Free Vibration Analysis of Functionally Graded Sandwich Circular Cylindrical Shells with Auxetic Honeycomb Core Layer and Partially Filled with Liquid

Abstract

This paper addresses the vibrational responses of functionally graded auxetic sandwich circular cylindrical shells. The sandwich shell comprises two face sheets of functionally graded material and an auxetic honeycomb material featuring a negative Poisson's ratio that forms a core layer. The sandwich shell is partially or fully liquid-filled and rests on an elastic foundation. Under the assumption that the liquid is ideal, its irrotational motion is described using the velocity potential function. In order to achieve this objective, a model based on Reddy's third-order shear deformation theory and the Rayleigh–Ritz method is developed. The present model's accuracy is confirmed through examples that compare the study results and those available in literature. Finally, some new numerical results are obtained that show the liquid level, the volume fraction index, the auxetic core geometric parameters and the mechanical boundary conditions strongly impact the vibration frequency of the sandwich circular cylindrical shells.

Appendix

$$k^{aa} = \int\limits_{0}^{L} {\left( {A_{11} \zeta^{{\prime\prime}2} + A_{66} \frac{{n^{2} }}{{R^{2} }}\zeta^{{\prime}2} } \right){\text{d}}x} ;k^{ab} = \frac{n}{R}\int\limits_{0}^{L} {\left( {A_{12} \zeta^{{\prime\prime}} \zeta - A_{66} \zeta^{{\prime}2} } \right){\text{d}}x} ;$$

$$k^{ac} = \int\limits_{0}^{L} {\left( {\left( {\frac{{A_{12} }}{R} + c_{1} D_{12} \frac{{n^{2} }}{{R^{2} }}} \right)\zeta^{{\prime\prime}} \zeta - c_{1} D_{11} \zeta^{{\prime\prime}2} - 2c_{1} D_{66} \frac{{n^{2} }}{{R^{2} }}\zeta^{{\prime}2} } \right){\text{d}}x} ;$$

$$k^{ad} = \int\limits_{0}^{L} {\left( {\left( {B_{11} - c_{1} D_{11} } \right)\zeta^{{\prime\prime}2} + \left( {B_{66} - c_{1} D_{66} } \right)\frac{{n^{2} }}{{R^{2} }}\zeta^{{\prime}2} } \right){\text{d}}x} ;$$

$$k^{ae} = \frac{n}{R}\int\limits_{0}^{L} {\left( {\left( {B_{12} - c_{1} D_{12} } \right)\zeta^{{\prime\prime}} \zeta - \left( {B_{66} - c_{1} D_{66} } \right)\zeta^{{\prime}2} } \right)} {\text{d}}x;$$

$$k^{bb} = \int\limits_{0}^{L} {\left( {A_{22} \frac{{n^{2} }}{{R^{2} }}\zeta^{2} + A_{66} \zeta^{{\prime}2} } \right){\text{d}}x} ;k^{bc} = \frac{n}{R}\int\limits_{0}^{L} {\left( {\left( {\frac{{A_{22} }}{R} + c_{1} D_{22} \frac{{n^{2} }}{{R^{2} }}} \right)\zeta^{2} - c_{1} D_{12} \zeta^{{\prime\prime}} \zeta + 2c_{1} D_{66} \zeta^{{\prime}2} } \right){\text{d}}x} ;$$

$$k^{bd} = \frac{n}{R}\int\limits_{0}^{L} {\left( {\left( {B_{12} - c_{1} D_{12} } \right)\zeta^{{\prime\prime}} \zeta - \left( {B_{66} - c_{1} D_{66} } \right)\zeta^{{\prime}2} } \right)} {\text{d}}x;$$

$$k^{be} = \int\limits_{0}^{L} {\left( {\left( {B_{22} - c_{1} D_{22} } \right)\frac{{n^{2} }}{{R^{2} }}\zeta^{2} + \left( {B_{66} - c_{1} D_{66} } \right)\zeta^{{\prime}2} } \right){\text{d}}x} ;$$

