Chaotic dynamics of a non-autonomous nonlinear system for a smart composite shell subjected to the hygro-thermal environment

  • Mahsa karimiaslEmail author
Technical Paper


In this research, nonlinear dynamic behaviors of multiscale composites doubly curved shells have been investigated by employing multiple scales Perturbation Method. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) according to Halpin–Tsai model have been assumed. The displacement- strain of nonlinear vibration of multiscale laminated doubly curved shells via higher order shear deformation (HSDT) theory and using Green–Lagrange nonlinear shell theory is obtained. The governing equations of composite doubly curved shell have been derived by implementing Hamilton’s principle and shell considered to be simply supported. For investigating correctness and accuracy, this paper is validated by other previous researches. Finally, bifurcation diagram, phase portraits and Poincare maps are investigated. The results indicate different dimensionless force; curvature ratio and kind of distribution pattern have strong influence on nonlinear vibration control of the composite multiscale doubly curved shell.



  1. Ahouel M, Houari MSA, Bedia EA, Tounsi A (2016) Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept. Steel Compos Struct 20(5):963–981Google Scholar
  2. Alijani F, Amabili M (2013) Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges. J Sound Vib 332(14):3564–3588Google Scholar
  3. Alijani F, Amabili M, Karagiozis K, Bakhtiari-Nejad F (2011) Nonlinear vibrations of functionally graded doubly curved shallow shells. J Sound Vib 330(7):1432–1454Google Scholar
  4. Amabili M, Reddy JN (2010) A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells. Int J Non-Linear Mech 45(4):409–418Google Scholar
  5. Ansari R, Torabi J (2016) Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading. Compos B Eng 95:196–208Google Scholar
  6. Belabed Z, Bousahla AA, Houari MSA, Tounsi A, Mahmoud SR (2018) A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate. Earthq Struct 14(2):103–115Google Scholar
  7. Beldjelili Y, Tounsi A, Mahmoud SR (2016) Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory. Smart Struct Syst 18(4):755–786Google Scholar
  8. Bellifa H, Benrahou KH, Hadji L, Houari MSA, Tounsi A (2016) Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position. J Braz Soc Mech Sci Eng 38(1):265–275Google Scholar
  9. Bellifa H, Bakora A, Tounsi A, Bousahla AA, Mahmoud SR (2017a) An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates. Steel Compos Struct 25(3):257–270Google Scholar
  10. Bellifa H, Benrahou KH, Bousahla AA, Tounsi A, Mahmoud SR (2017b) A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams. Struct Eng Mech 62(6):695–702Google Scholar
  11. Besseghier A, Houari MSA, Tounsi A, Mahmoud SR (2017) Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory. Smart Struct Syst 19(6):601–614Google Scholar
  12. Bouderba B, Houari MSA, Tounsi A, Mahmoud SR (2016) Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Struct Eng Mech 58(3):397–422Google Scholar
  13. Boukhari A, Atmane HA, Tounsi A, Adda B, Mahmoud SR (2016) An efficient shear deformation theory for wave propagation of functionally graded material plates. Struct Eng Mech 57(5):837–859Google Scholar
  14. Bounouara F, Benrahou KH, Belkorissat I, Tounsi A (2016) A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel Compos Struct 20(2):227–249Google Scholar
  15. Bourada M, Kaci A, Houari MSA, Tounsi A (2015) A new simple shear and normal deformations theory for functionally graded beams. Steel Compos Struct 18(2):409–423Google Scholar
  16. Bousahla AA, Benyoucef S, Tounsi A, Mahmoud SR (2016) On thermal stability of plates with functionally graded coefficient of thermal expansion. Struct Eng Mech 60(2):313–335Google Scholar
  17. Chen WQ, Cai JB, Ye GR (2003) Exact solutions of cross-ply laminates with bonding imperfections. AIAA J 41(11):2244–2250Google Scholar
  18. Chikh A, Tounsi A, Hebali H, Mahmoud SR (2017) Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT. Smart Struct Syst 19(3):289–297Google Scholar
  19. Chorfi SM, Houmat A (2010) Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form. Compos Struct 92(10):2573–2581Google Scholar
