Steel and Composite Structures

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Using 3D theory of elasticity for free vibration analysis of functionally graded laminated nanocomposite shells

  • R. Bina (Department of Mechanical Engineering, Shahid Chamran University) ;
  • M. Soltani Tehrani (Department of Civil Engineering, Najafabad Branch, Islamic Azad University) ;
  • A. Ahmadi (Faculty of Civil and Environmental Engineering, Tarbiat Modares University) ;
  • A. Ghanim Taki (Department of Radiology Techniques, health and medical techniques college, Alnoor University) ;
  • R. Akbarian (Islamic Azad University Branch of Islamshahr)
  • Received : 2024.03.26
  • Accepted : 2024.08.12
  • Published : 2024.08.25
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Abstract

The primary objective of this study is to analyze the free vibration behavior of a sandwich cylindrical shell with a defective core and wavy carbon nanotube (CNT)-enhanced face sheets, utilizing the three-dimensional theory of elasticity. The intricate equations of motion for the structure are solved semi-analytically using the generalized differential quadrature method. The shell structure consists of a damaged isotropic core and two external face sheets. The distributions of CNTs are either functionally graded (FG) or uniform across the thickness, with their mechanical properties determined through an extended rule of mixture. In this research, the conventional theory regarding the mechanical effectiveness of a matrix embedding finite-length fibers has been enhanced by introducing tube-to-tube random contact. This enhancement explicitly addresses the progressive reduction in the tubes' effective aspect ratio as the filler content increases. The study investigates the influence of a damaged matrix, CNT distribution, volume fraction, aspect ratio, and waviness on the free vibration characteristics of the sandwich cylindrical shell with wavy CNT-reinforced face sheets. Unlike two-dimensional theories such as classical and the first shear deformation plate theories, this inquiry is grounded in the three-dimensional theory of elasticity, which comprehensively accounts for transverse normal deformations.

Keywords

References

  1. Abrate, S. (1998), "Impact on composite structures", Cambridge UK: Cambridge University Press.
  2. Affdl Halpin, J.C. 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.
  3. Afrookhteh, S.S., Fathi, A., Naghdipour, M. and Alizadeh Sahraei, A. (2016), "An experimental investigation of the effects of weight fractions of reinforcement and timing of hardener addition on the strain sensitivity of carbon nanotube/polymer composites", U.P.B. Sci. Bull., Series B, 78(4), 121-130.
  4. Afrookhteh, S.S., Shakeri, M., Baniassadi, M. and Alizadeh Sahraei, A. (2018), "Microstructure Reconstruction and Characterization of the Porous GDLs for PEMFC Based on Fibers Orientation Distribution", Fuel Cells, 18(2), https://doi.org/10.1002/fuce.201700239.
  5. Al-Furjan, M.S.H. Yang, Y., Farrokhian, A., Shen, X., Kolahchi, R. and Rajak, D.K. (2022), "Dynamic instability of nanocomposite piezoelectric-leptadenia pyrotechnica rheological elastomer-porous functionally graded materials micro viscoelastic beams at various strain gradient higher-order theories", Polymer Compos., 43(1), 282-298, https://doi.org/10.1002/pc.26373.
  6. Al-Furjan, M.S.H., Qi, Z.H., Shan, L., Farrokhian, A., Shen, X. and Kolahchi, R. (2022), "Nano supercapacitors with practical application in aerospace technology: Vibration and wave propagation analysis", Aeros. Sci. Technol., 133, https://doi.org/10.1016/j.ast.2022.108082.
  7. Al-Furjan, M.S.H., Shan, L., Shen, X., Kolahchi, R. and Rajak, D.K. (2022), "Combination of FEM-DQM for nonlinear mechanics of porous GPL-reinforced sandwich nanoplates based on various theories", Thin-Wall. Struct., 178, https://doi.org/10.1016/j.tws.2022.109495.
  8. Alibeigloo, A. and Shaban, M. (2013), "Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity", Acta Mech, 224, 1415-1427, https://doi.org/10.1007/s00707-013-0817-2
  9. Anderson, T.A. (2003), "3D elasticity solution for a sandwich composite with functionally graded core subjected to transverse loading by a rigid sphere", Compos. Struct., 60(3), 265-274. https://doi.org/10.1016/S0263-8223(03)00013-8.
