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Frequency analysis of doubly curved functionally graded carbon nanotube-reinforced composite panels

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Abstract

Vibration characteristics of moderately thick doubly curved functionally graded composite panels reinforced by carbon nanotube are analyzed. Here, special cases of doubly curved shell panels such as spherical, cylindrical and hyperbolic paraboloid panels and five different distributions of carbon nanotubes through the thickness direction are considered. By utilizing the modified rule of mixture, mechanical properties are estimated. The equations of motion are derived via the first-order shear deformation theory, and non-dimensional frequencies are obtained by the use of Galerkin’s method. The suggested model is justified by a good agreement between the results given by present model and available data in the literature. The influences of volume fraction of carbon nanotubes, thickness ratio, aspect ratio, curvature ratio, and shallowness ratio on the frequencies of moderately thick doubly curved nanocomposite shell panels are also examined. Furthermore, the effect of various boundary conditions on the frequency analysis of doubly curved nanocomposite panels is studied, and the corresponding mode shapes are depicted.

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References

  1. Treacy, M.M.J., Ebbesen, T.W., Gibson, J.M.: Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 38, 678–680 (1996)

    Article  Google Scholar 

  2. Lu, J.P.: Elastic properties of carbon nanotubes and nanoropes. Phys. Rev. Lett. 79, 1297–1300 (1997)

    Article  Google Scholar 

  3. Dai, H.: Carbon nanotubes: opportunities and challenges. Surf. Sci. 500, 218–241 (2002)

    Article  Google Scholar 

  4. Cadek, M., Coleman, J.N., Barron, V., Hedicke, K., Blau, W.J.: Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites. Appl. Phys. Lett. 81, 5123–5125 (2002)

    Article  Google Scholar 

  5. Thostenson, E.T., Chou, T.W.: On the elastic properties of carbon nanotube-based composites: modelling and characterization. J. Phys. D Appl. Phys. 36, 573–582 (2003)

    Article  Google Scholar 

  6. Qian, D., Dickey, E.C., Andews, R., Rantell, T.: Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl. Phys. Lett. 76, 2868–2870 (2000)

    Article  Google Scholar 

  7. Griebel, M., Hamaekers, J.: Molecular dynamics simulations of the elastic moduli of polymer-carbon nanotube composites. Comput. Methods Appl. Mech. Eng. 193, 1773–1788 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu, N., Fukunaga, H., Lu, C., Kameyama, M., Yan, B.: Prediction of elastic properties of carbon nanotube reinforced composites. Proc. R. Soc. A Math. Phys. 461, 1685–1710 (2005)

    Article  Google Scholar 

  9. Song, Y.S., Youn, J.R.: Modeling of effective elastic properties for polymer based carbon nanotube composites. Polymer 47, 1741–1748 (2006)

    Article  Google Scholar 

  10. Han, Y., Elliott, J.: Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput. Mater. Sci. 39, 315–323 (2007)

    Article  Google Scholar 

  11. Ngabonziza, Y., Li, J., Barry, C.F.: Electrical conductivity and mechanical properties of multiwalled carbon nanotube-reinforced polypropylene nanocomposites. Acta Mech. 220, 289–298 (2011)

    Article  MATH  Google Scholar 

  12. Wernik, J.M., Meguid, S.A.: Multiscale modeling of the nonlinear response of nano-reinforced polymers. Acta Mech. 217, 1–16 (2011)

    Article  MATH  Google Scholar 

  13. Mehrabadi, S.J., Aragh, B.S., Khoshkhahesh, V., Taherpour, A.: Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes. Compos. Part B Eng. 43, 2031–2040 (2012)

    Article  Google Scholar 

  14. Bhardwaj, G., Upadhyay, A.K., Pandey, R., Shukla, K.K.: Non-linear flexural and dynamic response of CNT reinforced laminated composite plates. Compos. Part B Eng. 45, 89–100 (2013)

    Article  Google Scholar 

  15. Yanase, K., Moriyama, S., Ju, J.W.: Effects of CNT waviness on the effective elastic responses of CNT-reinforced polymer composites. Acta Mech. 224, 1351–1364 (2013)

    Article  MATH  Google Scholar 

  16. Zhu, P., Zhang, L.W., Liew, K.M.: Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation. Compos. Struct. 107, 298–314 (2014)

    Article  Google Scholar 

  17. Zhang, L.W., Zhu, P., Liew, K.M.: Thermal buckling of functionally graded plates using a local Kriging meshless method. Compos. Struct. 108, 472–492 (2014)

    Article  Google Scholar 

  18. Lei, Z.X., Zhang, L.W., Liew, K.M.: Elastodynamic analysis of carbon nanotube-reinforced functionally graded plates. Int. J. Mech. Sci. 99, 208–217 (2015)

    Article  Google Scholar 

  19. Zhang, L.W., Song, Z.G., Liew, K.M.: Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos. Struct. 128, 165–175 (2015)

