Geomechanics and Engineering

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Wave propagation of CNTRC beams resting on elastic foundation based on various higher-order beam theories

  • Yi-Wen Zhang (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Hao-Xuan Ding (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Gui-Lin She (College of Mechanical and Vehicle Engineering, Chongqing University) ;
  • Abdelouahed Tounsi (YFL (Yonsei Frontier Lab), Yonsei University)
  • Received : 2022.11.12
  • Accepted : 2023.03.21
  • Published : 2023.05.25
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Abstract

The aim of this work is to analyze and predict the wave propagation behavior of the carbon nanotube reinforced composites (CNTRC) beams within the framework of various higher order shear deformation beam theory. Using the Euler-Lagrange principle, the wave equations for CNTRC beams are derived, where the determining factor is to make the determinant equal to zero. Based on the eigenvalue method, the relationship between wave number and circular frequency is obtained. Furthermore, the phase and group velocities during wave propagation are obtained as a function of wave number, and the material properties of CNTRC beams are estimated by the mixture rule. In this paper, various higher order shear beam theory including Euler beam theory, Timoshenko beam theory and other beam theories are mainly adopted to analyze the wave propagation problem of the CNTRC beams, and by this way, we conduct a comparative analysis to verify the correctness of this paper. The mathematical model provided in this paper is verified numerically by comparing it with some existing results. We further investigate the effects of different enhancement modes of CNTs, volume fraction of CNTs, spring factor and other aspects on the wave propagation behaviors of the CNTRC beams.

Keywords

References

  1. Alazwari, M.A., Daikh, A.A., Houari, M.S., 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.
  2. Assie, A.E., Mohamed, S.A., Shanab, R.A., Abo-bakr, R.M. and Eltaher, M.A. (2023), "Static buckling of 2D FG porous plates resting on elastic foundation based on unified shear theories", J. Appl. Comput. Mech., 9(1), 239-258. https://doi.org/10.22055/jacm.2022.41265.3723.
  3. Babaei, H. (2022a), "Nonlinear analysis of size‑dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory", Eng. Struct., 38(3), 1717-1734. https://doi.org/10.1007/s00366-021-01317-7.
  4. Babaei, H. (2022b), "Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation", Appl. Math. Comput., 413, 126606. https://doi.org/10.1016/j.amc.2021.126606.
  5. Babaei, H. and Eslami, M.R. (2021a), "Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT", Appl. Math. Model., 91, 1061-1080. https://doi.org/10.1016/j.apm.2020.10.004.
  6. Babaei, H. and Eslami, M.R. (2021b), "Nonlinear analysis of thermal-mechanical coupling bending of clamped FG porous curved micro-tubes", J. Therm. Stresses, 44(4), 409-432. https://doi.org/10.1080/01495739.2020.1870417.
  7. Barretta, R., Ali Faghidian, S. and Marotti de Sciarra, F., Penna, R. and Pinnola F.P. (2020), "On torsion of nonlocal Lam strain gradient FG elastic beams", Compos. Struct., 233, 111550. https://doi.org/10.1016/j.compstruct.2019.111550
  8. Basha, M., Daikh, A.A., Melaibari, A., Wagih, A., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A. and Eltaher, M.A. (2022), "Nonlocal strain gradient theory for buckling and bending of FG-GRNC laminated sandwich plates", Steel Compos. Struct., 43(5), 639-660. https://doi.org/10.12989/scs.2022.43.5.639
  9. Bensattalah, T., Hamidi, A., Bouakkaz, K., Zidour, M. and Daouadji, T.H. (2020), "Critical buckling load of triple-walled carbon nanotube based on nonlocal elasticity theory", J. Nano Res., 62, 108-119. https://doi.org/10.4028/www.scientific.net/jnanor.62.108.
  10. Chen, X., Zhao, J.L., She, G.L., Jing, Y., Luo, J. and Pu, H.Y. (2022a), "On wave propagation of functionally graded CNT strengthened fluid-conveying pipe in thermal environment", Eur. Phys. J. Plus., 137(10), 1158. https://doi.org/10.1140/epjp/s13360-022-03234-0.