  10. Arefi, M. (2015), "Elastic solution of a curved beam made of functionally graded materials with different cross sections", Steel Compos. Struct., 18(3), 659-672. https://doi.org/10.12989/scs.2015.18.3.659.
  11. Bacciocchi, M. and Tarantino, A.M., (2019), "Time-dependent behavior of viscoelastic three-phase composite plates reinforced by carbon nanotubes", Compos. Struct., 216, 20-31. https://doi.org/10.1016/j.compstruct.2019.02.083.
  12. Barka, M., Benrahou, K.H., Bakora, A. and Tounsi, A. (2016), "Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation", Steel Compos. Struct., 22(1), 91-112. https://doi.org/10.12989/scs.2016.22.1.091.
  13. Bellman, R. and Casti, J. (1971), "Differential quadrature and long term integration", J. Math. Anal. Appl., 34(2), 235-238. https://doi.org/10.1016/0022-247X(71)90110-7.
  14. Beni, Y.T., Mehralian, F. and Zeighampour, H. (2016), "The modified couple stress functionally graded cylindrical thin shell formulation", Mech. Adv. Mater. Struct., http://dx.doi.org/10.1080/15376494.2015.1029167.
  15. Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higher-order shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546. DOI: https://doi.org/10.12989/scs.2015.19.3.521.
  16. Bouafia, H., Chikh, A., Bousahla,A.A., Bourada, F., Heireche, H., Tounsi, A., Benrahou, K.H., Tounsi,A., Al-Zahrani, M.M. and Hussain, M. (2021), "Natural frequencies of FGM nanoplates embedded in an elastic medium", Adv. Nano Res., 11(3), 239-249. https://doi.org/10.12989/anr.2021.11.3.239.
  17. Bouchafa, A., Bouiadjra, M.B., Houari, M.S.A. and Tounsi, A. (2015), "Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory", Steel Compos. Struct., 18(6), 1493-1515. https://doi.org/10.12989/scs.2015.18.6.1493.
  18. Bouguenina, O., Belakhdar, K., Tounsi, A. and Bedia, E.A.A. (2015), "Numerical analysis of FGM plates with variable thickness subjected to thermal buckling", Steel Compos. Struct., 19(3), 679-695. DOI: https://doi.org/10.12989/scs.2015.19.3.679.
  19. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 191-208. https://doi.org/10.12989/anr.2019.7.3.191.
  20. Brischetto, S., Tornabene, F., Fantuzzi, N., Bacciocchi, M. (2015), "Refined 2D and exact 3D shell models for the free vibration analysis of single- and double-walled carbon nanotubes", Technologies, 3(4), 259-284. https://doi.org/10.3390/technologies3040259.
  21. Chen, C.S., Liu, F.H. and Chen, W.R. (2017), "vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments", Steel and Composite Structures, An Int'l Journal, 23(3), 251-261. https://doi.org/10.12989/scs.2017.23.3.251.
  22. Chen, W.Q., Bian, Z.G. and Ding, H.U., (2004), "Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells", Int. J. Mech. Sci., 46(1), 159-171. https://doi.org/10.1016/j.ijmecsci.2003.12.005.
  23. Civalek, O. (2005), "Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of HDQ-FD methods", Int. J. Press Vessel Pip., 82(6), 470-479. https://doi.org/10.1016/j.ijpvp.2004.12.003.
  24. Dai, Z., Jiang, Z., Zhang, L. and Habibi, M. (2021), "Frequency characteristics and sensitivity analysis of a size-dependent laminated nanoshell", Adv. Nano Res., 10(2), 175-189. https://doi.org/10.12989/anr.2021.10.2.175.
  25. Ebrahimi, F., Fardshad, R.E. and Mahesh, V. (2019), "Frequency response analysis of curved embedded magneto-electro-viscoelastic functionally graded nanobeams", Adv. Nano Res., 7(6), 391-403. https://doi.org/10.12989/anr.2019.7.6.391.
  26. Eipakchi, H. and Nasrekani, F.M. (2021), "Geometrically nonlinear frequency analysis of composite cylinders with metamaterial honeycomb layer and adjustable Poisson's ratio using the multiple scale method", Thin-Wall. Struct., 169, https://doi.org/10.1016/j.tws.2021.108441.