    Article  Google Scholar 

  20. Zhang, L.W., Liew, K.M.: Large deflection analysis of FG-CNT reinforced composite skew plates resting on Pasternak foundations using an element-free approach. Compos. Struct. 132, 974–983 (2015)

    Article  Google Scholar 

  21. Zhang, L.W., Liew, K.M.: Geometrically nonlinear large deformation analysis of functionally graded carbon nanotube reinforced composite straight-sided quadrilateral plates. Comput. Method Appl. Mech. Eng. 295, 219–239 (2015)

    Article  MathSciNet  Google Scholar 

  22. Zhang, L.W., Lei, Z.X., Liew, K.M.: Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Compos. Part B Eng. 75, 36–46 (2015)

    Article  Google Scholar 

  23. Aragh, B.S., Barati, A.H.N., Hedayati, H.: Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Compos. Part B Eng. 43, 1943–1954 (2012)

    Article  Google Scholar 

  24. Yas, M.H., Pourasghar, A., Kamarian, S., Heshmati, M.: Three-dimensional free vibration analysis of functionally graded nanocomposite cylindrical panels reinforced by carbon nanotube. Mater. Des. 49, 583–590 (2013)

    Article  Google Scholar 

  25. Moradi-Dastjerdi, R., Pourasghar, A., Foroutan, M.: The effects of carbon nanotube orientation and aggregation on vibrational behavior of functionally graded nanocomposite cylinders by a mesh-free method. Acta Mech. 224, 2817–2832 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, H.S., Xiang, Y.: Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment. Compos. Part B Eng. 52, 311–322 (2013)

    Article  Google Scholar 

  27. Moradi-Dastjerdi, R., Foroutan, M., Pourasghar, A.: Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method. Mater. Des. 44, 256–266 (2013)

    Article  MATH  Google Scholar 

  28. Liew, K.M., Lei, Z.X., Yu, J.L., Zhang, L.W.: Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach. Comput. Method Appl. Mech. Eng. 268, 1–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Alibeigloo, A.: Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity. Eur. J. Mech. A Solids 44, 104–115 (2014)

    Article  MathSciNet  Google Scholar 

  30. Heydarpour, Y., Aghdam, M.M., Malekzadeh, P.: Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells. Compos. Struct. 117, 187–200 (2014)

    Article  MATH  Google Scholar 

  31. Zhang, L.W., Lei, Z.X., Liew, K.M., Yu, J.L.: Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels. Comput. Method Appl. Mech. Eng. 273, 1–18 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mehrabadi, S.J., Aragh, B.S.: Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes. Thin Wall. Struct. 80, 130–141 (2014)

    Article  Google Scholar 

  33. Zhang, L.W., Lei, Z.X., Liew, K.M., Yu, J.L.: Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Compos. Struct. 111, 205–212 (2014)

    Article  Google Scholar 

  34. Lei, Z.X., Zhang, L.W., Liew, K.M., Yu, J.L.: Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method. Compos. Struct. 113, 328–338 (2014)

    Article  Google Scholar 

  35. Shen, H.S., Xiang, Y.: Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos. Part B Eng. 67, 50–61 (2014)

    Article  Google Scholar 

  36. Zarouni, E., Rad, M.J., Tohidi, H.: Free vibration analysis of fiber reinforced composite conical shells resting on Pasternak-type elastic foundation using Ritz and Galerkin methods. Int. J. Mech. Mater. Des. 10, 421–438 (2014)

    Article  Google Scholar 

  37. Ghorbanpour Arani, A., Haghparast, E., Khoddami Maraghi, Z., Amir, S.: Static stress analysis of carbon nano-tube reinforced composite (CNTRC) cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields. Compos. Part B Eng. 68, 136–145 (2015)

    Article  Google Scholar 

  38. Jam, J.E., Kiani, Y.: Buckling of pressurized functionally graded carbon nanotube reinforced conical shells. Compos. Struct. 125, 586–595 (2015)

    Article  Google Scholar 

  39. Lei, Z.X., Zhang, L.W., Liew, K.M.: Vibration analysis of CNT-reinforced functionally graded rotating cylindrical panels using the element-free kp-Ritz method. Compos. Part B Eng. 77, 291–303 (2015)

    Article  Google Scholar 

  40. Zhang, L.W., Cui, W.C., Liew, K.M.: Vibration analysis of functionally graded carbon nanotube reinforced composite thick plates with elastically restrained edges. Int. J. Mech. Sci. 103, 9–21 (2015)

    Article  Google Scholar 

  41. Kundalwal, S.I., Meguid, S.A.: Effect of carbon nanotube waviness on active damping of laminated hybrid composite shells. Acta Mech. 226, 2035–2052 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, L.W., Lei, Z.X., Liew, K.M.: Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method. Appl. Math. Comput. 256, 488–504 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Lei, Z.X., Zhang, L.W., Liew, K.M.: Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Compos. Struct. 127, 245–259 (2015)