  11. Chen, X., Zhao, J.L., She, G.L., Jing, Y., Pu, H.Y. and Luo, J. (2022b), "Nonlinear free vibration analysis of functionally graded carbon nanotube reinforced fluid-conveying pipe in thermal environment", Steel Compos. Struct., 45(5), 641-652. https://doi.org/10.12989/scs.2022.45.5.641.
  12. Civalek, O., Akba, E.D., Akgz, B. and Dastjerdi, S. (2021a), "Forced vibration analysis of composite beams reinforced by carbon nanotubes", Nanomaterials, 11(3), 571. https://doi.org/10.3390/nano11030571.
  13. Civalek, O., Dastjerdi, S., Akbas, S.D. and Akgoz, B. (2021b), "Vibration Analysis of Carbon Nanotube-Reinforced Composite Microbeams", Math. Method. Appl. Sci., https://doi.org/10.1002/mma.7069.
  14. Daikh, A.A., Houari, M.S.A., Karami, B., Eltaher, M.A., Dimitri, R. and Tornabene, F. (2021), "Buckling analysis of CNTRC curved sandwich nanobeams in thermal environment", Appl. Sci., 11(7), 3250. https://doi.org/10.3390/app11073250.
  15. Ding, H.X. and She, G.L. (2021), "A higher-order beam model for the snap-buckling analysis of FG pipes conveying fluid", Struct. Eng. Mech., 80(1), 63-72. http://dx.doi.org/10.12989/sem.2021.80.1.063.
  16. Ding, H.X., She, G.L. and Zhang, Y.W. (2022a), "Nonlinear buckling and resonances of functionally graded fluid-conveying pipes with initial geometric imperfection", Eur. Phys. J. Plus, 137, 1329. https://doi.org/10.1140/epjp/s13360-022-03570-1.
  17. Ding, H.X., Zhang, Y.W. and She, G.L. (2022b), "On the resonance problems in FG-GPLRC beams with different boundary conditions resting on elastic foundations", Comput. Concrete, 30(6), 433-443. https://doi.org/10.12989/cac.2022.30.6.433.
  18. Ding, H.X. and She, G.L. (2023), "Nonlinear resonance of axially moving graphene platelet reinforced metal foam cylindrical shells with geometric imperfection", Archiv. Civil Mech. Eng., https://doi.org/10.1007/s43452-023-00634-6.
  19. Ebrahimi, F. and Farazmandnia, N. (2018), "Vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in thermal environment", Adv. Aircraft Spacecraft Sci., 5(1), 107-128. https://doi.org/10.12989/aas.2018.5.1.107.
  20. Emam, S.A., Eltaher, M.A., Khater, M.E. and Abdalla, W.S. (2018), "Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load", Appl. Sci., 8(11), 2238. https://doi.org/10.3390/app8112238.
  21. Esen, I., Daikh, A.A. and Eltaher, M.A. (2022b), "Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load", Eur. Phys. J. Plus, 136(4),1-22. https://doi.org/10.1140/epjp/s13360-021-01419-7.
  22. Faghidian, S.A. and Elishakoff, I. (2022a), "Wave propagation in Timoshenko-ehrenfest nanobeam: A mixture unified gradient theory", J. Vib. Acoust.-T. ASME., 144(6), 061005. https://doi.org/10.1115/1.4055805.
  23. 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.
  24. 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. with Bound. Elem., 134, 571-580. https://doi.org/10.1016/j.enganabound.2021.11.010.
  25. Faghidian, S.A. and Elishakoff, I. (2023a), "The tale of shear coefficients in Timoshenko-Ehrenfest beam theory: 130 years of progress", Meccanica., 58(1), 97-108. https://doi.org/10.1007/s11012-022-01618-1.
