Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

An efficient numerical model for free vibration of temperature-dependent porous FG nano-scale beams using a nonlocal strain gradient theory

  • Tarek Merzouki (LISV, University of Versailles Saint-Quentin) ;
  • Mohammed SidAhmed Houari (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, University Mustapha Stambouli of Mascara)
  • Received : 2023.12.15
  • Accepted : 2024.03.19
  • Published : 2024.04.10
⟨ Previous Next ⟩

Abstract

The present study conducts a thorough analysis of thermal vibrations in functionally graded porous nanocomposite beams within a thermal setting. Investigating the temperature-dependent material properties of these beams, which continuously vary across their thickness in accordance with a power-law function, a finite element approach is developed. This approach utilizes a nonlocal strain gradient theory and accounts for a linear temperature rise. The analysis employs four different patterns of porosity distribution to characterize the functionally graded porous materials. A novel two-variable shear deformation beam nonlocal strain gradient theory, based on trigonometric functions, is introduced to examine the combined effects of nonlocal stress and strain gradient on these beams. The derived governing equations are solved through a 3-nodes beam element. A comprehensive parametric study delves into the influence of structural parameters, such as thicknessratio, beam length, nonlocal scale parameter, and strain gradient parameter. Furthermore, the study explores the impact of thermal effects, porosity distribution forms, and material distribution profiles on the free vibration of temperature-dependent FG nanobeams. The results reveal the substantial influence of these effects on the vibration behavior of functionally graded nanobeams under thermal conditions. This research presents a finite element approach to examine the thermo-mechanical behavior of nonlocal temperature-dependent FG nanobeams, filling the gap where analytical results are unavailable.

