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Nonlinear thermal vibration of carbon nanotube polymer composite elliptical cylindrical shells

International Journal of Mechanics and Materials in Design volume 16, pages 331–350 (2020)Cite this article

Abstract

This paper investigated the nonlinear vibration and dynamic response of the carbon nanotube polymer composite elliptical cylindrical shells on elastic foundations in thermal environment. The material properties of the nanocomposite elliptical cylindrical shells are assumed to depend on temperature and graded in the thickness direction according to various linear functions. The shell is subjected to the combination of the uniformly distributed transverse load in harmonic form and the uniform temperature rise. The motion and geometrical compatibility equations are derived based on the Reddy’s higher order shear deformation shell theory. The natural frequencies and the deflection amplitude–time curves of the shell are determined by using the Galerkin method and fourth-order Runge–Kutta method. The numerical results show not only the positive influences of carbon nanotube volume fraction and elastic foundations but also the negative influences of initial imperfection and temperature increment on the nonlinear vibration and dynamic response of the carbon nanotube polymer composite elliptical cylindrical shells. The reliability of the present results is verified by comparing with other publications.

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Funding

This work has been supported/partly supported by Vietnam National University, Hanoi (VNU), under Project No. QG.18.37.

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Authors and Affiliations

  1. NTT Institute of High Technology, Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

    Ngo Dinh Dat

  2. Advanced Materials and Structures Laboratory, VNU – Hanoi – University of Engineering and Technology (UET), 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Tran Quoc Quan & Nguyen Dinh Duc

  3. National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul, 05006, Republic of Korea

    Nguyen Dinh Duc

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  1. Ngo Dinh Dat
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Correspondence to Tran Quoc Quan.

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Appendices

Appendix A

$$ \begin{aligned} \Delta & = A_{11} A_{22} - A_{12}^{2} ,\,\,I_{11}^{{}} = \frac{{A_{22} }}{\Delta },\,\,I_{12}^{{}} = \frac{{A_{12} }}{\Delta },\,\,I_{13}^{{}} = \frac{{B_{12} A_{12} - B_{11} A_{22} }}{\Delta },\,\,\,I_{14}^{{}} = \frac{{B_{22} A_{12} - B_{12} A_{22} }}{\Delta }, \\ I_{15}^{{}} & = \frac{{E_{12} A_{12} - E_{11} A_{22} }}{\Delta },\,\,I_{16}^{{}} = \frac{{E_{22} A_{12} - E_{12} A_{22} }}{\Delta },\,\,I_{17}^{{}} = \frac{{A_{22} }}{\Delta },\,I_{18}^{{}} = - \frac{{A_{12} }}{\Delta }, \\ I_{21}^{{}} & = \frac{{A_{11} }}{\Delta },\,\,I_{23}^{{}} = \frac{{B_{11} A_{12} - B_{12} A_{11} }}{\Delta },\,I_{24}^{{}} = \frac{{B_{12} A_{12} - B_{22} A_{11} }}{\Delta },\,\,I_{25}^{{}} = \frac{{E_{11} A_{12} - E_{12} A_{11} }}{\Delta }\,, \\ I_{26}^{{}} & = \frac{{E_{12} A_{12} - E_{22} A_{11} }}{\Delta },\,I_{27}^{{}} = - \frac{{A_{12} }}{\Delta },\,\,I_{28}^{{}} = \frac{{A_{11} }}{\Delta },\,\,\,I_{31}^{{}} = \frac{1}{{A_{66} }},\,\,I_{32}^{{}} = - \frac{{B_{66} }}{{A_{66} }},\,\,I_{33}^{{}} = - \frac{{E_{66} }}{{A_{66} }}. \\ \end{aligned} $$

Appendix B

$$ \begin{aligned} \overline{{I_{1} }} & = I_{1} ,\overline{{I_{1}^{*} }} = I_{1} + \frac{{2I_{2} }}{R},\;\overline{{I_{2} }} = I_{2} - c_{1} I_{4} ,\;\overline{{I_{2}^{*} }} = I_{2} + \frac{{I_{3} }}{R} - c_{1} I_{4} - \frac{{c_{1} I_{5} }}{R}, \\ \overline{{I_{3} }} & = c_{1} I_{4} ,\overline{{I_{3}^{*} }} = c_{1} I_{4} + \frac{{c_{1} I_{5} }}{R},\;\overline{{I_{4} }} = \overline{{I_{4}^{*} }} = I_{3} - 2c_{1} I_{5} + c_{1}^{2} I_{7} ,\;\overline{{I_{5} }} = \overline{{I_{5}^{*} }} = c_{1} I_{5} - c_{1}^{2} I_{7} . \\ \end{aligned} $$