  26. Faghidian, S.A., Zur, K.K. and Pan, E.R. (2023b), "Stationary variational principle of mixture unified gradient elasticity", Int. J. Eng. Sci., 182, 103786. https://doi.org/10.1016/j.ijengsci.2022.103786.
  27. Faghidian, S.A., Zur, K.K. and Elishakoff, I. (2023c), "Nonlinear flexure mechanics of mixture unified gradient nanobeams", Commun. Nonlinear Sci. Numer. Simul., 117, 106928. https://doi.org/10.1016/j.cnsns.2022.106928.
  28. Gan, L.L. and She, G.L. (2023), "Nonlinear snap-buckling and resonance of FG-GPLRC curved beams with different boundary conditions", Geomech. Eng., 32(5), 541-551. https://doi.org/10.12989/gae.2023.32.5.541.
  29. Gan, L.L., Xu, J.Q., and She, G.L. (2023), "Wave propagation of graphene platelets reinforced metal foams circular plates", Structural Engineering and Mechanics, 85(5), 645-654. https://doi.org/10.12989/sem.2023.85.5.645.
  30. Golmakani, M.E.,Malikan, M. and Pour, S.G. (2021), "Bending analysis of functionally graded nanoplates based on a higher-order shear deformation theory using dynamic relaxation method", Continuum Mech. Thermodynam., https://doi.org/10.1007/s00161-021-00995-4
  31. Hadji, L., Meziane, M. and Safa, A. (2018), "A new quasi-3d higher shear deformation theory for vibration of functionally graded carbon nanotube-reinforced composite beams resting on elastic foundation", Struct. Eng. Mech., 66(6), 771-781. https://doi.org/10.12989/sem.2018.66.6.771.
  32. Hendi, A., Eltaher, M.A., Mohamed, S.A. and Attia, M. (2022), "Nonlinear thermal vibration of pre/post-buckled two-dimensional FGM tapered microbeams based on a higher order shear deformation theory", Steel Compos. Struct., 41(6), 787-802. https://doi.org/10.12989/scs.2021.41.6.787.
  33. Heydari, A. (2018), "Exact vibration and buckling analyses ofarbitrary gradation of nano-higher order rectangular beam", Steel Compos. Struct., 28(5), 589-606. https://doi.org/10.12989/scs.2018.28.5.589.
  34. Karamanli, A. and Vo, T.P. (2021), "Finite element model for carbon nanotube-reinforced and graphene nanoplatelet-reinforced composite beams", Compos. Struct., 264, 113739. https://doi.org/10.1016/j.compstruct.2021.113739.
  35. Khelifa, Z., Hadji, L., Daouadji, T.H. and Bourada, M. (2018), "Buckling response with stretching effect of carbon nanotube-reinforced composite beams resting on elastic foundation", Struct. Eng. Mech., 67(2), 125-130. https://doi.org/10.12989/sem.2018.67.2.125.
  36. Ke, L.L., Yang, J. and Kitipornchai, S. (2010), "Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams", Compos. Struct., 92(3), 676-683. https://doi.org/10.1016/j.compstruct.2009.09.024.
  37. Li, Y.P., She, G.L., Gan, L.L. and Liu, H.B. (2023), "Nonlinear thermal post-buckling analysis of graphene platelets reinforced metal foams plates with initial geometrical imperfection", Steel. Compos. Struct., 46(5), 649-658. https://doi.org/10.12989/scs.2023.46.5.649.
  38. Lu, L., She, G.L. and Guo, X. (2021), "Size-dependent postbuckling analysis of graphene reinforced composite microtubes with geometrical imperfection", Int. J. Mech. Sci., 199, 106428. https://doi.org/10.1016/j.ijmecsci.2021.106428.
  39. Malikan, M. and Eremeyev, V.A. (2021), "Effect of surface on the flexomagnetic response of ferroic composite nanostructures; nonlinear bending analysis", Compos. Struct., 271, 114179. https://doi.org/10.1016/j.compstruct.2021.114179.
