Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Vibration analysis of FG reinforced porous nanobeams using two variables trigonometric shear deformation theory

  • Messai, Abderraouf (Department of Civil Engineering, University Ferhat Abbas SETIF 1) ;
  • Fortas, Lahcene (MN2I2S Laboratory, Faculty of Science and Technology, Biskra University) ;
  • Merzouki, Tarek (LISV, University of Versailles Saint-Quentin) ;
  • Houari, Mohammed Sid Ahmed (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, University Mustapha Stambouli of Mascara)
  • Received : 2021.04.28
  • Accepted : 2021.11.16
  • Published : 2022.02.25
⟨ Previous Next ⟩

Abstract

A finite element method analysis framework is introduced for the free vibration analyses of functionally graded porous beam structures by employing two variables trigonometric shear deformation theory. Both Young's modulus and material density of the FGP beam element are simultaneously considered as grading through the thickness of the beam. The finite element approach is developed using a nonlocal strain gradient theory. The governing equations derived here are solved introducing a 3-nodes beam element. A comprehensive parametric study is carried out, with a particular focus on the effects of various structural parameters such as the dispersion patterns of GPL reinforcements and porosity, thickness ratio, boundary conditions, nonlocal scale parameter and strain gradient parameters. The results indicate that porosity distribution and GPL pattern have significant effects on the response of the nanocomposite beams.

