical shell panel", Int. J. Mech. Mater. Des. 6(2), 175-188. https://doi.org/10.1007/s10999-010-9127-1
  97. Panda, S.K. and Singh, B.N. (2013a), "Thermal postbuckling behavior of laminated composite spherical shell panel using NFEM", Mech. Based Des. Struct., 41(4), 468-488. https://doi.org/10.1080/15397734.2013.797330
  98. Panda, S.K. and Singh, B.N. (2013b), "Post-buckling analysis of laminated composite doubly curved panel embedded with sma fibers subjected to thermal environment", Mech. Adv. Mater. Struct., 20(10), 842-853. https://doi.org/10.1080/15376494.2012.677097
  99. Park, S.H. (2013), "A design method of micro-perforated panel absorber at high sound pressure environment in launcher fairings", J. Sound Vib., 332(3), 521-535. https://doi.org/10.1016/J.JSV.2012.09.015
  100. Pelliciari, M. and Tarantino, A.M. (2021), "Equilibrium and stability of anisotropic hyperelastic graphene membranes", J. Elast., 144(2), 169-195. https://doi.org/10.1007/s10659-021-09837-5
  101. Pelliciari, M. and Tarantino, A.M. (2022), "A continuum model for circular graphene membranes under uniform lateral pressure", J. Elast., 151(2), 273-303. https://doi.org/10.1007/s10659-022-09937-w
  102. Pelliciari, M., Pasca, D.P., Aloisio, A. and Tarantino, A.M. (2022), "Size effect in single layer graphene sheets and transition from molecular mechanics to continuum theory", Int. J. Mech. Sci., 214, 106895. https://doi.org/10.1016/j.ijmecsci.2021.106895
  103. Pham, Q.H., Tran, T.T., Tran, V.K., Nguyen, P.C., Nguyen-Thoi, T. and Zenkour, A.M. (2021), "Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation", Mech. Adv. Mater. Struct., 1-21. https://doi.org/10.1080/15376494.2021.1968549.
  104. Reddy, J.N. (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.
  105. Reyntjens, S. and Puers, R. (2001), "A review of focused ion beam applications in microsystem technology", J. Micromech. Microeng., 11(4), 287-300. https://doi.org/10.1088/0960-1317/11/4/301.
  106. Sari, G. and Pakdemirli, M. (2013), "Non-linear vibrations of a microbeam resting on an elastic foundation", Arabian J. Sci. Eng., 38(5), 1191-1199. https://doi.org/10.1007/s13369-012-0533-6.
  107. Shariati, A., Barati, M.R., Ebrahimi, F., Singhal, A. and Toghroli, A. (2020), "Investigating vibrational behavior of graphene sheets under linearly varying in-plane bending load based on the nonlocal strain gradient theory", Adv. Nano Res., 8(4), 265-276. https://doi.org/10.12989/anr.2020.8.4.265.
  108. Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013.
  109. Soltani, M., Atoufi, F., Mohri, F., Dimitri, R. and Tornabene, F. (2021), "Nonlocal elasticity theory for lateral stability analysis of tapered thin-walled nanobeams with axially varying materials", Thin Wall. Struct., 159, 107268. https://doi.org/10.1016/j.tws.2020.107268.
  110. Tang, Y., Liu, Y. and Zhao, D. (2016), "Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory", Physica E, 84, 202-208. https://doi.org/10.1016/j.physe.2016.06.007.
  111. Tang, Y. and Qing, H. (2023), "Size-dependent nonlinear postbuckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models", Commun. Nonlinear Sci. Numer. Simul., 116. https://doi.org/10.1016/j.cnsns.2022.106808.
  112. Temiz, M.A., Tournadre, J., Arteaga, I.L. and Hirschberg, A. (2016), "Non-linear acoustic transfer impedance of microperforated plates with circular orifices", J. Sound Vib., 366, 418-428. https://doi.org/10.1016/J.JSV.2015.12.022.
  113. Toupin, R. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945.
  114. Tsuji, T., Kakita, T. and Tsuji, M. (2003), "Preparation of nanosize particles of silver with femtosecond laser ablation in water", Appl. Surface Sci., 206(1-4), 314-320. https://doi.org/10.1016/S0169-4332(02)01230-8.