  40. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2021), "Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model", Eng. with Comput., 37(4), 2823-2836. https://doi.org/10.1007/s00366-020-00976-2.
  41. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (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.
  42. Melaibari, A., Mohamed, S.A., Assie, A.E., Shanab, R.A. and Eltaher, M.A. (2023), "Static response of 2D FG porous plates resting on elastic foundation using midplane and neutral surfaces with movable constraints", Mathematics, 10(24), 4784. https://doi.org/10.3390/math10244784.
  43. Peng, X.B., Xu, J., Yang, E.C., Li, Y.H. and Yang, J. (2022), "Influence of the boundary relaxation on free vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections", Acta Mechanica, 233, 4161-4177. https://doi.org/10.1007/s00707-022-03320-5.
  44. Rafiee, M., Jie, Y. and Kitipornchai, S. (2013), "Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers", Compos. Struct., 96, 716-725. https://doi.org/10.1016/j.compstruct.2012.10.005.
  45. Selmi, A. (2021), "Nonlinear behavior of single walled carbon nanotube reinforced aluminium alloy beam", J. Nano Res., 69, 89-103. https://doi.org/10.4028/www.scientific.net/JNanoR.69.89.
  46. She, G.L. (2020), "Wave propagation of FG polymer composite nanoplates reinforced with GNPs", Steel Compos. Struct., 37(1), 27-35. https://doi.org/10.12989/scs.2020.37.1.027.
  47. She, G.L. (2021), "Guided wave propagation of porous functionally graded plates: The effect of thermal loadings", J. Therm. Stresses., 44(10), 1289-1305. https://doi.org/10.1080/01495739.2021.1974323
  48. She, G.L. and Ding, H.X. (2023), "Nonlinear primary resonance analysis of initially stressed graphene platelet reinforced metal foams doubly curved shells with geometric imperfection", Acta Mech. Sin., 39, 522392. https://doi.org/10.1007/s10409-022-22392-x.
  49. She, G.L., Ding, H.X. and Zhang, Y.W. (2022), "Wave propagation in a FG circular plate via the physical neutral surface concept", Struct. Eng. Mech., 82(2), 225-232. https://doi.org/10.12989/sem.2022.82.2.225.
  50. She, G.L. and Li, Y.P. (2022), "Wave propagation in an FG circular plate in thermal environment", Geomech. Eng., 31(6), 615-622. https://doi.org/10.12989/gae.2022.31.6.615.
  51. She, G.L., Liu, H.B. and Karami, B. (2021), "Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets", Thin Wall. Struct., 160, 107407. https://doi.org/10.1016/j.tws.2020.107407.
  52. 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), 368. https://doi.org/10.1140/epjp/i2018-12196-5.
  53. Shenas, A.G., Malekzadeh, P. and Ziaee, S. (2017), "Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment", Compos. Struct., 162, 325-340. https://doi.org/10.1016/j.compstruct.2018.03.104
  54. Talebizadehsardari, P., Eyvazian, A., and Mahani, R.B. (2020), "Static bending analysis of functionally graded polymer composite curved beams reinforced with carbon nanotubes", Thin-walled structures, 157, 107139. http://dx.doi.org/10.1016/j.tws.2020.107139
  55. Tayeb, B., Mohamed, Z., Tahar, H.D. and Khaled, B. (2019), "Theoretical analysis of chirality and scale effects on critical buckling load of zigzag triple walled carbon nanotubes under axial compression embedded in polymeric matrix", Struct. Eng. Mech., 70(3), 269-277. https://doi.org/10.12989/sem.2019.70.3.269.
  56. Timesli, A. (2020), "Buckling analysis of double walled carbon nanotubes embedded in Kerr elastic medium under axial compression using the nonlocal Donnell shell theory", Adv. Nano. Res., 9(2), 69-82. https://doi.org/110.12989/anr.2020.9.2.069.
  57. Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71, 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028.