    Article  Google Scholar 

  44. Zhang, L.W., Song, Z.G., Liew, K.M.: State-space Levy method for vibration analysis of FG-CNT composite plates subjected to in-plane loads based on higher-order shear deformation theory. Compos. Struct. 134, 989–1003 (2015)

    Article  Google Scholar 

  45. Thomas, B., Roy, T.: Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech. 227, 581–599 (2016)

    Article  MathSciNet  Google Scholar 

  46. Zhang, L.W., Song, Z.G., Liew, K.M.: Computation of aerothermoelastic properties and active flutter control of CNT reinforced functionally graded composite panels in supersonic airflow. Comput. Method Appl. Mech. Eng. 300, 427–441 (2016)

    Article  MathSciNet  Google Scholar 

  47. Song, Z.G., Zhang, L.W., Liew, K.M.: Active vibration control of CNT reinforced functionally graded plates based on a higher-order shear deformation theory. Int. J. Mech. Sci. 105, 90–101 (2016)

    Article  Google Scholar 

  48. Zhang, L.W., Liew, K.M., Reddy, J.N.: Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression. Comput. Method Appl. Mech. Eng. 298, 1–28 (2016)

    Article  MathSciNet  Google Scholar 

  49. Lei, Z.X., Zhang, L.W., Liew, K.M.: Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method. Compos. Part B Eng. 84, 211–221 (2016)

    Article  Google Scholar 

  50. Zhang, L.W., Song, Z.G., Liew, K.M.: Optimal shape control of CNT reinforced functionally graded composite plates using piezoelectric patches. Compos. Part B Eng. 85, 140–149 (2016)

    Article  Google Scholar 

  51. Seidel, G.D., Lagoudas, D.C.: Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mech. Mater. 38, 884–907 (2006)

    Article  Google Scholar 

  52. Fidelus, J.D., Wiesel, E., Gojny, F.H., Schulte, K., Wagner, H.D.: Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos. Part A Appl. Sci. Manuf. 36, 1555–1561 (2005)

    Article  Google Scholar 

  53. Esawi, A.M.K., Farag, M.M.: Carbon nanotube reinforced composites: potential and current challenges. Mater. Des. 28, 2394–2401 (2007)

    Article  Google Scholar 

  54. Reddy, J.N., Asce, M.: Exact solutions of moderately thick laminated shells. J. Eng. Mech. 110, 794–809 (1984)

    Article  Google Scholar 

  55. Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  56. Kiani, Y., Akbarzadeh, A.H., Chen, Z.T., Eslami, M.R.: Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation. Compos. Struct. 94, 474–484 (2012)

    Article  Google Scholar 

  57. Fazelzadeh, S.A., Pouresmaeeli, S., Ghavanloo, E.: Aeroelastic characteristics of functionally graded carbon nanotube-reinforced composite plates under a supersonic flow. Comput. Method. Appl. Mech. Eng. 285, 714–729 (2015)

    Article  MathSciNet  Google Scholar 

  58. Zhu, P., Lei, Z.X., Liew, K.M.: Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos. Struct. 94, 1450–1460 (2012)

    Article  Google Scholar 

  59. Alijani, F., Amabili, M., Bakhtiari-Nejad, F.: Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory. Compos. Struct. 93, 2541–2553 (2011)

    Article  MATH  Google Scholar 

  60. Matsunaga, H.: Free vibration and stability of functionally graded shallow shells according to a 2-D higher-order deformation theory. Compos. Struct. 84, 132–146 (2008)

    Article  Google Scholar 

  61. Chorfi, S.M., Houmat, A.: Nonlinear free vibration of a functionally graded doubly curved shallow shell of elliptical plan-form. Compos. Struct. 92, 2573–2581 (2010)

    Article  MATH  Google Scholar 

  62. Liew, K.M., Lim, C.W.: Vibration of doubly-curved shallow shells. Acta Mech. 114, 95–119 (1996)

    Article  MATH  Google Scholar 

  63. Kuttler, J.R., Sigillito, V.G.: On curve veering. J. Sound Vib. 75, 585–588 (1981)

    Article  Google Scholar 

  64. Perkins, N.C., Mote Jr., C.D.: Comments on curve veering in eigenvalue problems. J. Sound Vib. 106, 451–463 (1986)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical Engineering, Shiraz University, Shiraz, 71963-16548, Iran

    S. Pouresmaeeli & S. A. Fazelzadeh

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  1. S. Pouresmaeeli
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Correspondence to S. A. Fazelzadeh.

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Pouresmaeeli, S., Fazelzadeh, S.A. Frequency analysis of doubly curved functionally graded carbon nanotube-reinforced composite panels. Acta Mech 227, 2765–2794 (2016). https://doi.org/10.1007/s00707-016-1647-9

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  • DOI: https://doi.org/10.1007/s00707-016-1647-9

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