Keywords

References

  1. Affdl, J.H. and Kardos, J.L. (1976), "The Halpin-Tsai equations: A review", Polym. Eng. Sci., 16(5), 344-352. https://doi.org/10.1002/pen.760160512.
  2. 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.
  3. 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.
  4. Aifantis, K.E. and Willis, J.R. (2005), "The role of interfaces in enhancing the yield strength of composites and polycrystals", J. Mech. Phys. Solid., 53(5), 1047-1070. https://doi.org/10.1016/j.jmps.2004.12.003.
  5. Akgoz, B. and Civalek, O. (2015), "A novel microstructure-dependent shear deformable beam model", Int. J. Mech. Sci., 99, 10-20. https://doi.org/10.1016/j.ijmecsci.2015.05.003.
  6. Alam, M. and Mishra, S.K. (2021), "Nonlinear vibration of nonlocal strain gradient functionally graded beam on nonlinear compliant substrate", Compos. Struct., 263, 113447. https://doi.org/10.1016/j.compstruct.2020.113447.
  7. Alam, M., Mishra, S.K. and Kant, T. (2021), "Scale dependent critical external pressure for buckling of spherical shell based on nonlocal strain gradient theory", Int. J. Struct. Stab. Dyn., 21(01), 2150003. https://doi.org/10.1142/S0219455421500036.
  8. Ansari, R., Rouhi, H. and Sahmani, S. (2011), "Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics", Int. J. Mech. Sci., 53(9), 786-792. https://doi.org/10.1016/j.ijmecsci.2011.06.010.
  9. Arefi, M., Bidgoli, E.M.R., 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.
  10. 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.
  11. Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369.
  12. Banhart, J. (2001), "Manufacture, characterisation and application of cellular metals and metal foams", Prog. Mater. Sci., 46(6), 559-632. https://doi.org/10.1016/S0079-6425(00)00002-5.
  13. Bedia, W.A., Houari, M.S.A., Bessaim, A., Bousahla, A.A., Tounsi, A., Saeed, T. and Alhodaly, M.S. (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
  14. Chen, D., Yang, J. and Kitipornchai, S. (2015), "Elastic buckling and static bending of shear deformable functionally graded porous beam", Compos. Struct., 133, 54-61. https://doi.org/10.1016/j.compstruct.2015.07.052.
  15. Chen, D., Yang, J. and Kitipornchai, S. (2016), "Free and forced vibrations of shear deformable functionally graded porous beams", Int. J. Mech. Sci., 108, 14-22. https://doi.org/10.1016/j.ijmecsci.2016.01.025.
  16. Civalek, O ., Dastjerdi, S., Akbas, S.D. and Akgoz, B. (2021), "Vibration analysis of carbon nanotube-reinforced composite microbeams", Math. Meth. Appl. Sci., https://doi.org/10.1002/mma.7069.
  17. 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.
  18. Duan, W.H., Wang, C.M. and Zhang, Y.Y. (2007), "Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics", J. Appl. Phys., 101(2), 024305. https://doi.org/10.1063/1.2423140.
  19. Ebrahimi, F. and Barati, M.R. (2017), "Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory", Struct. Eng. Mech., 61(6), 721-736. https://doi.org/10.12989/sem.2017.61.6.721.
  20. Elmeiche, N., Abbad, H., Mechab, I. and Bernard, F. (2020), "Free vibration analysis of functionally graded beams with variable cross-section by the differential quadrature method based on the nonlocal theory", Struct. Eng. Mech., 75(6), 737-746. https://doi.org/10.12989/sem.2020.75.6.737.
  21. 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.
  22. 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.
  23. Eringen, A.C. and Edelen, D. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0.
  24. Fan, F., Safaei, B. and Sahmani, S. (2021), "Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis", Thin Wall. Struct., 159, 107231. https://doi.org/10.1016/j.tws.2020.107231.
  25. 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.
  26. Hangai, Y., Takahashi, K., Utsunomiya, T., Kitahara, S., Kuwazuru, O. and Yoshikawa, N. (2012), "Fabrication of functionally graded aluminum foam using aluminum alloy die castings by friction stir processing", Mater. Sci. Eng.: A, 534, 716-719. https://doi.org/10.1016/j.msea.2011.11.100.
  27. Hassani, A., Habibolahzadeh, A. and Bafti, H. (2012), "Production of graded aluminum foams via powder space holder technique", Mater. Des., 40, 510-515. https://doi.org/10.1016/j.matdes.2012.04.024.
  28. He, S.Y., Zhang, Y., Dai, G. and Jiang, J.Q. (2014), "Preparation of density-graded aluminum foam", Mater. Sci. Eng.: A, 618, 496-499. https://doi.org/10.1016/j.msea.2014.08.087.
  29. 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.
  30. Jalaei, M.H. and Civalek, O. (2019), "On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam", Int. J. Eng. Sci., 143, 14-32. https://doi.org/10.1016/j.ijengsci.2019.06.013.
  31. Jouneghani, F.Z., Dimitri, R. and Tornabene, F. (2018), "Structural response of porous FG nanobeams under hygrothermo-mechanical loadings", Compos. Part B: Eng., 152, 71-78. https://doi.org/10.1016/j.compositesb.2018.06.023.
  32. Kitipornchai, S., Chen, D. and Yang, J. (2017), "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Mater. Des., 116, 656-665. https://doi.org/10.1016/j.matdes.2016.12.061.
  33. 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.
