Nonlinear vibration of geometrically imperfect CNT-reinforced composite cylindrical panels exposed to thermal environments with elastically restrained edges

Abstract

This paper deals with the nonlinear free vibration of carbon nanotube (CNT)-reinforced composite cylindrical panels exposed to thermal environments. All boundary edges are assumed to be simply supported and tangentially restrained. CNTs are reinforced into isotropic matrix in such a way that their volume is uniform or varied across the thickness direction according to functional rules. In order to capture size effects, the effective properties of CNT-reinforced composite are estimated using an extended version of linear rule of mixture. Motion and compatibility equations are established within the framework of classical shell theory including von Kármán–Donnell nonlinearity, initial geometric imperfection and interactive pressure from elastic foundations. Analytical solutions are assumed, and Galerkin procedure is applied to derive a time differential equation containing quadratic and cubic nonlinear terms. Fourth-order Runge–Kutta integration scheme is employed to determine the frequencies of nonlinear vibration of the panels. Through a parametric study, numerous influences of CNT distribution, size of imperfection, degree of in-plane edge constraint, foundation stiffness and elevated temperature on the natural frequencies and nonlinear-to-linear frequency ratio are analyzed.

Appendix

The expressions of $a_{k3}$($k = 1, \ldots ,4$) in Eq. (20) are given as the following:

$$\begin{aligned} a_{13} = &\, \frac{{E_{0}^{m} }}{{B_{h}^{4} }}K_{1} + \frac{{E_{0}^{m} }}{{B_{h}^{4} }}\pi^{2} \left( {m^{2} B_{a}^{2} + n^{2} } \right)K_{2} + \frac{{\pi^{4} }}{{B_{h}^{4} }}\left[ {\overline{a}_{11} m^{4} B_{a}^{4} + \overline{a}_{21} n^{4} + \overline{a}_{31} m^{2} n^{2} B_{a}^{2} } \right. \\ & \quad + \left. {\frac{{\overline{a}_{41} m^{2} n^{2} B_{a}^{2} + m^{2} B_{a}^{2} B_{h} R_{b} }}{{\pi^{2} \left( {\overline{a}_{12} m^{4} B_{a}^{4} + \overline{a}_{22} m^{2} n^{2} B_{a}^{2} + \overline{a}_{32} n^{4} } \right)}}\left( {\frac{{m^{2} }}{{\pi^{2} }}B_{a}^{2} B_{h} R_{b} - \overline{a}_{42} m^{4} B_{a}^{4} - \overline{a}_{52} m^{2} n^{2} B_{a}^{2} - \overline{a}_{62} n^{4} } \right)} \right] \\ a_{23} = &\,\, \frac{{32mn\gamma_{m} \gamma_{n} \pi^{2} B_{a}^{2} }}{{3B_{h}^{4} \left( {\overline{a}_{12} m^{4} B_{a}^{4} + \overline{a}_{22} m^{2} n^{2} B_{a}^{2} + \overline{a}_{32} n^{4} } \right)}}\left( {\overline{a}_{42} m^{4} B_{a}^{4} + \overline{a}_{52} m^{2} n^{2} B_{a}^{2} + \overline{a}_{62} n^{4} - \frac{{m^{2} }}{{\pi^{2} }}B_{a}^{2} B_{h} R_{b} } \right), \\ a_{33} = &\,\, - \frac{{2n^{2} R_{b} \gamma_{m} \gamma_{n} }}{{3mn\overline{a}_{12} B_{h}^{3} }},\,a_{43} = \pi^{4} \frac{{\overline{a}_{12} m^{4} B_{a}^{4} + \overline{a}_{32} n^{4} }}{{16\overline{a}_{12} \overline{a}_{32} B_{h}^{4} }}. \\ \end{aligned}$$