Keywords

References

  1. Agrawal, R. Peng, B. Gdoutos, E.E. and Espinosa, H.D. (2008), "Elasticity size effects in ZnO nanowires-a combined experimental-computational approach", Nano Lett. 8(11), 3668-3674. https://doi.org/10.1021/nl801724b.
  2. Aifantis, E.C. (1992), "On the role of gradients in the localization of deformation and fracture", Int. J. Eng. Sci. 30(10), 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3.
  3. Akgoz, B. and Civalek, O. (2014), "Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium", Int. J. Eng. Sci. 85, 90-104. https://doi.org/10.1016/j.ijengsci.2014.08.011.
  4. Alazwari, M.A. Daikh, A.A. Houari, M.S.A. Tounsi, A. and Eltaher, M.A. (2021), "On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations", Steel Compos. Struct. 40(3), 389-404. https://doi.org/10.12989/scs.2021.40.3.389.
  5. Alazwari, M.A. Eltaher, M.A. and Abdelrahman, A.A. (2022), "On bending of cutout nanobeams based on nonlocal strain gradient elasticity theory", Steel Compos. Struct. 43(6), 707-723. https://doi.org/10.12989/scs.2022.43.6.707.
  6. Ansari, R. Gholami, R. Shojaei, M.F. Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory", Compos. Struct. 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048.
  7. Asghari, M. Kahrobaiyan, M.H. Rahaeifard, M. and Ahmadian, M.T. (2011), "Investigation of the size effects in Timoshenko beams based on the couple stress theory", Arch. Appl. Mech. 81, 863-874. https://doi.org/10.1007/s00419-010-0452-5.
  8. Askes, H. and Aifantis, E.C. (2009), "Gradient elasticity and flexural wave dispersion in carbon nanotubes", Phys. Rev. B, 80(19), 195412. https://doi.org/10.1103/PhysRevB.80.195412.
  9. Babaei Gavan, K. Westra, H.J. van der Drift, E.W. Venstra, W.J. and van der Zant, H.S. (2009), "Size-dependent effective Young's modulus of silicon nitride cantilevers", Appl. Phys. Lett. 94(23), 233108. https://doi.org/10.1063/1.3152772.
  10. Bessaim, A. Houari, M.S.A. Bezzina, S. Merdji, A. Daikh, A.A. Belarbi, M.O. and Tounsi, A. (2023), "Nonlocal strain gradient theory for bending analysis of 2D functionally graded nanobeams", Struct. Eng. Mech. 86(6), 731-738. https://doi.org/10.12989/sem.2023.86.6.731.
  11. Borjalilou, V. Taati, E. and Ahmadian, M.T. (2019), "Bending, buckling and free vibration of nonlocal FG-carbon nanotube-reinforced composite nanobeams: Exact solutions", SN Appl. Sci. 1, 1-15. https://doi.org/10.1007/s42452-019-1359-6.
  12. Daikh, A.A. Bachiri, A. Houari, M.S.A. and Tounsi, A. (2022), "Size dependent free vibration and buckling of multilayered carbon nanotubes reinforced composite nanoplates in thermal environment", Mech. Bas. Des. Struct. Mach. 50(4), 1371-1399. https://doi.org/10.1080/15397734.2020.1752232.
  13. Daikh, A.A. Bachiri, A. Houari, M.S.A. and Tounsi, A. (2022a), "Size dependent free vibration and buckling of multilayered carbon nanotubes reinforced composite nanoplates in thermal environment", Mech. Bas. Des. Struct. Mach. 50(4), 1371-1399. https://doi.org/10.1080/15397734.2020.1752232.
  14. Daikh, A.A. Drai, A. Houari, M.S.A. and 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.
  15. Daikh, A.A. Houari, M.S.A. Belarbi, M.O. Chakraverty, S. and Eltaher, M.A. (2022), "Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates", Eng. Comput. 38(Suppl 3), 2533-2554. https://doi.org/10.1007/s00366-021-01413-8.
  16. Daikh, A.A. Houari, M.S.A. Belarbi, M.O. Mohamed, S.A. and Eltaher, M.A. (2022b), "Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory", Defence Technol. 18(10), 1778-1809. https://doi.org/10.1016/j.dt.2021.09.011.
  17. Duan, W.H. and Wang, C.M. (2007), "Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory", Nanotechnol. 18(38), 385704. https://doi.org/10.1088/0957-4484/18/38/385704.
  18. Ebrahimi, F. and Reza Barati, M. (2016), "Vibration analysis of nonlocal beams made of functionally graded material in thermal environment", Eur. Phys. J. Plus, 131, 1-22. https://doi.org/10.1140/epjp/i2016-16279-y.
  19. Ebrahimi, F. and Salari, E. (2015), "Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment", Acta Astronautica, 113, 29-50. https://doi.org/10.1016/j.actaastro.2015.03.031.
  20. Ebrahimi, F. Ghasemi, F. and Salari, E. (2016), "Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities", Meccanica, 51, 223-249. https://doi.org/10.1007/s11012-015-0208-y.
  21. Ekinci, K.L. and Roukes, M.L. (2005), "Nanoelectromechanical systems", Rev. Scientif. Instrum. 76(6), 061101. https://doi.org/10.1063/1.1927327.
  22. 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. Model. 40(5-6), 4109-4128. https://doi.org/10.1016/j.apm.2015.11.026.
  23. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci. 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  24. 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.
  25. Esen, I. Alazwari, M.A. Eltaher, M.A. and Abdelrahman, A.A. (2022), "Dynamic response of FG porous nanobeams subjected thermal and magnetic fields under moving load", Steel Compos. Struct. 42(6), 805-826. https://doi.org/10.12989/scs.2022.42.6.805.
  26. Fang, J. Zheng, S. Xiao, J. and Zhang, X. (2020), "Vibration and thermal buckling analysis of rotating nonlocal functionally graded nanobeams in thermal environment", Aerosp. Sci. Technol. 106, 106146. https://doi.org/10.1016/j.ast.2020.106146.