  34. Li, L., Hu, Y. and Ling, L. (2016), "Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory", Physica E: Low Dimens. Syst. Nanostr., 75, 118-124. https://doi.org/10.1016/j.physe.2015.09.028.
  35. 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.
  36. Liu, F., Ming, P. and Li, J. (2007), "Ab initio calculation of ideal strength and phonon instability of graphene under tension", Phys. Rev. B, 76(6), 064120. https://doi.org/10.1103/PhysRevB.76.064120.
  37. Lu, L., Guo, X. and Zhao, J. (2017), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", Int. J. Eng. Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006.
  38. Merzouki, T., Houari, M.S.A., Haboussi, M., Bessaim, A. and Ganapathi, M. (2020), "Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory", Eng. Comput., 1-19. https://doi.org/10.1007/s00366-020-01156-y.
  39. Mindlin, R.D. (1963), "Microstructure in linear elasticity", Dept. of Civil Engineering and Engineering Mechanics, Columbia Univ. New York, USA.
  40. 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.
  41. 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.
  42. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018), "Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's nonlocal elasticity theory", Struct. Eng. Mech., 67(4), 417-425. https://doi.org/10.12989/sem.2018.67.4.417.
  43. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018), "Bending analysis of bi-directional functionally graded EulerBernoulli nano-beams using integral form of Eringen's nonlocal elasticity theory", Struct. Eng. Mech., 67(4), 417-425. https://doi.org/10.12989/sem.2018.67.4.417.
  44. Papargyri-Beskou, S., Tsepoura, K.G., Polyzos, D. and Beskos, D.E. (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. Polit, O., Anant, C., Anirudh, B. and Ganapathi, M. (2019), "Functionally graded graphene reinforced porous nanocomposite curved beams: Bending and elastic stability using a higher-order model with thickness stretch effect", Compos. Part B: Eng., 166, 310-327. https://doi.org/10.1016/j.compositesb.2018.11.074.
  46. Pollien, A., Conde, Y., Pambaguian, L. and Mortensen, A. (2005), "Graded open-cell aluminium foam core sandwich beams", Mater. Sci. Eng.: A, 404(1-2), 9-18. https://doi.org/10.1016/j.msea.2005.05.096.
  47. Qin, J., Chen, Q., Yang, C. and Huang, Y. (2016), "Research process on property and application of metal porous materials", J. Alloy. Compound., 654, 39-44. https://doi.org/10.1016/j.jallcom.2015.09.148.
  48. Rafiee, M.A., Rafiee, J., Wang, Z., Song, H., Yu, Z.Z. and Koratkar, N. (2009), "Enhanced mechanical properties of nanocomposites at low graphene content", ACS Nano, 3(12), 3884-3890. https://doi.org/10.1021/nn9010472.
  49. 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.
  50. Roberts, A.P. and Garboczi, E J. (2001), "Elastic moduli of model random three-dimensional closed-cell cellular solids", Acta Materialia, 49(2), 189-197. https://doi.org/10.1016/S1359-6454(00)00314-1.
  51. Sahmani, S., Aghdam, M.M. and Rabczuk, T. (2018), "Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory", Compos. Struct., 186, 68-78. https://doi.org/10.1016/j.compstruct.2017.11.082.
  52. Sears, A. and Batra, R.C. (2004), "Macroscopic properties of carbon nanotubes from molecular-mechanics simulations", Phys. Rev. B, 69(23), 235406. https://doi.org/10.1103/PhysRevB.69.235406.
  53. Setoodeh, A. and Rezaei, M. (2017), "Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation", Struct. Eng. Mech., 61(2), 209-220. https://doi.org/10.12989/sem.2017.61.2.209.
  54. Shafiei, N., Mousavi, A. and Ghadiri, M. (2016), "On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams", Int. J. Eng. Sci., 106, 42-56. https://doi.org/10.1016/j.ijengsci.2016.05.007.
  55. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133(9), 1-9. https://doi.org/10.1140/epjp/i2018-12196-5.
  56. Sudak, L.J. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281-7287. https://doi.org/10.1063/1.1625437.
  57. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A., ... & Mahmoud, S.R. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., 69(6), 637-649. https://doi.org/10.12989/sem.2019.69.6.637.
  58. Wang, L. and Hu, H. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B, 71(19), 195412. https://doi.org/10.1103/PhysRevB.71.195412.
  59. Wang, Q. and Wang, C.M. (2007), "The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes", Nanotechnol., 18(7), 075702. https://doi.org/10.1088/0957-4484/18/7/075702
  60. Wang, Y., Xie, K., Fu, T. and Zhang, W. (2021), "A third order shear deformable model and its applications for nonlinear dynamic response of graphene oxides reinforced curved beams resting on visco-elastic foundation and subjected to moving loads", Eng. Comput., 1-15. https://doi.org/10.1007/s00366-020-01238-x.
  61. Wu, D., Liu, A., Huang, Y., Huang, Y., Pi, Y. and Gao, W. (2018), "Dynamic analysis of functionally graded porous structures through finite element analysis", Eng. Struct., 165, 287-301. https://doi.org/10.1016/j.engstruct.2018.03.023.
  62. Zenkour, A.M. and Radwan, A.F. (2020), "A nonlocal strain gradient theory for porous functionally graded curved nanobeams under different boundary conditions", Физическая мезомеханика, 23(3), 77-92. https://doi.org/10.24411/1683-805X-2020-13008.