  115. Uzun, B., and Yayli, M. O . (2022), "Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory", Mater. Today Commun., 32, 103969. https://doi.org/10.1016/J.MTCOMM.2022.103969
  116. Uzun, B., Civalek, O . and Yayli, M. O . (2020a), "Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions", Mech. Based Des. Struct., 1-20. https://doi.org/10.1080/15397734.2020.1846560.
  117. Uzun, B., Kafkas, U. and Yayli, M.O . (2020b), "Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions", ZAMM J. Appl. Math. Mech., 100(12), e202000039. https://doi.org/10.1002/ZAMM.202000039.
  118. Uzun, B., Kafkas, U. and Yayli, M.O . (2020c), "Stability analysis of restrained nanotubes placed in electromagnetic field", Microsyst. Technol., 26(12), 3725-3736. https://doi.org/10.1007/s00542-020-04847-0
  119. Uzun, B., Kafkas, U., Deliktas, B. and Yayli, M.O . (2023), "Size-dependent vibration of porous bishop nanorod with arbitrary boundary conditions and nonlocal elasticity effects", J. Vib. Eng. Technol., 11, 809-826. https://doi.org/10.1007/s42417-022-00610-z
  120. Wang, B., Zhao, J. and Zhou, S. (2010), "A micro scale Timoshenko beam model based on strain gradient elasticity theory", Eur. J. Mech. A Solids, 29(4), 591-599. https://doi.org/10.1016/j.euromechsol.2009.12.005.
  121. Wang, J., Zhou, W., Huang, Y., Lyu, C., Chen, W. and Zhu, W. (2018), "Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control", Appl. Math. Mech., 39(8), 1059-1070. https://doi.org/10.1007/s10483-018-2360-6.
  122. Wang, Y.Q. and Liang, C. (2019), "Wave propagation characteristics in nanoporous metal foam nanobeams", Results Phys., 12, 287-297. https://doi.org/10.1016/J.RINP.2018.11.080.
  123. Xu, X.J., Wang, X.C., Zheng, M.L. and Ma, Z. (2017), "Bending and buckling of nonlocal strain gradient elastic beams", Compos. Struct., 160, 366-377. https://doi.org/10.1016/j.compstruct.2016.10.038.
  124. Yang, F., Chong, A.C.M., Lam, D.C.C. and 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.
  125. Yayli, M.O . (2017), "Buckling analysis of a cantilever single-walled carbon nanotube embedded in an elastic medium with an attached spring", Micro Nano Lett., 12(4), 255-259. https://doi.org/10.1049/mnl.2016.0662.
  126. Yayli, M.O . (2019), "Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity", Curve. Layer. Struct., 6(1), 1-10. https://doi.org/10.1515/cls-2019-0001.
  127. Yayli, M.O ., Uzun, B. and Deliktas, B. (2021), "Buckling analysis of restrained nanobeams using strain gradient elasticity", Waves Random Complex Med., 1-20. https://doi.org/10.1080/17455030.2020.1871112.
  128. Zeighampour, H., Tadi Beni, Y. and Botshekanan Dehkordi, M. (2018), "Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory", Thin Wall. Struct., 122, 378-386. https://doi.org/10.1016/j.tws.2017.10.037.
  129. Zeighampour, H. and Tadi Beni, Y. (2021), "Vibration analysis of boron nitride nanotubes by considering electric field and surface effect", Adv. Nano Res., 11(6), 607-620. https://doi.org/10.12989/anr.2021.11.6.607
  130. Zhang, W., Zhang, Y., Tang, J., Zhang, Y., Wang, L. and Ling, Q. (2002), "Study on preparation and optic properties of nano europium oxide-ethanol sol by pulsed laser ablation", Thin Solid Films, 417(1-2), 43-46. https://doi.org/10.1016/S0040-6090(02)00640-5.
  131. Zhu, X. and Li, L. (2017), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28. https://doi.org/10.1016/j.ijengsci.2017.06.019.
  132. Zur, K.K. and Faghidian, S.A. (2021), "Analytical and meshless numerical approaches to unified gradient elasticity theory", Eng. Anal. Bound. Elem., 130, 238-248. https://doi.org/10.1016/J.ENGANABOUND.2021.05.022