$$k^{cc} = \int\limits_{0}^{L} {\left( \begin{gathered} \left( {\frac{{A_{22} }}{{R^{2} }} + 2c_{1} D_{22} \frac{{n^{2} }}{{R^{3} }} + c_{1}^{2} G_{22} \frac{{n^{4} }}{{R^{4} }} + \left( {A_{44}^{s} - 6c_{1} C_{44}^{s} + 9c_{1}^{2} E_{44}^{s} } \right)\frac{{n^{2} }}{{R^{2} }} + k_{w} + k_{p} \frac{{n^{2} }}{{R^{2} }}} \right)\zeta^{2} \hfill \\ - 2\left( {c_{1} \frac{{D_{12} }}{R} + c_{1}^{2} G_{12} \frac{{n^{2} }}{{R^{2} }}} \right)\zeta^{{\prime\prime}} \zeta + c_{1}^{2} G_{11} \zeta^{{\prime\prime}2} \hfill \\ + \left( {4c_{1}^{2} G_{66} \frac{{n^{2} }}{{R^{2} }} + \left( {A_{55}^{s} - 6c_{1} C_{55}^{s} + 9c_{1}^{2} E_{55}^{s} } \right) + k_{p} } \right)\zeta^{{\prime}2} \hfill \\ \end{gathered} \right){\text{d}}x} ;$$

$$k^{cd} = \int\limits_{0}^{L} {\left\{ \begin{gathered} \left[ {\frac{{B_{12} - c_{1} D_{12} }}{R} + \left( {c_{1} E_{12} - c_{1}^{2} G_{12} } \right)\frac{{n^{2} }}{{R^{2} }}} \right]\zeta^{{\prime\prime}} \zeta - \left( {c_{1} E_{11} - c_{1}^{2} G_{11} } \right)\zeta^{{\prime\prime}2} \hfill \\ \left[ { - 2\left( {c_{1} E_{66} - c_{1}^{2} G_{66} } \right)\frac{{n^{2} }}{{R^{2} }} + \left( {A_{55}^{s} - 6c_{1} C_{55}^{s} + 9c_{1}^{2} E_{55}^{s} } \right)} \right]\zeta^{{\prime}2} \hfill \\ \end{gathered} \right\}{\text{d}}x} ;$$

$$k^{ce} = \frac{n}{R}\int\limits_{0}^{L} {\left\{ \begin{gathered} \left[ {\frac{{B_{22} - c_{1} D_{22} }}{R} + \left( {c_{1} E_{22} - c_{1}^{2} G_{22} } \right)\frac{{n^{2} }}{{R^{2} }} - \left( {A_{44}^{s} - 6c_{1} C_{44}^{s} + 9c_{1}^{2} E_{44}^{s} } \right)} \right]\zeta^{2} \hfill \\ - \left( {c_{1} E_{12} - c_{1}^{2} G_{12} } \right)\zeta^{{\prime\prime}} \zeta + 2\left( {c_{1} E_{66} - c_{1}^{2} G_{66} } \right)\zeta^{{\prime}2} \hfill \\ \end{gathered} \right\}{\text{d}}x} ;$$

$$\begin{aligned} k^{dd} & = \int\limits_{0}^{L} {\left\{ {\left( {C_{11} - 2c_{1} E_{11} + c_{1}^{2} G_{11} } \right)\zeta^{{\prime\prime}2} } \right.} \\ & \quad + \left. {\left[ {\left( {C_{66} - 2c_{1} E_{66} + G_{66} c_{1}^{2} } \right)\frac{{n^{2} }}{{R^{2} }} + \left( {A_{55}^{s} - 6c_{1} C_{55}^{s} + 9c_{1}^{2} E_{55}^{s} } \right)} \right]\zeta^{{\prime}2} } \right\}{\text{d}}x; \\ \end{aligned}$$