Appendix C

$$ \begin{aligned} X_{11} & = A_{44} - 6c_{1} D_{44} + 9c_{1}^{2} F_{44} ,\,\,X_{12} = A_{55} - 6c_{1} D_{55} + 9c_{1}^{2} F_{55} ,\,\,X_{13} = - c_{1}^{2} (E_{11} I_{15}^{{}} + E_{12} I_{25}^{{}} + H_{11} ),\,\, \\ X_{14} & = - c_{1}^{2} (4E_{66} I_{33}^{{}} + 4H_{66} + E_{11} I_{16} + E_{12} I_{26} + 2H_{12} + E_{12} I_{15} + E_{22} I_{25} ),\,\, \\ X_{15} & = - c_{1}^{2} (E_{12} I_{16} + E_{22} I_{26} + H_{22} ),\,\,X_{16} = c_{1} (E_{11} I_{13} - c_{1} E_{11} I_{15} + F_{11} - c_{1} H_{11} + E_{12} I_{23} - c_{1} E_{12} I_{25} ), \\ X_{17} & = c_{1} (2E_{66} I_{32} - 2c_{1} E_{66} I_{33} + 2F_{66} - 2c_{1} H_{66} + c_{1} E_{12} I_{13} - c_{1} E_{12} I_{15} + F_{12} - c_{1} H_{12} + E_{22} I_{23} - c_{1} E_{22} I_{25} ), \\ X_{18} & = c_{1} \left( {E_{12} I_{14} - c_{1} E_{12} I_{16} + E_{22} I_{24} - c_{1} E_{22} I_{26} + F_{22} - c_{1} H_{22} } \right),\,\, \\ X_{19} & = c_{1} (2E_{66} I_{32} - 2c_{1} E_{66} I_{33} + 2F_{66} - 2c_{1} H_{66} + E_{11} I_{14} - c_{1} E_{11} I_{16} + E_{12} I_{24} - c_{1} E_{12} I_{26} + F_{12} - c_{1} H_{12} ), \\ X_{110} & = - c_{1} (E_{11} I_{12} - E_{12} I_{21} ),\,\,X_{111} = - c_{1} (2E_{66} I_{31} - E_{11} I_{11} + 2E_{12} I_{12} - E_{22} I_{21} ),\,\,X_{112} = c_{1} (E_{12} I_{11} - E_{22} I_{12} ), \\ X_{21} & = - A_{44} + 6c_{1} D_{44} - 9c_{1}^{2} F_{44} ,\,\,X_{22} = - c_{1} (B_{11} I_{15} + F_{11} + B_{12} I_{25} - c_{1} E_{11} I_{15} - c_{1} H_{11} - c_{1} E_{12} I_{25} ), \\ X_{23} & = - c_{1} (B{}_{11}I_{16} + B_{12} I_{26} + F_{12} + 2B_{66} I_{33} + 2F_{66} - 2c_{1} E_{66} I_{33} - 2c_{1} H_{66} - c_{1} E_{11} I_{16} - c_{1} E_{12} I_{26} - c_{1} H_{12} ), \\ X_{24} & = B_{11} I_{13} - c_{1} B_{11} I_{15} + D_{11} - c_{1} F_{11} + B_{12} I_{23} - c_{1} B_{12} I_{25} - c_{1} E_{11} I_{13} + c_{1}^{2} E_{11} I_{15} - c_{1} F_{11} + c_{1}^{2} H_{11} \\ & \quad - \,c_{1} E_{12} I_{23} + c_{1}^{2} E_{12} I_{25} ,\,\,X_{25} = B_{66} I_{32} - c_{1} B_{66} I_{33} + D_{66} - c_{1} F_{66} - c_{1} E_{66} I_{32} + c_{1}^{2} E_{66} I_{33} - c_{1} F_{66} + c_{1}^{2} H_{66} , \\ X_{26} & = B_{11} I_{14} - c_{1} B_{11} I_{16} + B_{12} I_{24} - c_{1} B_{12} I_{26} + D_{12} - c_{1} F_{12} + B_{66} I_{32} - c_{1} B_{66} I_{33} + D_{66} - c_{1} F_{66} - c_{1} E_{66} I_{32} \\ & \quad + \,c_{1}^{2} E_{66} I_{33} - c_{1} F_{66} + c_{1}^{2} H_{66} - c_{1} E_{11} I_{14} + c_{1}^{2} E_{11} I_{16} - c_{1} E_{12} I_{24} + c_{1}^{2} E_{12} I_{26} - c_{1} F_{12} + c_{1}^{2} H_{12} , \\ X_{27} & = - B_{11} I_{12} + B_{12} I_{21} + c_{1} E_{11} I_{12} - c_{1} E_{12} I_{21} , \\ X_{28} & = B_{11} I_{11} - B_{12} I_{12} - B_{66} I_{31} - c_{1} E_{11} I_{11} + c_{1} E_{12} I_{12} + c_{1} E_{66} I_{31} ,\,\,X_{31} = - A_{55} + 6c_{1} D_{55} - 9c_{1}^{2} F_{55} , \\ X_{32} & = - c_{1} (2B_{66} I_{33} + 2F_{66} + B_{12} I_{15} + F_{12} + B_{22} I_{25} - 2c_{1} E_{66} I_{33} - 2c_{1} H_{66} - c_{1} E_{12} I_{15} - c_{1} H_{12} - c_{1} E_{22} I_{25} ), \\ X_{33} & = - c_{1} (B_{12} I_{16} + B_{22} I_{26} + F_{22} - c_{1} E_{12} I_{16} - \,c_{1} E_{22} I_{26} - c_{1} H_{22} ), \\ X_{34} & = B_{66} I_{32} - c_{1} B_{66} I_{33} + D_{66} - c_{1} F_{66} + B_{12} I_{13} - c_{1} B_{12} I_{15} + D_{12} - c_{1} F_{12} + B_{22} I_{23} - \,c_{1} B_{22} I_{25} - c_{1} E_{66} I_{32} \\ & \quad + \,c_{1}^{2} E_{66} I_{33} - c_{1} F_{66} + c_{1}^{2} H_{66} - c_{1} E_{12} I_{13} + c_{1}^{2} E_{12} I_{15} - c_{1} F_{12} + c_{1}^{2} H_{12} - c_{1} E_{22} I_{23} + \,c_{1}^{2} E_{22} I_{25} , \\ X_{35} & = B_{66} I_{32} - c_{1} B_{66} I_{33} + D_{66} - c_{1} F_{66} - c_{1} E_{66} I_{32} + c_{1}^{2} E_{66} I_{33} - c_{1} F_{66} + c_{1}^{2} H_{66} , \\ \end{aligned} $$
$$ \begin{aligned} X_{36} & = B_{12} I_{14} - c_{1} B_{12} I_{16} + B_{22} I_{24} - c_{1} B_{22} I_{26} + D_{22} - c_{1} F_{22} - c_{1} E_{12} I_{14} + c_{1}^{2} E_{12} I_{16} - c_{1} E_{22} I_{24} \\ & \quad + \,c_{1}^{2} E_{22} I_{26} - c_{1} F_{22} + c_{1}^{2} H_{22} ,\,\,X_{37} = - B_{66} I_{31} - B_{12} I_{12} + B_{22} I_{21} + c_{1} E_{66} I_{31} + c_{1} E_{12} I_{12} - c_{1} E_{22} I_{21} , \\ X_{38} & = B_{12} I_{11} - B_{22} I_{12} - c_{1} E_{12} I_{11} + c_{1} E_{22} I_{12} . \\ \end{aligned} $$