  27. Eipakchi, H., Nasrekani, F. M. (2020), "Response investigation of viscoelastic cylindrical shells with geometrical nonlinearity effect under moving pressure", Anal. Approach. Mech. Adv. Mater. Struct., 29(8), 1124-1137. https://doi.org/10.1080/15376494.2020.1808916.
  28. Eipakchi, H., Nasrekani, F.M. (2022), A "closed-form solution for asymmetric free vibration analysis of composite cylindrical shells with metamaterial honeycomb core layer based on shear deformation theory", Mech. Based Des. Struct. Machines, 51(11), 6513-6531. https://doi.org/10.1080/15397734.2022.2051183.
  29. Fantuzzi, N., Tornabene, F., Bacciocchi, M. and Dimitri, R., (2016), "Free vibration analysis of arbitrarily shaped functionally carbon nanotube-reinforced plates", Compos. Part B, 115(1), 384-408. https://doi.org/10.1016/j.compositesb.2016.09.021.
  30. Ghavamian, A., Rahmandoust, M. and Ochsner, A. (2012), "A numerical evaluation of the influence of defects on the elastic modulus of single and multi-walled carbon nanotubes", Comput. Mater. Sci., 62, 110-116. https://doi.org/10.1016/j.commatsci.2012.05.003.
  31. Hill, R. (1964a), "Theory ofmechanical properties of fibre-strengthened materials Elastic behavior, J. Mech. Phys. Solids, 12, 199-212. https://doi.org/10.1016/0022-5096(64)90019-5.
  32. Hill, R. (1964b), "Theory of mechanical properties of fibre-strengthened materials: II. Inelastic behavior", J. Mech. Phys. Solids, 12, 213-218. https://doi.org/10.1016/0022-5096(64)90020-1.
  33. Hong, M. and Lee, U. (2015), "Dynamics of a functionally graded material axial bar, Spectral element modeling and analysis", Composites: Part B, 69, 427-434. https://doi.org/10.1016/j.compositesb.2014.10.022.
  34. Hosseini, S.M. and Zhang, C. (2018), "Elastodynamic and wave propagation analysis in a FG Graphene platelets-reinforced nanocomposite cylinder using a modified nonlinear micromechanical model", Steel Compos. Struct., 27(3), 255-271.
  35. Jam, J.E., Noorabadi, M. and Namdaran, N. (2017), "Nonlinear free vibration analysis of micro-beams resting on viscoelastic foundation based on the modified couple stress theory", Archive Mech. Eng., https://doi.org/10.1515/meceng-2017-0015.
  36. Kashtalyan, M. and Menshykova, M. (2009), "Three-dimensional elasticity solution for sandwich panels with a functionally graded core", Compos. Struct., 87(1), 36-43. https://doi.org/10.1016/j.compstruct.2007.12.003.
  37. Khadir, A.I., Daikh, A.A. and Eltaher, M.A. (2021), "Novel four-unknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates", Adv. Nano Res., 11(6), 621-640. https://doi.org/10.12989/anr.2021.11.6.621.
  38. Kolahchi, R. and Kolahdouzan, F. (2021), "A numerical method for magneto-hygro-thermal dynamic stability analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions", Appl. Mathem. Modelling, 91, 458-475, https://doi.org/10.1016/j.apm.2020.09.060.
  39. Kolahchi, R., Behrooz Keshtegar, B. and Trung, N.T. (2021), "Optimization of dynamic properties for laminated multiphase nanocomposite sandwich conical shell in thermal and magnetic conditions", J. Sandw. Struct. Mater., 24(1), https://doi.org/10.1177/10996362211020388.
  40. Lanzoni, L. and Tarantino, A.M. (2014), "Damaged hyperelastic membranes", Int. J. Nonlinear Mech., 60, 9-22. https://doi.org/10.1016/j.ijnonlinmec.2013.12.001.
  41. Lanzoni, L. and Tarantino, A.M. (2015), "Equilibrium configurations and stability of a damaged body underuniaxial tractions", Z. Angew. Math. Phys., 66, 171-190. https://doi.org/10.1007/s00033-014-0397-6.
  42. Lemaitre, J. and Chaboche, J.L., (1990), "Mechanics of solid materials", Cambridge University Press: New York, NY, USA.