  27. Fortas, L. Messai, A. Merzouki, T. and Houari, M.S.A. (2022), "Elastic stability of functionally graded graphene reinforced porous nanocomposite beams using two variables shear deformation", Steel Compos. Struct. 43(1), 31-54. https://doi.org/10.12989/scs.2022.43.1.031.
  28. Hadji, L. and Avcar, M. (2021), "Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory", Adv. Nano Res. 10(3), 281. https://doi.org/10.12989/anr.2021.10.3.281.
  29. Hadji, L. Avcar, M. and Civalek, O. (2021), "An analytical solution for the free vibration of FG nanoplates", J. Brazil. Soc. Mech. Sci. Eng. 43(9), 418. https://doi.org/10.1007/s40430-021-03134-x
  30. Houari, M.S.A. Bessaim, A. Bernard, F. Tounsi, A. and Mahmoud, S.R. (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.
  31. Lam, D.C. Yang, F. Chong, A.C.M. Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid. 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  32. Li, X. Li, L. Hu, Y. Ding, Z. and Deng, W. (2017), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos. Struct. 165, 250-265. https://doi.org/10.1016/j.compstruct.2017.01.032.
  33. Lim, C.W. Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solid. 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  34. Long, C. Zhao, B. Chen, J. Liu, T. Peng, X. Peng, H. and Yang, X. (2021), "A size-dependent thermal buckling model for micro-beams based on modified gradient elasticity", Arch. Appl. Mech. 91, 3291-3302. https://doi.org/10.1007/s00419-021-01965-7.
  35. Ma, H.M. Gao, X.L. and Reddy, J. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid. 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007.
  36. Merzouki, T. Houari, M.S.A. Haboussi, M. Bessaim, A. and Ganapathi, M. (2022), "Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory", Eng. Comput. 38(Suppl 1), 647-665. https://doi.org/10.1007/s00366-020-01156-y.
  37. Messai, A. Fortas, L. Merzouki, T. and Houari, M.S.A. (2022), "Vibration analysis of FG reinforced porous nanobeams using two variables trigonometric shear deformation theory", Struct. Eng. Mech. 81(4), 461-479. https://doi.org/10.12989/sem.2022.81.4.461.
  38. Mindlin, R.D. (1963), Microstructure in Linear Elasticity, Columbia University, New York.
  39. Mindlin, R.D. (1965), "Second gradient of strain and surface-tension in linear elasticity", Int. J. Solid. Struct. 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5.
  40. Mouffoki, A. Bedia, E.A. Houari, M.S.A. Tounsi, A. and Mahmoud, S.R. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory", Smart Struct. Syst. 20(3), 369-383. https://doi.org/10.12989/sss.2017.20.3.369.
  41. Najafi, M. and Ahmadi, I. (2021), "A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation", Steel Compos. Struct. 40(1), 101-119. https://doi.org/10.12989/scs.2021.40.1.101.
  42. Nateghi, A. Salamat-talab, M. Rezapour, J. and Daneshian, B. (2012), "Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory", Appl. Math. Model. 36(10), 4971-4987. https://doi.org/10.1016/j.apm.2011.12.035.
  43. Ozmen, R. Kilic, R. and Esen, I. (2024), "Thermomechanical vibration and buckling response of nonlocal strain gradient porous FG nanobeams subjected to magnetic and thermal fields", Mech. Adv. Mater. Struct. 31(4), 834-853. https://doi.org/10.1080/15376494.2022.2124000.
  44. Papargyri-Beskou, S. Tsepoura, K.G. Polyzos, D. and Beskos, D. (2003), "Bending and stability analysis of gradient elastic beams", Int. J. Solid. Struct. 40(2), 385-400. https://doi.org/10.1016/S0020-7683(02)00522-X.
  45. Peddieson, J. Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci. 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0.
  46. Pham, Q.H. and Nguyen, P.C. (2022), "Effects of size-dependence on static and free vibration of FGP nanobeams using finite element method based on nonlocal strain gradient theory", Steel Compos. Struct. 45(3), 331-348. https://doi.org/10.12989/scs.2022.45.3.331.
  47. Rahmani, O. Refaeinejad, V. and Hosseini, S.A.H. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct. 23(3), 339-350. https://doi.org/10.12989/scs.2017.23.3.339.
  48. Reddy, J. (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.
  49. Simsek, M. and Reddy, J.N. (2013), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci. 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002.
  50. Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures", Phys. Lett. A, 363(3), 236-242. https://doi.org/10.1016/j.physleta.2006.10.093.
  51. 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.
  52. Yang, F.A.C.M. Chong, A.C.M. Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solid. Struct. 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  53. Yin, G.S. Deng, Q.T. and Yang, Z.C. (2015), "Bending and buckling of functionally graded Poisson's ratio nanoscale beam based on nonlocal theory", Iran. J. Sci. Technol. (Sci.), 39(4), 559-565. https://doi.org/10.22099/IJSTS.2015.3417.
  54. Zhang, Y.Y. Wang, Y.X. Zhang, X. Shen, H.M. and She, G.L. (2021), "On snap-buckling of FG-CNTR curved nanobeams considering surface effects", Steel Compos. Struct. 38(3), 293-304. https://doi.org/10.12989/scs.2021.38.3.293.