Appendix D

$$ \begin{aligned} J_{1} & = I_{31}^{{}} - 2I_{12}^{{}} ,\,\,J_{2} = I_{23}^{{}} - c_{1} I_{25}^{{}} ,\,\,J_{3} = I_{13}^{{}} - c_{1} I_{15}^{{}} - I_{32}^{{}} + c_{1} I_{33}^{{}} , \\ J_{4} & = I_{14}^{{}} - c_{1} I_{16}^{{}} ,\,J_{5} = I_{24}^{{}} - c_{1} I_{26}^{{}} - I_{32}^{{}} + c_{1} I_{33}^{{}} ,\,\,J_{6} = - c_{1} I_{15}^{{}} - c_{1} I_{26}^{{}} + 2c_{1} I_{33}^{{}} . \\ \end{aligned} $$

Appendix E

$$ \begin{aligned} & r_{11} = - k_{1} - k_{2} \left( {\lambda_{m}^{2} + \delta_{n}^{2} } \right) + X_{13} \lambda_{m}^{4} + X_{14} \lambda_{m}^{2} \delta_{n}^{2} + X_{15} \delta_{n}^{4} + X_{110} Q_{1} \lambda_{m}^{4} + X_{111} Q_{1} \lambda_{m}^{2} \delta_{n}^{2} + X_{112} Q_{1} \delta_{n}^{4} \\ & - \,Q_{1} \frac{{\lambda_{m}^{2} }}{R},\,\,r_{12} = - X_{11} \lambda_{m} + X_{16} \lambda_{m}^{3} + X_{17} \lambda_{m} \delta_{n}^{2} + X_{110} Q_{2} \lambda_{m}^{4} + X_{111} Q_{2} \lambda_{m}^{2} \delta_{n}^{2} + X_{112} Q_{2} \delta_{n}^{4} \\ & - \,Q_{2} \frac{{\lambda_{m}^{2} }}{R},\,\,r_{13} = - X_{12} \delta_{n} + X_{18} \delta_{n}^{3} + X_{19} \lambda_{m}^{2} \delta_{n}^{{}} + X_{110} Q_{3} \lambda_{m}^{4} + X_{111} Q_{3} \lambda_{m}^{2} \delta_{n}^{2} + X_{112} Q_{3} \delta_{n}^{4} \\ & - \,Q_{3} \frac{{\lambda_{m}^{2} }}{R},\,\,r_{14} = \frac{{32Q_{2} \lambda_{m} \delta_{n} }}{3LR\pi },\,\,r_{15} = \frac{{32Q_{3} \lambda_{m} \delta_{n} }}{3LR\pi },\,\,n_{1} = - X_{11} \lambda_{m}^{2} - X_{12} \delta_{n}^{2} ,\,\,n_{2} = \frac{{32Q_{1} \lambda_{m} \delta_{n} }}{3LR\pi }, \\ \end{aligned} $$
$$ \begin{aligned} n_{3} & = \frac{{2\delta_{n} }}{{3LR\pi I_{21}^{{}} \lambda_{m} R}} - \frac{{8X_{110} \lambda_{m} \delta_{n} }}{{3LR\pi I_{21}^{{}} }} - \frac{{8X_{112} \lambda_{m} \delta_{n} }}{{3LR\pi I_{11}^{{}} }},\,\,n_{4} = - \frac{{\lambda_{m}^{4} }}{{16I_{11}^{{}} }} - \frac{{\delta_{n}^{4} }}{{16I_{21}^{{}} }},\,\,n_{5} = \frac{16}{{mn\pi^{2} }}, \\ r_{21} & = - \lambda_{m}^{3} (X_{22} + Q_{1} X_{27} ) - \lambda_{m} \delta_{n}^{2} (X_{23} + Q_{1} X_{28} ),\,\,r_{22} = X_{21} - X_{24} \lambda_{m}^{2} - X_{25} \delta_{n}^{2} - X_{27} Q_{2} \lambda_{m}^{3} - X_{28} Q_{2} \lambda_{m} \delta_{n}^{2} , \\ r_{23} & = - X_{26} \lambda_{m} \delta_{n} - X_{27} Q_{3} \lambda_{m}^{3} - X_{28} Q_{3} \lambda_{m} \delta_{n}^{2} ,\,\,n_{6} = X_{21} \lambda_{m} ,\,\,n_{7} = \frac{{8X_{27} \delta_{n} }}{{3LR\pi I_{21}^{{}} }}, \\ r_{31} & = - \delta_{n}^{3} (X_{33} + Q_{1} X_{38} ) - \lambda_{m}^{2} \delta_{n}^{{}} (X_{32} + Q_{1} X_{37} ),\,\,r_{32} = - X_{34} \lambda_{m} \delta_{n} - X_{38} Q_{2} \delta_{n}^{3} - D_{37} Q_{2} \lambda_{m}^{2} \delta_{n}^{{}} , \\ r_{33} & = X_{31} - X_{35} \lambda_{m}^{2} - X_{36} \delta_{n}^{2} - \,X_{38} Q_{3} \delta_{n}^{3} - X_{37} Q_{3} \lambda_{m}^{2} \delta_{n}^{{}} ,\,\,n_{8} = X_{31} \delta_{n} ,\,\,n_{9} = \frac{{8X_{38} \lambda_{m} }}{{3LR\pi I_{11}^{{}} }}. \\ \end{aligned} $$

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Dat, N.D., Quan, T.Q. & Duc, N.D. Nonlinear thermal vibration of carbon nanotube polymer composite elliptical cylindrical shells. Int J Mech Mater Des 16, 331–350 (2020). https://doi.org/10.1007/s10999-019-09464-y

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Keywords

  • Nonlinear thermal vibration
  • The carbon nanotube polymer composite elliptical cylindrical shells
  • The Reddy’s higher order shear deformation shell theory
  • Elastic foundations