  43. Liu, D., Kitipornchai, S., Chen, W. and Yang, J. (2018), "Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell", Compos. Struct., 189, 560-569. https://doi.org/10.1016/j.compstruct.2018.01.106.
  44. Loy, C.T., Lam, K.Y. and Reddy, J.N. (1999), "Vibration of functionally graded cylindrical shells", Int. J. Mech. Sci., 41(3), 309-324.
  45. Marin, M. (1994), "The Lagrange identity method in thermoelasticity of bodies with microstructure", Int. J. Eng. Sci., 32(8), 1229-1240.
  46. Marin, M. (1995), "On existence and uniqueness in thermoelasticity of micropolar bodies", C. R. Acad. Sci. Paris, Ser. II, 321(12), 475-480.
  47. Marin, M. (1996), "Some basic theorems in elastostatics of micropolar materials with voids", J. Comput. Appl. Math., 70(1), 115-126.
  48. Marin, M., Hobiny, A. and Abbas, I. (2021), "Finite element analysis of nonlinear bioheat model in skin tissue due to external thermal sources", Mathematics, 9(13), 1459. https://doi.org/10.3390/math9131459.
  49. Marin, M., Ochsner, A. and Mubashir Bhatti, M. (2020), "Some results in Moore-Gibson-Thompson thermoelasticity of dipolar bodies", ZAMM-J. Appl. Mathem. Mech., ZAMM-Z. fur Angew. Math. Mech., 100(12), https://doi.org/10.1002/zamm.202000090.
  50. Martone, A., Faiella, G., Antonucci, V., Giordano, M. and Zarrelli, M. (2011), "The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix", Compos. Sci. Technol., 71(8), 1117-1123.
  51. Matsunaga, H. (2008), "Free vibration and stability of functionally graded shallow shells according to a 2-D higher-order deformation theory", Compos. Struct., 84(2), 132-146. https://doi.org/10.1016/j.compstruct.2007.07.006.
  52. Nasrekani, F.M. and Eipakchi, H. (2023), "Geometrically nonlinear effect on forced vibrational behavior of superlight composite beams with auxetic core layer under harmonic excitation based on FSDT", Mech. Based Des. Struct. Machines, 52(8), 5435-5456. https://doi.org/10.1080/15397734.2023.2255262.
  53. Paliwal, D.N., Kanagasabapathy, H. and Gupta, K.M. (1995), "The large deflection of an orthotropic cylindrical shell on a Pasternak foundation", Compos. Struct., 31, 31-37. https://doi.org/10.1016/0263-8223(94)00068-9.
  54. Paliwal, D.N., Pandey, R.K. and Nath, T. (1996), "Free vibration of circular cylindrical shell on Winkler and Pasternak foundation", Int. J. Press. Vessel Pip., 69(1), 79-89. https://doi.org/10.1016/0308-0161(95)00010-0.
  55. Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239.
  56. Patel, B.P., Gupta, S.S., Loknath, M.S.B. and Kadu, C.P. (2005), "Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory", Compos. Struct., 69(3), 259-270. https://doi.org/10.1016/j.compstruct.2004.07.002.
  57. Pelletier Jacob, L. and Vel Senthil,S. (2006), "An exact solution for the steady state thermo elastic response of functionally graded orthotropic cylindrical shells", Int. J. Solid Struct., 43(5), 1131-1158. https://doi.org/10.1016/j.ijsolstr.2005.03.079.
  58. Pradhan, S.C., Loy, C.T., Lam, K.Y., Reddy, J.N. (2000). "Vibration characteristic of functionally graded cylindrical shells under various boundary conditions", Appl. Acoust., 61(1), 119-129. https://doi.org/10.1016/S0003-682X(99)00063-8.
  59. Pradyumna, S. and Bandyopadhyay, J.N. (2008), "Free vibration analysis of functionally graded panels using higher-order finite-element formulation", J. Sound Vib., 318(1-2), 176-192. https://doi.org/10.1016/j.jsv.2008.03.056.
  60. Reddy, J.N. and Miravete, A. (1995), "Practical Analysis of Composite Laminates; CRC Press", Boca Raton, FL, USA.
  61. Shen H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19.
  62. Shen, H.S. and Zhang, C.L. (2010), "Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates", Mater. Des., 31(7), 3403-3411.