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where

$$\begin{aligned} \left( {\overline{a}_{11} ,\overline{a}_{21} ,\overline{a}_{31} } \right) = &\,\, \frac{1}{{h^{3} }}\left( {a_{11} ,a_{21} ,a_{31} } \right),\,\left( {\overline{a}_{41} ,\overline{a}_{42} ,\overline{a}_{52} ,\overline{a}_{62} } \right) = \frac{1}{h}\left( {a_{41} ,a_{42} ,a_{52} ,a_{62} } \right), \\ \left( {\overline{a}_{12} ,\overline{a}_{22} ,\overline{a}_{32} } \right) = &\,\, h\left( {a_{12} ,a_{22} ,a_{32} } \right),\,\left( {K_{1} ,K_{2} } \right) = \left( {k_{1} b^{2} ,k_{2} } \right)\frac{{b^{2} }}{{E_{0}^{m} h^{3} }} \\ \end{aligned}$$

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in which $E_{0}^{m}$ is value of $E^{m}$ computed at room temperature $T_{0} = 300{\text{K}}$.

The coefficients $a_{k6}$ and $a_{k7}$ ($k = 1,2,3$) in Eqs. (22a, 22b) are given as follows:

$$\begin{aligned} a_{{\mathop {16}}} = &\,\, \frac{{a_{34} }}{{a_{24} }}\left( {\overline{c}_{2} a_{25} - 1} \right)a_{46} - \overline{c}_{2} a_{35} a_{46} ,\quad a_{{\mathop {26}}} = \frac{{a_{44} }}{{a_{24} }}\left( {\overline{c}_{2} a_{25} - 1} \right)a_{46} - \overline{c}_{2} a_{45} a_{46} , \\ a_{36} = &\,\, \frac{{a_{54} }}{{a_{24} }}\left( {\overline{c}_{2} a_{25} - 1} \right)a_{46} - \overline{c}_{2} a_{55} a_{46} ,\quad a_{17} = \frac{{a_{35} }}{{a_{15} }}\left( {\overline{c}_{1} a_{14} - 1} \right)a_{47} - \overline{c}_{1} a_{34} a_{47} , \\ a_{27} = &\,\, \frac{{a_{45} }}{{a_{15} }}\left( {\overline{c}_{1} a_{14} - 1} \right)a_{47} - \overline{c}_{1} a_{44} a_{47} ,\quad a_{37} = \frac{{a_{55} }}{{a_{15} }}\left( {\overline{c}_{1} a_{14} - 1} \right)a_{47} - \overline{c}_{1} a_{54} a_{47} \\ \end{aligned}$$

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in which

$$\left( {a_{46} ,a_{47} } \right) = \frac{{\left( {\overline{c}_{1} a_{24} ,\overline{c}_{2} a_{15} } \right)}}{{\overline{c}_{1} \overline{c}_{2} a_{24} a_{15} + \left( {\overline{c}_{2} a_{25} - 1} \right)\left( {1 - \overline{c}_{1} a_{14} } \right)}},\quad \left( {\overline{c}_{1} ,\overline{c}_{2} } \right) = \frac{1}{h}\left( {c_{1} ,c_{2} } \right)$$

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and specific expressions of $a_{i4}$ and $a_{j5}$ ($i,j = 1, \ldots ,5$) were given in the work [26].

The detail of coefficients $r_{1} , \ldots ,r_{6}$ in Eq. (23) is defined as follows:

$$\begin{aligned} r_{1} = &\,\, a_{13} - \frac{{16\gamma_{m} \gamma_{n} R_{b} }}{{mn\pi^{2} B_{h} }}a_{17} ,\;r_{2} = a_{23} + \frac{{\pi^{2} }}{{B_{h}^{2} }}\left( {m^{2} B_{a}^{2} a_{16} + n^{2} a_{17} } \right),\;r_{3} = a_{33} - \frac{{16\gamma_{m} \gamma_{n} R_{b} }}{{mn\pi^{2} B_{h} }}a_{27} , \\ r_{4} = &\,\, a_{43} + \frac{{\pi^{2} }}{{B_{h}^{2} }}\left( {m^{2} B_{a}^{2} a_{26} + n^{2} a_{27} } \right),\;r_{5} = \frac{{\pi^{2} }}{{B_{h}^{2} }}\left( {m^{2} B_{a}^{2} a_{36} + n^{2} a_{37} } \right),\;r_{6} = \frac{{16\gamma_{m} \gamma_{n} R_{b} }}{{mn\pi^{2} B_{h} }}a_{37} \\ \end{aligned}$$

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