$$k^{de} = \frac{n}{R}\int\limits_{0}^{L} {\left[ {\left( {C_{12} - 2c_{1} E_{12} + c_{1}^{2} G_{12} } \right)\zeta^{{\prime\prime}} \zeta - \left( {C_{66} - 2c_{1} E_{66} + c_{1}^{2} G_{66} } \right)\zeta^{{\prime}2} } \right]{\text{d}}x} ;$$

$$k^{ee} = \int\limits_{0}^{L} {\left( \begin{gathered} \left[ {\left( {C_{22} - 2c_{1} E_{22} + c_{1}^{2} G_{22} } \right)\frac{{n^{2} }}{{R^{2} }} + \left( {A_{44}^{s} - 6c_{1} C_{44}^{s} + 9c_{1}^{2} E_{44}^{s} } \right)} \right]\zeta^{2} \hfill \\ + \left( {C_{66} - 2c_{1} E_{66} + c_{1}^{2} G_{66} } \right)\zeta^{{\prime}2} \hfill \\ \end{gathered} \right){\text{d}}x} ;$$

$$m^{aa} = I_{0} \int\limits_{0}^{L} {\zeta^{{\prime}2} {\text{d}}x} ;m^{ac} = - c_{1} I_{3} \int\limits_{0}^{L} {\zeta^{{\prime}2} {\text{d}}x} ;\;m^{ad} = \left( {I_{1} - c_{1} I_{3} } \right)\int\limits_{0}^{L} {\zeta^{{\prime}2} {\text{d}}x} ;\;m^{bb} = \left( {I_{0} + \frac{{2I_{1} }}{R} + \frac{{I_{2} }}{{R^{2} }}} \right)\int\limits_{0}^{L} {\zeta^{2} {\text{d}}x} ;$$

$$m^{bc} = c_{1} n\left( {\frac{{I_{3} }}{R} + \frac{{I_{4} }}{{R^{2} }}} \right)\int\limits_{0}^{L} {\zeta^{2} {\text{d}}x} ;\;m^{be} = \left( {I_{1} + \frac{{I_{2} }}{R} - c_{1} I_{3} - c_{1} \frac{{I_{4} }}{R}} \right)\int\limits_{0}^{L} {\zeta^{2} {\text{d}}x} ;$$

$$m^{cc} = \int\limits_{0}^{L} {\left[ {\left( {I_{0} + c_{1}^{2} I_{6} \frac{{n^{2} }}{{R^{2} }}} \right)\zeta^{2} + c_{1}^{2} I_{6} \zeta^{{\prime}2} } \right]{\text{d}}x} + m_{fluid} ;\;m^{cd} = \left( {c_{1}^{2} I_{6} - c_{1} I_{4} } \right)\int\limits_{0}^{L} {\zeta^{{\prime}2} {\text{d}}x} ;$$

$$m^{ce} = \frac{n}{R}\left( {c_{1} I_{4} - c_{1}^{2} I_{6} } \right)\int\limits_{0}^{L} {\zeta^{2} {\text{d}}x} ;\;m^{dd} = \left( {I_{2} - 2c_{1} I_{4} + c_{1}^{2} I_{6} } \right)\int\limits_{0}^{L} {\zeta^{{\prime}2} {\text{d}}x} ;\;m^{ee} = \left( {I_{2} - 2c_{1} I_{4} + c_{1}^{2} I_{6} } \right)\int\limits_{0}^{L} {\zeta^{2} {\text{d}}x} ;$$

$$m_{{{\text{fluid}}}} = \frac{{8\rho_{f} }}{\pi }\sum\limits_{k = 1}^{NF} {\frac{{I_{n} \left( {\frac{{\left( {2k - 1} \right)\pi R}}{2H}} \right)}}{{\left( {2k - 1} \right)\Theta_{k} }}} \left[ {\int\limits_{0}^{H} {\zeta \cos \left( {\frac{{\left( {2k - 1} \right)\pi x}}{2H}} \right){\text{d}}x} } \right]^{2}$$