  63. Shoaei, A.G., Eipakchi, H., Nasrekani, F.M. (2024), "Auxetic honeycomb core layer effect on vibrations of annular plates based on shear deformation theory", Eng. Struct., 306, https://doi.org/10.1016/j.engstruct.2024.117855.
  64. Shu, C. (2000), Differential Quadrature and its Application in Engineering. Springer, Berlin.
  65. Sobhani, B. and Yas, M.H. (2010), "Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels", Int. J. Mech. Sci., 52, 1047-1063.
  66. Tahouneh, V. (2014), "Free vibration analysis of bidirectional functionally graded annular plates resting on elastic foundations using differential quadrature method", Struct. Eng. Mech., 52(4), 663-686. https://doi.org/10.12989/sem.2014.52.4.663.
  67. Tahouneh, V. (2016), "Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates", Steel Compos. Struct., 20(3), 623-649. https://doi.org/10.12989/scs.2016.20.3.623.
  68. Tahouneh, V. and Naei, M.H. (2014), "A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation", Meccanica, 49(1), 91-109. https://10.1007/s11012-013-9776-x.
  69. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2020a), "Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches", Steel Compos. Struct., 34(2), 261-277. http://dx.doi.org/10.12989/scs.2020.34.2.261.
  70. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2020b), "Using IGA and trimming approaches for vibrational analysis of L-shape graphene sheets via nonlocal elasticity theory", Steel Compos. Struct., 33(5), 717-727. http://dx.doi.org/10.12989/scs.2019.33.5.717.
  71. Tarantino, A.M. (2014), "Equilibrium paths of a hyperelastic body under progressive damage", J. Elast., 114, 225-250, https://doi.org/10.1007/s10659-013-9439-0.
  72. Tornabene, F. (2009), "Free vibration analysis of functionally graded conical cylindrical shell and annular plate structures with a four-parameter power-law distribution", Comput. Meth. Appl. Mech. Eng., 198(37), 2911-2935. https://doi.org/10.1016/j.cma.2009.04.011.
  73. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2014), "Free vibrations of free-form doubly curved shells made of functionally graded materials using higher-order equivalent single layer theories", Compos. Part B, 67(1), 490-509. https://doi.org/10.1016/j.compositesb.2014.08.012.
  74. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2016b), "Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes", Compos. Part B, 115(1), 449-476. https://doi.org/10.1016/j.compositesb.2016.07.011.
  75. Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016a), "Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells", Compos. Part B, 89(1), 187-218. https://doi.org/10.1016/j.compositesb.2015.11.016.
  76. Vinson J .R. (2005), "Sandwich structures: past, present, and future", In: Proceedings of the 7th International Conference on Sandwich Structures, Aalborg, Denmark: Aalborg University, 3-12.
  77. Viola, E. and Tornabene, F. (2009), "Free vibrations of three-parameter functionally graded parabolic panels of revolution", Mech. Res. Commun., 36(5), 587-594. https://doi.org/10.1016/j.mechrescom.2009.02.001.
  78. Wagner, H.D., Lourie, O. and Feldman, Y. (1997), "Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix", Appl. Phys. Lett., 72(2), 188-190. https://doi.org/10.1063/1.120680.
  79. Wang, L. and Hu, H. (2014), "Thermal vibration single-walled carbon nanotubes with quantum effects", Proc. Math. Phys. Eng. Sci., 470(2168). https://doi.org/10.1098/rspa.2014.0087.
  80. Wang, Y.Q. and Liu, Y.F. (2019), "Free vibration and buckling of polymeric shells reinforced with 3D graphene foams", Results Phys., 14, https://doi.org/10.1016/j.rinp.2019.102510.
  81. Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22(1), 161-182. https://doi.org/10.12989/scs.2016.22.1.161.
  82. Yang, J. and Shen, S.H. (2003), "Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels", J. Sound Vib., 261(5), 871-893. https://doi.org/10.1016/S0022-460X(02)01015-5.
  83. Yang, R., Kameda, H. and Takada, S. (1998), "Shell model FEM analysis of buried pipelines under seismic loading", Bull Disaster Prev Res. Inst., 38, 115-146.
  84. Zhang, Y. and Wang L. (2020), "Effects of Van der Waals force on the vibration of typical Multi-layered Two-dimensional nanostructures", Scientific Reports-Natureresearch, 10(644). https://doi.org/10.1038/s41598-020-57522-9.