  20. El-Haina F et al (2017) A simple analytical approach for thermal buckling of thick functionally graded sandwich plates. Struct Eng Mech 63(5):585–595Google Scholar
  21. Fan Y, Wang H (2017) Nonlinear low-velocity impact on damped and matrix-cracked hybrid laminated beams containing carbon nanotube reinforced composite layers. Nonlinear Dyn 89(3):1863–1876MathSciNetGoogle Scholar
  22. Feng C, Kitipornchai S, Yang J (2017) Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos B Eng 110:132–140Google Scholar
  23. Fourn H, Atmane HA, Bourada M, Bousahla AA, Tounsi A, Mahmoud SR (2018) A novel four variable refined plate theory for wave propagation in functionally graded material plates. Steel Compos struct 27(1):109–122Google Scholar
  24. Gao XL, Li K (2005) A shear-lag model for carbon nanotube-reinforced polymer composites. Int J Solids Struct 42(5):1649–1667zbMATHGoogle Scholar
  25. Heydari MM, Bidgoli AH, Golshani HR, Beygipoor G, Davoodi A (2015) Nonlinear bending analysis of functionally graded CNT-reinforced composite Mindlin polymeric temperature-dependent plate resting on orthotropic elastomeric medium using GDQM. Nonlinear Dyn 79(2):1425–1441Google Scholar
  26. Houari MSA, Tounsi A, Bessaim A, Mahmoud SR (2016) A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates. Steel Compos Struct 22(2):257–276Google Scholar
  27. Hu N, Qiu J, Li Y, Chang C, Atobe S, Fukunaga H et al (2013) Multi-scale numerical simulations of thermal expansion properties of CNT-reinforced nanocomposites. Nanoscale Res Lett 8:1–8Google Scholar
  28. Kaci A, Houari MSA, Bousahla AA, Tounsi A, Mahmoud SR (2018) Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory. Struct Eng Mech 65(5):621–631Google Scholar
  29. Karami B, Janghorban M, Tounsi A (2017) Effects of triaxial magnetic field on the anisotropic nanoplates. Steel Compos Struct 25(3):361–374Google Scholar
  30. Karami B, Janghorban M, Tounsi A (2018a) Variational approach for wave dispersion in anisotropic doublycurved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin-Walled Struct 129:251–264Google Scholar
  31. Karami B, Janghorban M, Tounsi A (2018b) Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles. Steel Compos Struct 27(2):201–216Google Scholar
  32. Karami B, Janghorban M, Shahsavari D, Tounsi A (2018c) A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates. Steel Compos Struct 28(1):99–110Google Scholar
  33. Khetir H, Bouiadjra MB, Houari MSA, Tounsi A, Mahmoud SR (2017) A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates. Struct Eng Mech 64(4):391–402Google Scholar
  34. Kim M, Park YB, Okoli OI, Zhang C (2009) Processing, characterization, and modeling of carbon nanotube-reinforced multiscale composites. Compos Sci Technol 69(3):335–342Google Scholar
  35. Lee DM, Lee I (1997) Vibration behaviors of thermally postbuckled anisotropic plates using first-order shear deformable plate theory. Comput Struct 63(3):371–378zbMATHGoogle Scholar
  36. Mahapatra TR, Kar VR, Panda SK (2015) Nonlinear free vibration analysis of laminated composite doubly curved shell panel in hygrothermal environment. J Sandwich Struct Mater 17(5):511–545Google Scholar
  37. Mahapatra TR, Panda SK, Kar VR (2016a) Nonlinear flexural analysis of laminated composite panel under hygro-thermo-mechanical loading—a micromechanical approach. Int J Comput Methods 13(03):1650015MathSciNetzbMATHGoogle Scholar
  38. Mahapatra TR, Panda SK, Dash S (2016b) Effect of hygrothermal environment on the nonlinear free vibration responses of laminated composite plates: a nonlinear Unite element micromechanical approach. In: IOP conference series: materials science and engineering, vol 149, no. 1. IOP Publishing, BristolGoogle Scholar
  39. Mahi A, Tounsi A (2015) A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl Math Model 39(9):2489–2508MathSciNetGoogle Scholar
  40. Mehar K, Panda SK, Mahapatra TR (2017a) Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure. Eur J Mech-A/Solids 65:384–396MathSciNetzbMATHGoogle Scholar
  41. Mehar K, Panda SK, Bui TQ, Mahapatra TR (2017b) Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM. J Therm Stresses 40(7):899–916Google Scholar
  42. Mehar K, Panda SK, Mahapatra TR (2018a) Thermoelastic deflection responses of CNT reinforced sandwich shell structure using finite element method. Scientia Iranica 25(5):2722–2737Google Scholar