  58. Xu, J.Q. and She, G.L. (2022), "Thermal post-buckling analysis of porous functionally graded pipes with initial geometric imperfection", Geomech. Eng., 31(3), 329-337. https://doi.org/10.12989/gae.2022.31.3.329.
  59. Zhang, Y.W., Ding, H.X. and She, G.L. (2022), "Snap-buckling and resonance of functionally graded graphene reinforced composites curved beams resting on elastic foundations in thermal environment", J. Therm. Stresses, 45(12), 1029-1042. https://doi.org/10.1080/01495739.2022.2125137.
  60. Zhang, Y.W., Ding, H.X. and She, G.L. (2023a), "Wave propagation in spherical and cylindrical panels reinforced with carbon nanotubes", Steel Compos. Struct., 46(1), 133-141. https://doi.org/10.12989/scs.2023.46.1.133.
  61. Zhang, Y.W., She, G.L. and Ding, H.X. (2023b), "Nonlinear resonance of graphene platelets reinforced metal foams plates under axial motion with geometric imperfections", Eur. J. Mech. A-Solid., 98, 104887. https://doi.org/10.1016/j.euromechsol.2022.104887.
  62. Zhang, Y.W., She, G.L., Gan, L.L. and Li, Y.P. (2023c), "Thermal post-buckling behavior of GPLRMF cylindrical shells with initial geometrical imperfection", Geomech. Eng., 32(6), 615-625. https://doi.org/10.12989/gae.2023.32.6.615.
  63. Zhang, Y.W. and She, G.L. (2022), "Wave propagation and vibration of FG pipes conveying hot fluid", Steel Compos, Struct., 42(3) 397-405. https://doi.org/10.12989/scs.2022.42.3.397.
  64. Zhang, Y.W. and She, G.L. (2023a), "Nonlinear low-velocity impact response of graphene platelet-reinforced metal foam cylindrical shells under axial motion with geometrical imperfection", Nonlinear Dynam., 111, 6317-6334 https://doi.org/10.1007/s11071-022-08186-9.
  65. Zhang, Y.W. and She, G.L. (2023b), "Nonlinear primary resonance of axially moving functionally graded cylindrical shells in thermal environment", Mech. Adv. Mater. Struct., https://doi.org/10.1080/15376494.2023.2180556.
  66. Zhang, Y.Y., Wang, X.Y., Zhang, X., Shen, H.M. and She, G.L. (2021), "On snap-buckling of FG-CNTRC curved nanobeams considering surface effects", Steel Compos. Struct., 38(3), 293-304. https://doi.org/10.12989/scs.2021.38.3.293.
  67. Zhao, J.L., Chen, X., She, G.L., Jing, Y., Bai, R.Q., Yi, J., Pu, H.Y. and Luo, J. (2022a), "Vibration characteristics of functionally graded carbon nanotube-reinforced composite double-beams in thermal environments", Steel Compos. Struct., 43(6), 797-808. https://doi.org/10.12989/scs.2022.43.6.797.
  68. Zhao, J.L., She, G.L., Wu, F., Yuan, S.J., Bai, R.Q., Pu, H.Y., Wang, S.L. and Luo, J. (2022b), "Guided waves of porous FG nanoplates with four edges clamped", Adv. Nano. Res., 13(5), 465-474. https://doi.org/10.12989/anr.2022.13.5.465.
  69. Zouatnia, N., Hadji, L. and Kassoul, A. (2017), "A refined hyperbolic shear deformation theory for bending of functionally graded beams based on neutral surface position", Struct. Eng. Sci., 63(5), 683-689. http://doi.org/10.12989/sem.2017.63.5.683.
  70. Zenkour, A.M. (2018), "A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities", Compos. Struct., 201, 38-48. https://doi.org/10.1016/j.compstruct.2018.05.147.
  71. Zenkour, A.M. and Radwan, A.F. (2019), "Bending response of FG plates resting on elastic foundations in hygro thermal environment with porosities", Compos. Struct., 213, 133-143. https://doi.org/10.1016/j.compstruct.2019.01.065.