On the nonlinear dynamics of the multi-scale hybrid nanocomposite-reinforced annular plate under hygro-thermal environment

Abstract

In this article, the nonlinear free and forced vibration analysis of multi-scale hybrid nano-composites (multi-scale HNC) annular plate (multi-scale HNCAP) under hygro-thermal environment and subjected to mechanical loading is presented. The material of matrix composite is enhanced by either carbon fibers (CF) or carbon nanotubes (CNTs) at the small or macro-scale. The multi-scale laminated annular plate’s displacement fields are determined using third-order shear deformation theory (third-order SDT) and nonlinearity of vibration behavior of this structure is taken into account considering Von Karman nonlinear shell model. Energy method known as Hamilton principle is applied to create the motion equations governed to the multi-scale HNCAP, while they are solved using generalized differential quadrature method (GDQM) as well as multiple scale method. The results created from finite-element simulation illustrates a close agreement with the semi-numerical method results. Ultimately, the research’s outcomes reveal that increasing value of the moisture change ($\Delta H$) and orientation angle parameter ($\theta$), and the rigidity of the boundary conditions lead to an increase in the structure’s frequency. Besides, whenever the values of the nonlinear parameter ($\gamma$) are positive or negative, the dynamic behavior of the plate tends to have hardening or softening behaviors, respectively. Also, there are not any effects from $\gamma$ parameter on the maximum amplitudes of resonant vibration of the multi-scale HNCAP. Last but not least, by decreasing the structure’s flexibility, the plate can be susceptible to have unstable responses.

Appendix

In Eqs. (31a–31c), L_ij and M_ij are expressed as follows:

$$\begin{aligned} &\delta u_{o} : \hfill \\ &\quad L_{11} = A_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{A_{22} }}{{R^{2} }}u, \hfill \\ &\quad L_{12} = - D_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{D_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\ &\quad L_{13} = B_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - D_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}\phi + \frac{{D_{22} c_{1} }}{{R^{2} }}\phi \hfill \\& \quad M_{11} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,M_{12} = - I_{3} c_{1} \frac{{\partial^{3} w}}{{\partial R\partial t^{2} }},\,\,\,M_{13} = \left( {I_{1} - I_{3} c_{1} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }} \hfill \\ \end{aligned}$$

(70)

$$\begin{aligned}& \delta w_{0} : \hfill \\ &\quad L_{21} = c_{1} D_{11} \frac{{\partial^{3} u}}{{\partial R^{3} }} - \frac{{c_{1} D_{22} }}{{R^{2} }}\frac{\partial u}{{\partial R}}, \hfill \\ &\quad L_{22} = - G_{11} c_{1}^{2} \frac{{\partial^{4} w}}{{\partial R^{4} }} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial R^{2} }} + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\ &\quad - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} - q \, - (N^{T} + N^{H} )\frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\&\quad L_{23} = {\text{C}}\frac{\partial w}{{\partial t}},\,\,\,\,\,\,\,\,L_{24} = \frac{3}{2}{\text{A}}_{11} \frac{{\partial^{2} w}}{{\partial R^{2} }}\left( {\frac{\partial w}{{\partial R}}} \right)^{2} \hfill \\&\quad L_{25} = c_{1} E_{11} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - G_{11} c_{1}^{2} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - \frac{{c_{1} }}{R}\frac{{E_{22} \partial \phi }}{R\partial R} - \frac{{c_{1} }}{R}\frac{{G_{22} c_{1} }}{R}\frac{\partial \phi }{{\partial R}} \hfill \\&\quad + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} \hfill \\&\quad M_{21} = c_{1} I_{3} \frac{{\partial^{3} u}}{{\partial R\partial t^{2} }},\,\,M_{22} = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} - c_{1}^{2} I_{6} \frac{{\partial^{4} w}}{{\partial R^{2} \partial t^{2} }}, \hfill \\&\quad M_{23} = \left( {c_{1} I_{4} - c_{1}^{2} I_{6} } \right)\frac{{\partial^{3} \phi }}{{\partial R\partial t^{2} }}, \hfill \\ \, \hfill \\ \end{aligned}$$

(71)

$$\begin{aligned} &\delta \phi : \hfill \\&\quad L_{31} = B_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - c_{1} D_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}u + \frac{{D_{22} }}{{R^{2} }}u, \hfill \\&\quad L_{32} = - E_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + G_{11} c_{1}^{2} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{E_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\&\quad - \frac{\partial w}{{\partial R}}\left( {A_{55} - 3c_{1} C_{55} } \right) + 3\frac{\partial w}{{\partial R}}\left( {C_{55} - 3c_{1} E_{55} } \right)c_{1} \hfill \\&\quad L_{33} = C_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - E_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - c_{1} E_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} + G_{11} c_{1}^{2} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} \hfill \\&\quad - \frac{1}{{R^{2} }}\left\{ {C_{22} - E_{22} c_{1} } \right\}\phi + \frac{{c_{1} }}{{R^{2} }}\left\{ {E_{22} - G_{22} c_{1} } \right\}\phi \hfill \\ &\quad - \left( {A_{55} - 3c_{1} C_{55} } \right)\phi + 3c_{1} \left( {C_{55} - 3c_{1} E_{55} } \right)\phi \hfill \\&\quad M_{31} = \left( {I_{1} - c_{1} I_{3} } \right)\frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,\,M_{32} = \left( {I_{6} c_{1}^{2} - I_{4} c_{1} } \right)\frac{{\partial^{3} w}}{{\partial R\partial t^{2} }}, \hfill \\&\quad M_{33} = \left( {I_{6} c_{1}^{2} - 2c_{1} I_{4} + I_{2} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }}. \hfill \\ \end{aligned}$$

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