  43. Mehar K, Panda SK, Mahapatra TR (2018b) Large deformation bending responses of nanotube-reinforced polymer composite panel structure: Numerical and experimental analyses. Proc Inst Mech Eng Part G J Aerosp Eng (0954410018761192) Google Scholar
  44. Menasria A, Bouhadra A, Tounsi A, Bousahla AA, Mahmoud SR (2017) A new and simple HSDT for thermal stability analysis of FG sandwich plates. Steel Compos Struct 25(2):157–175Google Scholar
  45. Mokhtar Y, Heireche H, Bousahla AA, Houari MSA, Tounsi A, Mahmoud SR (2018) A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory. Smart Struct Syst 21(4):397–405Google Scholar
  46. Mouffoki A, Bedia EA, Houari MSA, Tounsi A, Mahmoud SR (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):369–383Google Scholar
  47. Naidu NS, Sinha PK (2007) Nonlinear free vibration analysis of laminated composite shells in hygrothermal environments. Compos Struct 77(4):475–483Google Scholar
  48. Odegard GM, Frankland SJV, Gates TS (2005) Effect of nanotube functionalization on the elastic properties of polyethylene nanotube composites. Aiaa J 43(8):1828Google Scholar
  49. Panda SK, Singh BN (2008) Non-linear free vibration analysis of laminated composite cylindrical/hyperboloid shell panels based on higher-order shear deformation theory using non-linear finite-element method. Proc Inst Mech Eng Part G J Aerosp Eng 222(7):993–1006Google Scholar
  50. Rafiee M, Yang J, Kitipornchai S (2013) Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Compos Struct 96:716–725Google Scholar
  51. Sahmani S, Aghdam MM (2017) Nonlinear instability of axially loaded functionally graded multilayer graphene platelet-reinforced nanoshells based on nonlocal strain gradient elasticity theory. Int J Mech Sci 131:95–106Google Scholar
  52. Shahsavari D, Karami B, Fahham HR, Li L (2018) On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory. Acta Mech 229(11):4549–4573MathSciNetGoogle Scholar
  53. Sharma N, Mahapatra TR, Panda SK (2018a) Numerical analysis of acoustic radiation responses of shear deformable laminated composite shell panel in hygrothermal environment. J Sound Vib 431:346–366Google Scholar
  54. Sharma N, Mahapatra TR, Panda SK (2018b) Thermoacoustic behavior of laminated composite curved panels using higher-order finite-boundary element model. Int J Appl Mech 10(02):1850017Google Scholar
  55. Shen HS (2009) A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators. Compos Struct 91(3):375–384Google Scholar
  56. Shen HS, Chen X, Guo L, Wu L, Huang XL (2015) Nonlinear vibration of FGM doubly curved panels resting on elastic foundations in thermal environments. Aerosp Sci Technol 47:434–446Google Scholar
  57. Shen HS, Lin F, Xiang Y (2017) Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments. Nonlinear Dyn 90(2):899–914Google Scholar
  58. Shiau LC, Kuo SY (2006) Free vibration of thermally buckled composite sandwich plates. J Vib Acoust 128(1):1–7Google Scholar
  59. Singh VK, Panda SK (2014) Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels. Thin-Walled Struct 85:341–349Google Scholar
  60. Singh VK, Panda SK (2015) Large amplitude free vibration analysis of laminated composite spherical shells embedded with piezoelectric layers. Smart Struct Syst 16(5):853–872Google Scholar
  61. Thostenson ET, Li WZ, Wang DZ, Ren ZF, Chou TW (2002) Carbon nanotube/carbon fiber hybrid multiscale composites. J Appl Phys 91(9):6034–6037Google Scholar
  62. Wu H, Yang J, Kitipornchai S (2017) Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment. Compos Struct 162:244–254Google Scholar
  63. Yazdi AA (2013) Applicability of homotopy perturbation method to study the nonlinear vibration of doubly curved cross-ply shells. Compos Struct 96:526–531Google Scholar
  64. Yazid M, Heireche H, Tounsi A, Bousahla AA, Houari MSA (2018) A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium. Smart Struct Syst 21(1):15–25Google Scholar
  65. Youcef DO, Kaci A, Benzair A, Bousahla AA, Tounsi A (2018) Dynamic analysis of nanoscale beams including surface stress effects. Smart Struct Syst 21(1):65–74Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Faculty of EngineeringImam Khomeini International UniversityQazvinIran

Personalised recommendations